Finite Group Databasebeta

Subgroups — #17994004

Order 4736.

176 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 51 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	#14557879	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557824	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557725	C₂	1	1	yes	maximal, characteristic
4	#14557598	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557283	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557281	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596939	—	2	1	no	—
9	C₂×C₈×D₃₇	—	2	1	no	—
10	#11924536	C₂²	1	1	yes	characteristic
11	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
12	#11924468	—	2	1	no	—
13	C₄⋊#64436	C₂²	1	1	yes	characteristic
14	#11924458	—	2	1	no	—
15	#11924457	—	2	1	no	—
16	#11924457	—	2	1	no	—
17	#11924457	—	2	1	no	—
18	#11924448	—	2	1	no	—
19	C₄×#64436	C₂²	1	1	yes	characteristic
20	#127×C₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₃₇	—	2	1	no	—
24	C₂×D₄×C₃₇	—	2	1	no	—
25	C₂×D₄×C₃₇	—	2	1	no	—
26	C₂×D₁₄₈	D₄	1	1	yes	characteristic
27	C₂×C₄×D₃₇	—	2	1	no	—
28	C₂×Dic₇₄	—	2	1	no	—
29	C₂×C₈×C₃₇	—	2	1	no	—
30	C₂×C₈×C₃₇	—	2	1	no	—
31	C₄²×C₃₇	C₂³	1	1	yes	characteristic
32	#10596939	—	4	2	no	—
33	#10596938	—	2	1	no	—
34	C₄⋊Dic₃₇	—	2	1	no	—
35	C₄⋊Dic₃₇	—	2	1	no	—
36	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
37	C₂×#64436	—	2	1	no	—
38	C₂×#64436	D₄	1	1	yes	characteristic
39	C₂×#64436	D₄	1	1	yes	characteristic
40	C₈×D₃₇	—	4	2	no	—
41	C₂³×C₃₇	—	4	1	no	—
42	C₂²×D₃₇	—	2	1	no	—
43	D₄×C₃₇	—	4	1	no	—
44	D₄×C₃₇	—	4	1	no	—
45	D₄×C₃₇	—	4	1	no	—
46	D₄×C₃₇	—	4	1	no	—
47	D₄×C₃₇	—	8	1	no	—
48	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
51	C₂×Dic₃₇	—	2	1	no	—
52	C₂×Dic₃₇	—	4	1	no	—
53	D₁₄₈	—	2	1	no	—
54	D₁₄₈	SD₁₆	1	2	yes	—
55	C₄×D₃₇	—	4	1	no	—
56	C₄×D₃₇	—	4	1	no	—
57	Dic₇₄	—	8	1	no	—
58	C₈×C₃₇	—	4	1	no	—
59	C₈×C₃₇	—	4	1	no	—
60	#64436	—	2	1	no	—
61	#64436	—	2	1	no	—
62	#64436	—	4	1	no	—
63	C₂²×C₃₇	—	8	1	no	—
64	C₂²×C₃₇	—	8	1	no	—
65	C₂²×C₃₇	—	8	1	no	—
66	C₂²×C₃₇	#121	1	1	yes	characteristic
67	C₂×D₃₇	—	2	2	no	—
68	C₂×D₃₇	—	4	1	no	—
69	C₂×D₃₇	—	4	1	no	—
70	C₄×C₃₇	—	2	1	no	—
71	C₄×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×C₃₇	#135	1	1	yes	characteristic
73	C₄×C₃₇	#135	1	1	yes	characteristic
74	C₄×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
75	Dic₃₇	—	4	1	no	—
76	Dic₃₇	—	8	1	no	—
77	#1677	—	37	1	no	maximal, Sylow
78	C₂×C₃₇	—	16	1	no	—
79	C₂×C₃₇	#463	1	1	yes	characteristic
80	C₂×C₃₇	#465	1	1	yes	characteristic
81	C₂×C₃₇	#472	1	1	yes	characteristic
82	D₃₇	—	4	2	no	—
83	#486	—	37	1	no	—
84	#476	—	37	1	no	—
85	#459	—	37	1	no	—
86	C₈×D₄	—	37	1	no	—
87	#334	—	37	1	no	—
88	#333	—	37	1	no	—
89	#331	D₃₇	1	1	yes	characteristic, complemented
90	C₃₇	#1677	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	74	1	no	—
92	C₂²×C₈	—	74	1	no	—
93	C₄⋊Q₈	—	37	1	no	—
94	#127	C₂×D₃₇	1	1	yes	characteristic
95	C₄×D₄	—	37	1	no	—
96	#106	—	74	1	no	—
97	C₄⋊C₈	—	37	1	no	—
98	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
99	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
100	#102	—	74	1	no	—
101	#102	—	74	1	no	—
102	#102	—	74	1	no	—
103	#102	—	74	1	no	—
104	C₂²⋊C₈	—	74	1	no	—
105	C₄×C₈	—	37	1	no	—
106	C₂×Q₈	—	74	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	37	1	no	—
111	C₂²×C₄	—	74	1	no	—
112	D₈	—	148	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	37	1	no	—
116	C₂×C₈	—	37	1	no	—
117	C₂×C₈	—	74	1	no	—
118	C₂×C₈	—	148	2	no	—
119	C₄⋊C₄	—	37	1	no	—
120	C₄⋊C₄	—	74	1	no	—
121	C₄⋊C₄	—	74	1	no	—
122	C₂²⋊C₄	—	74	1	no	—
123	C₄²	C₂²×D₃₇	1	1	yes	characteristic
124	C₂³	—	4	1	no	—
125	C₂³	—	74	1	no	—
126	Q₈	—	296	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	D₄	—	37	2	no	—
133	D₄	—	74	1	no	—
134	C₂×C₄	—	74	1	no	—
135	C₂×C₄	—	148	1	no	—
136	C₂×C₄	—	148	1	no	—
137	C₂×C₄	—	148	1	no	—
138	C₂×C₄	#10596962	1	1	yes	characteristic
139	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
140	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	4	1	no	—
143	C₈	—	74	1	no	—
144	C₈	—	74	1	no	—
145	C₈	—	148	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	74	2	no	—
150	C₂²	—	148	1	no	—
151	C₂²	—	148	1	no	—
152	C₂²	#11924509	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	296	1	no	—
155	C₄	—	148	1	no	—
156	C₄	SD₁₆×D₃₇	1	1	yes	characteristic
157	C₄	#11924555	1	1	yes	characteristic
158	C₄	#11924557	1	1	yes	characteristic
159	C₄	#11924560	1	1	yes	characteristic
160	C₂	—	16	1	no	—
161	C₂	—	148	2	no	—
162	C₂	#14557747	1	1	yes	characteristic
163	C₂	#14557761	1	1	yes	characteristic
164	C₂	#14557802	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11924509
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#331	C₂
socle	C₂²×C₃₇	#121
prime radical	#14557725	C₂
hypercenter	#331	D₃₇
hypocenter	C₃₇	#1677
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

Complements

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

N	H ≅ G/N	type
1. #14557879	C₂	semidirect
2. #14557824	C₂	semidirect
4. #14557598	C₂	semidirect
5. #14557283	C₂	semidirect
6. #14557281	C₂	semidirect
7. C₃₇×#331	C₂	semidirect
89. #331	D₃₇	semidirect
90. C₃₇	#1677	semidirect

Maximal subgroups

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

subgroup	index
#14557879	2
#14557824	2
#14557725	2
#14557598	2
#14557283	2
#14557281	2
C₃₇×#331	2
#1677	37