Subgroups — #18011368

Order 4736.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₇⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924623	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558580	C₂	1	2	yes	maximal, complemented
4	#14558578	C₂	1	2	yes	maximal, complemented
5	#14558576	C₂	1	2	yes	maximal, complemented
6	#14558575	C₂	1	2	yes	maximal, complemented
7	#14558548	C₂	1	1	yes	maximal, characteristic, complemented
8	#14558531	C₂	1	1	yes	maximal, characteristic, complemented
9	#14558530	C₂	1	1	yes	maximal, characteristic, complemented
10	#14558499	C₂	1	1	yes	maximal, characteristic, complemented
11	#14558329	C₂	1	1	yes	maximal, characteristic
12	#11924624	C₂²	1	1	yes	characteristic, complemented
13	#11924624	C₂²	1	1	yes	characteristic, complemented
14	#11924623	—	2	1	no	—
15	#11924623	—	2	1	no	—
16	#11924623	—	2	1	no	—
17	#11924623	—	2	1	no	—
18	#11924623	C₂²	1	1	yes	characteristic, complemented
19	#11924623	C₂²	1	2	yes	complemented
20	#11924623	C₂²	1	2	yes	complemented
21	#11924623	C₂²	1	1	yes	characteristic, complemented
22	C₂×#10596953	C₂²	1	1	yes	characteristic
23	D₃₇⋊M₁₆	—	2	1	no	—
24	D₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×#10596957	—	2	1	no	—
26	C₂×D₃₇⋊C₈	—	2	1	no	—
27	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²×#64438	—	2	1	no	—
29	#11924615	C₂²	1	1	yes	characteristic
30	#11924607	C₂²	1	2	yes	complemented
31	#11924606	C₂²	1	2	yes	complemented
32	#11924601	C₂²	1	2	yes	complemented
33	#11924600	C₂²	1	2	yes	complemented
34	#11924600	C₂²	1	2	yes	complemented
35	#11924600	C₂²	1	2	yes	complemented
36	#11924590	C₂²	1	2	yes	complemented
37	C₈×#3866	C₂²	1	2	yes	complemented
38	C₂×#10596964	C₂²	1	1	yes	characteristic
39	#41×D₃₇	C₂²	1	1	yes	characteristic
40	#11924569	C₄	1	1	yes	characteristic, complemented
41	#11924566	C₄	1	1	yes	characteristic, complemented
42	#11924565	C₄	1	2	yes	complemented
43	#11924562	C₄	1	1	yes	characteristic, complemented
44	#11924560	C₄	1	2	yes	complemented
45	#11924557	C₂²	1	2	yes	—
46	#11924555	C₂²	1	2	yes	—
47	M₁₆×D₃₇	C₂²	1	1	yes	characteristic
48	AGL(1,ℤ/8)×C₃₇	C₄	1	1	yes	characteristic, complemented
49	#41×C₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₃₇	—	2	1	no	—
52	C₂²×Dic₃₇	—	2	1	no	—
53	#10596966	C₂³	1	1	yes	characteristic
54	Q₈×D₃₇	C₂×C₄	1	1	yes	characteristic
55	#10596964	—	2	1	no	—
56	#10596964	—	2	1	no	—
57	#10596964	—	2	1	no	—
58	#10596964	C₂×C₄	1	2	yes	—
59	#10596964	C₂³	1	1	yes	characteristic
60	#10596964	C₂³	1	2	yes	—
61	D₄×D₃₇	—	2	1	no	—
62	D₄×D₃₇	C₂×C₄	1	1	yes	characteristic
63	#10596962	—	2	1	no	—
64	#10596962	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₃₇	—	2	1	no	—
66	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
67	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊#3866	—	2	1	no	—
69	#10596957	—	2	1	no	—
70	#10596957	—	2	2	no	—
71	#10596957	—	2	2	no	—
72	#10596957	—	2	1	no	—
73	#10596957	—	2	1	no	—
74	C₂×#64438	—	2	1	no	—
75	C₂×#64438	—	2	2	no	—
76	C₂×#64438	—	2	2	no	—
77	C₂×#64438	—	2	1	no	—
78	C₂×#64438	—	2	1	no	—
79	C₂×#64438	D₄	1	1	yes	characteristic
80	C₂×#64438	D₄	1	1	yes	characteristic
81	C₄⋊#3866	—	2	1	no	—
82	C₄×#3866	C₂³	1	2	yes	—
83	#10596953	—	2	1	no	—
84	#10596953	—	2	1	no	—
85	#10596953	—	2	1	no	—
86	#10596953	C₂³	1	1	yes	characteristic
87	#10596953	C₂³	1	1	yes	characteristic
88	#10596953	C₂³	1	2	yes	—
89	D₃₇⋊C₈	—	2	1	no	—
90	D₃₇⋊C₈	—	2	1	no	—
91	D₃₇⋊C₈	—	2	1	no	—
92	D₃₇⋊C₈	D₄	1	1	yes	characteristic
93	D₃₇⋊C₈	D₄	1	1	yes	characteristic
94	D₃₇⋊C₈	C₂³	1	1	yes	characteristic
95	D₃₇⋊C₈	C₂³	1	1	yes	characteristic
96	SD₁₆×C₃₇	C₂×C₄	1	2	yes	—
97	D₈×C₃₇	C₂×C₄	1	2	yes	—
98	M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
99	#10596942	C₂×C₄	1	2	yes	—
100	#10596940	C₂×C₄	1	2	yes	—
101	#10596940	C₂×C₄	1	2	yes	—
102	#10596939	C₂×C₄	1	2	yes	—
103	#10596934	C₂×C₄	1	1	yes	characteristic
104	Dic₁₄₈	C₂×C₄	1	2	yes	—
105	#10596930	C₂×C₄	1	2	yes	—
106	#10596929	C₂×C₄	1	2	yes	—
107	C₈×D₃₇	C₂³	1	2	yes	—
108	C₂³×C₃₇	—	2	1	no	—
109	C₂²×D₃₇	—	2	1	no	—
110	C₂²×D₃₇	#41	1	1	yes	characteristic
111	C₂×#3866	—	2	2	no	—
112	Q₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₃₇	—	2	1	no	—
114	D₄×C₃₇	—	2	1	no	—
115	D₄×C₃₇	C₂²×C₄	1	2	yes	—
116	D₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	—	2	1	no	—
118	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²⋊D₃₇	—	2	1	no	—
120	C₂²⋊D₃₇	—	2	2	no	—
121	C₂²⋊D₃₇	—	2	1	no	—
122	C₂²⋊D₃₇	—	2	2	no	—
123	C₂²⋊D₃₇	—	2	1	no	—
124	C₂×Dic₃₇	—	2	1	no	—
125	C₂×Dic₃₇	—	2	2	no	—
126	C₂×Dic₃₇	—	2	1	no	—
127	C₂×Dic₃₇	—	2	2	no	—
128	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
129	D₁₄₈	—	2	1	no	—
130	D₁₄₈	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₃₇	—	2	1	no	—
132	C₄×D₃₇	—	2	1	no	—
133	C₄×D₃₇	—	2	1	no	—
134	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
136	C₄×D₃₇	#41	1	1	yes	characteristic
137	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
138	Dic₇₄	—	2	1	no	—
139	Dic₇₄	—	2	1	no	—
140	Dic₇₄	C₂²×C₄	1	2	yes	—
141	Dic₇₄	C₂²×C₄	1	1	yes	characteristic
142	#64438	—	2	1	no	—
143	#64438	—	2	1	no	—
144	#64438	—	2	2	no	—
145	#64438	C₂×D₄	1	1	yes	characteristic
146	#64438	C₂×D₄	1	1	yes	characteristic
147	#64438	C₂×D₄	1	1	yes	characteristic
148	#64438	C₂×D₄	1	1	yes	characteristic
149	C₈×C₃₇	C₂²×C₄	1	2	yes	—
150	#64436	C₂²×C₄	1	2	yes	—
151	C₂²×C₃₇	—	2	2	no	—
152	C₂²×C₃₇	—	2	1	no	—
153	C₂²×C₃₇	—	4	2	no	—
154	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
155	C₂×D₃₇	—	2	1	no	—
156	C₂×D₃₇	—	4	1	no	—
157	C₂×D₃₇	—	4	2	no	—
158	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
159	C₂×D₃₇	C₂×#41	1	1	yes	characteristic
160	#3866	—	4	2	no	—
161	C₄×C₃₇	—	2	1	no	—
162	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
163	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
164	Dic₃₇	—	2	1	no	—
165	Dic₃₇	—	2	2	no	—
166	Dic₃₇	C₄×D₄	1	1	yes	characteristic
167	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
168	#2955	—	37	1	no	maximal, Sylow
169	C₂×C₃₇	—	2	1	no	—
170	C₂×C₃₇	—	4	1	no	—
171	C₂×C₃₇	—	4	2	no	—
172	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
173	D₃₇	—	2	1	no	—
174	D₃₇	—	2	1	no	—
175	D₃₇	—	4	1	no	—
176	#575	—	37	1	no	—
177	C₂×#131	—	37	1	no	—
178	C₂×#131	—	37	1	no	—
179	#444	—	37	2	no	—
180	#444	—	37	2	no	—
181	#443	—	37	2	no	—
182	#443	—	37	2	no	—
183	#430	—	37	1	no	—
184	#421	—	37	1	no	—
185	#421	—	37	1	no	—
186	#405	—	37	1	no	—
187	C₃₇	#2955	1	1	yes	characteristic, complemented, Sylow
188	C₂×#41	—	37	1	no	—
189	C₂×#41	—	37	1	no	—
190	#137	—	37	1	no	—
191	#137	—	37	2	no	—
192	#137	—	37	1	no	—
193	AGL(1,ℤ/8)	—	37	1	no	—
194	AGL(1,ℤ/8)	—	37	2	no	—
195	AGL(1,ℤ/8)	#3866	1	1	yes	characteristic, complemented
196	#135	—	37	2	no	—
197	#135	—	37	2	no	—
198	#131	—	37	2	no	—
199	#131	—	37	1	no	—
200	#131	—	37	2	no	—
201	#131	—	37	1	no	—
202	#131	—	37	1	no	—
203	#131	—	37	1	no	—
204	#131	—	74	1	no	—
205	#131	—	74	1	no	—
206	#131	—	74	1	no	—
207	#131	—	74	1	no	—
208	C₂×M₁₆	—	37	1	no	—
209	C₂×M₁₆	—	37	1	no	—
210	C₂×M₁₆	—	37	1	no	—
211	C₂×M₁₆	—	74	1	no	—
212	C₂×M₁₆	—	74	1	no	—
213	C₂²×C₈	—	37	1	no	—
214	C₂²×C₈	—	74	1	no	—
215	C₂²×C₈	—	74	1	no	—
216	#117	—	37	1	no	—
217	#108	—	37	2	no	—
218	#108	—	37	2	no	—
219	#104	—	37	2	no	—
220	#104	—	37	2	no	—
221	#104	—	37	2	no	—
222	#104	—	37	2	no	—
223	#97	—	37	2	no	—
224	C₄×C₈	—	37	2	no	—
225	#41	—	37	1	no	—
226	#41	—	37	2	no	—
227	#41	—	37	2	no	—
228	#41	—	37	1	no	—
229	#41	—	37	1	no	—
230	#41	—	74	1	no	—
231	#41	—	74	1	no	—
232	#41	—	74	1	no	—
233	#41	—	74	1	no	—
234	#41	C₂×#3866	1	1	yes	characteristic
235	C₂×Q₈	—	37	1	no	—
236	C₂×Q₈	—	37	1	no	—
237	C₂×D₄	—	37	1	no	—
238	C₂×D₄	—	74	1	no	—
239	C₂×D₄	—	74	1	no	—
240	C₂×D₄	C₂×#3866	1	1	yes	characteristic
241	C₂²×C₄	—	37	1	no	—
242	C₂²×C₄	—	74	1	no	—
243	C₂²×C₄	—	74	1	no	—
244	Q₁₆	—	37	2	no	—
245	Q₁₆	—	37	2	no	—
246	SD₁₆	—	37	2	no	—
247	SD₁₆	—	37	2	no	—
248	SD₁₆	—	37	2	no	—
249	SD₁₆	C₂×#3866	1	2	yes	—
250	D₈	—	37	2	no	—
251	D₈	C₂×#3866	1	2	yes	—
252	M₁₆	—	37	1	no	—
253	M₁₆	—	37	2	no	—
254	M₁₆	—	37	1	no	—
255	M₁₆	—	37	1	no	—
256	M₁₆	—	37	2	no	—
257	M₁₆	—	74	1	no	—
258	M₁₆	—	74	1	no	—
259	M₁₆	—	74	2	no	—
260	M₁₆	—	74	1	no	—
261	M₁₆	—	74	2	no	—
262	M₁₆	—	74	1	no	—
263	M₁₆	—	74	1	no	—
264	M₁₆	—	74	1	no	—
265	M₁₆	C₂×#3866	1	1	yes	characteristic
266	C₂×C₈	—	37	1	no	—
267	C₂×C₈	—	37	1	no	—
268	C₂×C₈	—	37	1	no	—
269	C₂×C₈	—	37	1	no	—
270	C₂×C₈	—	37	1	no	—
271	C₂×C₈	—	37	1	no	—
272	C₂×C₈	—	37	2	no	—
273	C₂×C₈	—	74	1	no	—
274	C₂×C₈	—	74	1	no	—
275	C₂×C₈	—	74	2	no	—
276	C₂×C₈	—	74	1	no	—
277	C₂×C₈	—	74	2	no	—
278	C₂×C₈	—	74	1	no	—
279	C₂×C₈	—	74	1	no	—
280	C₂×C₈	—	74	1	no	—
281	C₄⋊C₄	—	74	1	no	—
282	C₂²⋊C₄	—	74	1	no	—
283	C₄²	—	37	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	37	1	no	—
286	C₂³	—	74	1	no	—
287	Q₈	—	37	1	no	—
288	Q₈	—	37	2	no	—
289	Q₈	—	74	1	no	—
290	Q₈	—	74	1	no	—
291	Q₈	C₂²×#3866	1	1	yes	characteristic
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	37	1	no	—
295	D₄	—	74	1	no	—
296	D₄	—	74	1	no	—
297	D₄	—	74	1	no	—
298	D₄	—	74	2	no	—
299	D₄	—	74	2	no	—
300	D₄	—	74	1	no	—
301	D₄	C₂²×#3866	1	2	yes	—
302	D₄	C₂²×#3866	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	37	1	no	—
305	C₂×C₄	—	37	1	no	—
306	C₂×C₄	—	37	1	no	—
307	C₂×C₄	—	37	1	no	—
308	C₂×C₄	—	37	1	no	—
309	C₂×C₄	—	74	1	no	—
310	C₂×C₄	—	74	1	no	—
311	C₂×C₄	—	74	1	no	—
312	C₂×C₄	—	74	2	no	—
313	C₂×C₄	—	74	2	no	—
314	C₂×C₄	—	74	1	no	—
315	C₂×C₄	—	74	1	no	—
316	C₂×C₄	—	74	2	no	—
317	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
318	C₈	—	37	1	no	—
319	C₈	—	37	1	no	—
320	C₈	—	37	1	no	—
321	C₈	—	37	1	no	—
322	C₈	—	37	2	no	—
323	C₈	—	74	1	no	—
324	C₈	—	74	1	no	—
325	C₈	—	74	2	no	—
326	C₈	C₂²×#3866	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	37	1	no	—
331	C₂²	—	37	1	no	—
332	C₂²	—	74	1	no	—
333	C₂²	—	148	1	no	—
334	C₂²	—	148	2	no	—
335	C₂²	D₄×#3866	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	37	1	no	—
338	C₄	—	37	1	no	—
339	C₄	—	74	1	no	—
340	C₄	—	74	2	no	—
341	C₄	—	148	2	no	—
342	C₄	D₄×#3866	1	1	yes	characteristic
343	C₄	C₂³×#3866	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	74	1	no	—
348	C₂	—	74	1	no	—
349	C₂	—	148	1	no	—
350	C₂	C₂×D₄×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#3866
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₄	C₂³×#3866
Fitting	AGL(1,ℤ/8)×C₃₇	C₄
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	#14558329	C₂
cosocle	C₄×D₃₇	C₂⁴
hypercenter	AGL(1,ℤ/8)	#3866
hypocenter	C₃₇	#2955
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₃₇⋊#131	C₂	semidirect
2. C₂×#11924623	C₂	semidirect
3. #14558580	C₂	semidirect
4. #14558578	C₂	semidirect
5. #14558576	C₂	semidirect
6. #14558575	C₂	semidirect
7. #14558548	C₂	semidirect
8. #14558531	C₂	semidirect
9. #14558530	C₂	semidirect
10. #14558499	C₂	semidirect
12. #11924624	C₂²	semidirect
13. #11924624	C₂²	semidirect
18. #11924623	C₂²	semidirect
19. #11924623	C₂²	semidirect
20. #11924623	C₂²	semidirect
21. #11924623	C₂²	semidirect
30. #11924607	C₂²	semidirect
31. #11924606	C₂²	semidirect
32. #11924601	C₂²	semidirect
33. #11924600	C₂²	semidirect
34. #11924600	C₂²	semidirect
35. #11924600	C₂²	semidirect
36. #11924590	C₂²	semidirect
37. C₈×#3866	C₂²	semidirect
40. #11924569	C₄	semidirect
41. #11924566	C₄	semidirect
42. #11924565	C₄	semidirect
43. #11924562	C₄	semidirect
44. #11924560	C₄	semidirect
48. AGL(1,ℤ/8)×C₃₇	C₄	semidirect
187. C₃₇	#2955	semidirect
195. AGL(1,ℤ/8)	#3866	semidirect

Maximal subgroups

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

subgroup	index
D₃₇⋊#131	2
C₂×#11924623	2
#14558580	2
#14558578	2
#14558576	2
#14558575	2
#14558548	2
#14558531	2
#14558530	2
#14558499	2
#14558329	2
#2955	37