Subgroups — #18007186

Order 4736.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558247	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558183	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557930	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557855	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557854	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557850	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557849	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557821	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557811	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557783	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#14557609	C₂	1	1	yes	maximal, characteristic, complemented
13	#14557575	C₂	1	1	yes	maximal, characteristic, complemented
14	#14557572	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₇×#469	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10596964	—	2	1	no	—
17	C₂×#10596929	C₂²	1	1	yes	characteristic
18	#11924535	C₂²	1	1	yes	characteristic
19	#11924535	C₂²	1	1	yes	characteristic
20	#11924523	—	2	1	no	—
21	#11924522	—	2	1	no	—
22	#11924518	—	2	1	no	—
23	#11924517	—	2	1	no	—
24	#11924513	—	2	1	no	—
25	#11924508	—	2	1	no	—
26	#11924507	C₂²	1	1	yes	characteristic
27	#11924507	C₂²	1	1	yes	characteristic
28	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
29	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₃₇	C₂²	1	1	yes	characteristic
31	#11924495	C₂²	1	1	yes	characteristic
32	#11924494	C₂²	1	1	yes	characteristic
33	#11924492	—	2	1	no	—
34	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
36	#11924470	C₂²	1	1	yes	characteristic
37	#11924470	C₂²	1	1	yes	characteristic
38	#11924469	C₂²	1	1	yes	characteristic
39	#11924468	C₂²	1	1	yes	characteristic
40	#11924468	C₂²	1	1	yes	characteristic
41	#11924467	C₂²	1	1	yes	characteristic
42	#11924467	C₂²	1	1	yes	characteristic
43	#11924467	C₂²	1	1	yes	characteristic
44	#11924465	C₂²	1	1	yes	characteristic
45	#11924457	C₂²	1	1	yes	characteristic
46	#11924457	C₂²	1	1	yes	characteristic
47	#11924457	C₂²	1	1	yes	characteristic
48	#11924457	C₂²	1	1	yes	characteristic
49	C₂²⋊#64436	C₂²	1	1	yes	characteristic
50	#11924448	C₂²	1	1	yes	characteristic
51	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₃₇	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
54	#121×C₃₇	C₂²	1	1	yes	characteristic
55	#121×C₃₇	C₂²	1	1	yes	characteristic
56	#106×C₃₇	C₂²	1	1	yes	characteristic
57	#102×C₃₇	C₂²	1	1	yes	characteristic
58	#102×C₃₇	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₃₇	—	2	1	no	—
60	C₂×D₄×C₃₇	—	2	1	no	—
61	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₃₇	—	2	1	no	—
65	C₂×C₂²⋊D₃₇	—	2	1	no	—
66	C₂×C₂²⋊D₃₇	D₄	1	1	yes	characteristic
67	C₂²×Dic₃₇	—	2	1	no	—
68	C₂²×Dic₃₇	—	2	1	no	—
69	C₂²×Dic₃₇	—	2	1	no	—
70	#10596964	—	4	4	no	—
71	C₂×C₄×D₃₇	—	2	1	no	—
72	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₇₄	—	2	1	no	—
74	C₂×Dic₇₄	—	2	1	no	—
75	D₈×C₃₇	—	2	2	no	—
76	M₁₆×C₃₇	—	2	2	no	—
77	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₃₇	—	2	1	no	—
82	C₂²⋊C₄×C₃₇	—	2	1	no	—
83	C₂²⋊Dic₃₇	—	2	1	no	—
84	C₂²⋊Dic₃₇	—	2	1	no	—
85	C₂²⋊Dic₃₇	—	2	1	no	—
86	C₂²⋊Dic₃₇	—	2	1	no	—
87	C₂²⋊Dic₃₇	—	2	1	no	—
88	C₂²⋊Dic₃₇	D₄	1	1	yes	characteristic
89	#10596938	—	2	1	no	—
90	#10596938	D₄	1	1	yes	characteristic
91	C₄⋊Dic₃₇	—	2	1	no	—
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	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
96	#10596936	—	2	1	no	—
97	#10596936	—	2	1	no	—
98	#10596936	—	2	1	no	—
99	#10596936	D₄	1	1	yes	characteristic
100	C₄×Dic₃₇	—	2	1	no	—
101	C₄×Dic₃₇	—	2	1	no	—
102	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
103	C₂×#64436	C₂³	1	1	yes	characteristic
104	C₂×#64436	C₂³	1	1	yes	characteristic
105	#10596929	—	2	2	no	—
106	C₂³×C₃₇	—	2	1	no	—
107	C₂³×C₃₇	—	2	1	no	—
108	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
110	D₄×C₃₇	—	2	1	no	—
111	D₄×C₃₇	—	2	1	no	—
112	D₄×C₃₇	—	2	1	no	—
113	D₄×C₃₇	—	2	1	no	—
114	D₄×C₃₇	—	4	2	no	—
115	D₄×C₃₇	—	8	1	no	—
116	C₂×C₄×C₃₇	—	2	1	no	—
117	C₂×C₄×C₃₇	—	2	1	no	—
118	C₂×C₄×C₃₇	—	2	1	no	—
119	C₂×C₄×C₃₇	—	2	1	no	—
120	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₃₇	—	2	2	no	—
123	C₂²⋊D₃₇	—	4	2	no	—
124	C₂²⋊D₃₇	—	8	1	no	—
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	—	4	2	no	—
130	C₂×Dic₃₇	—	4	2	no	—
131	C₂×Dic₃₇	—	4	2	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	C₄×D₃₇	—	2	1	no	—
136	C₄×D₃₇	—	2	1	no	—
137	C₄×D₃₇	—	4	2	no	—
138	Dic₇₄	—	4	2	no	—
139	Dic₇₄	—	8	1	no	—
140	C₈×C₃₇	—	2	1	no	—
141	C₈×C₃₇	#41	1	2	yes	complemented
142	#64436	—	2	1	no	—
143	#64436	—	2	1	no	—
144	C₂²×C₃₇	—	2	1	no	—
145	C₂²×C₃₇	—	2	1	no	—
146	C₂²×C₃₇	—	2	1	no	—
147	C₂²×C₃₇	—	4	1	no	—
148	C₂²×C₃₇	—	4	1	no	—
149	C₂²×C₃₇	—	4	1	no	—
150	C₂²×C₃₇	—	4	1	no	—
151	C₂²×C₃₇	—	4	1	no	—
152	C₂²×C₃₇	—	4	1	no	—
153	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₃₇	—	2	1	no	—
155	C₂×D₃₇	—	2	1	no	—
156	C₂×D₃₇	—	2	1	no	—
157	C₄×C₃₇	—	2	1	no	—
158	C₄×C₃₇	—	4	1	no	—
159	C₄×C₃₇	—	4	1	no	—
160	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
161	C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₃₇	—	2	1	no	—
163	Dic₃₇	—	2	1	no	—
164	Dic₃₇	—	4	1	no	—
165	Dic₃₇	—	4	1	no	—
166	Dic₃₇	—	4	1	no	—
167	Dic₃₇	—	4	1	no	—
168	Dic₃₇	AGL(1,ℤ/8)	1	2	yes	—
169	#3354	—	37	1	no	maximal, Sylow
170	C₂×C₃₇	—	4	1	no	—
171	C₂×C₃₇	—	8	1	no	—
172	C₂×C₃₇	—	8	1	no	—
173	C₂×C₃₇	#547	1	1	yes	characteristic
174	C₂×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₃₇	#577	1	1	yes	characteristic
176	D₃₇	—	4	1	no	—
177	#555	—	37	1	no	—
178	#547	—	37	1	no	—
179	#498	—	37	1	no	—
180	#482	—	37	1	no	—
181	#482	—	37	1	no	—
182	#481	—	37	1	no	—
183	#481	—	37	1	no	—
184	#476	—	37	1	no	—
185	#474	—	37	1	no	—
186	#469	—	37	1	no	—
187	#469	D₃₇	1	1	yes	characteristic, complemented
188	C₄×D₈	—	37	1	no	—
189	#436	—	37	1	no	—
190	#428	—	37	1	no	—
191	#428	—	37	1	no	—
192	C₃₇	#3354	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	74	1	no	—
194	C₂×D₈	C₂×D₃₇	1	1	yes	characteristic
195	C₂×M₁₆	—	37	1	no	—
196	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
197	C₄⋊Q₈	—	37	1	no	—
198	C₄⋊Q₈	—	37	1	no	—
199	#124	—	74	1	no	—
200	#124	—	74	1	no	—
201	#123	—	74	1	no	—
202	#123	—	74	1	no	—
203	C₂²⋊Q₈	—	74	1	no	—
204	#121	—	37	1	no	—
205	#121	—	37	1	no	—
206	#121	—	74	1	no	—
207	#121	C₂×D₃₇	1	1	yes	characteristic
208	#121	C₂×D₃₇	1	1	yes	characteristic
209	C₄×D₄	—	37	1	no	—
210	C₄×D₄	—	37	1	no	—
211	C₄×D₄	—	37	1	no	—
212	#117	—	37	1	no	—
213	#117	—	37	1	no	—
214	#117	—	74	1	no	—
215	C₂×C₄⋊C₄	—	37	1	no	—
216	C₂×C₄⋊C₄	—	37	1	no	—
217	#107	—	37	1	no	—
218	#107	—	37	1	no	—
219	#107	—	37	1	no	—
220	#106	—	37	1	no	—
221	#106	—	37	1	no	—
222	#106	—	37	1	no	—
223	#106	—	37	1	no	—
224	#106	—	37	1	no	—
225	#106	C₂×D₃₇	1	1	yes	characteristic
226	C₄⋊C₈	—	37	1	no	—
227	#102	—	37	1	no	—
228	#102	—	37	1	no	—
229	#102	—	37	1	no	—
230	#102	—	37	1	no	—
231	#102	C₂×D₃₇	1	1	yes	characteristic
232	#102	C₂×D₃₇	1	1	yes	characteristic
233	C₂²⋊C₈	—	37	1	no	—
234	C₂²⋊C₈	—	37	1	no	—
235	C₄×C₈	—	37	1	no	—
236	#41	—	148	4	no	—
237	C₂×Q₈	—	74	1	no	—
238	C₂×Q₈	—	74	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	37	1	no	—
242	C₂×D₄	—	74	1	no	—
243	C₂×D₄	—	74	1	no	—
244	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
246	C₂²×C₄	—	37	1	no	—
247	C₂²×C₄	—	74	1	no	—
248	C₂²×C₄	—	74	1	no	—
249	C₂²×C₄	—	74	1	no	—
250	C₂²×C₄	—	74	1	no	—
251	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
252	D₈	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	74	2	no	—
255	C₂×C₈	—	37	1	no	—
256	C₂×C₈	—	37	1	no	—
257	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
259	C₄⋊C₄	—	37	1	no	—
260	C₄⋊C₄	—	37	1	no	—
261	C₄⋊C₄	—	37	1	no	—
262	C₄⋊C₄	—	37	1	no	—
263	C₄⋊C₄	—	37	1	no	—
264	C₄⋊C₄	—	74	1	no	—
265	C₄⋊C₄	—	74	1	no	—
266	C₄⋊C₄	—	74	1	no	—
267	C₄⋊C₄	—	74	1	no	—
268	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	37	1	no	—
273	C₂²⋊C₄	—	37	1	no	—
274	C₂²⋊C₄	—	74	1	no	—
275	C₂²⋊C₄	—	74	1	no	—
276	C₂²⋊C₄	—	74	1	no	—
277	C₂²⋊C₄	—	74	1	no	—
278	C₂²⋊C₄	—	74	1	no	—
279	C₂²⋊C₄	—	74	1	no	—
280	C₄²	—	37	1	no	—
281	C₄²	—	74	1	no	—
282	C₄²	—	74	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	37	1	no	—
286	C₂³	D₄×D₃₇	1	1	yes	characteristic
287	Q₈	—	296	1	no	—
288	Q₈	—	148	2	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	4	2	no	—
294	D₄	—	8	1	no	—
295	D₄	—	296	1	no	—
296	D₄	—	74	2	no	—
297	D₄	—	148	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	37	1	no	—
303	C₂×C₄	—	37	1	no	—
304	C₂×C₄	—	37	1	no	—
305	C₂×C₄	—	74	1	no	—
306	C₂×C₄	—	74	1	no	—
307	C₂×C₄	—	74	1	no	—
308	C₂×C₄	—	74	1	no	—
309	C₂×C₄	—	74	1	no	—
310	C₂×C₄	—	74	1	no	—
311	C₂×C₄	—	148	2	no	—
312	C₂×C₄	—	148	2	no	—
313	C₂×C₄	—	148	2	no	—
314	C₂×C₄	—	148	2	no	—
315	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	74	1	no	—
319	C₈	—	74	1	no	—
320	C₈	#10596964	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	74	1	no	—
331	C₂²	—	74	1	no	—
332	C₂²	—	74	1	no	—
333	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	37	2	no	—
338	C₄	—	74	1	no	—
339	C₄	—	74	1	no	—
340	C₄	—	148	1	no	—
341	C₄	—	148	1	no	—
342	C₄	—	148	1	no	—
343	C₄	—	148	1	no	—
344	C₄	C₂×#10596964	1	1	yes	characteristic
345	C₄	#11924583	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	8	1	no	—
348	C₂	—	8	1	no	—
349	C₂	—	148	1	no	—
350	C₂	#14558183	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₃₇	1	1	yes	characteristic
352	C₂	#14558343	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×#469	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	#469	D₃₇
hypocenter	C₃₇	#3354
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #14558247	C₂	semidirect
2. #14558183	C₂	semidirect
3. #14557930	C₂	semidirect
4. #14557855	C₂	semidirect
5. #14557854	C₂	semidirect
6. #14557850	C₂	semidirect
7. #14557849	C₂	semidirect
8. #14557821	C₂	semidirect
9. #14557811	C₂	semidirect
10. #14557783	C₂	semidirect
11. D₈×Dic₃₇	C₂	semidirect
12. #14557609	C₂	semidirect
13. #14557575	C₂	semidirect
14. #14557572	C₂	semidirect
15. C₃₇×#469	C₂	semidirect
141. C₈×C₃₇	#41	semidirect
187. #469	D₃₇	semidirect
192. C₃₇	#3354	semidirect
320. C₈	#10596964	semidirect

Maximal subgroups

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

subgroup	index
#14558247	2
#14558183	2
#14557930	2
#14557855	2
#14557854	2
#14557850	2
#14557849	2
#14557821	2
#14557811	2
#14557783	2
D₈×Dic₃₇	2
#14557609	2
#14557575	2
#14557572	2
C₃₇×#469	2
#3354	37