Subgroups — #31789734

Order 17536.

356 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577957	C₂	1	1	yes	maximal, characteristic
2	#22577534	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577532	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577532	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577531	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577531	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577527	C₂	1	1	yes	maximal, characteristic, complemented
9	#22577428	C₂	1	1	yes	maximal, characteristic
10	#22577428	C₂	1	1	yes	maximal, characteristic
11	C₄⋊C₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
12	#22577360	C₂	1	1	yes	maximal, characteristic
13	#22577357	C₂	1	1	yes	maximal, characteristic
14	#22577355	C₂	1	1	yes	maximal, characteristic
15	#446×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14555493	C₂²	1	1	yes	characteristic
17	C₂×#14555493	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
20	#17984087	C₄	1	1	yes	characteristic
21	#17984086	C₄	1	1	yes	characteristic
22	#17984078	C₄	1	1	yes	characteristic
23	#17984078	C₄	1	1	yes	characteristic
24	#17984068	C₄	1	1	yes	characteristic
25	#17984066	C₄	1	1	yes	characteristic
26	#17984066	C₄	1	1	yes	characteristic
27	#17984064	C₄	1	1	yes	characteristic
28	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
31	#17984045	C₂²	1	1	yes	characteristic
32	#17984045	C₂²	1	1	yes	characteristic
33	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
34	C₄⋊#11764040	C₂²	1	1	yes	characteristic
35	C₄⋊#11764040	C₂²	1	1	yes	characteristic
36	C₄⋊#11764040	C₂²	1	1	yes	characteristic
37	#17984017	C₂²	1	1	yes	characteristic
38	#17984017	C₂²	1	1	yes	characteristic
39	#17984017	C₂²	1	1	yes	characteristic
40	#17984017	C₂²	1	1	yes	characteristic
41	#17984017	C₂²	1	1	yes	characteristic
42	#17984017	C₂²	1	1	yes	characteristic
43	#17983999	C₂²	1	1	yes	characteristic
44	#17983999	C₂²	1	1	yes	characteristic
45	#17983998	C₂²	1	1	yes	characteristic
46	#17983998	C₂²	1	1	yes	characteristic
47	C₄×#11764040	C₂²	1	1	yes	characteristic
48	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
56	#97×C₁₃₇	C₂²	1	1	yes	characteristic
57	#97×C₁₃₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
60	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
61	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
65	#14555493	—	2	2	no	—
66	#14555493	Q₈	1	4	yes	complemented
67	C₂×#11764040	C₂³	1	1	yes	characteristic
68	C₂×#11764040	C₂³	1	1	yes	characteristic
69	C₂×#11764040	C₂³	1	1	yes	characteristic
70	C₂×#11764040	C₂³	1	1	yes	characteristic
71	C₈×D₁₃₇	—	2	2	no	—
72	C₈×D₁₃₇	—	2	2	no	—
73	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
74	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
75	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
76	#14555489	C₂×C₄	1	1	yes	characteristic
77	#14555489	C₂×C₄	1	1	yes	characteristic
78	#14555489	C₂×C₄	1	1	yes	characteristic
79	#14555489	C₂×C₄	1	1	yes	characteristic
80	#14555489	C₂×C₄	1	1	yes	characteristic
81	#14555489	C₂×C₄	1	1	yes	characteristic
82	#14555489	C₂×C₄	1	1	yes	characteristic
83	#14555489	C₂×C₄	1	1	yes	characteristic
84	#14555489	C₂×C₄	1	1	yes	characteristic
85	#14555489	C₂×C₄	1	1	yes	characteristic
86	#14555487	C₂×C₄	1	1	yes	characteristic
87	#14555487	C₂×C₄	1	1	yes	characteristic
88	#14555487	C₂×C₄	1	1	yes	characteristic
89	#14555487	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
103	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
104	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
105	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
106	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
107	Dic₂₇₄	—	2	2	no	—
108	Dic₂₇₄	—	2	2	no	—
109	Dic₂₇₄	—	4	1	no	—
110	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
117	C₄×D₁₃₇	—	2	2	no	—
118	C₄×D₁₃₇	—	4	1	no	—
119	C₄×D₁₃₇	C₂×Q₈	1	2	yes	—
120	C₄×D₁₃₇	#41	1	2	yes	—
121	#11764040	—	2	1	no	—
122	#11764040	—	2	1	no	—
123	#11764040	—	2	1	no	—
124	#11764040	C₂×Q₈	1	2	yes	—
125	Q₈×C₁₃₇	—	4	1	no	—
126	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
133	C₈×C₁₃₇	—	2	1	no	—
134	C₈×C₁₃₇	—	2	1	no	—
135	C₈×C₁₃₇	—	2	1	no	—
136	C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
137	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×D₁₃₇	—	2	1	no	—
139	C₂×D₁₃₇	—	2	1	no	—
140	C₂×D₁₃₇	C₄×Q₈	1	2	yes	—
141	C₄×C₁₃₇	—	2	1	no	—
142	C₄×C₁₃₇	—	2	1	no	—
143	C₄×C₁₃₇	—	2	1	no	—
144	C₄×C₁₃₇	—	2	1	no	—
145	C₄×C₁₃₇	—	2	1	no	—
146	C₄×C₁₃₇	—	2	1	no	—
147	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
148	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
149	Dic₁₃₇	—	2	1	no	—
150	Dic₁₃₇	—	2	1	no	—
151	Dic₁₃₇	—	2	1	no	—
152	Dic₁₃₇	—	2	1	no	—
153	Dic₁₃₇	—	2	1	no	—
154	Dic₁₃₇	C₄×Q₈	1	2	yes	—
155	Dic₁₃₇	#131	1	2	yes	—
156	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
158	C₂×C₁₃₇	#568	1	1	yes	characteristic
159	D₁₃₇	—	2	2	no	—
160	C₁₃₇	#2962	1	1	yes	characteristic, complemented, Sylow
161	#2962	—	137	1	no	maximal, Sylow
162	#533	—	137	1	no	—
163	#446	—	137	1	no	—
164	#446	—	137	1	no	—
165	#446	—	137	1	no	—
166	#446	—	137	1	no	—
167	#446	—	137	1	no	—
168	#446	D₁₃₇	1	1	yes	characteristic, complemented
169	C₈×Q₈	—	137	1	no	—
170	C₈×Q₈	—	137	1	no	—
171	#424	—	137	1	no	—
172	#424	—	137	1	no	—
173	C₂×C₄⋊C₈	—	137	1	no	—
174	#405	—	137	1	no	—
175	#405	—	137	1	no	—
176	C₄×M₁₆	—	137	1	no	—
177	C₂×M₁₆	—	137	1	no	—
178	C₂×M₁₆	—	137	1	no	—
179	C₂²×C₈	—	137	1	no	—
180	C₂²×C₈	—	137	1	no	—
181	C₄⋊Q₈	—	137	1	no	—
182	C₄⋊Q₈	—	137	1	no	—
183	#125	—	137	1	no	—
184	#125	—	137	1	no	—
185	C₂²⋊Q₈	—	137	1	no	—
186	C₂²⋊Q₈	—	137	1	no	—
187	C₂²⋊Q₈	—	137	1	no	—
188	C₂²⋊Q₈	—	137	1	no	—
189	C₄×Q₈	—	137	1	no	—
190	C₄×Q₈	—	137	1	no	—
191	C₄×Q₈	—	137	1	no	—
192	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
193	#117	—	137	1	no	—
194	#117	—	137	1	no	—
195	C₂×C₄²	—	137	1	no	—
196	C₄⋊C₈	—	137	1	no	—
197	C₄⋊C₈	—	137	1	no	—
198	C₄⋊C₈	—	137	1	no	—
199	C₄⋊C₈	—	137	1	no	—
200	C₄⋊C₈	—	137	1	no	—
201	C₄⋊C₈	—	137	1	no	—
202	C₄⋊C₈	—	137	1	no	—
203	C₄⋊C₈	—	137	1	no	—
204	C₄⋊C₈	—	137	1	no	—
205	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
206	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
207	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
208	#97	—	137	1	no	—
209	#97	—	137	1	no	—
210	#97	—	137	1	no	—
211	#97	—	137	1	no	—
212	#97	C₂×D₁₃₇	1	1	yes	characteristic
213	#97	C₂×D₁₃₇	1	1	yes	characteristic
214	C₄×C₈	—	137	1	no	—
215	C₄×C₈	—	137	1	no	—
216	C₄×C₈	—	137	1	no	—
217	C₄×C₈	—	137	1	no	—
218	C₄×C₈	—	137	1	no	—
219	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
220	C₂×Q₈	—	137	1	no	—
221	C₂×Q₈	—	137	1	no	—
222	C₂×Q₈	—	137	1	no	—
223	C₂×Q₈	C₄×D₁₃₇	1	1	yes	characteristic
224	C₂²×C₄	—	137	1	no	—
225	C₂²×C₄	—	137	1	no	—
226	C₂²×C₄	—	137	1	no	—
227	M₁₆	—	274	2	no	—
228	M₁₆	—	137	4	no	—
229	C₂×C₈	—	274	2	no	—
230	C₂×C₈	—	274	2	no	—
231	C₂×C₈	—	137	1	no	—
232	C₂×C₈	—	137	1	no	—
233	C₂×C₈	—	137	1	no	—
234	C₂×C₈	—	137	1	no	—
235	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
239	C₄⋊C₄	—	137	1	no	—
240	C₄⋊C₄	—	137	1	no	—
241	C₄⋊C₄	—	137	1	no	—
242	C₄⋊C₄	—	137	1	no	—
243	C₄⋊C₄	—	137	1	no	—
244	C₄⋊C₄	—	137	1	no	—
245	C₄⋊C₄	—	137	1	no	—
246	C₄⋊C₄	—	137	1	no	—
247	C₄⋊C₄	—	137	1	no	—
248	C₄⋊C₄	—	137	1	no	—
249	C₄⋊C₄	—	137	1	no	—
250	C₄⋊C₄	—	137	1	no	—
251	C₄⋊C₄	—	137	1	no	—
252	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
253	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
254	C₄⋊C₄	C₄×D₁₃₇	1	1	yes	characteristic
255	C₂²⋊C₄	—	137	1	no	—
256	C₂²⋊C₄	—	137	1	no	—
257	C₂²⋊C₄	—	137	1	no	—
258	C₂²⋊C₄	—	137	1	no	—
259	C₄²	—	137	1	no	—
260	C₄²	—	137	1	no	—
261	C₄²	—	137	1	no	—
262	C₄²	—	137	1	no	—
263	C₄²	—	137	1	no	—
264	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
265	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
266	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
267	C₂³	—	137	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	274	2	no	—
270	Q₈	—	274	2	no	—
271	Q₈	—	548	1	no	—
272	C₂×C₄	—	274	2	no	—
273	C₂×C₄	—	548	1	no	—
274	C₂×C₄	—	137	1	no	—
275	C₂×C₄	—	137	1	no	—
276	C₂×C₄	—	137	1	no	—
277	C₂×C₄	—	137	1	no	—
278	C₂×C₄	—	137	1	no	—
279	C₂×C₄	—	137	1	no	—
280	C₂×C₄	—	137	1	no	—
281	C₂×C₄	—	137	2	no	—
282	C₂×C₄	—	137	2	no	—
283	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	274	1	no	—
294	C₈	—	274	1	no	—
295	C₈	—	274	1	no	—
296	C₈	—	137	2	no	—
297	C₈	Q₈×D₁₃₇	1	2	yes	—
298	C₂²	—	274	1	no	—
299	C₂²	—	274	1	no	—
300	C₂²	—	137	2	no	—
301	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	274	1	no	—
309	C₄	—	274	1	no	—
310	C₄	—	274	1	no	—
311	C₄	—	274	1	no	—
312	C₄	—	274	1	no	—
313	C₄	—	137	2	no	—
314	C₄	—	137	2	no	—
315	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
316	C₄	#41×D₁₃₇	1	1	yes	characteristic
317	C₂	—	274	2	no	—
318	C₂	C₄×Q₈×D₁₃₇	1	1	yes	characteristic
319	C₂	#131×D₁₃₇	1	1	yes	characteristic
320	C₂	#22578169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#446×C₁₃₇	C₂
prime radical	C₂²×D₁₃₇	C₂²×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#446	D₁₃₇
hypocenter	C₁₃₇	#2962
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #22577534	C₂	semidirect
3. #22577532	C₂	semidirect
4. #22577532	C₂	semidirect
5. #22577531	C₂	semidirect
6. #22577531	C₂	semidirect
7. C₈×Dic₂₇₄	C₂	semidirect
8. #22577527	C₂	semidirect
15. #446×C₁₃₇	C₂	semidirect
66. #14555493	Q₈	semidirect
160. C₁₃₇	#2962	semidirect
168. #446	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577957	2
#22577534	2
#22577532	2
#22577531	2
C₈×Dic₂₇₄	2
#22577527	2
#22577428	2
C₄⋊C₈×D₁₃₇	2
#22577360	2
#22577357	2
#22577355	2
#446×C₁₃₇	2
#2962	137