Subgroups — #18008216

Order 4736.

1094 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558256	C₂	1	3	yes	maximal, complemented
2	#14558221	C₂	1	6	yes	maximal, complemented
3	Q₈×D₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#14558195	C₂	1	2	yes	maximal, complemented
5	#14558186	C₂	1	3	yes	maximal, complemented
6	#14558088	C₂	1	3	yes	maximal, complemented
7	#14558068	C₂	1	6	yes	maximal, complemented
8	C₄×#10596966	C₂	1	3	yes	maximal, complemented
9	C₄×#10596966	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×Q₈×D₃₇	C₂	1	2	yes	maximal, complemented
11	C₃₇×#557	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596966	C₂²	1	3	yes	—
13	C₂×#10596966	C₂²	1	1	yes	characteristic
14	C₂×Q₈×D₃₇	C₂²	1	2	yes	—
15	#11924536	C₂²	1	3	yes	—
16	#11924531	C₂²	1	12	yes	—
17	#11924527	C₂²	1	6	yes	—
18	#11924524	C₂²	1	6	yes	—
19	#11924521	C₂²	1	12	yes	—
20	#11924516	C₂²	1	6	yes	—
21	#11924516	C₂²	1	6	yes	—
22	#11924512	C₂²	1	6	yes	—
23	#11924511	C₂²	1	6	yes	—
24	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	Q₈×Dic₃₇	C₂²	1	3	yes	—
26	C₄×Dic₇₄	C₂²	1	6	yes	—
27	#11924501	C₂²	1	6	yes	—
28	C₄×D₁₄₈	C₂²	1	6	yes	—
29	C₄×D₁₄₈	C₂²	1	3	yes	—
30	C₄×D₁₄₈	C₂²	1	3	yes	—
31	#11924498	C₂²	1	1	yes	characteristic
32	#11924498	C₂²	1	3	yes	—
33	#11924498	C₂²	1	6	yes	—
34	#11924494	C₂²	1	3	yes	—
35	#11924494	C₂²	1	6	yes	—
36	#11924494	C₂²	1	3	yes	—
37	#11924494	C₂²	1	6	yes	—
38	C₄⋊C₄×D₃₇	C₂²	1	6	yes	—
39	C₄²×D₃₇	C₂²	1	6	yes	—
40	C₄²×D₃₇	C₂²	1	3	yes	—
41	C₄⋊Q₈×C₃₇	C₂²	1	3	yes	—
42	#125×C₃₇	C₂²	1	6	yes	—
43	C₄×Q₈×C₃₇	C₂²	1	2	yes	—
44	C₄×Q₈×C₃₇	C₂²	1	3	yes	—
45	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
46	C₂×Q₈×C₃₇	C₂³	1	3	yes	—
47	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
48	#10596966	—	2	4	no	—
49	#10596966	—	2	12	no	—
50	Q₈×D₃₇	—	2	8	no	—
51	C₂×D₁₄₈	C₂³	1	6	yes	—
52	C₂×D₁₄₈	C₂³	1	3	yes	—
53	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₃₇	C₂³	1	2	yes	—
55	C₂×C₄×D₃₇	C₂³	1	6	yes	—
56	C₂×C₄×D₃₇	C₂³	1	6	yes	—
57	C₂×C₄×D₃₇	C₂³	1	3	yes	—
58	C₂×C₄×D₃₇	C₂³	1	3	yes	—
59	C₂×C₄×D₃₇	C₂³	1	3	yes	—
60	C₂×Dic₇₄	C₂³	1	6	yes	—
61	C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
62	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
64	C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
65	C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
66	C₄²×C₃₇	C₂³	1	3	yes	—
67	C₄²×C₃₇	C₂³	1	6	yes	—
68	#10596938	C₂³	1	2	yes	—
69	#10596938	C₂³	1	6	yes	—
70	#10596938	C₂³	1	6	yes	—
71	#10596938	C₂³	1	6	yes	—
72	#10596938	C₂³	1	12	yes	—
73	C₄⋊Dic₃₇	C₂³	1	3	yes	—
74	C₄⋊Dic₃₇	C₂³	1	3	yes	—
75	C₄⋊Dic₃₇	C₂³	1	6	yes	—
76	C₄⋊Dic₃₇	C₂³	1	6	yes	—
77	#10596936	C₂³	1	12	yes	—
78	C₄×Dic₃₇	C₂³	1	3	yes	—
79	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
80	C₄×Dic₃₇	C₂³	1	3	yes	—
81	C₄×Dic₃₇	C₂³	1	6	yes	—
82	C₄×Dic₃₇	C₂³	1	6	yes	—
83	C₂²×D₃₇	C₂⁴	1	3	yes	—
84	C₂²×D₃₇	C₂⁴	1	2	yes	—
85	Q₈×C₃₇	—	2	6	no	—
86	Q₈×C₃₇	#41	1	4	yes	complemented
87	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
88	C₂×C₄×C₃₇	C₂⁴	1	6	yes	—
89	C₂×C₄×C₃₇	C₂⁴	1	3	yes	—
90	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₂⁴	1	3	yes	—
92	C₂×Dic₃₇	C₂⁴	1	3	yes	—
93	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₃₇	C₂⁴	1	6	yes	—
95	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
96	D₁₄₈	—	2	12	no	—
97	D₁₄₈	—	2	6	no	—
98	D₁₄₈	#41	1	4	yes	complemented
99	C₄×D₃₇	—	2	12	no	—
100	C₄×D₃₇	—	2	12	no	—
101	C₄×D₃₇	—	2	6	no	—
102	C₄×D₃₇	—	2	6	no	—
103	C₄×D₃₇	—	2	4	no	—
104	C₄×D₃₇	—	2	6	no	—
105	Dic₇₄	—	2	12	no	—
106	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
107	C₂×D₃₇	—	2	2	no	—
108	C₂×D₃₇	—	2	2	no	—
109	C₂×D₃₇	—	2	3	no	—
110	C₂×D₃₇	—	2	3	no	—
111	C₂×D₃₇	—	2	3	no	—
112	C₂×D₃₇	C₂×#41	1	4	yes	—
113	C₄×C₃₇	—	2	6	no	—
114	C₄×C₃₇	—	2	2	no	—
115	C₄×C₃₇	—	2	3	no	—
116	C₄×C₃₇	C₂×#41	1	2	yes	—
117	C₄×C₃₇	C₂×#41	1	6	yes	—
118	Dic₃₇	—	2	6	no	—
119	Dic₃₇	—	2	3	no	—
120	Dic₃₇	—	2	1	no	—
121	Dic₃₇	2₋¹⁺⁴	1	2	yes	—
122	#3482	—	37	1	no	maximal, Sylow
123	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
124	C₂×C₃₇	C₂²×#41	1	1	yes	characteristic
125	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
126	D₃₇	—	2	4	no	—
127	D₃₇	—	4	3	no	—
128	#557	D₃₇	1	1	yes	characteristic, complemented
129	#556	—	37	3	no	—
130	#552	—	37	6	no	—
131	D₄×Q₈	—	37	1	no	—
132	#548	—	37	2	no	—
133	#547	—	37	3	no	—
134	#533	—	37	3	no	—
135	#529	—	37	6	no	—
136	C₄×#41	—	37	3	no	—
137	C₄×#41	—	37	1	no	—
138	C₂×C₄×Q₈	—	37	2	no	—
139	C₃₇	#3482	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	37	3	no	—
141	C₂×#41	—	37	1	no	—
142	C₂²×Q₈	—	37	2	no	—
143	C₄⋊Q₈	—	37	3	no	—
144	C₄⋊Q₈	C₂×D₃₇	1	3	yes	—
145	#126	—	37	12	no	—
146	#125	—	37	6	no	—
147	#125	C₂×D₃₇	1	6	yes	—
148	#124	—	37	6	no	—
149	#123	—	37	12	no	—
150	C₂²⋊Q₈	—	37	6	no	—
151	C₂²⋊Q₈	—	37	6	no	—
152	C₂²⋊Q₈	—	37	6	no	—
153	#121	—	37	6	no	—
154	C₄×Q₈	—	37	1	no	—
155	C₄×Q₈	—	37	3	no	—
156	C₄×Q₈	—	37	6	no	—
157	C₄×Q₈	—	37	6	no	—
158	C₄×Q₈	C₂×D₃₇	1	2	yes	—
159	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
160	C₄×Q₈	C₂×D₃₇	1	3	yes	—
161	C₄×D₄	—	37	1	no	—
162	C₄×D₄	—	37	3	no	—
163	C₄×D₄	—	37	6	no	—
164	C₄×D₄	—	37	3	no	—
165	C₄×D₄	—	37	3	no	—
166	C₄×D₄	—	37	6	no	—
167	#117	—	37	3	no	—
168	#117	—	37	6	no	—
169	#117	—	37	3	no	—
170	#117	—	37	6	no	—
171	C₂×C₄⋊C₄	—	37	6	no	—
172	C₂×C₄²	—	37	6	no	—
173	C₂×C₄²	—	37	3	no	—
174	#41	—	74	12	no	—
175	#41	—	74	4	no	—
176	C₂×Q₈	—	37	6	no	—
177	C₂×Q₈	—	74	8	no	—
178	C₂×Q₈	C₂²×D₃₇	1	3	yes	—
179	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
180	C₂×D₄	—	37	1	no	—
181	C₂×D₄	—	37	6	no	—
182	C₂×D₄	—	37	3	no	—
183	C₂²×C₄	—	37	2	no	—
184	C₂²×C₄	—	37	6	no	—
185	C₂²×C₄	—	37	6	no	—
186	C₂²×C₄	—	37	3	no	—
187	C₂²×C₄	—	37	3	no	—
188	C₂²×C₄	—	37	3	no	—
189	C₄⋊C₄	—	37	12	no	—
190	C₄⋊C₄	—	37	3	no	—
191	C₄⋊C₄	—	37	6	no	—
192	C₄⋊C₄	—	37	3	no	—
193	C₄⋊C₄	—	37	6	no	—
194	C₄⋊C₄	C₂²×D₃₇	1	6	yes	—
195	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
196	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
197	C₄⋊C₄	C₂²×D₃₇	1	6	yes	—
198	C₄⋊C₄	C₂²×D₃₇	1	6	yes	—
199	C₂²⋊C₄	—	37	2	no	—
200	C₂²⋊C₄	—	37	6	no	—
201	C₂²⋊C₄	—	37	6	no	—
202	C₂²⋊C₄	—	37	6	no	—
203	C₂²⋊C₄	—	37	12	no	—
204	C₄²	—	37	1	no	—
205	C₄²	—	37	3	no	—
206	C₄²	—	37	3	no	—
207	C₄²	—	37	6	no	—
208	C₄²	—	37	6	no	—
209	C₄²	C₂²×D₃₇	1	3	yes	—
210	C₄²	C₂²×D₃₇	1	6	yes	—
211	C₂³	—	37	2	no	—
212	C₂³	—	37	3	no	—
213	Q₈	—	2	6	no	—
214	Q₈	—	74	12	no	—
215	Q₈	#10596966	1	4	yes	complemented
216	D₄	—	37	4	no	—
217	D₄	—	74	12	no	—
218	D₄	—	74	6	no	—
219	C₂×C₄	—	37	1	no	—
220	C₂×C₄	—	37	3	no	—
221	C₂×C₄	—	37	1	no	—
222	C₂×C₄	—	37	6	no	—
223	C₂×C₄	—	74	4	no	—
224	C₂×C₄	—	74	6	no	—
225	C₂×C₄	—	74	12	no	—
226	C₂×C₄	—	74	6	no	—
227	C₂×C₄	—	74	12	no	—
228	C₂×C₄	—	74	6	no	—
229	C₂×C₄	C₂³×D₃₇	1	2	yes	—
230	C₂×C₄	C₂³×D₃₇	1	3	yes	—
231	C₂×C₄	C₂³×D₃₇	1	6	yes	—
232	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂³×D₃₇	1	3	yes	—
234	C₂²	—	37	4	no	—
235	C₂²	—	74	2	no	—
236	C₂²	—	74	2	no	—
237	C₂²	—	74	3	no	—
238	C₂²	—	74	3	no	—
239	C₂²	—	74	3	no	—
240	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
241	C₄	—	2	6	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	3	no	—
244	C₄	—	37	2	no	—
245	C₄	—	74	3	no	—
246	C₄	—	74	1	no	—
247	C₄	—	74	6	no	—
248	C₄	#41×D₃₇	1	2	yes	—
249	C₄	C₂×#10596966	1	6	yes	—
250	C₂	—	74	4	no	—
251	C₂	—	148	3	no	—
252	C₂	C₂×#41×D₃₇	1	1	yes	characteristic
253	C₂	C₂²×#10596966	1	1	yes	characteristic
254	C₂	2₋¹⁺⁴×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁵
Fitting	C₃₇×#557	C₂
hypercenter	#557	D₃₇
hypocenter	C₃₇	#3482
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. #14558256	C₂	semidirect
2. #14558221	C₂	semidirect
3. Q₈×D₁₄₈	C₂	semidirect
4. #14558195	C₂	semidirect
5. #14558186	C₂	semidirect
6. #14558088	C₂	semidirect
7. #14558068	C₂	semidirect
8. C₄×#10596966	C₂	semidirect
9. C₄×#10596966	C₂	semidirect
10. C₄×Q₈×D₃₇	C₂	semidirect
11. C₃₇×#557	C₂	semidirect
86. Q₈×C₃₇	#41	semidirect
98. D₁₄₈	#41	semidirect
128. #557	D₃₇	semidirect
139. C₃₇	#3482	semidirect
215. Q₈	#10596966	semidirect

Maximal subgroups

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

subgroup	index
#14558256	2
#14558221	2
Q₈×D₁₄₈	2
#14558195	2
#14558186	2
#14558088	2
#14558068	2
C₄×#10596966	2
C₄×Q₈×D₃₇	2
C₃₇×#557	2
#3482	37