Subgroups — #18008001

Order 4736.

996 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558164	C₂	1	4	yes	maximal, complemented
2	#14558161	C₂	1	8	yes	maximal, complemented
3	#14558159	C₂	1	4	yes	maximal
4	#14558090	C₂	1	4	yes	maximal, complemented
5	#14558086	C₂	1	8	yes	maximal, complemented
6	C₂×#11924495	C₂	1	2	yes	maximal
7	C₂×#117×C₃₇	C₂	1	1	yes	maximal, characteristic
8	#11924536	C₂²	1	8	yes	complemented
9	#11924535	C₂²	1	16	yes	complemented
10	#11924528	C₂²	1	8	yes	complemented
11	#11924527	C₂²	1	16	yes	complemented
12	#11924515	C₂²	1	16	yes	—
13	#11924513	C₂²	1	32	yes	—
14	C₄×Dic₇₄	C₂²	1	16	yes	complemented
15	#11924495	C₂²	1	4	yes	—
16	#11924495	C₂²	1	4	yes	—
17	#11924495	C₂²	1	4	yes	—
18	#11924495	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊Dic₃₇	C₂²	1	4	yes	—
21	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
22	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
23	#117×C₃₇	C₂²	1	4	yes	—
24	#117×C₃₇	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₃₇	C₂²	1	2	yes	—
28	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₃₇	Q₈	1	8	yes	complemented
30	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
31	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃₇	C₂³	1	8	yes	—
36	C₂×Dic₇₄	C₂³	1	16	yes	—
37	C₄⋊C₄×C₃₇	C₂³	1	8	yes	—
38	C₂²⋊C₄×C₃₇	C₂³	1	8	yes	—
39	C₄²×C₃₇	C₂³	1	4	yes	—
40	C₄²×C₃₇	C₂³	1	4	yes	—
41	C₂²⋊Dic₃₇	C₂³	1	8	yes	—
42	C₂²⋊Dic₃₇	C₂³	1	8	yes	—
43	C₄⋊Dic₃₇	C₂³	1	8	yes	—
44	C₄⋊Dic₃₇	C₂³	1	8	yes	—
45	C₄⋊Dic₃₇	C₂³	1	16	yes	—
46	#10596936	C₂³	1	32	yes	—
47	C₄×Dic₃₇	C₂³	1	8	yes	—
48	C₄×Dic₃₇	C₂³	1	8	yes	—
49	C₂³×C₃₇	—	4	2	no	—
50	C₂³×C₃₇	C₂×Q₈	1	4	yes	—
51	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
52	C₂³×C₃₇	C₂⁴	1	2	yes	—
53	C₂×C₄×C₃₇	—	4	4	no	—
54	C₂×C₄×C₃₇	C₂×Q₈	1	8	yes	—
55	C₂×C₄×C₃₇	C₂×Q₈	1	8	yes	—
56	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
58	C₂×C₄×C₃₇	C₂⁴	1	8	yes	—
59	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
60	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
62	C₂×Dic₃₇	—	4	8	no	—
63	C₂×Dic₃₇	C₂⁴	1	16	yes	—
64	Dic₇₄	—	4	16	no	—
65	C₂²×C₃₇	—	2	4	no	—
66	C₂²×C₃₇	—	2	2	no	—
67	C₂²×C₃₇	—	4	4	no	—
68	C₂²×C₃₇	C₂²×Q₈	1	2	yes	—
69	C₂²×C₃₇	C₂²×Q₈	1	4	yes	—
70	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
71	C₄×C₃₇	—	2	8	no	—
72	C₄×C₃₇	C₂²×Q₈	1	4	yes	—
73	C₄×C₃₇	C₂²×Q₈	1	2	yes	—
74	C₄×C₃₇	C₂²×Q₈	1	2	yes	—
75	Dic₃₇	—	2	16	no	—
76	#3458	—	37	1	no	maximal, Sylow
77	C₂×C₃₇	—	2	4	no	—
78	C₂×C₃₇	—	2	2	no	—
79	C₂×C₃₇	C₂³×Q₈	1	1	yes	characteristic
80	C₂×C₃₇	#585	1	2	yes	—
81	#544	—	37	8	no	—
82	#544	—	37	4	no	—
83	#543	—	37	4	no	—
84	#533	—	37	8	no	—
85	#533	—	37	4	no	—
86	C₂×#117	—	37	2	no	—
87	C₂×#117	D₃₇	1	1	yes	characteristic
88	C₃₇	#3458	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
90	C₄⋊Q₈	—	37	16	no	—
91	C₄⋊Q₈	—	37	8	no	—
92	#125	—	37	16	no	—
93	#125	—	37	8	no	—
94	C₂²⋊Q₈	—	37	16	no	—
95	C₂²⋊Q₈	—	37	32	no	—
96	C₄×Q₈	—	37	16	no	—
97	#117	—	37	4	no	—
98	#117	—	37	4	no	—
99	#117	—	37	4	no	—
100	#117	—	37	4	no	—
101	#117	C₂×D₃₇	1	4	yes	—
102	#117	C₂×D₃₇	1	4	yes	—
103	C₂×C₄⋊C₄	—	37	4	no	—
104	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
105	C₂×C₂²⋊C₄	—	37	4	no	—
106	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
107	C₂×C₄²	—	37	4	no	—
108	C₂×C₄²	C₂×D₃₇	1	2	yes	—
109	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
110	C₂×Q₈	—	37	16	no	—
111	C₂²×C₄	—	37	8	no	—
112	C₂²×C₄	Dic₇₄	1	8	yes	complemented
113	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
114	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
115	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
117	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
118	C₄⋊C₄	—	37	32	no	—
119	C₄⋊C₄	—	37	8	no	—
120	C₄⋊C₄	—	37	8	no	—
121	C₄⋊C₄	—	37	16	no	—
122	C₄⋊C₄	C₂²×D₃₇	1	8	yes	—
123	C₂²⋊C₄	—	37	8	no	—
124	C₂²⋊C₄	—	37	8	no	—
125	C₂²⋊C₄	C₂²×D₃₇	1	8	yes	—
126	C₄²	—	37	8	no	—
127	C₄²	—	37	8	no	—
128	C₄²	C₂²×D₃₇	1	4	yes	—
129	C₄²	C₂²×D₃₇	1	4	yes	—
130	C₂³	—	4	2	no	—
131	C₂³	C₂×Dic₇₄	1	4	yes	—
132	C₂³	C₂³×D₃₇	1	1	yes	characteristic
133	C₂³	C₂³×D₃₇	1	2	yes	—
134	Q₈	—	148	16	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	37	16	no	—
137	C₂×C₄	—	148	8	no	—
138	C₂×C₄	C₂×Dic₇₄	1	8	yes	—
139	C₂×C₄	C₂×Dic₇₄	1	4	yes	—
140	C₂×C₄	C₂×Dic₇₄	1	8	yes	—
141	C₂×C₄	C₂×Dic₇₄	1	4	yes	—
142	C₂×C₄	C₂³×D₃₇	1	8	yes	—
143	C₂×C₄	C₂³×D₃₇	1	2	yes	—
144	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	C₂²×Dic₇₄	1	2	yes	—
150	C₂²	C₂²×Dic₇₄	1	4	yes	—
151	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	74	16	no	—
154	C₄	C₂²×Dic₇₄	1	4	yes	—
155	C₄	C₂²×Dic₇₄	1	2	yes	—
156	C₄	C₂²×Dic₇₄	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	2	4	no	—
159	C₂	C₂³×Dic₇₄	1	1	yes	characteristic
160	C₂	#14558369	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×#117×C₃₇	C₂
prime radical	C₂⁴×C₃₇	C₂³
hypercenter	C₂×#117	D₃₇
hypocenter	C₃₇	#3458
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #14558164	C₂	semidirect
2. #14558161	C₂	semidirect
4. #14558090	C₂	semidirect
5. #14558086	C₂	semidirect
8. #11924536	C₂²	semidirect
9. #11924535	C₂²	semidirect
10. #11924528	C₂²	semidirect
11. #11924527	C₂²	semidirect
14. C₄×Dic₇₄	C₂²	semidirect
29. C₂²×C₄×C₃₇	Q₈	semidirect
88. C₃₇	#3458	semidirect
112. C₂²×C₄	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
#14558164	2
#14558161	2
#14558159	2
#14558090	2
#14558086	2
C₂×#11924495	2
C₂×#117×C₃₇	2
#3458	37