Finite Group Databasebeta

Subgroups — #17994261

Order 4736.

242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14557909	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557909	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557905	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557880	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557879	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557255	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596940	—	2	2	no	—
9	C₂×#10596939	—	2	2	no	—
10	C₂×#10596939	—	2	2	no	—
11	#11924536	C₂²	1	1	yes	characteristic
12	#11924533	C₂²	1	1	yes	characteristic
13	#11924458	—	2	2	no	—
14	#11924457	—	2	2	no	—
15	C₄×#64436	C₂²	1	1	yes	characteristic
16	C₄×#64436	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₃₇	—	2	2	no	—
18	#127×C₃₇	C₂²	1	1	yes	characteristic
19	#127×C₃₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₃₇	—	2	1	no	—
22	C₂×D₄×C₃₇	—	2	1	no	—
23	C₂×D₄×C₃₇	—	2	1	no	—
24	C₂×D₄×C₃₇	—	2	1	no	—
25	C₂×D₄×C₃₇	—	2	1	no	—
26	C₂×D₄×C₃₇	—	2	1	no	—
27	C₂×D₁₄₈	—	2	1	no	—
28	C₂×D₁₄₈	—	2	1	no	—
29	C₂×D₁₄₈	—	2	1	no	—
30	C₂×Dic₇₄	—	2	1	no	—
31	D₈×C₃₇	—	4	4	no	—
32	C₂×C₈×C₃₇	D₄	1	2	yes	—
33	C₄²×C₃₇	C₂³	1	1	yes	characteristic
34	#10596940	—	4	4	no	—
35	#10596939	—	4	4	no	—
36	#10596939	—	4	4	no	—
37	C₄⋊Dic₃₇	—	2	1	no	—
38	C₄⋊Dic₃₇	—	2	1	no	—
39	C₂×#64436	D₄	1	2	yes	—
40	C₂×#64436	D₄	1	2	yes	—
41	C₂³×C₃₇	—	4	1	no	—
42	C₂³×C₃₇	—	4	1	no	—
43	C₂²×D₃₇	—	4	1	no	—
44	D₄×C₃₇	—	4	2	no	—
45	D₄×C₃₇	—	4	2	no	—
46	D₄×C₃₇	—	4	2	no	—
47	D₄×C₃₇	—	4	2	no	—
48	D₄×C₃₇	—	4	2	no	—
49	D₄×C₃₇	—	4	2	no	—
50	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₃₇	—	4	1	no	—
54	D₁₄₈	—	4	2	no	—
55	D₁₄₈	—	4	2	no	—
56	D₁₄₈	—	4	2	no	—
57	Dic₇₄	—	4	2	no	—
58	C₈×C₃₇	—	2	2	no	—
59	#64436	—	2	2	no	—
60	#64436	D₈	1	4	yes	complemented
61	C₂²×C₃₇	—	8	1	no	—
62	C₂²×C₃₇	—	8	1	no	—
63	C₂²×C₃₇	—	8	1	no	—
64	C₂²×C₃₇	—	8	1	no	—
65	C₂²×C₃₇	—	8	1	no	—
66	C₂²×C₃₇	—	8	1	no	—
67	C₂²×C₃₇	#127	1	1	yes	characteristic
68	C₂×D₃₇	—	8	1	no	—
69	C₂×D₃₇	—	8	1	no	—
70	C₂×D₃₇	—	8	1	no	—
71	C₄×C₃₇	C₂×D₈	1	2	yes	—
72	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
73	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
74	Dic₃₇	—	8	1	no	—
75	#1721	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	—	16	1	no	—
77	C₂×C₃₇	—	16	1	no	—
78	C₂×C₃₇	#493	1	1	yes	characteristic
79	C₂×C₃₇	#496	1	1	yes	characteristic
80	C₂×C₃₇	#496	1	1	yes	characteristic
81	D₃₇	—	16	1	no	—
82	#493	—	37	1	no	—
83	#493	—	37	1	no	—
84	#493	D₃₇	1	1	yes	characteristic, complemented
85	#492	—	37	1	no	—
86	#486	—	37	1	no	—
87	#486	—	37	1	no	—
88	#322	—	37	1	no	—
89	C₃₇	#1721	1	1	yes	characteristic, complemented, Sylow
90	C₂×SD₁₆	—	74	2	no	—
91	C₂×D₈	—	2	2	no	—
92	C₂×D₈	—	74	2	no	—
93	C₂×D₈	—	74	2	no	—
94	C₄⋊Q₈	—	37	1	no	—
95	#127	—	37	1	no	—
96	#127	C₂×D₃₇	1	1	yes	characteristic
97	#127	C₂×D₃₇	1	1	yes	characteristic
98	#102	—	74	2	no	—
99	#102	—	74	2	no	—
100	C₄×C₈	—	37	1	no	—
101	C₄×C₈	—	37	1	no	—
102	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
103	C₂×Q₈	—	74	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	74	1	no	—
111	C₂×D₄	—	74	1	no	—
112	C₂×D₄	—	74	1	no	—
113	SD₁₆	—	148	4	no	—
114	D₈	—	4	4	no	—
115	D₈	—	148	4	no	—
116	D₈	—	148	4	no	—
117	C₂×C₈	—	37	2	no	—
118	C₂×C₈	—	37	2	no	—
119	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
120	C₄⋊C₄	—	74	1	no	—
121	C₄⋊C₄	—	74	1	no	—
122	C₄²	C₂²×D₃₇	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	148	1	no	—
126	Q₈	—	148	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	2	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	2	no	—
133	D₄	—	148	2	no	—
134	D₄	—	148	2	no	—
135	D₄	—	148	2	no	—
136	C₂×C₄	—	148	1	no	—
137	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
138	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	37	4	no	—
142	C₈	—	74	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	296	1	no	—
150	C₂²	—	296	1	no	—
151	C₂²	—	296	1	no	—
152	C₂²	#11924532	1	1	yes	characteristic
153	C₄	—	296	1	no	—
154	C₄	D₈×D₃₇	1	2	yes	—
155	C₄	#11924560	1	2	yes	—
156	C₄	#11924562	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	16	1	no	—
159	C₂	—	592	1	no	—
160	C₂	#14557908	1	1	yes	characteristic
161	C₂	#14557920	1	1	yes	characteristic
162	C₂	#14557921	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#493	C₂
socle	C₂²×C₃₇	#127
hypercenter	#493	D₃₇
hypocenter	C₃₇	#1721
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #14557909	C₂	semidirect
2. #14557909	C₂	semidirect
3. #14557905	C₂	semidirect
4. #14557880	C₂	semidirect
5. #14557879	C₂	semidirect
6. #14557255	C₂	semidirect
7. C₃₇×#493	C₂	semidirect
60. #64436	D₈	semidirect
84. #493	D₃₇	semidirect
89. C₃₇	#1721	semidirect

Maximal subgroups

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

subgroup	index
#14557909	2
#14557905	2
#14557880	2
#14557879	2
#14557255	2
C₃₇×#493	2
#1721	37