Finite Group Databasebeta

Subgroups — #17996574

Order 4736.

206 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 53 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	#14557973	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11924482	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557808	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557349	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557340	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557317	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#578	C₂	1	1	yes	maximal, characteristic
8	#11924566	—	2	1	no	—
9	C₂×Dic₁₄₈	C₂²	1	1	yes	characteristic
10	#11924482	—	2	2	no	—
11	#11924482	—	2	1	no	—
12	#11924480	—	2	1	no	—
13	#11924476	C₂²	1	1	yes	characteristic
14	C₂×#10596925	C₂²	1	1	yes	characteristic
15	#11924472	C₂²	1	1	yes	characteristic
16	#11924454	—	2	1	no	—
17	2₋¹⁺⁴×C₃₇	—	2	1	no	—
18	#137×C₃₇	—	2	2	no	—
19	#137×C₃₇	—	2	1	no	—
20	#135×C₃₇	—	2	1	no	—
21	#135×C₃₇	C₂²	1	1	yes	characteristic
22	C₂×Q₁₆×C₃₇	—	2	1	no	—
23	C₂×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
24	#131×C₃₇	C₂²	1	1	yes	characteristic
25	#41×C₃₇	—	2	2	no	—
26	#41×C₃₇	—	2	1	no	—
27	#41×C₃₇	—	4	2	no	—
28	#41×C₃₇	—	4	1	no	—
29	#41×C₃₇	D₄	1	1	yes	characteristic
30	C₂×Q₈×C₃₇	—	2	2	no	—
31	C₂×Q₈×C₃₇	—	2	1	no	—
32	C₂×Q₈×C₃₇	—	4	1	no	—
33	C₂×Dic₇₄	—	2	1	no	—
34	Q₁₆×C₃₇	—	2	2	no	—
35	Q₁₆×C₃₇	—	2	1	no	—
36	Q₁₆×C₃₇	D₄	1	1	yes	characteristic
37	Q₁₆×C₃₇	D₄	1	2	yes	—
38	SD₁₆×C₃₇	—	2	2	no	—
39	SD₁₆×C₃₇	—	2	1	no	—
40	D₈×C₃₇	D₄	1	1	yes	characteristic
41	M₁₆×C₃₇	—	2	1	no	—
42	M₁₆×C₃₇	D₄	1	1	yes	characteristic
43	C₂×C₈×C₃₇	—	2	1	no	—
44	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
45	#10596942	—	4	1	no	—
46	#10596940	—	4	1	no	—
47	#10596934	—	2	1	no	—
48	Dic₁₄₈	—	2	1	no	—
49	Dic₁₄₈	—	2	1	no	—
50	#10596925	—	2	1	no	—
51	#10596925	—	2	1	no	—
52	Q₈×C₃₇	—	2	2	no	—
53	Q₈×C₃₇	—	2	1	no	—
54	Q₈×C₃₇	—	4	2	no	—
55	Q₈×C₃₇	—	4	1	no	—
56	Q₈×C₃₇	D₈	1	1	yes	characteristic
57	D₄×C₃₇	—	2	1	no	—
58	D₄×C₃₇	—	2	1	no	—
59	D₄×C₃₇	—	4	2	no	—
60	D₄×C₃₇	—	4	1	no	—
61	D₄×C₃₇	D₈	1	1	yes	characteristic
62	C₂×C₄×C₃₇	—	2	1	no	—
63	C₂×C₄×C₃₇	—	4	2	no	—
64	C₂×C₄×C₃₇	—	4	2	no	—
65	C₂×C₄×C₃₇	—	4	1	no	—
66	C₂×C₄×C₃₇	—	4	1	no	—
67	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂×Dic₃₇	—	4	1	no	—
69	Dic₇₄	—	4	1	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	—	4	1	no	—
77	C₂²×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₄×C₃₇	—	2	1	no	—
79	C₄×C₃₇	—	4	2	no	—
80	C₄×C₃₇	—	4	1	no	—
81	C₄×C₃₇	#120	1	1	yes	characteristic
82	C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
83	Dic₃₇	—	8	1	no	—
84	#2195	—	37	1	no	maximal, Sylow
85	C₂×C₃₇	—	2	1	no	—
86	C₂×C₃₇	—	4	1	no	—
87	C₂×C₃₇	—	8	1	no	—
88	C₂×C₃₇	#447	1	1	yes	characteristic
89	#578	D₃₇	1	1	yes	characteristic
90	#510	—	37	1	no	—
91	C₂×Q₃₂	—	37	1	no	—
92	#473	—	37	1	no	—
93	#362	—	37	1	no	—
94	#359	—	37	1	no	—
95	#350	—	37	1	no	—
96	C₃₇	#2195	1	1	yes	characteristic, complemented, Sylow
97	2₋¹⁺⁴	—	2	1	no	—
98	#137	—	2	2	no	—
99	#137	—	2	1	no	—
100	#137	—	74	1	no	—
101	#135	—	2	1	no	—
102	#135	C₂×D₃₇	1	1	yes	characteristic
103	C₂×Q₁₆	—	2	1	no	—
104	C₂×Q₁₆	—	37	1	no	—
105	C₂×Q₁₆	C₂×D₃₇	1	1	yes	characteristic
106	#131	C₂×D₃₇	1	1	yes	characteristic
107	Q₃₂	—	74	2	no	—
108	Q₃₂	—	74	1	no	—
109	SD₃₂	—	74	1	no	—
110	M₃₂	—	37	1	no	—
111	C₂×C₁₆	—	37	1	no	—
112	#108	—	37	1	no	—
113	#101	—	74	1	no	—
114	#41	—	2	2	no	—
115	#41	—	2	1	no	—
116	#41	—	4	2	no	—
117	#41	—	4	1	no	—
118	#41	C₂²⋊D₃₇	1	1	yes	characteristic
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	—	4	1	no	—
122	C₂×Q₈	—	74	1	no	—
123	Q₁₆	—	2	2	no	—
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	74	1	no	—
126	Q₁₆	—	74	1	no	—
127	Q₁₆	—	148	1	no	—
128	Q₁₆	C₂²⋊D₃₇	1	1	yes	characteristic
129	Q₁₆	C₂²⋊D₃₇	1	2	yes	—
130	SD₁₆	—	2	2	no	—
131	SD₁₆	—	2	1	no	—
132	SD₁₆	—	148	1	no	—
133	D₈	C₂²⋊D₃₇	1	1	yes	characteristic
134	M₁₆	—	2	1	no	—
135	M₁₆	—	74	1	no	—
136	M₁₆	C₂²⋊D₃₇	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
139	C₁₆	—	74	1	no	—
140	C₁₆	—	74	1	no	—
141	Q₈	—	2	2	no	—
142	Q₈	—	2	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	148	1	no	—
146	Q₈	—	148	1	no	—
147	Q₈	#10596939	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	4	2	no	—
151	D₄	—	4	1	no	—
152	D₄	#10596939	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	148	1	no	—
159	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	148	1	no	—
162	C₈	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
163	C₈	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	4	1	no	—
166	C₂²	#11924562	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	4	1	no	—
170	C₄	—	296	1	no	—
171	C₄	#11924505	1	1	yes	characteristic
172	C₄	C₂×#10596939	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	8	1	no	—
176	C₂	#14557667	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14557667
commutator, cosocle	C₂×C₈×C₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₃₇
Fitting, prime radical	C₃₇×#578	C₂
socle	C₂×C₃₇	#447
hypercenter	#578	D₃₇
hypocenter	C₃₇	#2195
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #578

Complements

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

N	H ≅ G/N	type
1. #14557973	C₂	semidirect
2. C₂×#11924482	C₂	semidirect
3. #14557808	C₂	semidirect
4. #14557349	C₂	semidirect
5. #14557340	C₂	semidirect
6. #14557317	C₂	semidirect
96. C₃₇	#2195	semidirect

Maximal subgroups

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

subgroup	index
#14557973	2
C₂×#11924482	2
#14557808	2
#14557349	2
#14557340	2
#14557317	2
C₃₇×#578	2
#2195	37