Subgroups — #18011043

Order 4736.

568 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558478	C₂	1	2	yes	maximal, complemented
2	#14558477	C₂	1	2	yes	maximal, complemented
3	#14558476	C₂	1	2	yes	maximal
4	#14558475	C₂	1	2	yes	maximal
5	C₄×C₄⋊#3866	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊#3866	C₂	1	2	yes	maximal
7	#14558469	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#3866	C₂²	1	4	yes	—
10	C₂×C₂²⋊#3866	C₂²	1	4	yes	—
11	C₂×C₄×#3866	C₂²	1	4	yes	—
12	(C₂×#3866)⋊C₄	C₂²	1	4	yes	—
13	(C₂×#3866)⋊C₄	C₂²	1	4	yes	—
14	(C₂×#3866)⋊C₄	C₂²	1	4	yes	—
15	(C₂×#3866)⋊C₄	C₂²	1	4	yes	—
16	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×Dic₃₇	C₄	1	1	yes	characteristic, complemented
22	C₄²×D₃₇	C₄	1	1	yes	characteristic, complemented
23	C₄²×D₃₇	C₄	1	1	yes	characteristic, complemented
24	C₄²×D₃₇	C₄	1	1	yes	characteristic, complemented
25	C₄²×D₃₇	C₄	1	1	yes	characteristic, complemented
26	C₄²×D₃₇	C₂²	1	1	yes	characteristic
27	C₄²×D₃₇	C₂²	1	1	yes	characteristic
28	C₄²×D₃₇	C₂²	1	1	yes	characteristic
29	C₄²×D₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₃₇	C₄	1	1	yes	characteristic, complemented
31	C₂³×D₃₇	C₂³	1	1	yes	characteristic
32	C₂²×#3866	C₂³	1	8	yes	—
33	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₃₇	—	2	4	no	—
40	C₂×C₄×D₃₇	—	2	4	no	—
41	C₂×C₄×D₃₇	—	2	4	no	—
42	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
54	C₂²⋊#3866	—	2	8	no	—
55	C₄⋊#3866	—	2	8	no	—
56	C₄×#3866	—	2	8	no	—
57	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
73	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₃₇	—	2	2	no	—
75	C₂²×D₃₇	—	2	2	no	—
76	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₃₇	#41	1	4	yes	—
78	C₂²×D₃₇	C₂⁴	1	1	yes	characteristic
79	C₂×#3866	—	2	8	no	—
80	C₂×#3866	—	2	8	no	—
81	C₂×#3866	—	2	8	no	—
82	C₂×C₄×C₃₇	—	2	2	no	—
83	C₂×C₄×C₃₇	—	2	2	no	—
84	C₂×C₄×C₃₇	—	2	2	no	—
85	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₃₇	—	2	2	no	—
92	C₂×Dic₃₇	—	2	2	no	—
93	C₂×Dic₃₇	—	2	2	no	—
94	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
100	C₄×D₃₇	—	2	2	no	—
101	C₄×D₃₇	—	2	2	no	—
102	C₄×D₃₇	—	2	2	no	—
103	C₄×D₃₇	—	2	2	no	—
104	C₄×D₃₇	—	2	2	no	—
105	C₄×D₃₇	—	2	2	no	—
106	C₄×D₃₇	—	2	2	no	—
107	C₄×D₃₇	—	2	2	no	—
108	C₄×D₃₇	—	2	2	no	—
109	C₄×D₃₇	—	2	2	no	—
110	C₄×D₃₇	#41	1	4	yes	—
111	C₄×D₃₇	#41	1	4	yes	—
112	C₂²×C₃₇	—	2	1	no	—
113	C₂²×C₃₇	—	2	1	no	—
114	C₂²×C₃₇	#117	1	2	yes	—
115	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₃₇	—	2	1	no	—
117	C₂×D₃₇	—	2	1	no	—
118	C₂×D₃₇	—	2	8	no	—
119	C₂×D₃₇	#117	1	2	yes	—
120	C₂×D₃₇	C₂×#41	1	2	yes	—
121	C₂×D₃₇	C₂×#41	1	2	yes	—
122	C₂×D₃₇	C₂×#41	1	2	yes	—
123	#3866	—	4	8	no	—
124	C₄×C₃₇	—	2	1	no	—
125	C₄×C₃₇	—	2	1	no	—
126	C₄×C₃₇	—	2	1	no	—
127	C₄×C₃₇	—	2	1	no	—
128	C₄×C₃₇	#117	1	2	yes	—
129	C₄×C₃₇	#117	1	2	yes	—
130	Dic₃₇	—	2	1	no	—
131	Dic₃₇	—	2	1	no	—
132	Dic₃₇	—	2	1	no	—
133	Dic₃₇	—	2	1	no	—
134	Dic₃₇	#117	1	2	yes	—
135	Dic₃₇	#117	1	2	yes	—
136	#2283	—	37	1	no	maximal, Sylow
137	C₂×C₃₇	—	2	2	no	—
138	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
139	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
140	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
141	D₃₇	—	2	2	no	—
142	D₃₇	#529	1	4	yes	—
143	C₂²×C₄²	—	37	1	no	—
144	#383	—	37	2	no	—
145	#382	—	37	2	no	—
146	#381	—	37	2	no	—
147	#380	—	37	2	no	—
148	C₄×C₄⋊C₄	—	37	2	no	—
149	C₄×C₂²⋊C₄	—	37	2	no	—
150	#376	—	37	2	no	—
151	C₃₇	#2283	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	—	37	1	no	—
153	C₂³×C₄	—	37	1	no	—
154	C₂³×C₄	—	37	1	no	—
155	C₂×C₄⋊C₄	—	37	4	no	—
156	C₂×C₂²⋊C₄	—	37	4	no	—
157	C₂×C₄²	—	37	1	no	—
158	C₂×C₄²	—	37	1	no	—
159	C₂×C₄²	—	37	1	no	—
160	C₂×C₄²	—	37	1	no	—
161	C₂×C₄²	—	37	1	no	—
162	C₂×C₄²	—	37	1	no	—
163	C₂×C₄²	—	37	1	no	—
164	C₂×C₄²	—	37	1	no	—
165	C₂×C₄²	—	37	1	no	—
166	C₂×C₄²	—	37	1	no	—
167	C₂×C₄²	—	37	1	no	—
168	C₂×C₄²	—	37	4	no	—
169	C₂×C₄²	#3866	1	1	yes	characteristic, complemented
170	#95	—	37	4	no	—
171	#95	—	37	4	no	—
172	#95	—	37	4	no	—
173	#95	—	37	4	no	—
174	C₂⁴	—	37	1	no	—
175	C₂²×C₄	—	37	1	no	—
176	C₂²×C₄	—	37	1	no	—
177	C₂²×C₄	—	37	1	no	—
178	C₂²×C₄	—	37	1	no	—
179	C₂²×C₄	—	37	1	no	—
180	C₂²×C₄	—	37	1	no	—
181	C₂²×C₄	—	37	1	no	—
182	C₂²×C₄	—	37	1	no	—
183	C₂²×C₄	—	37	1	no	—
184	C₂²×C₄	—	37	1	no	—
185	C₂²×C₄	—	37	1	no	—
186	C₂²×C₄	—	37	1	no	—
187	C₂²×C₄	—	37	1	no	—
188	C₂²×C₄	—	37	1	no	—
189	C₂²×C₄	—	37	1	no	—
190	C₂²×C₄	—	37	8	no	—
191	C₂²×C₄	—	74	4	no	—
192	C₂²×C₄	—	74	4	no	—
193	C₂²×C₄	—	74	4	no	—
194	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
195	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
196	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
197	C₄⋊C₄	—	74	8	no	—
198	C₂²⋊C₄	—	74	8	no	—
199	C₄²	—	37	1	no	—
200	C₄²	—	37	1	no	—
201	C₄²	—	37	1	no	—
202	C₄²	—	37	1	no	—
203	C₄²	—	37	1	no	—
204	C₄²	—	37	1	no	—
205	C₄²	—	37	1	no	—
206	C₄²	—	37	1	no	—
207	C₄²	—	37	1	no	—
208	C₄²	—	37	1	no	—
209	C₄²	—	37	1	no	—
210	C₄²	—	37	1	no	—
211	C₄²	—	74	8	no	—
212	C₄²	C₂×#3866	1	1	yes	characteristic
213	C₄²	C₂×#3866	1	1	yes	characteristic
214	C₄²	C₂×#3866	1	1	yes	characteristic
215	C₄²	C₂×#3866	1	1	yes	characteristic
216	C₂³	—	37	1	no	—
217	C₂³	—	37	4	no	—
218	C₂³	—	37	1	no	—
219	C₂³	—	74	2	no	—
220	C₂³	—	74	2	no	—
221	C₂³	C₂²×#3866	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	37	1	no	—
226	C₂×C₄	—	37	1	no	—
227	C₂×C₄	—	37	4	no	—
228	C₂×C₄	—	37	4	no	—
229	C₂×C₄	—	37	1	no	—
230	C₂×C₄	—	37	1	no	—
231	C₂×C₄	—	37	1	no	—
232	C₂×C₄	—	37	1	no	—
233	C₂×C₄	—	74	2	no	—
234	C₂×C₄	—	74	2	no	—
235	C₂×C₄	—	74	2	no	—
236	C₂×C₄	—	74	2	no	—
237	C₂×C₄	—	74	2	no	—
238	C₂×C₄	—	74	2	no	—
239	C₂×C₄	—	74	2	no	—
240	C₂×C₄	—	74	2	no	—
241	C₂×C₄	—	74	2	no	—
242	C₂×C₄	—	74	2	no	—
243	C₂×C₄	—	74	2	no	—
244	C₂×C₄	—	74	2	no	—
245	C₂×C₄	—	74	2	no	—
246	C₂×C₄	—	74	8	no	—
247	C₂×C₄	—	74	8	no	—
248	C₂×C₄	—	74	8	no	—
249	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
250	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
251	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
252	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
253	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
254	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	37	2	no	—
258	C₂²	—	37	2	no	—
259	C₂²	—	37	2	no	—
260	C₂²	—	37	2	no	—
261	C₂²	—	74	8	no	—
262	C₂²	—	74	1	no	—
263	C₂²	—	74	1	no	—
264	C₂²	#11924615	1	2	yes	—
265	C₂²	C₂³×#3866	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	37	2	no	—
271	C₄	—	37	2	no	—
272	C₄	—	74	1	no	—
273	C₄	—	74	1	no	—
274	C₄	—	74	1	no	—
275	C₄	—	74	1	no	—
276	C₄	—	148	8	no	—
277	C₄	#11924615	1	2	yes	—
278	C₄	#11924615	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	37	4	no	—
281	C₂	—	74	2	no	—
282	C₂	C₂×#11924615	1	1	yes	characteristic
283	C₂	C₂×#11924615	1	1	yes	characteristic
284	C₂	C₂×#11924615	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Frattini	C₂²	C₂³×#3866
Fitting	C₂×C₄²×C₃₇	C₄
prime radical	C₂³×D₃₇	C₂³
cosocle	C₂²×D₃₇	C₂⁴
hypercenter	C₂×C₄²	#3866
hypocenter	C₃₇	#2283
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #14558478	C₂	semidirect
2. #14558477	C₂	semidirect
5. C₄×C₄⋊#3866	C₂	semidirect
7. #14558469	C₂	semidirect
19. C₂×C₄×Dic₃₇	C₄	semidirect
20. C₂×C₄×Dic₃₇	C₄	semidirect
21. C₂×C₄×Dic₃₇	C₄	semidirect
22. C₄²×D₃₇	C₄	semidirect
23. C₄²×D₃₇	C₄	semidirect
24. C₄²×D₃₇	C₄	semidirect
25. C₄²×D₃₇	C₄	semidirect
30. C₂×C₄²×C₃₇	C₄	semidirect
151. C₃₇	#2283	semidirect
169. C₂×C₄²	#3866	semidirect

Maximal subgroups

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

subgroup	index
#14558478	2
#14558477	2
#14558476	2
#14558475	2
C₄×C₄⋊#3866	2
C₄×C₂²⋊#3866	2
#14558469	2
C₂×C₄²×D₃₇	2
#2283	37