Subgroups — C₂×C₆₇×#3136 #30739861

Order 17152.

2262 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3136	C₂	1	16	yes	maximal, direct product factor
4	C₂×#490×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#489×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#488×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#486×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#544×C₆₇	—	2	4	no	—
18	#544×C₆₇	C₂²	1	8	yes	complemented
19	#535×C₆₇	—	2	4	no	—
20	#535×C₆₇	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
23	C₂×#127×C₆₇	C₂²	1	2	yes	—
24	C₂×#125×C₆₇	—	2	1	no	—
25	C₂×#125×C₆₇	C₂²	1	2	yes	complemented
26	C₂×#124×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
28	C₂×#121×C₆₇	—	2	1	no	—
29	C₂×#121×C₆₇	—	2	1	no	—
30	C₂×#120×C₆₇	—	2	2	no	—
31	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	#490×C₆₇	C₂²	1	16	yes	complemented
35	#489×C₆₇	C₂²	1	16	yes	complemented
36	#488×C₆₇	C₂²	1	16	yes	complemented
37	#486×C₆₇	C₂²	1	16	yes	complemented
38	#418×C₆₇	C₂²	1	8	yes	complemented
39	#417×C₆₇	C₂²	1	8	yes	complemented
40	#416×C₆₇	C₂²	1	8	yes	complemented
41	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
43	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	C₂×#102×C₆₇	C₂²	1	2	yes	—
45	C₂×#102×C₆₇	C₂²	1	2	yes	—
46	C₂×#102×C₆₇	C₂²	1	2	yes	—
47	C₂×#102×C₆₇	C₂²	1	2	yes	—
48	C₂×#102×C₆₇	C₂²	1	8	yes	complemented
49	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
50	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
51	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
52	#405×C₆₇	C₂²	1	8	yes	complemented
53	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
54	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
55	C₂⁵×C₆₇	—	2	1	no	—
56	C₂×#41×C₆₇	—	2	2	no	—
57	C₂×#41×C₆₇	—	2	4	no	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	C₂³	1	4	yes	—
60	C₂²×Q₈×C₆₇	—	2	2	no	—
61	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	—	2	1	no	—
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	—	2	2	no	—
65	C₂²×D₄×C₆₇	—	2	2	no	—
66	C₂²×D₄×C₆₇	—	2	2	no	—
67	C₂²×D₄×C₆₇	—	2	4	no	—
68	C₂²×D₄×C₆₇	—	2	4	no	—
69	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
70	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
72	C₂³×C₄×C₆₇	—	2	1	no	—
73	C₂³×C₄×C₆₇	—	2	1	no	—
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₆₇	—	2	4	no	—
76	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
77	C₂²×C₈×C₆₇	—	2	4	no	—
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
81	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
82	C₄⋊Q₈×C₆₇	—	2	8	no	—
83	C₄⋊Q₈×C₆₇	—	2	4	no	—
84	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
85	#127×C₆₇	—	2	4	no	—
86	#127×C₆₇	C₂³	1	8	yes	—
87	#125×C₆₇	—	2	8	no	—
88	#125×C₆₇	—	2	4	no	—
89	#125×C₆₇	C₂³	1	8	yes	complemented
90	#124×C₆₇	—	2	4	no	—
91	#124×C₆₇	C₂³	1	8	yes	—
92	C₂²⋊Q₈×C₆₇	—	2	16	no	—
93	#121×C₆₇	—	2	8	no	—
94	#121×C₆₇	—	2	8	no	—
95	#120×C₆₇	—	2	16	no	—
96	#117×C₆₇	—	2	2	no	—
97	#117×C₆₇	D₄	1	4	yes	—
98	#117×C₆₇	C₂³	1	4	yes	—
99	#117×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
115	C₂×C₄²×C₆₇	—	2	1	no	—
116	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
117	#103×C₆₇	C₂³	1	4	yes	complemented
118	#103×C₆₇	C₂³	1	8	yes	complemented
119	#103×C₆₇	C₂³	1	4	yes	complemented
120	#102×C₆₇	C₂³	1	8	yes	—
121	#102×C₆₇	C₂³	1	8	yes	—
122	#102×C₆₇	C₂³	1	8	yes	—
123	#102×C₆₇	C₂³	1	8	yes	—
124	#102×C₆₇	C₂³	1	4	yes	complemented
125	#102×C₆₇	C₂³	1	8	yes	complemented
126	#102×C₆₇	C₂³	1	4	yes	complemented
127	#97×C₆₇	C₂³	1	8	yes	complemented
128	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
129	C₂⁴×C₆₇	—	2	2	no	—
130	C₂⁴×C₆₇	—	2	1	no	—
131	C₂⁴×C₆₇	—	2	4	no	—
132	C₂⁴×C₆₇	—	4	4	no	—
133	C₂⁴×C₆₇	—	8	2	no	—
134	C₂⁴×C₆₇	—	8	2	no	—
135	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
136	#41×C₆₇	—	2	8	no	—
137	#41×C₆₇	—	2	8	no	—
138	#41×C₆₇	—	4	8	no	—
139	C₂×Q₈×C₆₇	—	2	1	no	—
140	C₂×Q₈×C₆₇	—	2	8	no	—
141	C₂×Q₈×C₆₇	—	2	2	no	—
142	C₂×Q₈×C₆₇	—	2	2	no	—
143	C₂×Q₈×C₆₇	—	8	4	no	—
144	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
145	C₂×D₄×C₆₇	—	2	4	no	—
146	C₂×D₄×C₆₇	—	2	4	no	—
147	C₂×D₄×C₆₇	—	2	2	no	—
148	C₂×D₄×C₆₇	—	2	1	no	—
149	C₂×D₄×C₆₇	—	2	8	no	—
150	C₂×D₄×C₆₇	—	2	8	no	—
151	C₂×D₄×C₆₇	—	2	4	no	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	8	no	—
154	C₂×D₄×C₆₇	—	2	8	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	2	2	no	—
157	C₂×D₄×C₆₇	—	4	4	no	—
158	C₂×D₄×C₆₇	—	4	4	no	—
159	C₂×D₄×C₆₇	—	4	8	no	—
160	C₂×D₄×C₆₇	—	8	4	no	—
161	C₂×D₄×C₆₇	—	8	4	no	—
162	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₆₇	—	2	1	no	—
165	C₂²×C₄×C₆₇	—	2	1	no	—
166	C₂²×C₄×C₆₇	—	2	1	no	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	4	4	no	—
171	C₂²×C₄×C₆₇	—	8	2	no	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	#41	1	4	yes	—
177	C₂²×C₄×C₆₇	#41	1	4	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	M₁₆×C₆₇	—	2	8	no	—
181	C₂×C₈×C₆₇	—	2	2	no	—
182	C₂×C₈×C₆₇	—	2	2	no	—
183	C₂×C₈×C₆₇	—	2	8	no	—
184	C₂×C₈×C₆₇	—	2	4	no	—
185	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
186	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
187	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	8	no	—
191	C₄⋊C₄×C₆₇	—	2	8	no	—
192	C₄⋊C₄×C₆₇	—	2	8	no	—
193	C₄⋊C₄×C₆₇	—	2	1	no	—
194	C₄⋊C₄×C₆₇	—	2	1	no	—
195	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
196	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
199	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	8	no	—
202	C₂²⋊C₄×C₆₇	—	2	8	no	—
203	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
204	C₄²×C₆₇	—	2	4	no	—
205	C₄²×C₆₇	C₂×D₄	1	4	yes	—
206	C₄²×C₆₇	C₂⁴	1	4	yes	—
207	C₂³×C₆₇	—	2	2	no	—
208	C₂³×C₆₇	—	2	2	no	—
209	C₂³×C₆₇	—	2	4	no	—
210	C₂³×C₆₇	—	2	2	no	—
211	C₂³×C₆₇	—	4	2	no	—
212	C₂³×C₆₇	—	4	8	no	—
213	C₂³×C₆₇	—	4	4	no	—
214	C₂³×C₆₇	—	4	2	no	—
215	C₂³×C₆₇	—	4	4	no	—
216	C₂³×C₆₇	—	4	2	no	—
217	C₂³×C₆₇	—	4	4	no	—
218	C₂³×C₆₇	—	8	4	no	—
219	C₂³×C₆₇	—	8	4	no	—
220	C₂³×C₆₇	—	8	16	no	—
221	C₂³×C₆₇	#124	1	4	yes	—
222	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
223	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₇	—	2	2	no	—
225	Q₈×C₆₇	—	2	2	no	—
226	Q₈×C₆₇	—	4	2	no	—
227	Q₈×C₆₇	—	8	8	no	—
228	D₄×C₆₇	—	2	4	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	2	2	no	—
231	D₄×C₆₇	—	2	2	no	—
232	D₄×C₆₇	—	4	8	no	—
233	D₄×C₆₇	—	4	8	no	—
234	D₄×C₆₇	—	4	4	no	—
235	D₄×C₆₇	—	4	2	no	—
236	D₄×C₆₇	—	8	8	no	—
237	D₄×C₆₇	—	8	8	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	4	2	no	—
245	C₂×C₄×C₆₇	—	4	4	no	—
246	C₂×C₄×C₆₇	—	4	2	no	—
247	C₂×C₄×C₆₇	—	4	2	no	—
248	C₂×C₄×C₆₇	—	4	8	no	—
249	C₂×C₄×C₆₇	—	8	4	no	—
250	C₂×C₄×C₆₇	#124	1	4	yes	—
251	C₂×C₄×C₆₇	#124	1	8	yes	—
252	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
253	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
257	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
258	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
259	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
260	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
261	C₈×C₆₇	—	2	2	no	—
262	C₈×C₆₇	—	2	2	no	—
263	C₈×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	8	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	4	2	no	—
268	C₂²×C₆₇	—	4	4	no	—
269	C₂²×C₆₇	—	4	2	no	—
270	C₂²×C₆₇	—	4	4	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	4	4	no	—
273	C₂²×C₆₇	—	8	16	no	—
274	C₂²×C₆₇	—	8	4	no	—
275	C₂²×C₆₇	—	8	8	no	—
276	C₂²×C₆₇	C₂×#124	1	4	yes	—
277	C₂²×C₆₇	C₂×#124	1	2	yes	—
278	C₂²×C₆₇	#576	1	2	yes	—
279	C₂²×C₆₇	#577	1	2	yes	—
280	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	2	4	no	—
283	C₄×C₆₇	—	4	2	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	—	4	2	no	—
286	C₄×C₆₇	—	4	2	no	—
287	C₄×C₆₇	C₂×#124	1	4	yes	—
288	C₄×C₆₇	C₂×#124	1	2	yes	—
289	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
290	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
291	C₂×#3136	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₆₇	—	2	4	no	—
293	C₂×C₆₇	—	8	2	no	—
294	C₂×C₆₇	—	8	4	no	—
295	C₂×C₆₇	#3136	1	4	yes	direct product factor
296	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
297	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
298	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
299	C₂×#544	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₂×#535	C₂×C₆₇	1	1	yes	characteristic
301	#3136	C₂×C₆₇	1	16	yes	direct product factor
302	C₂×#490	C₂×C₆₇	1	2	yes	complemented
303	C₂×#489	C₂×C₆₇	1	2	yes	complemented
304	C₂×#488	C₂×C₆₇	1	2	yes	complemented
305	C₂×#486	C₂×C₆₇	1	2	yes	complemented
306	C₂×#417	C₂×C₆₇	1	1	yes	characteristic, complemented
307	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
308	C₂×#416	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₂²×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
310	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
311	C₆₇	C₂×#3136	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
313	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
314	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
315	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
316	#544	—	2	4	no	—
317	#544	C₂²×C₆₇	1	8	yes	complemented
318	#535	—	2	4	no	—
319	#535	C₂²×C₆₇	1	8	yes	—
320	C₂×C₄⋊Q₈	—	2	1	no	—
321	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
322	C₂×#127	C₂²×C₆₇	1	2	yes	—
323	C₂×#125	—	2	1	no	—
324	C₂×#125	C₂²×C₆₇	1	2	yes	complemented
325	C₂×#124	C₂²×C₆₇	1	2	yes	—
326	C₂×C₂²⋊Q₈	—	2	2	no	—
327	C₂×#121	—	2	1	no	—
328	C₂×#121	—	2	1	no	—
329	C₂×#120	—	2	2	no	—
330	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
332	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
333	#490	C₂²×C₆₇	1	16	yes	complemented
334	#489	C₂²×C₆₇	1	16	yes	complemented
335	#488	C₂²×C₆₇	1	16	yes	complemented
336	#486	C₂²×C₆₇	1	16	yes	complemented
337	#418	C₂²×C₆₇	1	8	yes	complemented
338	#417	C₂²×C₆₇	1	8	yes	complemented
339	#416	C₂²×C₆₇	1	8	yes	complemented
340	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
341	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
342	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
343	C₂×#102	C₂²×C₆₇	1	2	yes	—
344	C₂×#102	C₂²×C₆₇	1	2	yes	—
345	C₂×#102	C₂²×C₆₇	1	2	yes	—
346	C₂×#102	C₂²×C₆₇	1	2	yes	—
347	C₂×#102	C₂²×C₆₇	1	8	yes	complemented
348	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
349	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
350	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
351	#405	C₂²×C₆₇	1	8	yes	complemented
352	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
353	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
354	C₂⁵	—	2	1	no	—
355	C₂×#41	—	2	2	no	—
356	C₂×#41	—	2	4	no	—
357	C₂×#41	—	2	4	no	—
358	C₂×#41	C₂³×C₆₇	1	4	yes	—
359	C₂²×Q₈	—	2	2	no	—
360	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
361	C₂²×D₄	—	2	1	no	—
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	—	2	2	no	—
364	C₂²×D₄	—	2	2	no	—
365	C₂²×D₄	—	2	2	no	—
366	C₂²×D₄	—	2	4	no	—
367	C₂²×D₄	—	2	4	no	—
368	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
369	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
370	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
371	C₂³×C₄	—	2	1	no	—
372	C₂³×C₄	—	2	1	no	—
373	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
374	C₂×M₁₆	—	2	4	no	—
375	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
376	C₂²×C₈	—	2	4	no	—
377	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
380	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
381	C₄⋊Q₈	—	2	8	no	—
382	C₄⋊Q₈	—	2	4	no	—
383	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
384	#127	—	2	4	no	—
385	#127	C₂³×C₆₇	1	8	yes	—
386	#125	—	2	8	no	—
387	#125	—	2	4	no	—
388	#125	C₂³×C₆₇	1	8	yes	complemented
389	#124	—	2	4	no	—
390	#124	C₂³×C₆₇	1	8	yes	—
391	C₂²⋊Q₈	—	2	16	no	—
392	#121	—	2	8	no	—
393	#121	—	2	8	no	—
394	#120	—	2	16	no	—
395	#117	—	2	2	no	—
396	#117	D₄×C₆₇	1	4	yes	—
397	#117	C₂³×C₆₇	1	4	yes	—
398	#117	C₂³×C₆₇	1	4	yes	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	2	no	—
401	C₂×C₄⋊C₄	—	2	2	no	—
402	C₂×C₄⋊C₄	—	2	2	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
405	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	—	2	2	no	—
413	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
414	C₂×C₄²	—	2	1	no	—
415	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
416	#103	C₂³×C₆₇	1	4	yes	complemented
417	#103	C₂³×C₆₇	1	4	yes	complemented
418	#103	C₂³×C₆₇	1	8	yes	complemented
419	#102	C₂³×C₆₇	1	8	yes	—
420	#102	C₂³×C₆₇	1	8	yes	—
421	#102	C₂³×C₆₇	1	8	yes	—
422	#102	C₂³×C₆₇	1	8	yes	—
423	#102	C₂³×C₆₇	1	4	yes	complemented
424	#102	C₂³×C₆₇	1	4	yes	complemented
425	#102	C₂³×C₆₇	1	8	yes	complemented
426	#97	C₂³×C₆₇	1	8	yes	complemented
427	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
428	C₂⁴	—	2	2	no	—
429	C₂⁴	—	2	1	no	—
430	C₂⁴	—	2	4	no	—
431	C₂⁴	—	4	4	no	—
432	C₂⁴	—	8	2	no	—
433	C₂⁴	—	8	2	no	—
434	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
435	#41	—	2	8	no	—
436	#41	—	2	8	no	—
437	#41	—	4	8	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	8	no	—
440	C₂×Q₈	—	2	2	no	—
441	C₂×Q₈	—	2	2	no	—
442	C₂×Q₈	—	8	4	no	—
443	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	8	no	—
449	C₂×D₄	—	2	8	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	4	8	no	—
459	C₂×D₄	—	8	4	no	—
460	C₂×D₄	—	8	4	no	—
461	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
462	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	—	8	2	no	—
471	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
473	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
475	C₂²×C₄	#41×C₆₇	1	4	yes	—
476	C₂²×C₄	#41×C₆₇	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
479	M₁₆	—	2	8	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	2	2	no	—
482	C₂×C₈	—	2	8	no	—
483	C₂×C₈	—	2	4	no	—
484	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
485	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
486	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	8	no	—
491	C₄⋊C₄	—	2	8	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
495	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
496	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
497	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
498	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	—	2	8	no	—
502	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
503	C₄²	—	2	4	no	—
504	C₄²	C₂×D₄×C₆₇	1	4	yes	—
505	C₄²	C₂⁴×C₆₇	1	4	yes	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	8	no	—
512	C₂³	—	4	4	no	—
513	C₂³	—	4	2	no	—
514	C₂³	—	4	4	no	—
515	C₂³	—	4	2	no	—
516	C₂³	—	4	4	no	—
517	C₂³	—	8	4	no	—
518	C₂³	—	8	4	no	—
519	C₂³	—	8	16	no	—
520	C₂³	#124×C₆₇	1	4	yes	—
521	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
522	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
523	Q₈	—	2	2	no	—
524	Q₈	—	2	2	no	—
525	Q₈	—	4	2	no	—
526	Q₈	—	8	8	no	—
527	D₄	—	2	4	no	—
528	D₄	—	2	4	no	—
529	D₄	—	2	2	no	—
530	D₄	—	2	2	no	—
531	D₄	—	4	8	no	—
532	D₄	—	4	8	no	—
533	D₄	—	4	4	no	—
534	D₄	—	4	2	no	—
535	D₄	—	8	8	no	—
536	D₄	—	8	8	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	4	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	8	no	—
548	C₂×C₄	—	8	4	no	—
549	C₂×C₄	#124×C₆₇	1	4	yes	—
550	C₂×C₄	#124×C₆₇	1	8	yes	—
551	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
552	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
553	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
555	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
556	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
557	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
558	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
559	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
560	C₈	—	2	2	no	—
561	C₈	—	2	2	no	—
562	C₈	—	2	4	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	8	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	2	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	8	16	no	—
573	C₂²	—	8	8	no	—
574	C₂²	—	8	4	no	—
575	C₂²	C₂×#124×C₆₇	1	4	yes	—
576	C₂²	C₂×#124×C₆₇	1	2	yes	—
577	C₂²	#576×C₆₇	1	2	yes	—
578	C₂²	#577×C₆₇	1	2	yes	—
579	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
580	C₄	—	2	4	no	—
581	C₄	—	2	4	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	2	no	—
586	C₄	C₂×#124×C₆₇	1	4	yes	—
587	C₄	C₂×#124×C₆₇	1	2	yes	—
588	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
589	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
590	C₂	—	2	4	no	—
591	C₂	—	8	4	no	—
592	C₂	—	8	2	no	—
593	C₂	C₆₇×#3136	1	4	yes	direct product factor
594	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
595	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
596	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#535×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#544×C₆₇	C₂	semidirect
3. C₆₇×#3136	C₂	direct (N × H)
4. C₂×#490×C₆₇	C₂	semidirect
5. C₂×#489×C₆₇	C₂	semidirect
6. C₂×#488×C₆₇	C₂	semidirect
7. C₂×#486×C₆₇	C₂	semidirect
8. C₂×#417×C₆₇	C₂	semidirect
9. C₂×#418×C₆₇	C₂	semidirect
10. C₂×#416×C₆₇	C₂	semidirect
11. C₂²×#102×C₆₇	C₂	semidirect
12. C₂×#405×C₆₇	C₂	semidirect
18. #544×C₆₇	C₂²	semidirect
22. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
25. C₂×#125×C₆₇	C₂²	semidirect
34. #490×C₆₇	C₂²	semidirect
35. #489×C₆₇	C₂²	semidirect
36. #488×C₆₇	C₂²	semidirect
37. #486×C₆₇	C₂²	semidirect
38. #418×C₆₇	C₂²	semidirect
39. #417×C₆₇	C₂²	semidirect
40. #416×C₆₇	C₂²	semidirect
41. C₂×#103×C₆₇	C₂²	semidirect
42. C₂×#103×C₆₇	C₂²	semidirect
43. C₂×#103×C₆₇	C₂²	semidirect
48. C₂×#102×C₆₇	C₂²	semidirect
49. C₂×#102×C₆₇	C₂²	semidirect
50. C₂×#102×C₆₇	C₂²	semidirect
51. C₂×#102×C₆₇	C₂²	semidirect
52. #405×C₆₇	C₂²	semidirect
53. C₂×#97×C₆₇	C₂²	semidirect
54. C₂×C₄×C₈×C₆₇	C₂²	semidirect
84. C₄⋊Q₈×C₆₇	C₂³	semidirect
89. #125×C₆₇	C₂³	semidirect
117. #103×C₆₇	C₂³	semidirect
118. #103×C₆₇	C₂³	semidirect
119. #103×C₆₇	C₂³	semidirect
124. #102×C₆₇	C₂³	semidirect
125. #102×C₆₇	C₂³	semidirect
126. #102×C₆₇	C₂³	semidirect
127. #97×C₆₇	C₂³	semidirect
128. C₄×C₈×C₆₇	C₂³	semidirect
291. C₂×#3136	C₆₇	direct (N × H)
295. C₂×C₆₇	#3136	direct (N × H)
299. C₂×#544	C₂×C₆₇	semidirect
301. #3136	C₂×C₆₇	direct (N × H)
302. C₂×#490	C₂×C₆₇	semidirect
303. C₂×#489	C₂×C₆₇	semidirect
304. C₂×#488	C₂×C₆₇	semidirect
305. C₂×#486	C₂×C₆₇	semidirect
306. C₂×#417	C₂×C₆₇	semidirect
307. C₂×#418	C₂×C₆₇	semidirect
308. C₂×#416	C₂×C₆₇	semidirect
309. C₂²×#102	C₂×C₆₇	semidirect
310. C₂×#405	C₂×C₆₇	semidirect
311. C₆₇	C₂×#3136	direct (N × H)
317. #544	C₂²×C₆₇	semidirect
321. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
324. C₂×#125	C₂²×C₆₇	semidirect
333. #490	C₂²×C₆₇	semidirect
334. #489	C₂²×C₆₇	semidirect
335. #488	C₂²×C₆₇	semidirect
336. #486	C₂²×C₆₇	semidirect
337. #418	C₂²×C₆₇	semidirect
338. #417	C₂²×C₆₇	semidirect
339. #416	C₂²×C₆₇	semidirect
340. C₂×#103	C₂²×C₆₇	semidirect
341. C₂×#103	C₂²×C₆₇	semidirect
342. C₂×#103	C₂²×C₆₇	semidirect
347. C₂×#102	C₂²×C₆₇	semidirect
348. C₂×#102	C₂²×C₆₇	semidirect
349. C₂×#102	C₂²×C₆₇	semidirect
350. C₂×#102	C₂²×C₆₇	semidirect
351. #405	C₂²×C₆₇	semidirect
352. C₂×#97	C₂²×C₆₇	semidirect
353. C₂×C₄×C₈	C₂²×C₆₇	semidirect
383. C₄⋊Q₈	C₂³×C₆₇	semidirect
388. #125	C₂³×C₆₇	semidirect
416. #103	C₂³×C₆₇	semidirect
417. #103	C₂³×C₆₇	semidirect
418. #103	C₂³×C₆₇	semidirect
423. #102	C₂³×C₆₇	semidirect
424. #102	C₂³×C₆₇	semidirect
425. #102	C₂³×C₆₇	semidirect
426. #97	C₂³×C₆₇	semidirect
427. C₄×C₈	C₂³×C₆₇	semidirect
593. C₂	C₆₇×#3136	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#544×C₆₇	2
C₂×#535×C₆₇	2
C₆₇×#3136	2
C₂×#490×C₆₇	2
C₂×#489×C₆₇	2
C₂×#488×C₆₇	2
C₂×#486×C₆₇	2
C₂×#417×C₆₇	2
C₂×#418×C₆₇	2
C₂×#416×C₆₇	2
C₂²×#102×C₆₇	2
C₂×#405×C₆₇	2
C₂×#3136	67

Subgroups — C2×C67×#3136 #30739861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3136 #30739861