Subgroups — #31009201

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569077	C₂	1	1	yes	maximal, characteristic, complemented
2	#22569061	C₂	1	1	yes	maximal, characteristic
3	#22562074	C₂	1	4	yes	maximal, complemented
4	SD₃₂×Dic₆₇	C₂	1	4	yes	maximal, complemented
5	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
6	#22561929	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561925	C₂	1	1	yes	maximal, characteristic
8	#22561805	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
10	#16863874	—	2	1	no	—
11	#16863871	—	2	1	no	—
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	#16863508	—	2	2	no	—
14	#16863503	—	2	2	no	—
15	Q₁₆×Dic₆₇	C₂²	1	4	yes	complemented
16	D₈×Dic₆₇	C₂²	1	4	yes	—
17	#16863434	C₂²	1	1	yes	characteristic
18	#16863434	C₂²	1	1	yes	characteristic
19	#16863405	—	2	1	no	—
20	#16863401	—	2	1	no	—
21	#16863351	C₂²	1	1	yes	characteristic
22	#16863224	C₂²	1	2	yes	complemented
23	#16863222	C₂²	1	2	yes	complemented
24	#16863200	C₂²	1	4	yes	complemented
25	#16863197	C₂²	1	4	yes	—
26	#16863171	C₂²	1	2	yes	complemented
27	C₁₆×Dic₆₇	C₂²	1	2	yes	complemented
28	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×SD₃₂×C₆₇	C₄	1	8	yes	complemented
31	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
32	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	D₄	1	1	yes	characteristic
35	Q₈×Dic₆₇	—	2	4	no	—
36	Q₈×Dic₆₇	—	4	2	no	—
37	D₄×Dic₆₇	—	2	4	no	—
38	D₄×Dic₆₇	—	4	2	no	—
39	#14524002	—	2	1	no	—
40	#14524002	—	2	1	no	—
41	#14524002	—	4	2	no	—
42	#14524002	D₄	1	1	yes	characteristic
43	C₂×C₂²⋊Dic₆₇	—	4	2	no	—
44	#14523983	—	2	2	no	—
45	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
46	#14523977	C₂³	1	2	yes	—
47	#14523977	C₂³	1	2	yes	—
48	#14523975	—	2	1	no	—
49	#14523975	—	2	1	no	—
50	#14523967	—	2	4	no	—
51	#14523964	—	2	4	no	—
52	#14523953	D₄	1	2	yes	—
53	C₈×Dic₆₇	C₂³	1	2	yes	—
54	C₂²×Q₈×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	—	2	1	no	—
56	C₂×Q₁₆×C₆₇	—	2	1	no	—
57	C₂×Q₁₆×C₆₇	—	2	2	no	—
58	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
59	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
60	C₂×D₈×C₆₇	—	2	1	no	—
61	C₂×D₈×C₆₇	—	2	2	no	—
62	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×D₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	SD₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
66	SD₃₂×C₆₇	C₂×C₄	1	8	yes	complemented
67	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
68	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×Dic₆₇	—	4	4	no	—
71	#11761612	—	2	2	no	—
72	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
73	C₄⋊Dic₆₇	—	2	2	no	—
74	C₄⋊Dic₆₇	—	2	2	no	—
75	C₄⋊Dic₆₇	—	4	4	no	—
76	C₄⋊Dic₆₇	D₈	1	2	yes	—
77	C₂²⋊Dic₆₇	—	4	1	no	—
78	C₂²⋊Dic₆₇	—	4	1	no	—
79	C₂²⋊Dic₆₇	—	4	4	no	—
80	C₂²⋊Dic₆₇	D₈	1	2	yes	—
81	C₄×Dic₆₇	—	4	1	no	—
82	C₄×Dic₆₇	—	4	1	no	—
83	C₄×Dic₆₇	—	4	4	no	—
84	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
85	C₆₇⋊C₁₆	—	2	2	no	—
86	C₂⁴×C₆₇	—	4	1	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	4	1	no	—
90	C₂×Q₈×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	—	4	1	no	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	Q₁₆×C₆₇	—	2	4	no	—
98	Q₁₆×C₆₇	—	2	2	no	—
99	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	—
100	D₈×C₆₇	—	2	4	no	—
101	D₈×C₆₇	—	2	2	no	—
102	D₈×C₆₇	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	#41	1	2	yes	—
106	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
108	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂³×C₆₇	—	4	2	no	—
110	C₂³×C₆₇	—	4	4	no	—
111	C₂³×C₆₇	—	16	2	no	—
112	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₆₇	—	2	4	no	—
114	Q₈×C₆₇	—	4	2	no	—
115	Q₈×C₆₇	—	4	4	no	—
116	D₄×C₆₇	—	2	4	no	—
117	D₄×C₆₇	—	4	2	no	—
118	D₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#135	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₆₇	—	4	2	no	—
126	C₂×Dic₆₇	—	4	2	no	—
127	C₂×Dic₆₇	—	16	4	no	—
128	C₂×Dic₆₇	C₂×D₈	1	4	yes	—
129	C₈×C₆₇	C₄×D₄	1	2	yes	—
130	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
132	C₆₇⋊C₈	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	4	4	no	—
135	C₂²×C₆₇	—	8	4	no	—
136	C₂²×C₆₇	—	16	4	no	—
137	C₂²×C₆₇	C₄×D₈	1	2	yes	—
138	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₆₇	—	4	4	no	—
140	C₄×C₆₇	C₄×D₈	1	2	yes	—
141	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
143	Dic₆₇	—	2	4	no	—
144	Dic₆₇	—	8	2	no	—
145	Dic₆₇	—	8	2	no	—
146	#33457	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	—	8	4	no	—
149	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
150	C₂×C₆₇	#3416	1	2	yes	—
151	C₂²×SD₃₂	D₆₇	1	1	yes	characteristic
152	#2950	—	67	1	no	—
153	#2948	—	67	1	no	—
154	#2175	—	67	4	no	—
155	C₄×SD₃₂	—	67	4	no	—
156	#2158	—	67	1	no	—
157	#2141	—	67	1	no	—
158	#2140	—	67	1	no	—
159	#2108	—	67	1	no	—
160	C₆₇	#33457	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
162	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
163	#519	—	134	1	no	—
164	#518	—	134	1	no	—
165	C₂×SD₃₂	Dic₆₇	1	8	yes	complemented
166	C₂×SD₃₂	C₂×D₆₇	1	2	yes	—
167	C₂×SD₃₂	C₂×D₆₇	1	2	yes	—
168	C₂×M₃₂	—	67	1	no	—
169	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
170	#442	—	134	2	no	—
171	#441	—	134	2	no	—
172	C₄×Q₁₆	—	67	4	no	—
173	C₄×D₈	—	67	4	no	—
174	#427	—	67	1	no	—
175	#427	—	67	1	no	—
176	#419	—	134	1	no	—
177	#418	—	134	1	no	—
178	#405	—	67	1	no	—
179	#367	—	67	2	no	—
180	#366	—	67	2	no	—
181	#358	—	67	4	no	—
182	#357	—	67	4	no	—
183	#346	—	67	2	no	—
184	C₄×C₁₆	—	67	2	no	—
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	complemented
190	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	C₂×Dic₆₇	1	4	yes	—
194	C₂×D₈	C₂²×D₆₇	1	2	yes	—
195	C₂×M₁₆	—	67	1	no	—
196	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
197	C₄×Q₈	—	268	2	no	—
198	C₄×Q₈	—	134	4	no	—
199	C₄×D₄	—	268	2	no	—
200	C₄×D₄	—	134	4	no	—
201	#117	—	268	2	no	—
202	#117	—	67	1	no	—
203	#117	—	134	1	no	—
204	#117	—	134	1	no	—
205	C₂×C₂²⋊C₄	—	268	2	no	—
206	SD₃₂	C₂×Dic₆₇	1	8	yes	complemented
207	SD₃₂	C₂×Dic₆₇	1	8	yes	complemented
208	M₃₂	—	134	2	no	—
209	C₂×C₁₆	—	67	2	no	—
210	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	complemented
211	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
212	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
213	#107	—	67	2	no	—
214	#107	—	67	2	no	—
215	#106	—	134	1	no	—
216	#106	—	134	1	no	—
217	#103	—	134	4	no	—
218	#102	—	134	4	no	—
219	#97	—	67	2	no	—
220	C₄×C₈	—	67	2	no	—
221	C₂⁴	—	4	1	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	268	4	no	—
232	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
233	Q₁₆	—	2	4	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	C₂²×Dic₆₇	1	4	yes	—
236	D₈	—	2	4	no	—
237	D₈	—	2	2	no	—
238	D₈	C₂²×Dic₆₇	1	4	yes	—
239	M₁₆	—	134	2	no	—
240	C₂×C₈	—	67	2	no	—
241	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
242	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
243	C₂×C₈	#11761627	1	2	yes	—
244	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
245	C₄⋊C₄	—	268	4	no	—
246	C₄⋊C₄	—	67	2	no	—
247	C₄⋊C₄	—	134	2	no	—
248	C₄⋊C₄	—	134	2	no	—
249	C₂²⋊C₄	—	268	1	no	—
250	C₂²⋊C₄	—	268	1	no	—
251	C₂²⋊C₄	—	268	4	no	—
252	C₂²⋊C₄	—	67	2	no	—
253	C₄²	—	268	4	no	—
254	C₄²	—	268	1	no	—
255	C₄²	—	268	1	no	—
256	C₄²	—	67	2	no	—
257	C₁₆	—	134	2	no	—
258	C₁₆	C₂²×Dic₆₇	1	2	yes	—
259	C₁₆	C₂²×Dic₆₇	1	2	yes	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	16	2	no	—
263	C₂³	D₈×D₆₇	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	4	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	268	2	no	—
273	C₂×C₄	—	268	2	no	—
274	C₂×C₄	—	1072	4	no	—
275	C₂×C₄	—	67	4	no	—
276	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
277	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
278	C₂×C₄	#14524062	1	2	yes	—
279	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
280	C₈	—	134	2	no	—
281	C₈	D₄×Dic₆₇	1	2	yes	—
282	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
283	C₈	C₂×#11761627	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	16	4	no	—
288	C₂²	D₈×Dic₆₇	1	2	yes	—
289	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
290	C₄	—	4	4	no	—
291	C₄	—	536	2	no	—
292	C₄	—	536	2	no	—
293	C₄	—	134	4	no	—
294	C₄	D₈×Dic₆₇	1	2	yes	—
295	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
296	C₄	C₂×#14524062	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	8	4	no	—
299	C₂	C₂×D₈×Dic₆₇	1	1	yes	characteristic
300	C₂	#22573351	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×SD₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂²×D₈×C₆₇	C₂²
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×SD₃₂	D₆₇
hypocenter	C₆₇	#33457
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #22569077	C₂	semidirect
3. #22562074	C₂	semidirect
4. SD₃₂×Dic₆₇	C₂	semidirect
5. #22561993	C₂	semidirect
6. #22561929	C₂	semidirect
8. #22561805	C₂	semidirect
15. Q₁₆×Dic₆₇	C₂²	semidirect
22. #16863224	C₂²	semidirect
23. #16863222	C₂²	semidirect
24. #16863200	C₂²	semidirect
26. #16863171	C₂²	semidirect
27. C₁₆×Dic₆₇	C₂²	semidirect
30. C₂×SD₃₂×C₆₇	C₄	semidirect
58. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
65. SD₃₂×C₆₇	C₂×C₄	semidirect
66. SD₃₂×C₆₇	C₂×C₄	semidirect
67. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
160. C₆₇	#33457	semidirect
165. C₂×SD₃₂	Dic₆₇	semidirect
189. C₂×Q₁₆	C₂×Dic₆₇	semidirect
206. SD₃₂	C₂×Dic₆₇	semidirect
207. SD₃₂	C₂×Dic₆₇	semidirect
210. C₂×C₁₆	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569077	2
#22569061	2
#22562074	2
SD₃₂×Dic₆₇	2
#22561993	2
#22561929	2
#22561925	2
#22561805	2
C₂²×SD₃₂×C₆₇	2
#33457	67