Subgroups — C₂×C₆₇×#3009 #30739612

Order 17152.

5062 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#137×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3009	C₂	1	16	yes	maximal, direct product factor
6	C₂×#452×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#450×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×H(5,𝔽₂)×C₆₇	—	2	8	no	—
12	C₂×H(5,𝔽₂)×C₆₇	—	2	4	no	—
13	C₂×H(5,𝔽₂)×C₆₇	—	2	8	no	—
14	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	8	yes	—
15	C₂²×#41×C₆₇	—	2	1	no	—
16	C₂²×#41×C₆₇	C₂²	1	2	yes	—
17	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂³×D₄×C₆₇	—	2	2	no	—
21	C₂³×D₄×C₆₇	—	2	1	no	—
22	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#137×C₆₇	—	2	8	no	—
25	C₂×#137×C₆₇	C₂²	1	8	yes	complemented
26	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
27	C₂²×SD₁₆×C₆₇	—	2	8	no	—
28	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
29	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
30	C₂²×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
31	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#536×C₆₇	—	2	4	no	—
34	#536×C₆₇	C₂²	1	8	yes	complemented
35	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
36	C₂×#124×C₆₇	—	2	1	no	—
37	C₂×#123×C₆₇	—	2	2	no	—
38	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
39	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#452×C₆₇	C₂²	1	32	yes	complemented
41	#450×C₆₇	C₂²	1	32	yes	complemented
42	#418×C₆₇	C₂²	1	8	yes	complemented
43	#416×C₆₇	C₂²	1	8	yes	complemented
44	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
45	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
46	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
47	#408×C₆₇	C₂²	1	8	yes	complemented
48	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
49	C₂⁵×C₆₇	—	2	2	no	—
50	C₂⁵×C₆₇	—	2	1	no	—
51	H(5,𝔽₂)×C₆₇	—	2	64	no	—
52	H(5,𝔽₂)×C₆₇	—	2	32	no	—
53	H(5,𝔽₂)×C₆₇	—	2	32	no	—
54	C₂×#41×C₆₇	—	2	16	no	—
55	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	2	8	no	—
57	C₂×#41×C₆₇	—	2	8	no	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	—	2	4	no	—
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×#41×C₆₇	—	2	4	no	—
62	C₂×#41×C₆₇	—	2	4	no	—
63	C₂×#41×C₆₇	D₄	1	8	yes	—
64	C₂×#41×C₆₇	D₄	1	4	yes	—
65	C₂×#41×C₆₇	C₂³	1	8	yes	—
66	C₂×#41×C₆₇	C₂³	1	4	yes	—
67	C₂×#41×C₆₇	C₂³	1	4	yes	—
68	C₂²×Q₈×C₆₇	—	2	1	no	—
69	C₂²×Q₈×C₆₇	—	2	2	no	—
70	C₂²×Q₈×C₆₇	—	2	4	no	—
71	C₂²×Q₈×C₆₇	—	2	4	no	—
72	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
73	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×Q₈×C₆₇	C₂³	1	4	yes	complemented
76	C₂²×D₄×C₆₇	—	2	32	no	—
77	C₂²×D₄×C₆₇	—	2	16	no	—
78	C₂²×D₄×C₆₇	—	2	2	no	—
79	C₂²×D₄×C₆₇	—	2	1	no	—
80	C₂²×D₄×C₆₇	—	2	16	no	—
81	C₂²×D₄×C₆₇	—	2	8	no	—
82	C₂²×D₄×C₆₇	—	2	8	no	—
83	C₂²×D₄×C₆₇	—	2	8	no	—
84	C₂²×D₄×C₆₇	—	2	2	no	—
85	C₂²×D₄×C₆₇	—	2	1	no	—
86	C₂²×D₄×C₆₇	—	2	4	no	—
87	C₂²×D₄×C₆₇	—	2	2	no	—
88	C₂²×D₄×C₆₇	—	2	4	no	—
89	C₂²×D₄×C₆₇	D₄	1	2	yes	—
90	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
91	C₂²×D₄×C₆₇	D₄	1	8	yes	—
92	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
93	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
95	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
96	C₂³×C₄×C₆₇	—	2	2	no	—
97	C₂³×C₄×C₆₇	—	2	1	no	—
98	C₂³×C₄×C₆₇	—	2	1	no	—
99	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
100	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
101	#137×C₆₇	—	2	32	no	—
102	C₂×Q₁₆×C₆₇	—	2	8	no	—
103	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	complemented
104	C₂×SD₁₆×C₆₇	—	2	16	no	—
105	C₂×SD₁₆×C₆₇	—	2	32	no	—
106	C₂×SD₁₆×C₆₇	—	2	8	no	—
107	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	complemented
108	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	complemented
109	C₂×M₁₆×C₆₇	—	2	4	no	—
110	C₂×M₁₆×C₆₇	C₂³	1	4	yes	complemented
111	C₂²×C₈×C₆₇	—	2	4	no	—
112	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
113	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
114	C₂²×C₈×C₆₇	C₂³	1	4	yes	complemented
115	C₄⋊Q₈×C₆₇	—	2	8	no	—
116	#124×C₆₇	—	2	8	no	—
117	#123×C₆₇	—	2	16	no	—
118	C₂²⋊Q₈×C₆₇	—	2	8	no	—
119	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
120	#117×C₆₇	—	2	2	no	—
121	#117×C₆₇	C₂³	1	4	yes	complemented
122	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
126	C₂×C₄²×C₆₇	—	2	1	no	—
127	#103×C₆₇	C₂³	1	8	yes	complemented
128	#102×C₆₇	C₂³	1	8	yes	complemented
129	#102×C₆₇	C₂³	1	16	yes	complemented
130	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
131	C₂⁴×C₆₇	—	2	16	no	—
132	C₂⁴×C₆₇	—	2	2	no	—
133	C₂⁴×C₆₇	—	2	1	no	—
134	C₂⁴×C₆₇	—	2	8	no	—
135	C₂⁴×C₆₇	—	2	8	no	—
136	C₂⁴×C₆₇	—	2	4	no	—
137	C₂⁴×C₆₇	—	8	4	no	—
138	C₂⁴×C₆₇	—	8	8	no	—
139	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
140	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
141	#41×C₆₇	—	2	64	no	—
142	#41×C₆₇	—	2	32	no	—
143	#41×C₆₇	—	2	16	no	—
144	#41×C₆₇	—	2	16	no	—
145	#41×C₆₇	—	2	16	no	—
146	#41×C₆₇	—	2	16	no	—
147	#41×C₆₇	—	2	8	no	—
148	#41×C₆₇	—	2	8	no	—
149	#41×C₆₇	—	2	8	no	—
150	#41×C₆₇	C₂×D₄	1	16	yes	—
151	C₂×Q₈×C₆₇	—	2	4	no	—
152	C₂×Q₈×C₆₇	—	2	2	no	—
153	C₂×Q₈×C₆₇	—	2	1	no	—
154	C₂×Q₈×C₆₇	—	2	2	no	—
155	C₂×Q₈×C₆₇	—	2	4	no	—
156	C₂×Q₈×C₆₇	—	2	4	no	—
157	C₂×Q₈×C₆₇	—	2	8	no	—
158	C₂×Q₈×C₆₇	—	2	4	no	—
159	C₂×Q₈×C₆₇	—	2	8	no	—
160	C₂×Q₈×C₆₇	—	4	4	no	—
161	C₂×Q₈×C₆₇	—	4	8	no	—
162	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
163	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
164	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
165	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
166	C₂×D₄×C₆₇	—	2	16	no	—
167	C₂×D₄×C₆₇	—	2	16	no	—
168	C₂×D₄×C₆₇	—	2	128	no	—
169	C₂×D₄×C₆₇	—	2	64	no	—
170	C₂×D₄×C₆₇	—	2	32	no	—
171	C₂×D₄×C₆₇	—	2	8	no	—
172	C₂×D₄×C₆₇	—	2	8	no	—
173	C₂×D₄×C₆₇	—	2	8	no	—
174	C₂×D₄×C₆₇	—	2	4	no	—
175	C₂×D₄×C₆₇	—	2	16	no	—
176	C₂×D₄×C₆₇	—	2	16	no	—
177	C₂×D₄×C₆₇	—	2	4	no	—
178	C₂×D₄×C₆₇	—	2	8	no	—
179	C₂×D₄×C₆₇	—	2	16	no	—
180	C₂×D₄×C₆₇	—	2	8	no	—
181	C₂×D₄×C₆₇	—	2	4	no	—
182	C₂×D₄×C₆₇	—	2	1	no	—
183	C₂×D₄×C₆₇	—	2	2	no	—
184	C₂×D₄×C₆₇	—	2	4	no	—
185	C₂×D₄×C₆₇	—	2	2	no	—
186	C₂×D₄×C₆₇	—	2	8	no	—
187	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
188	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
189	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
190	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
191	C₂×D₄×C₆₇	C₂×D₄	1	16	yes	—
192	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
193	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
194	C₂²×C₄×C₆₇	—	2	16	no	—
195	C₂²×C₄×C₆₇	—	2	1	no	—
196	C₂²×C₄×C₆₇	—	2	8	no	—
197	C₂²×C₄×C₆₇	—	2	8	no	—
198	C₂²×C₄×C₆₇	—	2	4	no	—
199	C₂²×C₄×C₆₇	—	2	4	no	—
200	C₂²×C₄×C₆₇	—	2	4	no	—
201	C₂²×C₄×C₆₇	—	2	4	no	—
202	C₂²×C₄×C₆₇	—	2	2	no	—
203	C₂²×C₄×C₆₇	—	2	2	no	—
204	C₂²×C₄×C₆₇	—	4	4	no	—
205	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
206	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
207	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
208	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
210	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
211	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
212	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
214	Q₁₆×C₆₇	—	2	16	no	—
215	SD₁₆×C₆₇	—	2	32	no	—
216	SD₁₆×C₆₇	—	2	16	no	—
217	M₁₆×C₆₇	—	2	8	no	—
218	C₂×C₈×C₆₇	—	2	4	no	—
219	C₂×C₈×C₆₇	—	2	8	no	—
220	C₂×C₈×C₆₇	—	2	4	no	—
221	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
222	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
223	C₄⋊C₄×C₆₇	—	2	16	no	—
224	C₄⋊C₄×C₆₇	—	2	2	no	—
225	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
226	C₂²⋊C₄×C₆₇	—	2	4	no	—
227	C₂²⋊C₄×C₆₇	—	2	16	no	—
228	C₄²×C₆₇	—	2	4	no	—
229	C₂³×C₆₇	—	2	4	no	—
230	C₂³×C₆₇	—	2	32	no	—
231	C₂³×C₆₇	—	2	4	no	—
232	C₂³×C₆₇	—	2	8	no	—
233	C₂³×C₆₇	—	2	4	no	—
234	C₂³×C₆₇	—	2	8	no	—
235	C₂³×C₆₇	—	2	4	no	—
236	C₂³×C₆₇	—	2	2	no	—
237	C₂³×C₆₇	—	2	16	no	—
238	C₂³×C₆₇	—	4	8	no	—
239	C₂³×C₆₇	—	4	4	no	—
240	C₂³×C₆₇	—	8	16	no	—
241	C₂³×C₆₇	—	8	8	no	—
242	C₂³×C₆₇	—	8	16	no	—
243	C₂³×C₆₇	—	8	32	no	—
244	C₂³×C₆₇	#120	1	4	yes	—
245	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
246	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
247	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
248	Q₈×C₆₇	—	2	4	no	—
249	Q₈×C₆₇	—	2	8	no	—
250	Q₈×C₆₇	—	2	2	no	—
251	Q₈×C₆₇	—	2	4	no	—
252	Q₈×C₆₇	—	2	4	no	—
253	Q₈×C₆₇	—	4	4	no	—
254	Q₈×C₆₇	—	4	8	no	—
255	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
256	D₄×C₆₇	—	2	32	no	—
257	D₄×C₆₇	—	2	32	no	—
258	D₄×C₆₇	—	2	16	no	—
259	D₄×C₆₇	—	2	16	no	—
260	D₄×C₆₇	—	2	8	no	—
261	D₄×C₆₇	—	2	4	no	—
262	D₄×C₆₇	—	2	16	no	—
263	D₄×C₆₇	—	2	8	no	—
264	D₄×C₆₇	—	2	4	no	—
265	D₄×C₆₇	—	2	2	no	—
266	D₄×C₆₇	C₂²×D₄	1	4	yes	—
267	D₄×C₆₇	C₂²×D₄	1	8	yes	—
268	C₂×C₄×C₆₇	—	2	32	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	16	no	—
273	C₂×C₄×C₆₇	—	2	8	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	4	no	—
276	C₂×C₄×C₆₇	—	2	4	no	—
277	C₂×C₄×C₆₇	—	2	4	no	—
278	C₂×C₄×C₆₇	—	4	8	no	—
279	C₂×C₄×C₆₇	—	4	4	no	—
280	C₂×C₄×C₆₇	—	4	4	no	—
281	C₂×C₄×C₆₇	#120	1	4	yes	—
282	C₂×C₄×C₆₇	#120	1	8	yes	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
284	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
289	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
290	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
291	C₈×C₆₇	—	2	4	no	—
292	C₈×C₆₇	—	2	4	no	—
293	C₂²×C₆₇	—	2	8	no	—
294	C₂²×C₆₇	—	2	4	no	—
295	C₂²×C₆₇	—	2	4	no	—
296	C₂²×C₆₇	—	2	8	no	—
297	C₂²×C₆₇	—	2	4	no	—
298	C₂²×C₆₇	—	2	8	no	—
299	C₂²×C₆₇	—	4	8	no	—
300	C₂²×C₆₇	—	4	16	no	—
301	C₂²×C₆₇	—	4	8	no	—
302	C₂²×C₆₇	—	4	8	no	—
303	C₂²×C₆₇	—	4	4	no	—
304	C₂²×C₆₇	—	8	16	no	—
305	C₂²×C₆₇	—	8	32	no	—
306	C₂²×C₆₇	C₂×#120	1	4	yes	—
307	C₂²×C₆₇	C₂×#120	1	2	yes	—
308	C₂²×C₆₇	#577	1	4	yes	—
309	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
310	C₄×C₆₇	—	2	4	no	—
311	C₄×C₆₇	—	2	4	no	—
312	C₄×C₆₇	—	4	4	no	—
313	C₄×C₆₇	—	4	4	no	—
314	C₄×C₆₇	C₂×#120	1	4	yes	—
315	C₄×C₆₇	C₂×#120	1	2	yes	—
316	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
317	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
318	C₂×#3009	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
319	C₂×C₆₇	—	2	4	no	—
320	C₂×C₆₇	—	4	4	no	—
321	C₂×C₆₇	—	4	8	no	—
322	C₂×C₆₇	—	4	4	no	—
323	C₂×C₆₇	#3009	1	4	yes	direct product factor
324	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
325	C₂×C₆₇	C₂×#577	1	2	yes	—
326	C₂²×H(5,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic
327	C₂²×#137	C₂×C₆₇	1	1	yes	characteristic, complemented
328	C₂³×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
329	C₂×#536	C₂×C₆₇	1	1	yes	characteristic, complemented
330	#3009	C₂×C₆₇	1	16	yes	direct product factor
331	C₂×#452	C₂×C₆₇	1	4	yes	complemented
332	C₂×#450	C₂×C₆₇	1	4	yes	complemented
333	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
334	C₂×#416	C₂×C₆₇	1	1	yes	characteristic, complemented
335	C₂×#408	C₂×C₆₇	1	1	yes	characteristic, complemented
336	C₆₇	C₂×#3009	1	1	yes	characteristic, direct product factor, Sylow
337	C₂×H(5,𝔽₂)	—	2	8	no	—
338	C₂×H(5,𝔽₂)	—	2	4	no	—
339	C₂×H(5,𝔽₂)	—	2	8	no	—
340	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	8	yes	—
341	C₂²×#41	—	2	1	no	—
342	C₂²×#41	C₂²×C₆₇	1	2	yes	—
343	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
344	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
345	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
346	C₂³×D₄	—	2	2	no	—
347	C₂³×D₄	—	2	1	no	—
348	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
349	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
350	C₂×#137	—	2	8	no	—
351	C₂×#137	C₂²×C₆₇	1	8	yes	complemented
352	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
353	C₂²×SD₁₆	—	2	8	no	—
354	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
355	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
356	C₂²×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
357	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
358	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
359	#536	—	2	4	no	—
360	#536	C₂²×C₆₇	1	8	yes	complemented
361	C₂×C₄⋊Q₈	—	2	1	no	—
362	C₂×#124	—	2	1	no	—
363	C₂×#123	—	2	2	no	—
364	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
365	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic, complemented
366	#452	C₂²×C₆₇	1	32	yes	complemented
367	#450	C₂²×C₆₇	1	32	yes	complemented
368	#418	C₂²×C₆₇	1	8	yes	complemented
369	#416	C₂²×C₆₇	1	8	yes	complemented
370	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
371	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
372	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
373	#408	C₂²×C₆₇	1	8	yes	complemented
374	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
375	C₂⁵	—	2	2	no	—
376	C₂⁵	—	2	1	no	—
377	H(5,𝔽₂)	—	2	64	no	—
378	H(5,𝔽₂)	—	2	32	no	—
379	H(5,𝔽₂)	—	2	32	no	—
380	C₂×#41	—	2	16	no	—
381	C₂×#41	—	2	8	no	—
382	C₂×#41	—	2	8	no	—
383	C₂×#41	—	2	8	no	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	—	2	4	no	—
386	C₂×#41	—	2	2	no	—
387	C₂×#41	—	2	4	no	—
388	C₂×#41	—	2	4	no	—
389	C₂×#41	D₄×C₆₇	1	8	yes	—
390	C₂×#41	D₄×C₆₇	1	4	yes	—
391	C₂×#41	C₂³×C₆₇	1	8	yes	—
392	C₂×#41	C₂³×C₆₇	1	4	yes	—
393	C₂×#41	C₂³×C₆₇	1	4	yes	—
394	C₂²×Q₈	—	2	1	no	—
395	C₂²×Q₈	—	2	2	no	—
396	C₂²×Q₈	—	2	4	no	—
397	C₂²×Q₈	—	2	4	no	—
398	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
399	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
400	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₆₇	1	4	yes	complemented
402	C₂²×D₄	—	2	32	no	—
403	C₂²×D₄	—	2	16	no	—
404	C₂²×D₄	—	2	2	no	—
405	C₂²×D₄	—	2	1	no	—
406	C₂²×D₄	—	2	16	no	—
407	C₂²×D₄	—	2	8	no	—
408	C₂²×D₄	—	2	8	no	—
409	C₂²×D₄	—	2	8	no	—
410	C₂²×D₄	—	2	2	no	—
411	C₂²×D₄	—	2	1	no	—
412	C₂²×D₄	—	2	4	no	—
413	C₂²×D₄	—	2	2	no	—
414	C₂²×D₄	—	2	4	no	—
415	C₂²×D₄	D₄×C₆₇	1	2	yes	—
416	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
417	C₂²×D₄	D₄×C₆₇	1	8	yes	—
418	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
419	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
420	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
421	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
422	C₂³×C₄	—	2	2	no	—
423	C₂³×C₄	—	2	1	no	—
424	C₂³×C₄	—	2	1	no	—
425	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
426	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
427	#137	—	2	32	no	—
428	C₂×Q₁₆	—	2	8	no	—
429	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	complemented
430	C₂×SD₁₆	—	2	16	no	—
431	C₂×SD₁₆	—	2	32	no	—
432	C₂×SD₁₆	—	2	8	no	—
433	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	complemented
434	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	complemented
435	C₂×M₁₆	—	2	4	no	—
436	C₂×M₁₆	C₂³×C₆₇	1	4	yes	complemented
437	C₂²×C₈	—	2	4	no	—
438	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
439	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
440	C₂²×C₈	C₂³×C₆₇	1	4	yes	complemented
441	C₄⋊Q₈	—	2	8	no	—
442	#124	—	2	8	no	—
443	#123	—	2	16	no	—
444	C₂²⋊Q₈	—	2	8	no	—
445	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	complemented
446	#117	—	2	2	no	—
447	#117	C₂³×C₆₇	1	4	yes	complemented
448	C₂×C₄⋊C₄	—	2	4	no	—
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
450	C₂×C₂²⋊C₄	—	2	1	no	—
451	C₂×C₂²⋊C₄	—	2	4	no	—
452	C₂×C₄²	—	2	1	no	—
453	#103	C₂³×C₆₇	1	8	yes	complemented
454	#102	C₂³×C₆₇	1	8	yes	complemented
455	#102	C₂³×C₆₇	1	16	yes	complemented
456	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
457	C₂⁴	—	2	16	no	—
458	C₂⁴	—	2	2	no	—
459	C₂⁴	—	2	1	no	—
460	C₂⁴	—	2	8	no	—
461	C₂⁴	—	2	8	no	—
462	C₂⁴	—	2	4	no	—
463	C₂⁴	—	8	4	no	—
464	C₂⁴	—	8	8	no	—
465	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
466	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
467	#41	—	2	64	no	—
468	#41	—	2	32	no	—
469	#41	—	2	16	no	—
470	#41	—	2	16	no	—
471	#41	—	2	16	no	—
472	#41	—	2	16	no	—
473	#41	—	2	8	no	—
474	#41	—	2	8	no	—
475	#41	—	2	8	no	—
476	#41	C₂×D₄×C₆₇	1	16	yes	—
477	C₂×Q₈	—	2	4	no	—
478	C₂×Q₈	—	2	2	no	—
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	2	2	no	—
481	C₂×Q₈	—	2	4	no	—
482	C₂×Q₈	—	2	4	no	—
483	C₂×Q₈	—	2	8	no	—
484	C₂×Q₈	—	2	4	no	—
485	C₂×Q₈	—	2	8	no	—
486	C₂×Q₈	—	4	4	no	—
487	C₂×Q₈	—	4	8	no	—
488	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
489	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
490	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
491	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
492	C₂×D₄	—	2	16	no	—
493	C₂×D₄	—	2	16	no	—
494	C₂×D₄	—	2	128	no	—
495	C₂×D₄	—	2	64	no	—
496	C₂×D₄	—	2	32	no	—
497	C₂×D₄	—	2	8	no	—
498	C₂×D₄	—	2	8	no	—
499	C₂×D₄	—	2	8	no	—
500	C₂×D₄	—	2	4	no	—
501	C₂×D₄	—	2	16	no	—
502	C₂×D₄	—	2	16	no	—
503	C₂×D₄	—	2	4	no	—
504	C₂×D₄	—	2	8	no	—
505	C₂×D₄	—	2	16	no	—
506	C₂×D₄	—	2	8	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	—	2	4	no	—
511	C₂×D₄	—	2	2	no	—
512	C₂×D₄	—	2	8	no	—
513	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
514	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
515	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
516	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
517	C₂×D₄	C₂×D₄×C₆₇	1	16	yes	—
518	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
519	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
520	C₂²×C₄	—	2	16	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	8	no	—
523	C₂²×C₄	—	2	8	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	—	2	4	no	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	4	no	—
530	C₂²×C₄	—	4	4	no	—
531	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
532	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
533	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
534	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
535	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
536	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
537	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
538	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
539	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
540	Q₁₆	—	2	16	no	—
541	SD₁₆	—	2	32	no	—
542	SD₁₆	—	2	16	no	—
543	M₁₆	—	2	8	no	—
544	C₂×C₈	—	2	4	no	—
545	C₂×C₈	—	2	8	no	—
546	C₂×C₈	—	2	4	no	—
547	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
548	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
549	C₄⋊C₄	—	2	16	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	16	no	—
554	C₄²	—	2	4	no	—
555	C₂³	—	2	4	no	—
556	C₂³	—	2	32	no	—
557	C₂³	—	2	4	no	—
558	C₂³	—	2	8	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	2	8	no	—
561	C₂³	—	2	4	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	16	no	—
564	C₂³	—	4	8	no	—
565	C₂³	—	4	4	no	—
566	C₂³	—	8	16	no	—
567	C₂³	—	8	8	no	—
568	C₂³	—	8	16	no	—
569	C₂³	—	8	32	no	—
570	C₂³	#120×C₆₇	1	4	yes	—
571	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
572	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
573	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
574	Q₈	—	2	4	no	—
575	Q₈	—	2	8	no	—
576	Q₈	—	2	2	no	—
577	Q₈	—	2	4	no	—
578	Q₈	—	2	4	no	—
579	Q₈	—	4	4	no	—
580	Q₈	—	4	8	no	—
581	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
582	D₄	—	2	32	no	—
583	D₄	—	2	32	no	—
584	D₄	—	2	16	no	—
585	D₄	—	2	16	no	—
586	D₄	—	2	8	no	—
587	D₄	—	2	4	no	—
588	D₄	—	2	16	no	—
589	D₄	—	2	8	no	—
590	D₄	—	2	4	no	—
591	D₄	—	2	2	no	—
592	D₄	C₂²×D₄×C₆₇	1	4	yes	—
593	D₄	C₂²×D₄×C₆₇	1	8	yes	—
594	C₂×C₄	—	2	32	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	16	no	—
599	C₂×C₄	—	2	8	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	4	8	no	—
605	C₂×C₄	—	4	4	no	—
606	C₂×C₄	—	4	4	no	—
607	C₂×C₄	#120×C₆₇	1	4	yes	—
608	C₂×C₄	#120×C₆₇	1	8	yes	—
609	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
610	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
611	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
612	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
613	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
614	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
615	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
616	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
617	C₈	—	2	4	no	—
618	C₈	—	2	4	no	—
619	C₂²	—	2	8	no	—
620	C₂²	—	2	4	no	—
621	C₂²	—	2	8	no	—
622	C₂²	—	2	4	no	—
623	C₂²	—	2	8	no	—
624	C₂²	—	2	4	no	—
625	C₂²	—	4	8	no	—
626	C₂²	—	4	16	no	—
627	C₂²	—	4	8	no	—
628	C₂²	—	4	8	no	—
629	C₂²	—	4	4	no	—
630	C₂²	—	8	16	no	—
631	C₂²	—	8	32	no	—
632	C₂²	C₂×#120×C₆₇	1	4	yes	—
633	C₂²	C₂×#120×C₆₇	1	2	yes	—
634	C₂²	#577×C₆₇	1	4	yes	—
635	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
636	C₄	—	2	4	no	—
637	C₄	—	2	4	no	—
638	C₄	—	4	4	no	—
639	C₄	—	4	4	no	—
640	C₄	C₂×#120×C₆₇	1	4	yes	—
641	C₄	C₂×#120×C₆₇	1	2	yes	—
642	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
643	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
644	C₂	—	2	4	no	—
645	C₂	—	4	4	no	—
646	C₂	—	4	8	no	—
647	C₂	—	4	4	no	—
648	C₂	C₆₇×#3009	1	4	yes	direct product factor
649	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
650	C₂	C₂×#577×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×H(5,𝔽₂)×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×#137×C₆₇	C₂	semidirect
3. C₂³×SD₁₆×C₆₇	C₂	semidirect
4. C₂×#536×C₆₇	C₂	semidirect
5. C₆₇×#3009	C₂	direct (N × H)
6. C₂×#452×C₆₇	C₂	semidirect
7. C₂×#450×C₆₇	C₂	semidirect
8. C₂×#418×C₆₇	C₂	semidirect
9. C₂×#416×C₆₇	C₂	semidirect
10. C₂×#408×C₆₇	C₂	semidirect
19. C₂³×Q₈×C₆₇	C₂²	semidirect
25. C₂×#137×C₆₇	C₂²	semidirect
26. C₂²×Q₁₆×C₆₇	C₂²	semidirect
28. C₂²×SD₁₆×C₆₇	C₂²	semidirect
29. C₂²×SD₁₆×C₆₇	C₂²	semidirect
30. C₂²×SD₁₆×C₆₇	C₂²	semidirect
31. C₂²×M₁₆×C₆₇	C₂²	semidirect
32. C₂³×C₈×C₆₇	C₂²	semidirect
34. #536×C₆₇	C₂²	semidirect
38. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
39. C₂×#117×C₆₇	C₂²	semidirect
40. #452×C₆₇	C₂²	semidirect
41. #450×C₆₇	C₂²	semidirect
42. #418×C₆₇	C₂²	semidirect
43. #416×C₆₇	C₂²	semidirect
44. C₂×#103×C₆₇	C₂²	semidirect
45. C₂×#102×C₆₇	C₂²	semidirect
46. C₂×#102×C₆₇	C₂²	semidirect
47. #408×C₆₇	C₂²	semidirect
48. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
75. C₂²×Q₈×C₆₇	C₂³	semidirect
103. C₂×Q₁₆×C₆₇	C₂³	semidirect
107. C₂×SD₁₆×C₆₇	C₂³	semidirect
108. C₂×SD₁₆×C₆₇	C₂³	semidirect
110. C₂×M₁₆×C₆₇	C₂³	semidirect
114. C₂²×C₈×C₆₇	C₂³	semidirect
119. C₂²⋊Q₈×C₆₇	C₂³	semidirect
121. #117×C₆₇	C₂³	semidirect
127. #103×C₆₇	C₂³	semidirect
128. #102×C₆₇	C₂³	semidirect
129. #102×C₆₇	C₂³	semidirect
130. C₂²⋊C₈×C₆₇	C₂³	semidirect
318. C₂×#3009	C₆₇	direct (N × H)
323. C₂×C₆₇	#3009	direct (N × H)
327. C₂²×#137	C₂×C₆₇	semidirect
328. C₂³×SD₁₆	C₂×C₆₇	semidirect
329. C₂×#536	C₂×C₆₇	semidirect
330. #3009	C₂×C₆₇	direct (N × H)
331. C₂×#452	C₂×C₆₇	semidirect
332. C₂×#450	C₂×C₆₇	semidirect
333. C₂×#418	C₂×C₆₇	semidirect
334. C₂×#416	C₂×C₆₇	semidirect
335. C₂×#408	C₂×C₆₇	semidirect
336. C₆₇	C₂×#3009	direct (N × H)
345. C₂³×Q₈	C₂²×C₆₇	semidirect
351. C₂×#137	C₂²×C₆₇	semidirect
352. C₂²×Q₁₆	C₂²×C₆₇	semidirect
354. C₂²×SD₁₆	C₂²×C₆₇	semidirect
355. C₂²×SD₁₆	C₂²×C₆₇	semidirect
356. C₂²×SD₁₆	C₂²×C₆₇	semidirect
357. C₂²×M₁₆	C₂²×C₆₇	semidirect
358. C₂³×C₈	C₂²×C₆₇	semidirect
360. #536	C₂²×C₆₇	semidirect
364. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
365. C₂×#117	C₂²×C₆₇	semidirect
366. #452	C₂²×C₆₇	semidirect
367. #450	C₂²×C₆₇	semidirect
368. #418	C₂²×C₆₇	semidirect
369. #416	C₂²×C₆₇	semidirect
370. C₂×#103	C₂²×C₆₇	semidirect
371. C₂×#102	C₂²×C₆₇	semidirect
372. C₂×#102	C₂²×C₆₇	semidirect
373. #408	C₂²×C₆₇	semidirect
374. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
401. C₂²×Q₈	C₂³×C₆₇	semidirect
429. C₂×Q₁₆	C₂³×C₆₇	semidirect
433. C₂×SD₁₆	C₂³×C₆₇	semidirect
434. C₂×SD₁₆	C₂³×C₆₇	semidirect
436. C₂×M₁₆	C₂³×C₆₇	semidirect
440. C₂²×C₈	C₂³×C₆₇	semidirect
445. C₂²⋊Q₈	C₂³×C₆₇	semidirect
447. #117	C₂³×C₆₇	semidirect
453. #103	C₂³×C₆₇	semidirect
454. #102	C₂³×C₆₇	semidirect
455. #102	C₂³×C₆₇	semidirect
456. C₂²⋊C₈	C₂³×C₆₇	semidirect
648. C₂	C₆₇×#3009	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×H(5,𝔽₂)×C₆₇	2
C₂²×#137×C₆₇	2
C₂³×SD₁₆×C₆₇	2
C₂×#536×C₆₇	2
C₆₇×#3009	2
C₂×#452×C₆₇	2
C₂×#450×C₆₇	2
C₂×#418×C₆₇	2
C₂×#416×C₆₇	2
C₂×#408×C₆₇	2
C₂×#3009	67

Subgroups — C2×C67×#3009 #30739612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3009 #30739612