Subgroups — #31010266

Order 17152.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569786	C₂	1	2	yes	maximal, complemented
3	#22562302	C₂	1	4	yes	maximal, complemented
4	#22562297	C₂	1	2	yes	maximal, complemented
5	#22562292	C₂	1	2	yes	maximal, complemented
6	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561916	C₂	1	2	yes	maximal, complemented
8	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
10	#16863903	—	2	2	no	—
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₆₇	—	2	4	no	—
13	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
14	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
15	#16863634	C₂²	1	4	yes	—
16	#16863629	C₂²	1	4	yes	—
17	#16863622	—	2	4	no	—
18	#16863434	C₂²	1	2	yes	—
19	#16863392	—	2	2	no	—
20	#16863224	C₂²	1	2	yes	complemented
21	#16863222	C₂²	1	2	yes	complemented
22	#16863200	C₂²	1	4	yes	complemented
23	#16863197	C₂²	1	4	yes	complemented
24	C₂×#135×C₆₇	C₂²	1	2	yes	—
25	#508×C₆₇	—	2	4	no	—
26	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	complemented
28	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₈×D₆₇	—	2	2	no	—
33	C₂×C₈×D₆₇	—	2	2	no	—
34	C₂×C₈×D₆₇	D₄	1	2	yes	—
35	C₂×C₈×D₆₇	C₂³	1	2	yes	—
36	#14524024	—	4	2	no	—
37	#14524019	—	2	4	no	—
38	#14524014	—	2	4	no	—
39	#14524011	—	4	2	no	—
40	C₄×C₂²⋊D₆₇	—	4	4	no	—
41	#14524002	—	2	2	no	—
42	C₂×C₆₇⋊C₁₆	—	2	2	no	—
43	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
44	C₁₆×D₆₇	—	2	4	no	—
45	C₁₆×D₆₇	—	2	4	no	—
46	#14523977	C₂³	1	4	yes	—
47	#14523975	—	2	2	no	—
48	#14523967	—	2	4	no	—
49	#14523964	—	2	4	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	#135×C₆₇	—	2	4	no	—
52	#135×C₆₇	—	2	4	no	—
53	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
54	C₂×SD₁₆×C₆₇	—	2	2	no	—
55	C₂×D₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	Q₃₂×C₆₇	—	2	2	no	—
58	SD₃₂×C₆₇	—	2	4	no	—
59	D₁₆×C₆₇	—	2	2	no	—
60	C₂×C₁₆×C₆₇	D₄	1	4	yes	complemented
61	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×D₆₇	D₈	1	1	yes	characteristic
64	C₂×C₂²⋊D₆₇	—	4	4	no	—
65	C₂²×Dic₆₇	D₈	1	1	yes	characteristic
66	C₂×C₄×D₆₇	—	2	2	no	—
67	C₂×C₄×D₆₇	—	2	2	no	—
68	C₂×C₄×D₆₇	D₈	1	2	yes	—
69	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
72	C₈×D₆₇	—	2	4	no	—
73	C₈×D₆₇	—	2	2	no	—
74	C₈×D₆₇	—	2	2	no	—
75	C₄⋊Dic₆₇	—	2	4	no	—
76	Dic₆₇⋊C₄	—	4	4	no	—
77	C₂²⋊Dic₆₇	—	4	2	no	—
78	C₂²⋊Dic₆₇	—	4	4	no	—
79	(C₂×D₆₇)⋊C₄	—	4	4	no	—
80	C₄×Dic₆₇	—	4	2	no	—
81	C₆₇⋊C₁₆	—	2	2	no	—
82	#41×C₆₇	—	4	4	no	—
83	#41×C₆₇	—	4	4	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₆₇	—	2	2	no	—
90	Q₁₆×C₆₇	—	2	2	no	—
91	SD₁₆×C₆₇	—	4	4	no	—
92	D₈×C₆₇	—	2	2	no	—
93	D₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₆₇	#41	1	2	yes	—
97	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
99	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
100	C₂³×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₆₇	—	2	2	no	—
103	C₂²×D₆₇	—	2	4	no	—
104	C₂²×D₆₇	C₂×D₈	1	2	yes	—
105	Q₈×C₆₇	—	4	2	no	—
106	Q₈×C₆₇	—	4	2	no	—
107	D₄×C₆₇	—	4	2	no	—
108	D₄×C₆₇	—	4	2	no	—
109	D₄×C₆₇	—	8	4	no	—
110	C₂×C₄×C₆₇	—	4	2	no	—
111	C₂×C₄×C₆₇	—	8	4	no	—
112	C₂×C₄×C₆₇	#121	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#135	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
116	C₂²⋊D₆₇	—	8	8	no	—
117	C₂×Dic₆₇	—	2	2	no	—
118	C₂×Dic₆₇	—	4	4	no	—
119	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
120	C₄×D₆₇	—	2	2	no	—
121	C₄×D₆₇	—	2	2	no	—
122	C₄×D₆₇	—	2	4	no	—
123	C₈×C₆₇	#121	1	2	yes	—
124	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₆₇⋊C₈	—	2	2	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	8	4	no	—
129	C₂²×C₆₇	—	8	2	no	—
130	C₂²×C₆₇	#466	1	2	yes	—
131	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
132	C₂×D₆₇	—	2	8	no	—
133	C₂×D₆₇	—	2	2	no	—
134	C₂×D₆₇	—	2	2	no	—
135	C₂×D₆₇	#508	1	4	yes	complemented
136	C₄×C₆₇	—	8	2	no	—
137	C₄×C₆₇	#466	1	2	yes	—
138	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
139	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
140	Dic₆₇	—	2	2	no	—
141	Dic₆₇	—	8	4	no	—
142	#33600	—	67	1	no	maximal, Sylow
143	C₂×C₆₇	—	2	2	no	—
144	C₂×C₆₇	—	16	2	no	—
145	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
146	C₂×C₆₇	C₂×#508	1	2	yes	—
147	D₆₇	—	2	4	no	—
148	C₂×#508	D₆₇	1	1	yes	characteristic, complemented
149	C₂³×C₁₆	—	67	1	no	—
150	#3050	—	67	2	no	—
151	#2217	—	67	4	no	—
152	#2216	—	67	2	no	—
153	#2215	—	67	2	no	—
154	#2158	—	67	1	no	—
155	#2138	—	67	2	no	—
156	C₆₇	#33600	1	1	yes	characteristic, complemented, Sylow
157	C₂×#135	C₂×D₆₇	1	2	yes	—
158	C₂³×C₈	—	67	1	no	—
159	#525	—	134	2	no	—
160	#508	—	2	4	no	—
161	C₂×Q₃₂	C₂×D₆₇	1	1	yes	characteristic, complemented
162	C₂×SD₃₂	C₂×D₆₇	1	2	yes	complemented
163	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
164	C₂²×C₁₆	—	67	1	no	—
165	C₂²×C₁₆	—	67	2	no	—
166	C₂²×C₁₆	—	67	2	no	—
167	C₂²×C₁₆	—	134	4	no	—
168	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
169	#467	—	67	4	no	—
170	#466	—	67	4	no	—
171	#465	—	134	4	no	—
172	#427	—	67	2	no	—
173	#416	—	134	2	no	—
174	#367	—	67	2	no	—
175	#366	—	67	2	no	—
176	#358	—	67	4	no	—
177	#357	—	67	4	no	—
178	C₂×#41	—	2	2	no	—
179	C₂³×C₄	—	67	1	no	—
180	#135	—	2	4	no	—
181	#135	—	2	4	no	—
182	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×D₈	C₂²×D₆₇	1	2	yes	—
185	C₂²×C₈	—	67	1	no	—
186	C₂²×C₈	—	67	2	no	—
187	C₂²×C₈	—	67	2	no	—
188	C₂²×C₈	—	134	2	no	—
189	C₂²×C₈	—	134	2	no	—
190	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
191	#123	—	268	2	no	—
192	C₂²⋊Q₈	—	134	4	no	—
193	#121	—	134	4	no	—
194	#120	—	268	2	no	—
195	C₄×D₄	—	268	4	no	—
196	#117	—	134	2	no	—
197	Q₃₂	—	2	2	no	—
198	SD₃₂	—	2	4	no	—
199	D₁₆	—	2	2	no	—
200	C₂×C₁₆	—	67	2	no	—
201	C₂×C₁₆	—	134	2	no	—
202	C₂×C₁₆	—	134	4	no	—
203	C₂×C₁₆	—	134	4	no	—
204	C₂×C₁₆	C₂²⋊D₆₇	1	4	yes	complemented
205	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
207	#107	—	67	4	no	—
208	#106	—	134	2	no	—
209	#103	—	134	4	no	—
210	#102	—	134	4	no	—
211	C₂⁴	—	67	1	no	—
212	#41	—	4	4	no	—
213	#41	—	4	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	268	4	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	67	1	no	—
220	C₂²×C₄	—	67	2	no	—
221	C₂²×C₄	—	67	2	no	—
222	C₂²×C₄	—	134	2	no	—
223	C₂²×C₄	—	134	2	no	—
224	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	D₈	—	2	2	no	—
229	D₈	—	2	2	no	—
230	C₂×C₈	—	67	2	no	—
231	C₂×C₈	—	134	2	no	—
232	C₂×C₈	—	134	4	no	—
233	C₂×C₈	—	134	2	no	—
234	C₂×C₈	—	134	2	no	—
235	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
237	C₂×C₈	#11761627	1	2	yes	—
238	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
239	C₄⋊C₄	—	268	4	no	—
240	C₄⋊C₄	—	134	4	no	—
241	C₂²⋊C₄	—	268	4	no	—
242	C₂²⋊C₄	—	268	2	no	—
243	C₂²⋊C₄	—	268	4	no	—
244	C₄²	—	268	2	no	—
245	C₁₆	—	134	2	no	—
246	C₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
247	C₁₆	C₂×C₂²⋊D₆₇	1	2	yes	—
248	C₂³	—	4	2	no	—
249	C₂³	—	67	2	no	—
250	C₂³	—	134	4	no	—
251	C₂³	—	134	2	no	—
252	C₂³	D₈×D₆₇	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	4	no	—
258	D₄	—	536	8	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	268	4	no	—
262	C₂×C₄	—	67	2	no	—
263	C₂×C₄	—	134	2	no	—
264	C₂×C₄	—	134	4	no	—
265	C₂×C₄	—	134	2	no	—
266	C₂×C₄	—	134	2	no	—
267	C₂×C₄	#14524014	1	2	yes	—
268	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
269	C₂×C₄	#14524062	1	2	yes	—
270	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
271	C₈	—	134	2	no	—
272	C₈	#14524014	1	2	yes	—
273	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
274	C₈	C₂×#11761627	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	8	4	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	67	4	no	—
279	C₂²	—	134	8	no	—
280	C₂²	—	134	2	no	—
281	C₂²	—	134	2	no	—
282	C₂²	#16863629	1	2	yes	—
283	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
284	C₄	—	8	2	no	—
285	C₄	—	536	4	no	—
286	C₄	—	134	2	no	—
287	C₄	#16863629	1	2	yes	—
288	C₄	C₂×#14524014	1	1	yes	characteristic
289	C₄	C₂×#14524062	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	16	2	no	—
292	C₂	—	134	4	no	—
293	C₂	C₂×#16863629	1	1	yes	characteristic
294	C₂	#508×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#508×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	C₂×#508	D₆₇
hypocenter	C₆₇	#33600
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₆₇	C₂	semidirect
2. #22569786	C₂	semidirect
3. #22562302	C₂	semidirect
4. #22562297	C₂	semidirect
5. #22562292	C₂	semidirect
6. #22561993	C₂	semidirect
7. #22561916	C₂	semidirect
8. C₂×#508×C₆₇	C₂	semidirect
20. #16863224	C₂²	semidirect
21. #16863222	C₂²	semidirect
22. #16863200	C₂²	semidirect
23. #16863197	C₂²	semidirect
26. C₂×Q₃₂×C₆₇	C₂²	semidirect
27. C₂×SD₃₂×C₆₇	C₂²	semidirect
28. C₂×D₁₆×C₆₇	C₂²	semidirect
60. C₂×C₁₆×C₆₇	D₄	semidirect
135. C₂×D₆₇	#508	semidirect
148. C₂×#508	D₆₇	semidirect
156. C₆₇	#33600	semidirect
161. C₂×Q₃₂	C₂×D₆₇	semidirect
162. C₂×SD₃₂	C₂×D₆₇	semidirect
163. C₂×D₁₆	C₂×D₆₇	semidirect
204. C₂×C₁₆	C₂²⋊D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₆₇	2
#22569786	2
#22562302	2
#22562297	2
#22562292	2
#22561993	2
#22561916	2
C₂×#508×C₆₇	2
#33600	67