Subgroups — C₂×C₆₇×#3007 #30739609

Order 17152.

5078 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#137×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3007	C₂	1	16	yes	maximal, direct product factor
5	C₂×#452×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#451×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#450×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#417×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	—	2	16	no	—
11	C₂²×#41×C₆₇	—	2	1	no	—
12	C₂²×#41×C₆₇	—	2	4	no	—
13	C₂²×#41×C₆₇	—	2	1	no	—
14	C₂²×#41×C₆₇	C₂²	1	2	yes	—
15	C₂²×#41×C₆₇	C₂²	1	8	yes	—
16	C₂²×#41×C₆₇	C₂²	1	2	yes	—
17	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₄×C₆₇	—	2	1	no	—
20	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄×C₆₇	—	2	1	no	—
22	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#137×C₆₇	—	2	16	no	—
24	C₂×#137×C₆₇	C₂²	1	16	yes	complemented
25	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
26	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
27	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
29	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
30	C₂×C₂²⋊Q₈×C₆₇	—	2	4	no	—
31	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
32	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
33	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
34	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
37	#452×C₆₇	C₂²	1	32	yes	complemented
38	#451×C₆₇	C₂²	1	16	yes	complemented
39	#450×C₆₇	C₂²	1	16	yes	complemented
40	#417×C₆₇	C₂²	1	16	yes	complemented
41	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
42	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
43	#407×C₆₇	C₂²	1	8	yes	complemented
44	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
45	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
46	C₂⁵×C₆₇	—	2	1	no	—
47	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
48	C₂×#41×C₆₇	—	2	16	no	—
49	C₂×#41×C₆₇	—	2	64	no	—
50	C₂×#41×C₆₇	—	2	16	no	—
51	C₂×#41×C₆₇	—	2	64	no	—
52	C₂×#41×C₆₇	—	2	8	no	—
53	C₂×#41×C₆₇	—	2	8	no	—
54	C₂×#41×C₆₇	—	2	8	no	—
55	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	—	2	8	no	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	D₄	1	16	yes	—
60	C₂×#41×C₆₇	C₂³	1	8	yes	—
61	C₂×#41×C₆₇	C₂³	1	8	yes	—
62	C₂²×Q₈×C₆₇	—	2	8	no	—
63	C₂²×Q₈×C₆₇	—	2	2	no	—
64	C₂²×Q₈×C₆₇	—	2	1	no	—
65	C₂²×Q₈×C₆₇	—	2	2	no	—
66	C₂²×Q₈×C₆₇	—	2	1	no	—
67	C₂²×Q₈×C₆₇	—	2	8	no	—
68	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
69	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
70	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×Q₈×C₆₇	C₂³	1	4	yes	complemented
72	C₂²×D₄×C₆₇	—	2	16	no	—
73	C₂²×D₄×C₆₇	—	2	8	no	—
74	C₂²×D₄×C₆₇	—	2	2	no	—
75	C₂²×D₄×C₆₇	—	2	2	no	—
76	C₂²×D₄×C₆₇	—	2	8	no	—
77	C₂²×D₄×C₆₇	—	2	2	no	—
78	C₂²×D₄×C₆₇	—	2	1	no	—
79	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
80	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	—	2	16	no	—
82	C₂³×C₄×C₆₇	—	2	1	no	—
83	C₂³×C₄×C₆₇	—	2	8	no	—
84	C₂³×C₄×C₆₇	—	2	2	no	—
85	C₂³×C₄×C₆₇	—	2	2	no	—
86	C₂³×C₄×C₆₇	—	2	8	no	—
87	C₂³×C₄×C₆₇	—	2	1	no	—
88	C₂³×C₄×C₆₇	—	2	1	no	—
89	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
90	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
91	C₂³×C₄×C₆₇	D₄	1	4	yes	—
92	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
93	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
95	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
96	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
97	#137×C₆₇	—	2	64	no	—
98	C₂×Q₁₆×C₆₇	—	2	16	no	—
99	C₂×Q₁₆×C₆₇	C₂³	1	16	yes	complemented
100	C₂×SD₁₆×C₆₇	—	2	16	no	—
101	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	complemented
102	C₂×M₁₆×C₆₇	—	2	8	no	—
103	C₂×M₁₆×C₆₇	C₂³	1	8	yes	complemented
104	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
105	C₂²⋊Q₈×C₆₇	—	2	4	no	—
106	C₂²⋊Q₈×C₆₇	—	2	8	no	—
107	C₂²⋊Q₈×C₆₇	—	2	8	no	—
108	C₂²⋊Q₈×C₆₇	—	2	4	no	—
109	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
110	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
111	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
113	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	complemented
118	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
120	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
121	#103×C₆₇	C₂³	1	16	yes	complemented
122	#102×C₆₇	C₂³	1	16	yes	complemented
123	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
124	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	complemented
125	C₂⁴×C₆₇	—	2	8	no	—
126	C₂⁴×C₆₇	—	2	4	no	—
127	C₂⁴×C₆₇	—	2	4	no	—
128	C₂⁴×C₆₇	—	2	4	no	—
129	C₂⁴×C₆₇	—	2	2	no	—
130	C₂⁴×C₆₇	—	4	2	no	—
131	C₂⁴×C₆₇	—	4	4	no	—
132	C₂⁴×C₆₇	—	4	4	no	—
133	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
135	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
137	#41×C₆₇	—	2	64	no	—
138	#41×C₆₇	—	2	32	no	—
139	#41×C₆₇	—	2	64	no	—
140	#41×C₆₇	—	2	16	no	—
141	#41×C₆₇	—	2	16	no	—
142	#41×C₆₇	—	2	32	no	—
143	#41×C₆₇	—	2	16	no	—
144	#41×C₆₇	C₂×D₄	1	32	yes	—
145	C₂×Q₈×C₆₇	—	2	16	no	—
146	C₂×Q₈×C₆₇	—	2	8	no	—
147	C₂×Q₈×C₆₇	—	2	16	no	—
148	C₂×Q₈×C₆₇	—	2	4	no	—
149	C₂×Q₈×C₆₇	—	2	4	no	—
150	C₂×Q₈×C₆₇	—	2	2	no	—
151	C₂×Q₈×C₆₇	—	2	2	no	—
152	C₂×Q₈×C₆₇	—	2	4	no	—
153	C₂×Q₈×C₆₇	—	2	8	no	—
154	C₂×Q₈×C₆₇	—	2	16	no	—
155	C₂×Q₈×C₆₇	—	4	4	no	—
156	C₂×Q₈×C₆₇	—	4	8	no	—
157	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
158	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
159	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
160	C₂×D₄×C₆₇	—	2	64	no	—
161	C₂×D₄×C₆₇	—	2	16	no	—
162	C₂×D₄×C₆₇	—	2	8	no	—
163	C₂×D₄×C₆₇	—	2	16	no	—
164	C₂×D₄×C₆₇	—	2	16	no	—
165	C₂×D₄×C₆₇	—	2	8	no	—
166	C₂×D₄×C₆₇	—	2	8	no	—
167	C₂×D₄×C₆₇	—	2	16	no	—
168	C₂×D₄×C₆₇	—	2	4	no	—
169	C₂×D₄×C₆₇	—	2	2	no	—
170	C₂×D₄×C₆₇	—	2	4	no	—
171	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
172	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₇	—	2	8	no	—
174	C₂²×C₄×C₆₇	—	2	64	no	—
175	C₂²×C₄×C₆₇	—	2	16	no	—
176	C₂²×C₄×C₆₇	—	2	4	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	2	2	no	—
179	C₂²×C₄×C₆₇	—	2	4	no	—
180	C₂²×C₄×C₆₇	—	2	16	no	—
181	C₂²×C₄×C₆₇	—	2	4	no	—
182	C₂²×C₄×C₆₇	—	2	16	no	—
183	C₂²×C₄×C₆₇	—	2	8	no	—
184	C₂²×C₄×C₆₇	—	2	8	no	—
185	C₂²×C₄×C₆₇	—	2	4	no	—
186	C₂²×C₄×C₆₇	—	2	4	no	—
187	C₂²×C₄×C₆₇	—	2	16	no	—
188	C₂²×C₄×C₆₇	—	2	4	no	—
189	C₂²×C₄×C₆₇	—	2	4	no	—
190	C₂²×C₄×C₆₇	—	2	2	no	—
191	C₂²×C₄×C₆₇	—	2	2	no	—
192	C₂²×C₄×C₆₇	—	4	4	no	—
193	C₂²×C₄×C₆₇	—	4	4	no	—
194	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
196	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
198	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
199	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
204	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
208	Q₁₆×C₆₇	—	2	32	no	—
209	SD₁₆×C₆₇	—	2	32	no	—
210	M₁₆×C₆₇	—	2	16	no	—
211	C₂×C₈×C₆₇	—	2	8	no	—
212	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
213	C₄⋊C₄×C₆₇	—	2	8	no	—
214	C₄⋊C₄×C₆₇	—	2	8	no	—
215	C₄⋊C₄×C₆₇	—	2	2	no	—
216	C₄⋊C₄×C₆₇	—	2	4	no	—
217	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
218	C₂²⋊C₄×C₆₇	—	2	8	no	—
219	C₂²⋊C₄×C₆₇	—	2	8	no	—
220	C₂³×C₆₇	—	2	2	no	—
221	C₂³×C₆₇	—	2	2	no	—
222	C₂³×C₆₇	—	2	2	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	16	no	—
225	C₂³×C₆₇	—	2	8	no	—
226	C₂³×C₆₇	—	2	2	no	—
227	C₂³×C₆₇	—	2	8	no	—
228	C₂³×C₆₇	—	2	2	no	—
229	C₂³×C₆₇	—	2	8	no	—
230	C₂³×C₆₇	—	2	2	no	—
231	C₂³×C₆₇	—	2	4	no	—
232	C₂³×C₆₇	—	4	4	no	—
233	C₂³×C₆₇	—	4	16	no	—
234	C₂³×C₆₇	—	4	32	no	—
235	C₂³×C₆₇	—	4	8	no	—
236	C₂³×C₆₇	—	4	4	no	—
237	C₂³×C₆₇	—	4	8	no	—
238	C₂³×C₆₇	—	4	4	no	—
239	C₂³×C₆₇	#120	1	4	yes	—
240	C₂³×C₆₇	#137	1	4	yes	complemented
241	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
242	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
243	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
244	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
245	Q₈×C₆₇	—	2	16	no	—
246	Q₈×C₆₇	—	2	8	no	—
247	Q₈×C₆₇	—	2	4	no	—
248	Q₈×C₆₇	—	2	8	no	—
249	Q₈×C₆₇	—	4	8	no	—
250	Q₈×C₆₇	—	4	4	no	—
251	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
252	D₄×C₆₇	—	2	32	no	—
253	D₄×C₆₇	—	2	16	no	—
254	D₄×C₆₇	—	2	32	no	—
255	D₄×C₆₇	—	2	8	no	—
256	D₄×C₆₇	—	2	4	no	—
257	D₄×C₆₇	C₂²×D₄	1	8	yes	—
258	C₂×C₄×C₆₇	—	2	16	no	—
259	C₂×C₄×C₆₇	—	2	32	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	8	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	8	no	—
264	C₂×C₄×C₆₇	—	2	4	no	—
265	C₂×C₄×C₆₇	—	2	8	no	—
266	C₂×C₄×C₆₇	—	2	32	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	8	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	4	no	—
274	C₂×C₄×C₆₇	—	4	8	no	—
275	C₂×C₄×C₆₇	—	4	8	no	—
276	C₂×C₄×C₆₇	—	4	4	no	—
277	C₂×C₄×C₆₇	—	4	4	no	—
278	C₂×C₄×C₆₇	#120	1	4	yes	—
279	C₂×C₄×C₆₇	#120	1	8	yes	—
280	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
286	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
287	C₈×C₆₇	—	2	8	no	—
288	C₂²×C₆₇	—	2	2	no	—
289	C₂²×C₆₇	—	2	8	no	—
290	C₂²×C₆₇	—	2	8	no	—
291	C₂²×C₆₇	—	2	2	no	—
292	C₂²×C₆₇	—	2	2	no	—
293	C₂²×C₆₇	—	2	2	no	—
294	C₂²×C₆₇	—	2	2	no	—
295	C₂²×C₆₇	—	2	8	no	—
296	C₂²×C₆₇	—	2	2	no	—
297	C₂²×C₆₇	—	2	2	no	—
298	C₂²×C₆₇	—	4	16	no	—
299	C₂²×C₆₇	—	4	32	no	—
300	C₂²×C₆₇	—	4	4	no	—
301	C₂²×C₆₇	—	4	16	no	—
302	C₂²×C₆₇	—	4	4	no	—
303	C₂²×C₆₇	—	4	4	no	—
304	C₂²×C₆₇	C₂×#120	1	4	yes	—
305	C₂²×C₆₇	C₂×#120	1	2	yes	—
306	C₂²×C₆₇	C₂×#137	1	2	yes	—
307	C₂²×C₆₇	C₂×#137	1	4	yes	complemented
308	C₂²×C₆₇	#575	1	2	yes	—
309	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
310	C₄×C₆₇	—	2	2	no	—
311	C₄×C₆₇	—	2	8	no	—
312	C₄×C₆₇	—	2	2	no	—
313	C₄×C₆₇	—	4	4	no	—
314	C₄×C₆₇	—	4	4	no	—
315	C₄×C₆₇	C₂×#120	1	4	yes	—
316	C₄×C₆₇	C₂×#120	1	2	yes	—
317	C₄×C₆₇	C₂³×D₄	1	2	yes	—
318	C₂×#3007	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
319	C₂×C₆₇	—	2	4	no	—
320	C₂×C₆₇	—	2	4	no	—
321	C₂×C₆₇	—	4	2	no	—
322	C₂×C₆₇	—	4	8	no	—
323	C₂×C₆₇	#3007	1	4	yes	direct product factor
324	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
325	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
326	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
327	C₂³×#41	C₂×C₆₇	1	1	yes	characteristic
328	C₂²×#137	C₂×C₆₇	1	2	yes	complemented
329	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
330	#3007	C₂×C₆₇	1	16	yes	direct product factor
331	C₂×#452	C₂×C₆₇	1	4	yes	complemented
332	C₂×#451	C₂×C₆₇	1	2	yes	complemented
333	C₂×#450	C₂×C₆₇	1	2	yes	complemented
334	C₂×#417	C₂×C₆₇	1	2	yes	complemented
335	C₂×#407	C₂×C₆₇	1	1	yes	characteristic, complemented
336	C₆₇	C₂×#3007	1	1	yes	characteristic, direct product factor, Sylow
337	C₂²×#41	—	2	16	no	—
338	C₂²×#41	—	2	1	no	—
339	C₂²×#41	—	2	4	no	—
340	C₂²×#41	—	2	1	no	—
341	C₂²×#41	C₂²×C₆₇	1	2	yes	—
342	C₂²×#41	C₂²×C₆₇	1	8	yes	—
343	C₂²×#41	C₂²×C₆₇	1	2	yes	—
344	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
345	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
346	C₂³×D₄	—	2	1	no	—
347	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
348	C₂⁴×C₄	—	2	1	no	—
349	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
350	C₂×#137	—	2	16	no	—
351	C₂×#137	C₂²×C₆₇	1	16	yes	complemented
352	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
353	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
354	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
355	C₂×C₂²⋊Q₈	—	2	1	no	—
356	C₂×C₂²⋊Q₈	—	2	1	no	—
357	C₂×C₂²⋊Q₈	—	2	4	no	—
358	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
359	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
360	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
361	C₂²×C₄⋊C₄	—	2	1	no	—
362	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
363	C₂²×C₂²⋊C₄	—	2	1	no	—
364	#452	C₂²×C₆₇	1	32	yes	complemented
365	#451	C₂²×C₆₇	1	16	yes	complemented
366	#450	C₂²×C₆₇	1	16	yes	complemented
367	#417	C₂²×C₆₇	1	16	yes	complemented
368	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
369	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
370	#407	C₂²×C₆₇	1	8	yes	complemented
371	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
372	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
373	C₂⁵	—	2	1	no	—
374	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
375	C₂×#41	—	2	16	no	—
376	C₂×#41	—	2	64	no	—
377	C₂×#41	—	2	16	no	—
378	C₂×#41	—	2	64	no	—
379	C₂×#41	—	2	8	no	—
380	C₂×#41	—	2	8	no	—
381	C₂×#41	—	2	8	no	—
382	C₂×#41	—	2	8	no	—
383	C₂×#41	—	2	4	no	—
384	C₂×#41	—	2	8	no	—
385	C₂×#41	—	2	4	no	—
386	C₂×#41	D₄×C₆₇	1	16	yes	—
387	C₂×#41	C₂³×C₆₇	1	8	yes	—
388	C₂×#41	C₂³×C₆₇	1	8	yes	—
389	C₂²×Q₈	—	2	8	no	—
390	C₂²×Q₈	—	2	1	no	—
391	C₂²×Q₈	—	2	2	no	—
392	C₂²×Q₈	—	2	1	no	—
393	C₂²×Q₈	—	2	2	no	—
394	C₂²×Q₈	—	2	8	no	—
395	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
396	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
397	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
398	C₂²×Q₈	C₂³×C₆₇	1	4	yes	complemented
399	C₂²×D₄	—	2	16	no	—
400	C₂²×D₄	—	2	2	no	—
401	C₂²×D₄	—	2	8	no	—
402	C₂²×D₄	—	2	2	no	—
403	C₂²×D₄	—	2	8	no	—
404	C₂²×D₄	—	2	1	no	—
405	C₂²×D₄	—	2	2	no	—
406	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
407	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
408	C₂³×C₄	—	2	16	no	—
409	C₂³×C₄	—	2	1	no	—
410	C₂³×C₄	—	2	2	no	—
411	C₂³×C₄	—	2	8	no	—
412	C₂³×C₄	—	2	2	no	—
413	C₂³×C₄	—	2	8	no	—
414	C₂³×C₄	—	2	1	no	—
415	C₂³×C₄	—	2	1	no	—
416	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
417	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
418	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
419	C₂³×C₄	D₄×C₆₇	1	4	yes	—
420	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
421	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
422	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
423	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
424	#137	—	2	64	no	—
425	C₂×Q₁₆	—	2	16	no	—
426	C₂×Q₁₆	C₂³×C₆₇	1	16	yes	complemented
427	C₂×SD₁₆	—	2	16	no	—
428	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	complemented
429	C₂×M₁₆	—	2	8	no	—
430	C₂×M₁₆	C₂³×C₆₇	1	8	yes	complemented
431	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
432	C₂²⋊Q₈	—	2	4	no	—
433	C₂²⋊Q₈	—	2	8	no	—
434	C₂²⋊Q₈	—	2	8	no	—
435	C₂²⋊Q₈	—	2	4	no	—
436	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
437	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
438	C₂×C₄⋊C₄	—	2	2	no	—
439	C₂×C₄⋊C₄	—	2	8	no	—
440	C₂×C₄⋊C₄	—	2	2	no	—
441	C₂×C₄⋊C₄	—	2	2	no	—
442	C₂×C₄⋊C₄	—	2	1	no	—
443	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	complemented
445	C₂×C₂²⋊C₄	—	2	2	no	—
446	C₂×C₂²⋊C₄	—	2	8	no	—
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	#103	C₂³×C₆₇	1	16	yes	complemented
449	#102	C₂³×C₆₇	1	16	yes	complemented
450	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
451	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	complemented
452	C₂⁴	—	2	8	no	—
453	C₂⁴	—	2	4	no	—
454	C₂⁴	—	2	4	no	—
455	C₂⁴	—	2	4	no	—
456	C₂⁴	—	2	2	no	—
457	C₂⁴	—	4	2	no	—
458	C₂⁴	—	4	4	no	—
459	C₂⁴	—	4	4	no	—
460	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
462	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
464	#41	—	2	64	no	—
465	#41	—	2	32	no	—
466	#41	—	2	64	no	—
467	#41	—	2	16	no	—
468	#41	—	2	16	no	—
469	#41	—	2	32	no	—
470	#41	—	2	16	no	—
471	#41	C₂×D₄×C₆₇	1	32	yes	—
472	C₂×Q₈	—	2	16	no	—
473	C₂×Q₈	—	2	8	no	—
474	C₂×Q₈	—	2	16	no	—
475	C₂×Q₈	—	2	4	no	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×Q₈	—	2	2	no	—
478	C₂×Q₈	—	2	2	no	—
479	C₂×Q₈	—	2	4	no	—
480	C₂×Q₈	—	2	8	no	—
481	C₂×Q₈	—	2	16	no	—
482	C₂×Q₈	—	4	4	no	—
483	C₂×Q₈	—	4	8	no	—
484	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
485	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
486	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
487	C₂×D₄	—	2	64	no	—
488	C₂×D₄	—	2	16	no	—
489	C₂×D₄	—	2	8	no	—
490	C₂×D₄	—	2	16	no	—
491	C₂×D₄	—	2	16	no	—
492	C₂×D₄	—	2	8	no	—
493	C₂×D₄	—	2	8	no	—
494	C₂×D₄	—	2	16	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
499	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
500	C₂²×C₄	—	2	8	no	—
501	C₂²×C₄	—	2	64	no	—
502	C₂²×C₄	—	2	16	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	4	no	—
507	C₂²×C₄	—	2	16	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	16	no	—
510	C₂²×C₄	—	2	8	no	—
511	C₂²×C₄	—	2	8	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	4	no	—
514	C₂²×C₄	—	2	16	no	—
515	C₂²×C₄	—	2	2	no	—
516	C₂²×C₄	—	2	4	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	4	4	no	—
520	C₂²×C₄	—	4	4	no	—
521	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
522	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
523	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
524	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
525	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
526	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
527	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
528	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
529	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
530	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
531	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
532	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
533	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
534	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
535	Q₁₆	—	2	32	no	—
536	SD₁₆	—	2	32	no	—
537	M₁₆	—	2	16	no	—
538	C₂×C₈	—	2	8	no	—
539	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
540	C₄⋊C₄	—	2	8	no	—
541	C₄⋊C₄	—	2	8	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
545	C₂²⋊C₄	—	2	8	no	—
546	C₂²⋊C₄	—	2	8	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	16	no	—
552	C₂³	—	2	8	no	—
553	C₂³	—	2	2	no	—
554	C₂³	—	2	8	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	8	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	4	no	—
559	C₂³	—	4	4	no	—
560	C₂³	—	4	16	no	—
561	C₂³	—	4	8	no	—
562	C₂³	—	4	32	no	—
563	C₂³	—	4	4	no	—
564	C₂³	—	4	8	no	—
565	C₂³	—	4	4	no	—
566	C₂³	#120×C₆₇	1	4	yes	—
567	C₂³	#137×C₆₇	1	4	yes	complemented
568	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
569	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
570	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
571	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
572	Q₈	—	2	16	no	—
573	Q₈	—	2	8	no	—
574	Q₈	—	2	4	no	—
575	Q₈	—	2	8	no	—
576	Q₈	—	4	8	no	—
577	Q₈	—	4	4	no	—
578	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
579	D₄	—	2	32	no	—
580	D₄	—	2	16	no	—
581	D₄	—	2	32	no	—
582	D₄	—	2	8	no	—
583	D₄	—	2	4	no	—
584	D₄	C₂²×D₄×C₆₇	1	8	yes	—
585	C₂×C₄	—	2	16	no	—
586	C₂×C₄	—	2	32	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	8	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	8	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	8	no	—
593	C₂×C₄	—	2	32	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	8	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	4	8	no	—
602	C₂×C₄	—	4	8	no	—
603	C₂×C₄	—	4	4	no	—
604	C₂×C₄	—	4	4	no	—
605	C₂×C₄	#120×C₆₇	1	4	yes	—
606	C₂×C₄	#120×C₆₇	1	8	yes	—
607	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
608	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
609	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
610	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
611	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
612	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
613	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
614	C₈	—	2	8	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	8	no	—
620	C₂²	—	2	8	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	8	no	—
625	C₂²	—	4	16	no	—
626	C₂²	—	4	32	no	—
627	C₂²	—	4	4	no	—
628	C₂²	—	4	4	no	—
629	C₂²	—	4	4	no	—
630	C₂²	—	4	16	no	—
631	C₂²	C₂×#120×C₆₇	1	4	yes	—
632	C₂²	C₂×#120×C₆₇	1	2	yes	—
633	C₂²	C₂×#137×C₆₇	1	2	yes	—
634	C₂²	C₂×#137×C₆₇	1	4	yes	complemented
635	C₂²	#575×C₆₇	1	2	yes	—
636	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
637	C₄	—	2	2	no	—
638	C₄	—	2	8	no	—
639	C₄	—	2	2	no	—
640	C₄	—	4	4	no	—
641	C₄	—	4	4	no	—
642	C₄	C₂×#120×C₆₇	1	4	yes	—
643	C₄	C₂×#120×C₆₇	1	2	yes	—
644	C₄	C₂³×D₄×C₆₇	1	2	yes	—
645	C₂	—	2	4	no	—
646	C₂	—	2	4	no	—
647	C₂	—	4	2	no	—
648	C₂	—	4	8	no	—
649	C₂	C₆₇×#3007	1	4	yes	direct product factor
650	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
651	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
652	C₂	C₂×#575×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂³×#41×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×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. C₂²×C₂²⋊Q₈×C₆₇	C₂	semidirect
4. C₆₇×#3007	C₂	direct (N × H)
5. C₂×#452×C₆₇	C₂	semidirect
6. C₂×#451×C₆₇	C₂	semidirect
7. C₂×#450×C₆₇	C₂	semidirect
8. C₂×#417×C₆₇	C₂	semidirect
9. C₂×#407×C₆₇	C₂	semidirect
18. C₂³×Q₈×C₆₇	C₂²	semidirect
24. C₂×#137×C₆₇	C₂²	semidirect
25. C₂²×Q₁₆×C₆₇	C₂²	semidirect
26. C₂²×SD₁₆×C₆₇	C₂²	semidirect
27. C₂²×M₁₆×C₆₇	C₂²	semidirect
32. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
33. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
35. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
37. #452×C₆₇	C₂²	semidirect
38. #451×C₆₇	C₂²	semidirect
39. #450×C₆₇	C₂²	semidirect
40. #417×C₆₇	C₂²	semidirect
41. C₂×#103×C₆₇	C₂²	semidirect
42. C₂×#102×C₆₇	C₂²	semidirect
43. #407×C₆₇	C₂²	semidirect
45. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
71. C₂²×Q₈×C₆₇	C₂³	semidirect
99. C₂×Q₁₆×C₆₇	C₂³	semidirect
101. C₂×SD₁₆×C₆₇	C₂³	semidirect
103. C₂×M₁₆×C₆₇	C₂³	semidirect
110. C₂²⋊Q₈×C₆₇	C₂³	semidirect
117. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
121. #103×C₆₇	C₂³	semidirect
122. #102×C₆₇	C₂³	semidirect
124. C₂²⋊C₈×C₆₇	C₂³	semidirect
240. C₂³×C₆₇	#137	semidirect
307. C₂²×C₆₇	C₂×#137	semidirect
318. C₂×#3007	C₆₇	direct (N × H)
323. C₂×C₆₇	#3007	direct (N × H)
328. C₂²×#137	C₂×C₆₇	semidirect
329. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
330. #3007	C₂×C₆₇	direct (N × H)
331. C₂×#452	C₂×C₆₇	semidirect
332. C₂×#451	C₂×C₆₇	semidirect
333. C₂×#450	C₂×C₆₇	semidirect
334. C₂×#417	C₂×C₆₇	semidirect
335. C₂×#407	C₂×C₆₇	semidirect
336. C₆₇	C₂×#3007	direct (N × H)
345. C₂³×Q₈	C₂²×C₆₇	semidirect
351. C₂×#137	C₂²×C₆₇	semidirect
352. C₂²×Q₁₆	C₂²×C₆₇	semidirect
353. C₂²×SD₁₆	C₂²×C₆₇	semidirect
354. C₂²×M₁₆	C₂²×C₆₇	semidirect
359. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
360. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
362. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
364. #452	C₂²×C₆₇	semidirect
365. #451	C₂²×C₆₇	semidirect
366. #450	C₂²×C₆₇	semidirect
367. #417	C₂²×C₆₇	semidirect
368. C₂×#103	C₂²×C₆₇	semidirect
369. C₂×#102	C₂²×C₆₇	semidirect
370. #407	C₂²×C₆₇	semidirect
372. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
398. C₂²×Q₈	C₂³×C₆₇	semidirect
426. C₂×Q₁₆	C₂³×C₆₇	semidirect
428. C₂×SD₁₆	C₂³×C₆₇	semidirect
430. C₂×M₁₆	C₂³×C₆₇	semidirect
437. C₂²⋊Q₈	C₂³×C₆₇	semidirect
444. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
448. #103	C₂³×C₆₇	semidirect
449. #102	C₂³×C₆₇	semidirect
451. C₂²⋊C₈	C₂³×C₆₇	semidirect
567. C₂³	#137×C₆₇	semidirect
634. C₂²	C₂×#137×C₆₇	semidirect
649. C₂	C₆₇×#3007	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₆₇	2
C₂²×#137×C₆₇	2
C₂²×C₂²⋊Q₈×C₆₇	2
C₆₇×#3007	2
C₂×#452×C₆₇	2
C₂×#451×C₆₇	2
C₂×#450×C₆₇	2
C₂×#417×C₆₇	2
C₂×#407×C₆₇	2
C₂×#3007	67

Subgroups — C2×C67×#3007 #30739609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3007 #30739609