Subgroups — #18004811

Order 4736.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558143	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558016	C₂	1	1	yes	maximal, characteristic
3	#14557869	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557861	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557842	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557831	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557741	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557736	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557695	C₂	1	1	yes	maximal, characteristic
10	#14557637	C₂	1	1	yes	maximal, characteristic, complemented
11	#14557634	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557624	C₂	1	1	yes	maximal, characteristic, complemented
14	#14557591	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#452	C₂	1	1	yes	maximal, characteristic
16	C₂×#10596942	C₂²	1	1	yes	characteristic
17	C₂×#10596940	C₂²	1	1	yes	characteristic
18	#11924534	C₂²	1	1	yes	characteristic
19	#11924529	—	2	1	no	—
20	#11924529	—	2	1	no	—
21	#11924527	—	2	1	no	—
22	#11924527	C₂²	1	1	yes	characteristic
23	#11924526	—	2	1	no	—
24	#11924515	C₂²	1	1	yes	characteristic
25	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
26	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
27	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
28	#11924501	C₂²	1	1	yes	characteristic
29	D₄×Dic₃₇	—	2	1	no	—
30	D₄×Dic₃₇	—	2	1	no	—
31	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
32	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
33	#11924495	—	2	1	no	—
34	#11924495	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₃₇	—	2	1	no	—
36	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
37	#11924470	C₂²	1	1	yes	characteristic
38	#11924470	C₂²	1	1	yes	characteristic
39	#11924467	C₂²	1	1	yes	characteristic
40	#11924467	C₂²	1	1	yes	characteristic
41	#11924465	C₂²	1	1	yes	characteristic
42	#11924465	C₂²	1	1	yes	characteristic
43	#11924461	C₂²	1	1	yes	characteristic
44	#11924461	C₂²	1	1	yes	characteristic
45	#11924460	C₂²	1	1	yes	characteristic
46	#11924460	C₂²	1	1	yes	characteristic
47	#11924460	C₂²	1	1	yes	characteristic
48	#11924457	C₂²	1	1	yes	characteristic
49	C₂²⋊#64436	C₂²	1	1	yes	characteristic
50	#11924446	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
52	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₃₇	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
56	#103×C₃₇	C₂²	1	1	yes	characteristic
57	#102×C₃₇	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
59	#41×C₃₇	—	2	2	no	—
60	#41×C₃₇	—	2	2	no	—
61	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₃₇	—	2	1	no	—
64	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	—	2	1	no	—
66	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
67	C₂²×Dic₃₇	—	2	1	no	—
68	C₂²×Dic₃₇	—	2	1	no	—
69	C₂²×Dic₃₇	—	2	1	no	—
70	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
71	Q₁₆×C₃₇	—	2	2	no	—
72	SD₁₆×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₃₇	—	2	1	no	—
76	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₃₇	—	2	1	no	—
78	C₂²⋊Dic₃₇	—	2	1	no	—
79	C₂²⋊Dic₃₇	—	2	1	no	—
80	C₂²⋊Dic₃₇	—	2	1	no	—
81	C₂²⋊Dic₃₇	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₃₇	D₄	1	1	yes	characteristic
83	#10596942	—	2	2	no	—
84	#10596940	—	2	2	no	—
85	C₄⋊Dic₃₇	—	2	1	no	—
86	C₄⋊Dic₃₇	—	2	1	no	—
87	C₄⋊Dic₃₇	—	2	1	no	—
88	C₄⋊Dic₃₇	—	2	1	no	—
89	C₄⋊Dic₃₇	—	2	1	no	—
90	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
91	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
92	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
95	#10596936	—	2	1	no	—
96	#10596936	—	2	1	no	—
97	#10596936	—	2	1	no	—
98	#10596936	—	2	1	no	—
99	#10596936	—	2	1	no	—
100	C₄×Dic₃₇	—	2	1	no	—
101	C₄×Dic₃₇	—	2	1	no	—
102	C₄×Dic₃₇	—	2	1	no	—
103	C₄×Dic₃₇	—	2	1	no	—
104	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
105	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
106	C₂×#64436	C₂³	1	1	yes	characteristic
107	C₂×#64436	C₂³	1	1	yes	characteristic
108	C₂³×C₃₇	—	2	1	no	—
109	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₃₇	—	2	1	no	—
111	Q₈×C₃₇	—	2	1	no	—
112	Q₈×C₃₇	—	2	1	no	—
113	Q₈×C₃₇	#41	1	2	yes	—
114	D₄×C₃₇	—	2	4	no	—
115	D₄×C₃₇	—	2	1	no	—
116	D₄×C₃₇	#41	1	2	yes	—
117	C₂×C₄×C₃₇	—	2	4	no	—
118	C₂×C₄×C₃₇	—	2	1	no	—
119	C₂×C₄×C₃₇	—	2	1	no	—
120	C₂×C₄×C₃₇	—	2	1	no	—
121	C₂×C₄×C₃₇	—	2	1	no	—
122	C₂×C₄×C₃₇	—	2	1	no	—
123	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₃₇	—	2	1	no	—
126	C₂×Dic₃₇	—	2	1	no	—
127	C₂×Dic₃₇	—	2	1	no	—
128	C₂×Dic₃₇	—	2	1	no	—
129	C₂×Dic₃₇	—	8	1	no	—
130	C₂×Dic₃₇	—	8	1	no	—
131	C₂×Dic₃₇	—	8	1	no	—
132	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
136	Dic₇₄	—	2	1	no	—
137	Dic₇₄	—	2	1	no	—
138	C₈×C₃₇	—	2	1	no	—
139	C₈×C₃₇	—	2	1	no	—
140	#64436	—	2	1	no	—
141	#64436	—	2	1	no	—
142	C₂²×C₃₇	—	2	1	no	—
143	C₂²×C₃₇	—	2	2	no	—
144	C₂²×C₃₇	—	2	1	no	—
145	C₂²×C₃₇	—	2	1	no	—
146	C₂²×C₃₇	—	4	1	no	—
147	C₂²×C₃₇	—	4	1	no	—
148	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₃₇	—	2	1	no	—
150	C₄×C₃₇	—	2	2	no	—
151	C₄×C₃₇	—	4	1	no	—
152	C₄×C₃₇	—	4	1	no	—
153	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
154	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
155	Dic₃₇	—	2	1	no	—
156	Dic₃₇	—	2	1	no	—
157	Dic₃₇	—	2	1	no	—
158	Dic₃₇	—	2	1	no	—
159	Dic₃₇	—	4	1	no	—
160	Dic₃₇	—	4	1	no	—
161	Dic₃₇	—	4	1	no	—
162	Dic₃₇	—	4	1	no	—
163	#3120	—	37	1	no	maximal, Sylow
164	C₂×C₃₇	—	4	1	no	—
165	C₂×C₃₇	—	4	2	no	—
166	C₂×C₃₇	#525	1	1	yes	characteristic
167	C₂×C₃₇	#577	1	1	yes	characteristic
168	C₂×C₃₇	#578	1	1	yes	characteristic
169	#541	—	37	1	no	—
170	#520	—	37	1	no	—
171	#484	—	37	1	no	—
172	#483	—	37	1	no	—
173	#479	—	37	1	no	—
174	#478	—	37	1	no	—
175	#462	—	37	1	no	—
176	#461	—	37	1	no	—
177	#452	—	37	1	no	—
178	#452	D₃₇	1	1	yes	characteristic
179	#441	—	37	1	no	—
180	#440	—	37	1	no	—
181	C₄×Q₁₆	—	37	1	no	—
182	C₄×SD₁₆	—	37	1	no	—
183	#433	—	37	1	no	—
184	C₃₇	#3120	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
186	C₂×Q₁₆	—	37	1	no	—
187	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
188	C₂×SD₁₆	—	37	1	no	—
189	C₂×SD₁₆	C₂×D₃₇	1	1	yes	characteristic
190	C₄⋊Q₈	—	37	1	no	—
191	#126	—	74	1	no	—
192	#126	—	74	1	no	—
193	#125	—	37	1	no	—
194	#125	—	74	1	no	—
195	#125	—	74	1	no	—
196	C₂²⋊Q₈	—	37	1	no	—
197	C₂²⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
198	C₄×Q₈	—	37	1	no	—
199	C₄×Q₈	—	37	1	no	—
200	C₄×Q₈	—	37	1	no	—
201	C₄×Q₈	—	37	1	no	—
202	C₄×D₄	—	37	1	no	—
203	C₄×D₄	—	37	1	no	—
204	C₄×D₄	—	74	1	no	—
205	C₄×D₄	—	74	1	no	—
206	#117	—	37	1	no	—
207	#117	—	74	1	no	—
208	C₂×C₄⋊C₄	—	37	1	no	—
209	C₂×C₄⋊C₄	—	74	1	no	—
210	#107	—	37	1	no	—
211	#107	—	37	1	no	—
212	#106	—	37	1	no	—
213	#106	—	37	1	no	—
214	C₄⋊C₈	—	37	1	no	—
215	C₄⋊C₈	—	37	1	no	—
216	#103	—	37	1	no	—
217	#103	—	37	1	no	—
218	#103	—	37	1	no	—
219	#103	—	37	1	no	—
220	#103	—	37	1	no	—
221	#103	C₂×D₃₇	1	1	yes	characteristic
222	#102	—	37	1	no	—
223	#102	C₂×D₃₇	1	1	yes	characteristic
224	C₂²⋊C₈	—	37	1	no	—
225	C₂²⋊C₈	C₂×D₃₇	1	1	yes	characteristic
226	#97	—	37	1	no	—
227	C₄×C₈	—	37	1	no	—
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	C₂×Q₈	—	37	1	no	—
231	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	74	1	no	—
237	C₂²×C₄	—	74	1	no	—
238	C₂²×C₄	—	74	1	no	—
239	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	74	2	no	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	74	2	no	—
244	C₂×C₈	—	37	1	no	—
245	C₂×C₈	—	37	1	no	—
246	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	37	1	no	—
250	C₄⋊C₄	—	37	1	no	—
251	C₄⋊C₄	—	37	1	no	—
252	C₄⋊C₄	—	37	1	no	—
253	C₄⋊C₄	—	37	1	no	—
254	C₄⋊C₄	—	74	1	no	—
255	C₄⋊C₄	—	74	1	no	—
256	C₄⋊C₄	—	74	1	no	—
257	C₄⋊C₄	—	74	1	no	—
258	C₄⋊C₄	—	74	1	no	—
259	C₄⋊C₄	—	74	1	no	—
260	C₄⋊C₄	—	74	1	no	—
261	C₄⋊C₄	—	74	1	no	—
262	C₄⋊C₄	—	74	1	no	—
263	C₄⋊C₄	—	74	1	no	—
264	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	37	1	no	—
267	C₂²⋊C₄	—	37	1	no	—
268	C₂²⋊C₄	—	74	1	no	—
269	C₂²⋊C₄	—	74	1	no	—
270	C₂²⋊C₄	—	74	1	no	—
271	C₄²	—	37	1	no	—
272	C₄²	—	37	1	no	—
273	C₄²	—	74	1	no	—
274	C₄²	—	74	1	no	—
275	C₄²	—	74	1	no	—
276	C₄²	—	74	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	D₄×D₃₇	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	74	1	no	—
283	Q₈	—	74	1	no	—
284	Q₈	#10596964	1	2	yes	—
285	D₄	—	2	4	no	—
286	D₄	—	2	1	no	—
287	D₄	#10596964	1	2	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	37	1	no	—
295	C₂×C₄	—	37	1	no	—
296	C₂×C₄	—	37	1	no	—
297	C₂×C₄	—	37	1	no	—
298	C₂×C₄	—	296	1	no	—
299	C₂×C₄	—	296	1	no	—
300	C₂×C₄	—	296	1	no	—
301	C₂×C₄	—	74	1	no	—
302	C₂×C₄	—	74	1	no	—
303	C₂×C₄	—	74	1	no	—
304	C₂×C₄	—	74	1	no	—
305	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	74	1	no	—
310	C₈	—	74	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	74	1	no	—
323	C₄	—	74	1	no	—
324	C₄	—	74	1	no	—
325	C₄	—	74	1	no	—
326	C₄	—	148	1	no	—
327	C₄	—	148	1	no	—
328	C₄	—	148	1	no	—
329	C₄	—	148	1	no	—
330	C₄	C₂×#10596964	1	1	yes	characteristic
331	C₄	C₂×#10596964	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	4	2	no	—
334	C₂	#14558039	1	1	yes	characteristic
335	C₂	#14558342	1	1	yes	characteristic
336	C₂	#14558347	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#452	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#41×C₃₇	C₂²
hypercenter	#452	D₃₇
hypocenter	C₃₇	#3120
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #14558143	C₂	semidirect
3. #14557869	C₂	semidirect
4. #14557861	C₂	semidirect
5. #14557842	C₂	semidirect
6. #14557831	C₂	semidirect
7. #14557741	C₂	semidirect
8. #14557736	C₂	semidirect
10. #14557637	C₂	semidirect
11. #14557634	C₂	semidirect
12. Q₁₆×Dic₃₇	C₂	semidirect
13. #14557624	C₂	semidirect
14. #14557591	C₂	semidirect
184. C₃₇	#3120	semidirect

Maximal subgroups

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

subgroup	index
#14558143	2
#14558016	2
#14557869	2
#14557861	2
#14557842	2
#14557831	2
#14557741	2
#14557736	2
#14557695	2
#14557637	2
#14557634	2
Q₁₆×Dic₃₇	2
#14557624	2
#14557591	2
C₃₇×#452	2
#3120	37