Subgroups — C₂×C₆₇×#3011 #30739614

Order 17152.

4718 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#137×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3011	C₂	1	16	yes	maximal, direct product factor
6	C₂×#452×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#451×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#419×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	C₂×2₋¹⁺⁴×C₆₇	—	2	8	no	—
12	C₂×2₋¹⁺⁴×C₆₇	—	2	4	no	—
13	C₂×2₋¹⁺⁴×C₆₇	—	2	8	no	—
14	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	8	yes	—
15	C₂²×#41×C₆₇	—	2	2	no	—
16	C₂²×#41×C₆₇	—	2	1	no	—
17	C₂²×#41×C₆₇	C₂²	1	2	yes	—
18	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×Q₈×C₆₇	—	2	1	no	—
21	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂³×Q₈×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	—	2	8	no	—
27	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
28	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
29	C₂²×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
30	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
31	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
32	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
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	—
39	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
40	#452×C₆₇	C₂²	1	32	yes	complemented
41	#451×C₆₇	C₂²	1	32	yes	complemented
42	#419×C₆₇	C₂²	1	8	yes	complemented
43	#416×C₆₇	C₂²	1	8	yes	complemented
44	C₂×#103×C₆₇	C₂²	1	2	yes	—
45	C₂×#103×C₆₇	C₂²	1	4	yes	—
46	C₂×#102×C₆₇	C₂²	1	2	yes	—
47	#408×C₆₇	C₂²	1	8	yes	complemented
48	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
49	2₋¹⁺⁴×C₆₇	—	2	64	no	—
50	2₋¹⁺⁴×C₆₇	—	2	32	no	—
51	2₋¹⁺⁴×C₆₇	—	2	32	no	—
52	C₂×#41×C₆₇	—	2	32	no	—
53	C₂×#41×C₆₇	—	2	16	no	—
54	C₂×#41×C₆₇	—	2	8	no	—
55	C₂×#41×C₆₇	—	2	16	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	16	no	—
69	C₂²×Q₈×C₆₇	—	2	8	no	—
70	C₂²×Q₈×C₆₇	—	2	8	no	—
71	C₂²×Q₈×C₆₇	—	2	8	no	—
72	C₂²×Q₈×C₆₇	—	2	2	no	—
73	C₂²×Q₈×C₆₇	—	2	1	no	—
74	C₂²×Q₈×C₆₇	—	2	4	no	—
75	C₂²×Q₈×C₆₇	—	2	4	no	—
76	C₂²×Q₈×C₆₇	—	2	2	no	—
77	C₂²×Q₈×C₆₇	—	2	1	no	—
78	C₂²×Q₈×C₆₇	—	2	2	no	—
79	C₂²×Q₈×C₆₇	—	2	4	no	—
80	C₂²×Q₈×C₆₇	—	2	4	no	—
81	C₂²×Q₈×C₆₇	D₄	1	8	yes	—
82	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
83	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
85	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
86	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
88	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
89	C₂²×D₄×C₆₇	—	2	2	no	—
90	C₂²×D₄×C₆₇	—	2	1	no	—
91	C₂²×D₄×C₆₇	D₄	1	2	yes	—
92	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
93	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₂³×C₄×C₆₇	—	2	2	no	—
95	C₂³×C₄×C₆₇	—	2	1	no	—
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	16	no	—
103	C₂×Q₁₆×C₆₇	—	2	32	no	—
104	C₂×Q₁₆×C₆₇	—	2	8	no	—
105	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
106	C₂×Q₁₆×C₆₇	C₂³	1	16	yes	—
107	C₂×SD₁₆×C₆₇	—	2	8	no	—
108	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
109	C₂×M₁₆×C₆₇	—	2	4	no	—
110	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
111	C₂²×C₈×C₆₇	—	2	4	no	—
112	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
113	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
114	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
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	—
120	#117×C₆₇	—	2	2	no	—
121	#117×C₆₇	C₂³	1	4	yes	—
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	—
128	#103×C₆₇	C₂³	1	16	yes	—
129	#102×C₆₇	C₂³	1	8	yes	—
130	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	—
131	C₂⁴×C₆₇	—	2	1	no	—
132	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
133	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
134	#41×C₆₇	—	2	128	no	—
135	#41×C₆₇	—	2	64	no	—
136	#41×C₆₇	—	2	32	no	—
137	#41×C₆₇	—	2	16	no	—
138	#41×C₆₇	—	2	16	no	—
139	#41×C₆₇	—	2	16	no	—
140	#41×C₆₇	—	2	16	no	—
141	#41×C₆₇	—	2	8	no	—
142	#41×C₆₇	—	2	8	no	—
143	#41×C₆₇	—	2	8	no	—
144	#41×C₆₇	C₂×D₄	1	16	yes	—
145	C₂×Q₈×C₆₇	—	2	16	no	—
146	C₂×Q₈×C₆₇	—	2	64	no	—
147	C₂×Q₈×C₆₇	—	2	32	no	—
148	C₂×Q₈×C₆₇	—	2	16	no	—
149	C₂×Q₈×C₆₇	—	2	16	no	—
150	C₂×Q₈×C₆₇	—	2	8	no	—
151	C₂×Q₈×C₆₇	—	2	4	no	—
152	C₂×Q₈×C₆₇	—	2	16	no	—
153	C₂×Q₈×C₆₇	—	2	2	no	—
154	C₂×Q₈×C₆₇	—	2	4	no	—
155	C₂×Q₈×C₆₇	—	2	8	no	—
156	C₂×Q₈×C₆₇	—	2	4	no	—
157	C₂×Q₈×C₆₇	—	2	1	no	—
158	C₂×Q₈×C₆₇	—	2	2	no	—
159	C₂×Q₈×C₆₇	—	2	8	no	—
160	C₂×Q₈×C₆₇	—	2	2	no	—
161	C₂×Q₈×C₆₇	—	2	4	no	—
162	C₂×Q₈×C₆₇	—	2	4	no	—
163	C₂×Q₈×C₆₇	—	2	4	no	—
164	C₂×Q₈×C₆₇	—	2	8	no	—
165	C₂×Q₈×C₆₇	—	2	8	no	—
166	C₂×Q₈×C₆₇	—	4	4	no	—
167	C₂×Q₈×C₆₇	—	4	8	no	—
168	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
169	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
170	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	—
171	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
172	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
173	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
174	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
175	C₂×D₄×C₆₇	—	2	16	no	—
176	C₂×D₄×C₆₇	—	2	8	no	—
177	C₂×D₄×C₆₇	—	2	8	no	—
178	C₂×D₄×C₆₇	—	2	8	no	—
179	C₂×D₄×C₆₇	—	2	4	no	—
180	C₂×D₄×C₆₇	—	2	4	no	—
181	C₂×D₄×C₆₇	—	2	1	no	—
182	C₂×D₄×C₆₇	—	2	2	no	—
183	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
184	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
185	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
186	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₆₇	—	2	16	no	—
188	C₂²×C₄×C₆₇	—	2	16	no	—
189	C₂²×C₄×C₆₇	—	2	8	no	—
190	C₂²×C₄×C₆₇	—	2	8	no	—
191	C₂²×C₄×C₆₇	—	2	4	no	—
192	C₂²×C₄×C₆₇	—	2	2	no	—
193	C₂²×C₄×C₆₇	—	2	1	no	—
194	C₂²×C₄×C₆₇	—	2	8	no	—
195	C₂²×C₄×C₆₇	—	2	8	no	—
196	C₂²×C₄×C₆₇	—	2	4	no	—
197	C₂²×C₄×C₆₇	—	2	4	no	—
198	C₂²×C₄×C₆₇	—	2	4	no	—
199	C₂²×C₄×C₆₇	—	2	4	no	—
200	C₂²×C₄×C₆₇	—	2	2	no	—
201	C₂²×C₄×C₆₇	—	2	2	no	—
202	C₂²×C₄×C₆₇	—	4	4	no	—
203	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
204	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
205	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
206	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
208	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
209	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
210	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
211	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
212	Q₁₆×C₆₇	—	2	32	no	—
213	Q₁₆×C₆₇	—	2	16	no	—
214	SD₁₆×C₆₇	—	2	16	no	—
215	M₁₆×C₆₇	—	2	8	no	—
216	C₂×C₈×C₆₇	—	2	4	no	—
217	C₂×C₈×C₆₇	—	2	8	no	—
218	C₂×C₈×C₆₇	—	2	4	no	—
219	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
220	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
221	C₄⋊C₄×C₆₇	—	2	16	no	—
222	C₄⋊C₄×C₆₇	—	2	2	no	—
223	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
224	C₂²⋊C₄×C₆₇	—	2	4	no	—
225	C₂²⋊C₄×C₆₇	—	2	16	no	—
226	C₄²×C₆₇	—	2	4	no	—
227	C₂³×C₆₇	—	2	4	no	—
228	C₂³×C₆₇	—	2	4	no	—
229	C₂³×C₆₇	—	2	4	no	—
230	C₂³×C₆₇	—	2	8	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	4	4	no	—
233	C₂³×C₆₇	#120	1	4	yes	—
234	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
235	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
236	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
237	Q₈×C₆₇	—	2	32	no	—
238	Q₈×C₆₇	—	2	4	no	—
239	Q₈×C₆₇	—	2	16	no	—
240	Q₈×C₆₇	—	2	8	no	—
241	Q₈×C₆₇	—	2	8	no	—
242	Q₈×C₆₇	—	2	2	no	—
243	Q₈×C₆₇	—	2	4	no	—
244	Q₈×C₆₇	—	2	4	no	—
245	Q₈×C₆₇	—	2	4	no	—
246	Q₈×C₆₇	—	4	4	no	—
247	Q₈×C₆₇	—	4	8	no	—
248	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
249	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
250	D₄×C₆₇	—	2	32	no	—
251	D₄×C₆₇	—	2	16	no	—
252	D₄×C₆₇	—	2	16	no	—
253	D₄×C₆₇	—	2	8	no	—
254	D₄×C₆₇	—	2	4	no	—
255	D₄×C₆₇	—	2	2	no	—
256	D₄×C₆₇	C₂²×D₄	1	4	yes	—
257	C₂×C₄×C₆₇	—	2	32	no	—
258	C₂×C₄×C₆₇	—	2	32	no	—
259	C₂×C₄×C₆₇	—	2	8	no	—
260	C₂×C₄×C₆₇	—	2	4	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	16	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	16	no	—
266	C₂×C₄×C₆₇	—	2	8	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	4	8	no	—
272	C₂×C₄×C₆₇	—	4	4	no	—
273	C₂×C₄×C₆₇	—	4	4	no	—
274	C₂×C₄×C₆₇	#120	1	4	yes	—
275	C₂×C₄×C₆₇	#120	1	8	yes	—
276	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
277	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
282	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
283	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
284	C₈×C₆₇	—	2	4	no	—
285	C₈×C₆₇	—	2	4	no	—
286	C₂²×C₆₇	—	2	8	no	—
287	C₂²×C₆₇	—	2	4	no	—
288	C₂²×C₆₇	—	2	4	no	—
289	C₂²×C₆₇	—	2	4	no	—
290	C₂²×C₆₇	—	2	8	no	—
291	C₂²×C₆₇	—	4	8	no	—
292	C₂²×C₆₇	—	4	8	no	—
293	C₂²×C₆₇	—	4	4	no	—
294	C₂²×C₆₇	C₂×#120	1	4	yes	—
295	C₂²×C₆₇	C₂×#120	1	2	yes	—
296	C₂²×C₆₇	#578	1	4	yes	—
297	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
298	C₄×C₆₇	—	2	8	no	—
299	C₄×C₆₇	—	2	4	no	—
300	C₄×C₆₇	—	2	4	no	—
301	C₄×C₆₇	—	4	4	no	—
302	C₄×C₆₇	—	4	4	no	—
303	C₄×C₆₇	C₂×#120	1	4	yes	—
304	C₄×C₆₇	C₂×#120	1	2	yes	—
305	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
306	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
307	C₂×#3011	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂×C₆₇	—	2	4	no	—
309	C₂×C₆₇	—	4	4	no	—
310	C₂×C₆₇	—	4	4	no	—
311	C₂×C₆₇	#3011	1	4	yes	direct product factor
312	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
313	C₂×C₆₇	C₂×#578	1	2	yes	—
314	C₂²×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic
315	C₂²×#137	C₂×C₆₇	1	1	yes	characteristic, complemented
316	C₂³×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
317	C₂×#536	C₂×C₆₇	1	1	yes	characteristic, complemented
318	#3011	C₂×C₆₇	1	16	yes	direct product factor
319	C₂×#452	C₂×C₆₇	1	4	yes	complemented
320	C₂×#451	C₂×C₆₇	1	4	yes	complemented
321	C₂×#419	C₂×C₆₇	1	1	yes	characteristic, complemented
322	C₂×#416	C₂×C₆₇	1	1	yes	characteristic, complemented
323	C₂×#408	C₂×C₆₇	1	1	yes	characteristic, complemented
324	C₆₇	C₂×#3011	1	1	yes	characteristic, direct product factor, Sylow
325	C₂×2₋¹⁺⁴	—	2	8	no	—
326	C₂×2₋¹⁺⁴	—	2	4	no	—
327	C₂×2₋¹⁺⁴	—	2	8	no	—
328	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	8	yes	—
329	C₂²×#41	—	2	2	no	—
330	C₂²×#41	—	2	1	no	—
331	C₂²×#41	C₂²×C₆₇	1	2	yes	—
332	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
333	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
334	C₂³×Q₈	—	2	1	no	—
335	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
336	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
337	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×#137	—	2	8	no	—
339	C₂×#137	C₂²×C₆₇	1	8	yes	complemented
340	C₂²×Q₁₆	—	2	8	no	—
341	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
342	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
343	C₂²×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
344	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
345	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
346	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
347	#536	—	2	4	no	—
348	#536	C₂²×C₆₇	1	8	yes	complemented
349	C₂×C₄⋊Q₈	—	2	1	no	—
350	C₂×#124	—	2	1	no	—
351	C₂×#123	—	2	2	no	—
352	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
353	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
354	#452	C₂²×C₆₇	1	32	yes	complemented
355	#451	C₂²×C₆₇	1	32	yes	complemented
356	#419	C₂²×C₆₇	1	8	yes	complemented
357	#416	C₂²×C₆₇	1	8	yes	complemented
358	C₂×#103	C₂²×C₆₇	1	2	yes	—
359	C₂×#103	C₂²×C₆₇	1	4	yes	—
360	C₂×#102	C₂²×C₆₇	1	2	yes	—
361	#408	C₂²×C₆₇	1	8	yes	complemented
362	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
363	2₋¹⁺⁴	—	2	64	no	—
364	2₋¹⁺⁴	—	2	32	no	—
365	2₋¹⁺⁴	—	2	32	no	—
366	C₂×#41	—	2	32	no	—
367	C₂×#41	—	2	16	no	—
368	C₂×#41	—	2	8	no	—
369	C₂×#41	—	2	16	no	—
370	C₂×#41	—	2	8	no	—
371	C₂×#41	—	2	8	no	—
372	C₂×#41	—	2	4	no	—
373	C₂×#41	—	2	4	no	—
374	C₂×#41	—	2	2	no	—
375	C₂×#41	—	2	4	no	—
376	C₂×#41	—	2	4	no	—
377	C₂×#41	D₄×C₆₇	1	8	yes	—
378	C₂×#41	D₄×C₆₇	1	4	yes	—
379	C₂×#41	C₂³×C₆₇	1	8	yes	—
380	C₂×#41	C₂³×C₆₇	1	4	yes	—
381	C₂×#41	C₂³×C₆₇	1	4	yes	—
382	C₂²×Q₈	—	2	16	no	—
383	C₂²×Q₈	—	2	8	no	—
384	C₂²×Q₈	—	2	8	no	—
385	C₂²×Q₈	—	2	8	no	—
386	C₂²×Q₈	—	2	2	no	—
387	C₂²×Q₈	—	2	1	no	—
388	C₂²×Q₈	—	2	4	no	—
389	C₂²×Q₈	—	2	4	no	—
390	C₂²×Q₈	—	2	2	no	—
391	C₂²×Q₈	—	2	1	no	—
392	C₂²×Q₈	—	2	2	no	—
393	C₂²×Q₈	—	2	4	no	—
394	C₂²×Q₈	—	2	4	no	—
395	C₂²×Q₈	D₄×C₆₇	1	8	yes	—
396	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
397	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
398	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
399	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
400	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
402	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
403	C₂²×D₄	—	2	2	no	—
404	C₂²×D₄	—	2	1	no	—
405	C₂²×D₄	D₄×C₆₇	1	2	yes	—
406	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
407	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂³×C₄	—	2	2	no	—
409	C₂³×C₄	—	2	1	no	—
410	C₂³×C₄	—	2	2	no	—
411	C₂³×C₄	—	2	1	no	—
412	C₂³×C₄	—	2	1	no	—
413	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
414	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
415	#137	—	2	32	no	—
416	C₂×Q₁₆	—	2	16	no	—
417	C₂×Q₁₆	—	2	32	no	—
418	C₂×Q₁₆	—	2	8	no	—
419	C₂×Q₁₆	C₂³×C₆₇	1	16	yes	—
420	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
421	C₂×SD₁₆	—	2	8	no	—
422	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
423	C₂×M₁₆	—	2	4	no	—
424	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
425	C₂²×C₈	—	2	4	no	—
426	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
427	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
428	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
429	C₄⋊Q₈	—	2	8	no	—
430	#124	—	2	8	no	—
431	#123	—	2	16	no	—
432	C₂²⋊Q₈	—	2	8	no	—
433	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
434	#117	—	2	2	no	—
435	#117	C₂³×C₆₇	1	4	yes	—
436	C₂×C₄⋊C₄	—	2	4	no	—
437	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	4	no	—
440	C₂×C₄²	—	2	1	no	—
441	#103	C₂³×C₆₇	1	8	yes	—
442	#103	C₂³×C₆₇	1	16	yes	—
443	#102	C₂³×C₆₇	1	8	yes	—
444	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	—
445	C₂⁴	—	2	1	no	—
446	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
447	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
448	#41	—	2	128	no	—
449	#41	—	2	64	no	—
450	#41	—	2	32	no	—
451	#41	—	2	16	no	—
452	#41	—	2	16	no	—
453	#41	—	2	16	no	—
454	#41	—	2	16	no	—
455	#41	—	2	8	no	—
456	#41	—	2	8	no	—
457	#41	—	2	8	no	—
458	#41	C₂×D₄×C₆₇	1	16	yes	—
459	C₂×Q₈	—	2	16	no	—
460	C₂×Q₈	—	2	64	no	—
461	C₂×Q₈	—	2	32	no	—
462	C₂×Q₈	—	2	16	no	—
463	C₂×Q₈	—	2	16	no	—
464	C₂×Q₈	—	2	8	no	—
465	C₂×Q₈	—	2	4	no	—
466	C₂×Q₈	—	2	16	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	2	4	no	—
469	C₂×Q₈	—	2	8	no	—
470	C₂×Q₈	—	2	4	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	2	8	no	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	—	2	4	no	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×Q₈	—	2	4	no	—
478	C₂×Q₈	—	2	8	no	—
479	C₂×Q₈	—	2	8	no	—
480	C₂×Q₈	—	4	4	no	—
481	C₂×Q₈	—	4	8	no	—
482	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
483	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
484	C₂×Q₈	C₂×D₄×C₆₇	1	16	yes	—
485	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
486	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
487	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
488	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
489	C₂×D₄	—	2	16	no	—
490	C₂×D₄	—	2	8	no	—
491	C₂×D₄	—	2	8	no	—
492	C₂×D₄	—	2	8	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
498	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
499	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
500	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
501	C₂²×C₄	—	2	16	no	—
502	C₂²×C₄	—	2	16	no	—
503	C₂²×C₄	—	2	8	no	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	8	no	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	—	2	8	no	—
509	C₂²×C₄	—	2	8	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	4	4	no	—
517	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
518	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
519	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
520	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
521	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
522	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
523	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
524	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
525	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
526	Q₁₆	—	2	32	no	—
527	Q₁₆	—	2	16	no	—
528	SD₁₆	—	2	16	no	—
529	M₁₆	—	2	8	no	—
530	C₂×C₈	—	2	4	no	—
531	C₂×C₈	—	2	8	no	—
532	C₂×C₈	—	2	4	no	—
533	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
534	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
535	C₄⋊C₄	—	2	16	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	16	no	—
540	C₄²	—	2	4	no	—
541	C₂³	—	2	4	no	—
542	C₂³	—	2	4	no	—
543	C₂³	—	2	4	no	—
544	C₂³	—	2	8	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	4	4	no	—
547	C₂³	#120×C₆₇	1	4	yes	—
548	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
549	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
550	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	Q₈	—	2	32	no	—
552	Q₈	—	2	4	no	—
553	Q₈	—	2	16	no	—
554	Q₈	—	2	8	no	—
555	Q₈	—	2	8	no	—
556	Q₈	—	2	2	no	—
557	Q₈	—	2	4	no	—
558	Q₈	—	2	4	no	—
559	Q₈	—	2	4	no	—
560	Q₈	—	4	4	no	—
561	Q₈	—	4	8	no	—
562	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
563	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
564	D₄	—	2	32	no	—
565	D₄	—	2	16	no	—
566	D₄	—	2	16	no	—
567	D₄	—	2	8	no	—
568	D₄	—	2	4	no	—
569	D₄	—	2	2	no	—
570	D₄	C₂²×D₄×C₆₇	1	4	yes	—
571	C₂×C₄	—	2	32	no	—
572	C₂×C₄	—	2	32	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	16	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	16	no	—
580	C₂×C₄	—	2	8	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	4	8	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	#120×C₆₇	1	4	yes	—
589	C₂×C₄	#120×C₆₇	1	8	yes	—
590	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
591	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
592	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
596	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
597	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
598	C₈	—	2	4	no	—
599	C₈	—	2	4	no	—
600	C₂²	—	2	8	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	2	8	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	2	4	no	—
605	C₂²	—	4	8	no	—
606	C₂²	—	4	8	no	—
607	C₂²	—	4	4	no	—
608	C₂²	C₂×#120×C₆₇	1	4	yes	—
609	C₂²	C₂×#120×C₆₇	1	2	yes	—
610	C₂²	#578×C₆₇	1	4	yes	—
611	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
612	C₄	—	2	8	no	—
613	C₄	—	2	4	no	—
614	C₄	—	2	4	no	—
615	C₄	—	4	4	no	—
616	C₄	—	4	4	no	—
617	C₄	C₂×#120×C₆₇	1	4	yes	—
618	C₄	C₂×#120×C₆₇	1	2	yes	—
619	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
620	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
621	C₂	—	2	4	no	—
622	C₂	—	4	4	no	—
623	C₂	—	4	4	no	—
624	C₂	C₆₇×#3011	1	4	yes	direct product factor
625	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
626	C₂	C₂×#578×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×2₋¹⁺⁴×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×Q₁₆×C₆₇	C₂	semidirect
4. C₂×#536×C₆₇	C₂	semidirect
5. C₆₇×#3011	C₂	direct (N × H)
6. C₂×#452×C₆₇	C₂	semidirect
7. C₂×#451×C₆₇	C₂	semidirect
8. C₂×#419×C₆₇	C₂	semidirect
9. C₂×#416×C₆₇	C₂	semidirect
10. C₂×#408×C₆₇	C₂	semidirect
25. C₂×#137×C₆₇	C₂²	semidirect
29. C₂²×Q₁₆×C₆₇	C₂²	semidirect
34. #536×C₆₇	C₂²	semidirect
40. #452×C₆₇	C₂²	semidirect
41. #451×C₆₇	C₂²	semidirect
42. #419×C₆₇	C₂²	semidirect
43. #416×C₆₇	C₂²	semidirect
47. #408×C₆₇	C₂²	semidirect
307. C₂×#3011	C₆₇	direct (N × H)
311. C₂×C₆₇	#3011	direct (N × H)
315. C₂²×#137	C₂×C₆₇	semidirect
316. C₂³×Q₁₆	C₂×C₆₇	semidirect
317. C₂×#536	C₂×C₆₇	semidirect
318. #3011	C₂×C₆₇	direct (N × H)
319. C₂×#452	C₂×C₆₇	semidirect
320. C₂×#451	C₂×C₆₇	semidirect
321. C₂×#419	C₂×C₆₇	semidirect
322. C₂×#416	C₂×C₆₇	semidirect
323. C₂×#408	C₂×C₆₇	semidirect
324. C₆₇	C₂×#3011	direct (N × H)
339. C₂×#137	C₂²×C₆₇	semidirect
343. C₂²×Q₁₆	C₂²×C₆₇	semidirect
348. #536	C₂²×C₆₇	semidirect
354. #452	C₂²×C₆₇	semidirect
355. #451	C₂²×C₆₇	semidirect
356. #419	C₂²×C₆₇	semidirect
357. #416	C₂²×C₆₇	semidirect
361. #408	C₂²×C₆₇	semidirect
624. C₂	C₆₇×#3011	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₆₇	2
C₂²×#137×C₆₇	2
C₂³×Q₁₆×C₆₇	2
C₂×#536×C₆₇	2
C₆₇×#3011	2
C₂×#452×C₆₇	2
C₂×#451×C₆₇	2
C₂×#419×C₆₇	2
C₂×#416×C₆₇	2
C₂×#408×C₆₇	2
C₂×#3011	67

Subgroups — C2×C67×#3011 #30739614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3011 #30739614