Finite Group Databasebeta

Subgroups — #17996544

Order 4736.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14558333	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557971	C₂	1	2	yes	maximal, complemented
3	#14557786	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557344	C₂	1	2	yes	maximal, complemented
5	C₃₇×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596962	—	2	1	no	—
7	#11924569	—	2	2	no	—
8	#11924564	—	2	2	no	—
9	#11924558	—	2	1	no	—
10	#11924558	C₂²	1	2	yes	—
11	C₂×D₂₉₆	C₂²	1	1	yes	characteristic, complemented
12	#11924510	—	2	1	no	—
13	#11924481	—	2	2	no	—
14	D₅₉₂	—	2	2	no	—
15	#11924468	C₂²	1	1	yes	characteristic, complemented
16	#11924458	—	2	1	no	—
17	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
18	M₃₂×C₃₇	C₂²	1	2	yes	complemented
19	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₃₇	—	4	1	no	—
21	C₂×C₂²⋊D₃₇	—	4	1	no	—
22	#10596962	—	2	2	no	—
23	#10596962	—	2	2	no	—
24	#10596962	—	4	1	no	—
25	#10596962	—	4	1	no	—
26	C₂×D₁₄₈	—	2	1	no	—
27	C₂×D₁₄₈	—	2	1	no	—
28	C₂×C₄×D₃₇	—	4	1	no	—
29	C₂×Dic₇₄	—	2	1	no	—
30	M₁₆×C₃₇	—	2	2	no	—
31	C₂×C₈×C₃₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
33	#10596938	—	4	1	no	—
34	C₄⋊Dic₃₇	—	2	1	no	—
35	Dic₁₄₈	—	2	1	no	—
36	Dic₁₄₈	D₄	1	2	yes	complemented
37	D₂₉₆	—	2	1	no	—
38	D₂₉₆	—	2	2	no	—
39	D₂₉₆	D₄	1	2	yes	complemented
40	#10596930	—	2	2	no	—
41	#10596930	—	2	2	no	—
42	C₁₆×C₃₇	—	2	2	no	—
43	C₂³×C₃₇	D₈	1	1	yes	characteristic
44	C₂²×D₃₇	—	4	1	no	—
45	C₂²×D₃₇	—	4	1	no	—
46	C₂×C₄×C₃₇	—	2	1	no	—
47	C₂×C₄×C₃₇	—	2	1	no	—
48	C₂×C₄×C₃₇	D₈	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂²⋊D₃₇	—	4	2	no	—
51	C₂²⋊D₃₇	—	4	2	no	—
52	C₂²⋊D₃₇	—	8	2	no	—
53	C₂×Dic₃₇	—	4	1	no	—
54	C₂×Dic₃₇	—	4	1	no	—
55	D₁₄₈	—	2	2	no	—
56	D₁₄₈	—	4	1	no	—
57	D₁₄₈	—	4	1	no	—
58	D₁₄₈	—	4	1	no	—
59	C₄×D₃₇	—	4	2	no	—
60	C₄×D₃₇	—	4	2	no	—
61	Dic₇₄	—	2	2	no	—
62	Dic₇₄	—	4	1	no	—
63	C₈×C₃₇	—	2	1	no	—
64	C₈×C₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₃₇	—	2	1	no	—
66	C₂²×C₃₇	—	4	1	no	—
67	C₂²×C₃₇	C₂×D₈	1	1	yes	characteristic
68	C₂×D₃₇	—	4	2	no	—
69	C₂×D₃₇	—	8	2	no	—
70	C₂×D₃₇	—	8	1	no	—
71	C₂×D₃₇	—	8	1	no	—
72	C₂×D₃₇	—	8	1	no	—
73	C₄×C₃₇	—	2	1	no	—
74	C₄×C₃₇	#120	1	1	yes	characteristic
75	C₄×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₃₇	—	4	2	no	—
77	Dic₃₇	—	8	1	no	—
78	#2190	—	37	1	no	maximal, Sylow
79	C₂×C₃₇	—	2	1	no	—
80	C₂×C₃₇	—	4	1	no	—
81	C₂×C₃₇	#447	1	1	yes	characteristic
82	D₃₇	—	8	2	no	—
83	D₃₇	—	16	1	no	—
84	#575	—	37	1	no	—
85	#509	—	37	2	no	—
86	#469	—	37	1	no	—
87	#360	—	37	2	no	—
88	#349	D₃₇	1	1	yes	characteristic, complemented
89	C₃₇	#2190	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	74	1	no	—
91	#137	—	74	2	no	—
92	AGL(1,ℤ/8)	—	74	2	no	—
93	#135	—	37	2	no	—
94	#135	—	74	1	no	—
95	C₂×D₈	—	37	1	no	—
96	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
97	#121	—	74	1	no	—
98	SD₃₂	—	74	2	no	—
99	D₁₆	—	74	2	no	—
100	M₃₂	C₂×D₃₇	1	2	yes	complemented
101	#106	—	37	1	no	—
102	#102	—	74	1	no	—
103	#41	—	74	2	no	—
104	#41	—	74	2	no	—
105	#41	—	148	1	no	—
106	#41	—	148	1	no	—
107	C₂×Q₈	—	74	1	no	—
108	C₂×D₄	—	74	1	no	—
109	C₂×D₄	—	74	1	no	—
110	C₂×D₄	—	148	1	no	—
111	C₂×D₄	—	148	1	no	—
112	C₂²×C₄	—	148	1	no	—
113	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
114	Q₁₆	—	37	2	no	—
115	Q₁₆	—	74	1	no	—
116	SD₁₆	—	74	2	no	—
117	SD₁₆	—	74	2	no	—
118	D₈	—	37	2	no	—
119	D₈	—	74	1	no	—
120	D₈	—	74	2	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	D₁₄₈	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
124	C₄⋊C₄	—	74	1	no	—
125	C₂²⋊C₄	—	148	1	no	—
126	C₁₆	—	2	2	no	—
127	C₂³	—	148	1	no	—
128	C₂³	—	148	1	no	—
129	C₂³	D₂₉₆	1	1	yes	characteristic
130	Q₈	—	74	2	no	—
131	Q₈	—	148	1	no	—
132	D₄	—	296	2	no	—
133	D₄	—	74	2	no	—
134	D₄	—	148	1	no	—
135	D₄	—	148	2	no	—
136	D₄	—	148	2	no	—
137	D₄	—	148	1	no	—
138	D₄	—	148	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	148	2	no	—
142	C₂×C₄	—	148	1	no	—
143	C₂×C₄	—	148	1	no	—
144	C₂×C₄	—	148	2	no	—
145	C₂×C₄	D₂₉₆	1	1	yes	characteristic
146	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	D₄×D₃₇	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	296	1	no	—
152	C₂²	—	296	1	no	—
153	C₂²	—	296	2	no	—
154	C₂²	—	296	1	no	—
155	C₂²	—	148	2	no	—
156	C₂²	C₂×D₂₉₆	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	296	1	no	—
159	C₄	—	148	2	no	—
160	C₄	#11924506	1	1	yes	characteristic
161	C₄	#11924564	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	296	2	no	—
165	C₂	—	592	1	no	—
166	C₂	#14557669	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14557669
commutator, cosocle	C₂×C₈×C₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₃₇
Fitting	C₃₇×#349	C₂
socle	C₂×C₃₇	#447
hypercenter	#349	D₃₇
hypocenter	C₃₇	#2190
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
1. #14558333	C₂	semidirect
2. #14557971	C₂	semidirect
3. #14557786	C₂	semidirect
4. #14557344	C₂	semidirect
5. C₃₇×#349	C₂	semidirect
11. C₂×D₂₉₆	C₂²	semidirect
15. #11924468	C₂²	semidirect
18. M₃₂×C₃₇	C₂²	semidirect
36. Dic₁₄₈	D₄	semidirect
39. D₂₉₆	D₄	semidirect
88. #349	D₃₇	semidirect
89. C₃₇	#2190	semidirect
100. M₃₂	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#14558333	2
#14557971	2
#14557786	2
#14557344	2
C₃₇×#349	2
#2190	37