Finite Group Databasebeta

Subgroups — #17996551

Order 4736.

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

normal subgroups only

This group has 220 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	#14558338	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557969	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924477	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557345	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557339	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557317	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#472	C₂	1	1	yes	maximal, characteristic, complemented
8	#11924582	—	2	1	no	—
9	#11924564	—	2	1	no	—
10	#11924564	—	2	2	no	—
11	#11924558	—	2	1	no	—
12	#11924558	C₂²	1	1	yes	characteristic
13	C₂×D₂₉₆	—	2	1	no	—
14	C₂×D₂₉₆	C₂²	1	1	yes	characteristic
15	#11924480	—	2	1	no	—
16	#11924477	—	2	2	no	—
17	#11924477	—	2	1	no	—
18	#11924476	C₂²	1	1	yes	characteristic, complemented
19	C₂×#10596925	C₂²	1	1	yes	characteristic, complemented
20	AGL(1,ℤ/8)×C₃₇	—	2	1	no	—
21	C₂×D₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
22	#131×C₃₇	C₂²	1	1	yes	characteristic
23	#108×C₃₇	C₂²	1	1	yes	characteristic, complemented
24	#100×C₃₇	—	2	1	no	—
25	#41×C₃₇	D₄	1	1	yes	characteristic
26	C₂×D₄×C₃₇	—	2	1	no	—
27	#10596966	—	4	1	no	—
28	D₄×D₃₇	—	4	2	no	—
29	D₄×D₃₇	—	4	1	no	—
30	#10596962	—	2	2	no	—
31	#10596962	—	2	1	no	—
32	C₂×D₁₄₈	—	2	2	no	—
33	C₂×D₁₄₈	—	2	1	no	—
34	SD₁₆×C₃₇	—	4	1	no	—
35	D₈×C₃₇	—	2	1	no	—
36	D₈×C₃₇	—	2	1	no	—
37	D₈×C₃₇	—	4	1	no	—
38	M₁₆×C₃₇	—	2	1	no	—
39	M₁₆×C₃₇	—	2	1	no	—
40	M₁₆×C₃₇	D₄	1	1	yes	characteristic
41	C₂×C₈×C₃₇	—	2	1	no	—
42	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
43	Dic₁₄₈	D₄	1	1	yes	characteristic
44	D₂₉₆	—	2	2	no	—
45	D₂₉₆	—	2	1	no	—
46	D₂₉₆	D₄	1	1	yes	characteristic
47	D₂₉₆	D₄	1	2	yes	—
48	#10596930	—	2	2	no	—
49	#10596930	—	2	1	no	—
50	#10596925	—	2	1	no	—
51	#10596925	—	2	1	no	—
52	C₂³×C₃₇	—	4	1	no	—
53	C₂²×D₃₇	—	4	2	no	—
54	C₂²×D₃₇	—	4	1	no	—
55	Q₈×C₃₇	D₈	1	1	yes	characteristic
56	D₄×C₃₇	—	2	1	no	—
57	D₄×C₃₇	—	4	1	no	—
58	D₄×C₃₇	—	4	1	no	—
59	D₄×C₃₇	D₈	1	1	yes	characteristic
60	C₂×C₄×C₃₇	—	2	1	no	—
61	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊D₃₇	—	4	2	no	—
63	C₂²⋊D₃₇	—	4	1	no	—
64	D₁₄₈	—	2	2	no	—
65	D₁₄₈	—	2	1	no	—
66	D₁₄₈	—	4	2	no	—
67	D₁₄₈	—	4	1	no	—
68	C₄×D₃₇	—	4	2	no	—
69	C₄×D₃₇	—	4	1	no	—
70	Dic₇₄	—	2	1	no	—
71	C₈×C₃₇	—	2	1	no	—
72	C₈×C₃₇	—	4	1	no	—
73	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
74	C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₃₇	—	2	1	no	—
76	C₂²×C₃₇	—	8	1	no	—
77	C₂²×C₃₇	—	16	1	no	—
78	C₂²×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×D₃₇	—	4	2	no	—
80	C₂×D₃₇	—	4	1	no	—
81	C₂×D₃₇	—	16	1	no	—
82	C₂×D₃₇	—	16	2	no	—
83	C₄×C₃₇	—	2	1	no	—
84	C₄×C₃₇	#120	1	1	yes	characteristic
85	C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
86	Dic₃₇	—	4	1	no	—
87	#2192	—	37	1	no	maximal, Sylow
88	C₂×C₃₇	—	2	1	no	—
89	C₂×C₃₇	—	4	1	no	—
90	C₂×C₃₇	—	16	1	no	—
91	C₂×C₃₇	#447	1	1	yes	characteristic
92	D₃₇	—	8	2	no	—
93	D₃₇	—	8	1	no	—
94	#576	—	37	1	no	—
95	#509	—	37	1	no	—
96	C₂×D₁₆	—	37	1	no	—
97	#472	D₃₇	1	1	yes	characteristic, complemented
98	#361	—	37	1	no	—
99	#359	—	37	1	no	—
100	#350	—	37	1	no	—
101	C₃₇	#2192	1	1	yes	characteristic, complemented, Sylow
102	H(5,𝔽₂)	—	74	1	no	—
103	AGL(1,ℤ/8)	—	2	1	no	—
104	AGL(1,ℤ/8)	—	74	2	no	—
105	AGL(1,ℤ/8)	—	74	1	no	—
106	#135	—	37	1	no	—
107	#135	—	74	1	no	—
108	C₂×D₈	—	37	1	no	—
109	C₂×D₈	—	74	1	no	—
110	C₂×D₈	C₂×D₃₇	1	1	yes	characteristic, complemented
111	#131	C₂×D₃₇	1	1	yes	characteristic
112	SD₃₂	—	74	1	no	—
113	D₁₆	—	74	2	no	—
114	D₁₆	—	74	1	no	—
115	M₃₂	—	37	1	no	—
116	C₂×C₁₆	—	37	1	no	—
117	#108	C₂×D₃₇	1	1	yes	characteristic, complemented
118	#100	—	2	1	no	—
119	#41	—	74	2	no	—
120	#41	—	74	1	no	—
121	#41	—	148	1	no	—
122	#41	C₂²⋊D₃₇	1	1	yes	characteristic
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	74	2	no	—
125	C₂×D₄	—	74	1	no	—
126	C₂×D₄	—	148	2	no	—
127	C₂×D₄	—	148	1	no	—
128	Q₁₆	—	37	1	no	—
129	SD₁₆	—	4	1	no	—
130	SD₁₆	—	74	2	no	—
131	SD₁₆	—	74	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	2	1	no	—
134	D₈	—	4	1	no	—
135	D₈	—	37	2	no	—
136	D₈	—	37	1	no	—
137	D₈	—	74	2	no	—
138	D₈	—	74	1	no	—
139	M₁₆	—	2	1	no	—
140	M₁₆	—	2	1	no	—
141	M₁₆	C₂²⋊D₃₇	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
144	C₁₆	—	74	1	no	—
145	C₁₆	—	74	1	no	—
146	C₂³	—	4	1	no	—
147	C₂³	—	148	2	no	—
148	C₂³	—	148	1	no	—
149	Q₈	—	74	1	no	—
150	Q₈	#10596939	1	1	yes	characteristic
151	D₄	—	2	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	74	2	no	—
155	D₄	—	74	1	no	—
156	D₄	—	148	2	no	—
157	D₄	—	148	2	no	—
158	D₄	—	148	1	no	—
159	D₄	—	148	1	no	—
160	D₄	#10596939	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	148	2	no	—
163	C₂×C₄	—	148	1	no	—
164	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	4	1	no	—
167	C₈	D₄×D₃₇	1	1	yes	characteristic
168	C₈	D₄×D₃₇	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	8	1	no	—
171	C₂²	—	16	1	no	—
172	C₂²	—	592	1	no	—
173	C₂²	—	592	2	no	—
174	C₂²	—	148	2	no	—
175	C₂²	—	148	1	no	—
176	C₂²	#11924561	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	148	1	no	—
179	C₄	#11924504	1	1	yes	characteristic
180	C₄	C₂×#10596939	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	16	1	no	—
184	C₂	—	296	2	no	—
185	C₂	—	296	1	no	—
186	C₂	#14557666	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14557666
commutator, cosocle	C₂×C₈×C₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₃₇
Fitting	C₃₇×#472	C₂
socle	C₂×C₃₇	#447
hypercenter	#472	D₃₇
hypocenter	C₃₇	#2192
LCS	G ▷ C₂×C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #472

Complements

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

N	H ≅ G/N	type
1. #14558338	C₂	semidirect
2. #14557969	C₂	semidirect
3. C₂×#11924477	C₂	semidirect
4. #14557345	C₂	semidirect
5. #14557339	C₂	semidirect
6. #14557317	C₂	semidirect
7. C₃₇×#472	C₂	semidirect
18. #11924476	C₂²	semidirect
19. C₂×#10596925	C₂²	semidirect
21. C₂×D₈×C₃₇	C₂²	semidirect
23. #108×C₃₇	C₂²	semidirect
97. #472	D₃₇	semidirect
101. C₃₇	#2192	semidirect
110. C₂×D₈	C₂×D₃₇	semidirect
117. #108	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#14558338	2
#14557969	2
C₂×#11924477	2
#14557345	2
#14557339	2
#14557317	2
C₃₇×#472	2
#2192	37