Finite Group Databasebeta

Subgroups — #18010917

Order 4736.

224 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₇⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558543	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558449	C₂	1	4	yes	maximal, complemented
4	C₂×#11924555	C₂	1	1	yes	maximal, characteristic
5	#11924615	—	2	1	no	—
6	#11924611	C₂²	1	2	yes	complemented
7	#11924610	—	2	1	no	—
8	D₃₇⋊C₁₆	—	2	1	no	—
9	#11924605	—	2	1	no	—
10	#11924604	C₂²	1	2	yes	complemented
11	C₂×#10596964	—	2	1	no	—
12	#11924555	—	2	1	no	—
13	#11924555	—	2	1	no	—
14	#11924555	C₄	1	2	yes	complemented
15	#11924555	C₂²	1	2	yes	—
16	C₂×Dic₁₄₈	C₄	1	1	yes	characteristic, complemented
17	C₂×#10596940	—	2	1	no	—
18	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₃₇	C₄	1	1	yes	characteristic, complemented
20	C₂×D₄×C₃₇	—	2	1	no	—
21	C₂×C₂²⋊D₃₇	—	4	1	no	—
22	C₂²×Dic₃₇	—	4	1	no	—
23	#10596964	—	2	2	no	—
24	#10596964	—	2	2	no	—
25	#10596964	—	4	1	no	—
26	#10596964	—	4	1	no	—
27	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
28	C₂×Dic₇₄	—	2	1	no	—
29	C₂²⋊#3866	—	4	1	no	—
30	C₄⋊#3866	—	2	2	no	—
31	C₄×#3866	—	4	1	no	—
32	D₈×C₃₇	—	2	1	no	—
33	D₈×C₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
35	#10596940	—	2	2	no	—
36	#10596940	—	2	2	no	—
37	C₂×#64436	D₄	1	1	yes	characteristic
38	Dic₁₄₈	—	2	1	no	—
39	Dic₁₄₈	C₂×C₄	1	2	yes	—
40	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
41	C₈×D₃₇	D₄	1	1	yes	characteristic
42	C₈×D₃₇	D₄	1	1	yes	characteristic
43	C₈×D₃₇	C₂³	1	1	yes	characteristic
44	#10596927	—	2	2	no	—
45	C₂³×C₃₇	—	4	1	no	—
46	C₂²×D₃₇	D₈	1	1	yes	characteristic
47	C₂×#3866	—	4	2	no	—
48	D₄×C₃₇	—	2	2	no	—
49	D₄×C₃₇	—	4	1	no	—
50	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²⋊D₃₇	—	4	2	no	—
52	C₂²⋊D₃₇	—	4	2	no	—
53	C₂×Dic₃₇	—	4	1	no	—
54	C₂×Dic₃₇	—	4	2	no	—
55	C₂×Dic₃₇	—	4	2	no	—
56	C₂×Dic₃₇	SD₁₆	1	1	yes	characteristic
57	C₄×D₃₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₃₇	D₈	1	1	yes	characteristic
59	C₄×D₃₇	SD₁₆	1	1	yes	characteristic
60	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
61	Dic₇₄	—	2	2	no	—
62	Dic₇₄	—	4	1	no	—
63	C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
65	#64436	C₂²⋊C₄	1	1	yes	characteristic
66	#64436	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₃₇	—	4	2	no	—
68	C₂²×C₃₇	—	8	2	no	—
69	C₂²×C₃₇	#102	1	1	yes	characteristic
70	C₂×D₃₇	—	4	1	no	—
71	C₂×D₃₇	#102	1	1	yes	characteristic
72	C₂×D₃₇	C₂×D₈	1	1	yes	characteristic
73	#3866	—	8	2	no	—
74	C₄×C₃₇	#102	1	1	yes	characteristic
75	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Dic₃₇	—	4	2	no	—
77	Dic₃₇	#102	1	1	yes	characteristic
78	Dic₃₇	C₂×SD₁₆	1	1	yes	characteristic
79	#2145	—	37	1	no	maximal, Sylow
80	C₂×C₃₇	—	2	1	no	—
81	C₂×C₃₇	—	8	2	no	—
82	C₂×C₃₇	C₂×#102	1	1	yes	characteristic
83	D₃₇	—	2	1	no	—
84	D₃₇	—	2	1	no	—
85	C₂×#135	—	37	1	no	—
86	C₂×M₃₂	—	37	1	no	—
87	#427	—	37	1	no	—
88	#360	—	37	4	no	—
89	C₃₇	#2145	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	74	1	no	—
91	#135	—	37	2	no	—
92	#135	—	37	2	no	—
93	#135	—	74	1	no	—
94	#135	—	74	1	no	—
95	C₂×Q₁₆	—	37	1	no	—
96	C₂×SD₁₆	—	74	1	no	—
97	C₂×D₈	#3866	1	1	yes	characteristic, complemented
98	C₂²×C₈	—	37	1	no	—
99	#117	—	74	1	no	—
100	M₃₂	—	37	2	no	—
101	M₃₂	—	74	1	no	—
102	C₂×C₁₆	—	74	1	no	—
103	#107	—	74	1	no	—
104	#106	—	37	2	no	—
105	#41	—	74	2	no	—
106	#41	—	74	2	no	—
107	#41	—	148	1	no	—
108	#41	—	148	1	no	—
109	C₂×Q₈	—	74	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	148	1	no	—
112	C₂²×C₄	—	37	1	no	—
113	C₂²×C₄	—	148	1	no	—
114	Q₁₆	—	37	2	no	—
115	Q₁₆	—	74	1	no	—
116	SD₁₆	—	74	2	no	—
117	SD₁₆	—	74	2	no	—
118	D₈	—	2	1	no	—
119	D₈	C₂×#3866	1	2	yes	—
120	C₂×C₈	—	37	1	no	—
121	C₂×C₈	—	37	1	no	—
122	C₂×C₈	—	37	1	no	—
123	C₂×C₈	—	37	1	no	—
124	C₂×C₈	—	37	1	no	—
125	C₂×C₈	C₂×#3866	1	1	yes	characteristic
126	C₄⋊C₄	—	74	2	no	—
127	C₂²⋊C₄	—	148	1	no	—
128	C₄²	—	148	1	no	—
129	C₁₆	—	74	2	no	—
130	C₂³	—	4	1	no	—
131	C₂³	—	37	1	no	—
132	Q₈	—	74	2	no	—
133	Q₈	—	148	1	no	—
134	D₄	—	2	2	no	—
135	D₄	—	4	1	no	—
136	D₄	—	148	2	no	—
137	D₄	—	148	2	no	—
138	C₂×C₄	—	37	1	no	—
139	C₂×C₄	—	37	1	no	—
140	C₂×C₄	—	37	1	no	—
141	C₂×C₄	—	37	1	no	—
142	C₂×C₄	—	37	1	no	—
143	C₂×C₄	—	148	2	no	—
144	C₂×C₄	—	148	2	no	—
145	C₂×C₄	—	148	2	no	—
146	C₂×C₄	—	148	1	no	—
147	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
148	C₈	—	37	1	no	—
149	C₈	—	37	1	no	—
150	C₈	C₂²⋊#3866	1	1	yes	characteristic
151	C₈	C₂²×#3866	1	1	yes	characteristic
152	C₂²	—	4	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	37	1	no	—
155	C₂²	—	37	1	no	—
156	C₂²	—	148	1	no	—
157	C₂²	#11924598	1	1	yes	characteristic
158	C₄	—	37	1	no	—
159	C₄	—	37	1	no	—
160	C₄	—	296	2	no	—
161	C₄	—	148	2	no	—
162	C₄	#11924598	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊#3866	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	8	2	no	—
166	C₂	—	74	1	no	—
167	C₂	—	74	1	no	—
168	C₂	C₂×#11924598	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11924598
commutator	C₈×C₃₇	C₂²×C₄
Frattini	C₈	C₂²×#3866
Fitting	C₂×D₈×C₃₇	C₄
socle	C₂×C₃₇	C₂×#102
prime radical	C₂×#11924555	C₂
cosocle	C₈×D₃₇	C₂³
hypercenter	C₂×D₈	#3866
hypocenter	C₃₇	#2145
LCS	G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₃₇⋊M₃₂	C₂	semidirect
2. #14558543	C₂	semidirect
3. #14558449	C₂	semidirect
6. #11924611	C₂²	semidirect
10. #11924604	C₂²	semidirect
14. #11924555	C₄	semidirect
16. C₂×Dic₁₄₈	C₄	semidirect
19. C₂×D₈×C₃₇	C₄	semidirect
89. C₃₇	#2145	semidirect
97. C₂×D₈	#3866	semidirect

Maximal subgroups

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

subgroup	index
D₃₇⋊M₃₂	2
#14558543	2
#14558449	2
C₂×#11924555	2
#2145	37