Finite Group Databasebeta

Subgroups — #17996562

Order 4736.

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

normal subgroups only

This group has 210 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	#14558350	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924481	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557803	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557350	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	#11924583	—	2	1	no	—
9	#11924569	—	2	1	no	—
10	#11924569	—	2	2	no	—
11	#11924561	—	2	1	no	—
12	#11924558	—	2	1	no	—
13	#11924558	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₄₈	—	2	1	no	—
15	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
16	C₂×D₂₉₆	C₂²	1	1	yes	characteristic
17	#11924481	—	2	2	no	—
18	#11924481	—	2	1	no	—
19	D₅₉₂	—	2	1	no	—
20	#11924472	C₂²	1	1	yes	characteristic
21	#11924452	—	2	1	no	—
22	#131×C₃₇	C₂²	1	1	yes	characteristic
23	M₃₂×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
25	#41×C₃₇	D₄	1	1	yes	characteristic
26	Q₈×D₃₇	—	4	1	no	—
27	#10596964	—	4	2	no	—
28	#10596964	—	4	1	no	—
29	#10596962	—	2	2	no	—
30	#10596962	—	2	1	no	—
31	C₂×D₁₄₈	—	2	1	no	—
32	C₂×Dic₇₄	—	2	2	no	—
33	C₂×Dic₇₄	—	2	1	no	—
34	M₁₆×C₃₇	—	2	1	no	—
35	M₁₆×C₃₇	D₄	1	1	yes	characteristic
36	C₂×C₈×C₃₇	—	2	1	no	—
37	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	#10596941	—	4	1	no	—
39	#10596939	—	4	1	no	—
40	#10596934	—	2	1	no	—
41	Dic₁₄₈	—	2	2	no	—
42	Dic₁₄₈	—	2	1	no	—
43	Dic₁₄₈	D₄	1	1	yes	characteristic
44	Dic₁₄₈	D₄	1	2	yes	—
45	D₂₉₆	—	2	1	no	—
46	D₂₉₆	—	2	1	no	—
47	D₂₉₆	D₄	1	1	yes	characteristic
48	#10596930	—	2	2	no	—
49	#10596930	—	2	1	no	—
50	C₁₆×C₃₇	—	2	1	no	—
51	C₁₆×C₃₇	—	2	1	no	—
52	C₂²×D₃₇	—	4	1	no	—
53	Q₈×C₃₇	D₈	1	1	yes	characteristic
54	D₄×C₃₇	—	2	1	no	—
55	D₄×C₃₇	D₈	1	1	yes	characteristic
56	C₂×C₄×C₃₇	—	2	1	no	—
57	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂²⋊D₃₇	—	4	2	no	—
59	C₂²⋊D₃₇	—	4	1	no	—
60	C₂×Dic₃₇	—	4	2	no	—
61	C₂×Dic₃₇	—	4	1	no	—
62	D₁₄₈	—	2	1	no	—
63	D₁₄₈	—	4	1	no	—
64	D₁₄₈	—	4	1	no	—
65	C₄×D₃₇	—	4	2	no	—
66	C₄×D₃₇	—	4	1	no	—
67	Dic₇₄	—	2	2	no	—
68	Dic₇₄	—	2	1	no	—
69	Dic₇₄	—	4	2	no	—
70	Dic₇₄	—	4	1	no	—
71	C₈×C₃₇	—	2	1	no	—
72	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
74	#64436	—	4	1	no	—
75	C₂²×C₃₇	—	2	1	no	—
76	C₂²×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₃₇	—	4	1	no	—
78	C₂×D₃₇	—	8	1	no	—
79	C₂×D₃₇	—	16	1	no	—
80	C₄×C₃₇	—	2	1	no	—
81	C₄×C₃₇	#120	1	1	yes	characteristic
82	C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
83	Dic₃₇	—	4	1	no	—
84	Dic₃₇	—	4	2	no	—
85	#2193	—	37	1	no	maximal, Sylow
86	C₂×C₃₇	—	2	1	no	—
87	C₂×C₃₇	—	4	1	no	—
88	C₂×C₃₇	#447	1	1	yes	characteristic
89	D₃₇	—	8	1	no	—
90	D₃₇	—	16	1	no	—
91	#578	—	37	1	no	—
92	#509	—	37	1	no	—
93	C₂×SD₃₂	—	37	1	no	—
94	#472	—	37	1	no	—
95	#362	—	37	1	no	—
96	#359	—	37	1	no	—
97	#350	D₃₇	1	1	yes	characteristic, complemented
98	C₃₇	#2193	1	1	yes	characteristic, complemented, Sylow
99	2₋¹⁺⁴	—	74	1	no	—
100	#137	—	74	2	no	—
101	#137	—	74	1	no	—
102	AGL(1,ℤ/8)	—	74	1	no	—
103	#135	—	37	1	no	—
104	#135	—	74	1	no	—
105	C₂×Q₁₆	—	37	1	no	—
106	C₂×Q₁₆	—	74	1	no	—
107	C₂×D₈	—	37	1	no	—
108	#131	C₂×D₃₇	1	1	yes	characteristic
109	SD₃₂	—	74	2	no	—
110	SD₃₂	—	74	1	no	—
111	D₁₆	—	74	1	no	—
112	M₃₂	C₂×D₃₇	1	1	yes	characteristic
113	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
114	#108	—	37	1	no	—
115	#100	—	74	1	no	—
116	#41	—	74	2	no	—
117	#41	—	74	1	no	—
118	#41	—	148	2	no	—
119	#41	—	148	1	no	—
120	#41	D₁₄₈	1	1	yes	characteristic
121	C₂×Q₈	—	74	2	no	—
122	C₂×Q₈	—	74	1	no	—
123	C₂×Q₈	—	148	1	no	—
124	C₂×D₄	—	74	1	no	—
125	Q₁₆	—	37	2	no	—
126	Q₁₆	—	37	1	no	—
127	Q₁₆	—	74	2	no	—
128	Q₁₆	—	74	1	no	—
129	SD₁₆	—	74	2	no	—
130	SD₁₆	—	74	1	no	—
131	SD₁₆	—	148	1	no	—
132	D₈	—	37	1	no	—
133	D₈	—	74	1	no	—
134	D₈	—	74	1	no	—
135	D₈	—	148	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	74	1	no	—
138	M₁₆	D₁₄₈	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	2	1	no	—
143	C₂³	—	148	1	no	—
144	Q₈	—	74	2	no	—
145	Q₈	—	74	1	no	—
146	Q₈	—	148	2	no	—
147	Q₈	—	148	1	no	—
148	Q₈	D₂₉₆	1	1	yes	characteristic
149	D₄	—	2	1	no	—
150	D₄	—	74	1	no	—
151	D₄	—	148	2	no	—
152	D₄	—	148	1	no	—
153	D₄	—	148	1	no	—
154	D₄	—	148	1	no	—
155	D₄	D₂₉₆	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	148	2	no	—
158	C₂×C₄	—	148	2	no	—
159	C₂×C₄	—	148	1	no	—
160	C₂×C₄	—	148	1	no	—
161	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	148	1	no	—
164	C₈	D₄×D₃₇	1	1	yes	characteristic
165	C₈	D₄×D₃₇	1	1	yes	characteristic
166	C₂²	—	2	1	no	—
167	C₂²	—	296	1	no	—
168	C₂²	—	592	1	no	—
169	C₂²	—	148	1	no	—
170	C₂²	#11924564	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	148	2	no	—
173	C₄	—	148	1	no	—
174	C₄	#11924506	1	1	yes	characteristic
175	C₄	C₂×D₂₉₆	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	4	1	no	—
178	C₂	—	296	1	no	—
179	C₂	—	592	1	no	—
180	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×#350	C₂
socle	C₂×C₃₇	#447
hypercenter	#350	D₃₇
hypocenter	C₃₇	#2193
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #350

Complements

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

N	H ≅ G/N	type
1. #14558350	C₂	semidirect
2. #14557971	C₂	semidirect
3. C₂×#11924481	C₂	semidirect
4. #14557803	C₂	semidirect
5. #14557350	C₂	semidirect
6. #14557341	C₂	semidirect
7. C₃₇×#350	C₂	semidirect
97. #350	D₃₇	semidirect
98. C₃₇	#2193	semidirect

Maximal subgroups

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

subgroup	index
#14558350	2
#14557971	2
C₂×#11924481	2
#14557803	2
#14557350	2
#14557341	2
C₃₇×#350	2
#2193	37