Finite Group Databasebeta

Subgroups — #18006389

Order 4736.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14558339	C₂	1	4	yes	maximal, complemented
2	#14557793	C₂	1	4	yes	maximal, complemented
3	#14557715	C₂	1	2	yes	maximal, complemented
4	#14557700	C₂	1	2	yes	maximal, complemented
5	#14557651	C₂	1	2	yes	maximal, complemented
6	C₃₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
7	#11924584	C₂²	1	2	yes	—
8	#11924581	C₂²	1	2	yes	complemented
9	#11924567	C₂²	1	4	yes	complemented
10	#11924565	—	2	4	no	—
11	#11924563	—	2	4	no	—
12	#11924562	C₂²	1	4	yes	—
13	#11924558	C₂²	1	2	yes	—
14	#11924557	—	2	4	no	—
15	SD₁₆×D₃₇	—	2	4	no	—
16	#11924542	C₂²	1	4	yes	complemented
17	#11924472	C₂²	1	2	yes	complemented
18	#11924464	C₂²	1	4	yes	complemented
19	#11924455	C₂²	1	2	yes	complemented
20	#11924453	C₂²	1	2	yes	complemented
21	C₂×SD₁₆×C₃₇	C₂²	1	4	yes	complemented
22	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
23	#127×C₃₇	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
25	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
26	C₂×D₄×C₃₇	—	4	1	no	—
27	C₂×D₄×C₃₇	C₂³	1	2	yes	—
28	C₂×C₂²⋊D₃₇	—	4	2	no	—
29	#10596966	—	2	4	no	—
30	Q₈×D₃₇	—	2	4	no	—
31	#10596964	—	2	4	no	—
32	D₄×D₃₇	—	2	4	no	—
33	#10596962	—	4	2	no	—
34	#10596962	C₂³	1	4	yes	—
35	SD₁₆×C₃₇	—	2	4	no	—
36	SD₁₆×C₃₇	—	2	4	no	—
37	C₂×C₈×C₃₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₃₇	—	4	1	no	—
39	C₄²×C₃₇	C₂³	1	1	yes	characteristic
40	#10596942	—	2	4	no	—
41	#10596941	—	2	4	no	—
42	#10596940	—	2	4	no	—
43	#10596939	—	2	4	no	—
44	#10596934	C₂³	1	4	yes	—
45	Dic₁₄₈	D₄	1	2	yes	complemented
46	D₂₉₆	D₄	1	2	yes	complemented
47	#10596930	D₄	1	4	yes	complemented
48	#10596929	—	2	4	no	—
49	C₈×D₃₇	—	2	4	no	—
50	C₂³×C₃₇	—	2	2	no	—
51	C₂²×D₃₇	—	4	2	no	—
52	Q₈×C₃₇	—	2	4	no	—
53	D₄×C₃₇	—	2	4	no	—
54	D₄×C₃₇	—	8	2	no	—
55	C₂×C₄×C₃₇	—	2	1	no	—
56	C₂×C₄×C₃₇	—	2	2	no	—
57	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
58	C₂²⋊D₃₇	—	2	4	no	—
59	C₂²⋊D₃₇	—	4	4	no	—
60	C₂×Dic₃₇	—	4	2	no	—
61	D₁₄₈	—	4	2	no	—
62	D₁₄₈	C₂×D₄	1	4	yes	—
63	C₄×D₃₇	—	2	4	no	—
64	C₄×D₃₇	—	4	4	no	—
65	Dic₇₄	—	4	2	no	—
66	Dic₇₄	C₂×D₄	1	4	yes	—
67	C₈×C₃₇	C₂×D₄	1	2	yes	—
68	C₈×C₃₇	C₂×D₄	1	2	yes	—
69	#64436	—	2	4	no	—
70	C₂²×C₃₇	—	4	2	no	—
71	C₂²×C₃₇	—	8	2	no	—
72	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×D₃₇	—	2	4	no	—
74	C₂×D₃₇	—	16	2	no	—
75	C₄×C₃₇	—	2	2	no	—
76	C₄×C₃₇	—	4	2	no	—
77	C₄×C₃₇	C₂²×D₄	1	2	yes	—
78	Dic₃₇	—	2	4	no	—
79	#3291	—	37	1	no	maximal, Sylow
80	C₂×C₃₇	—	2	1	no	—
81	C₂×C₃₇	—	8	2	no	—
82	C₂×C₃₇	D₄²	1	1	yes	characteristic
83	D₃₇	—	4	4	no	—
84	#577	—	37	4	no	—
85	#492	D₃₇	1	1	yes	characteristic, complemented
86	#471	—	37	4	no	—
87	#456	—	37	2	no	—
88	#453	—	37	2	no	—
89	#443	—	37	2	no	—
90	C₃₇	#3291	1	1	yes	characteristic, complemented, Sylow
91	2₋¹⁺⁴	—	37	2	no	—
92	H(5,𝔽₂)	—	37	2	no	—
93	#137	—	37	4	no	—
94	#137	—	74	4	no	—
95	AGL(1,ℤ/8)	—	37	4	no	—
96	AGL(1,ℤ/8)	—	74	4	no	—
97	#135	—	37	2	no	—
98	#135	—	74	4	no	—
99	C₂×SD₁₆	—	74	4	no	—
100	C₂×SD₁₆	C₂×D₃₇	1	4	yes	complemented
101	#131	—	37	4	no	—
102	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic, complemented
103	#127	C₂×D₃₇	1	1	yes	characteristic
104	#108	—	37	2	no	—
105	#104	—	37	4	no	—
106	#101	—	37	2	no	—
107	#100	—	37	2	no	—
108	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
109	#41	—	37	4	no	—
110	#41	—	74	4	no	—
111	#41	—	74	4	no	—
112	#41	—	148	2	no	—
113	C₂×Q₈	—	74	4	no	—
114	C₂×Q₈	C₂²×D₃₇	1	2	yes	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	74	4	no	—
117	C₂×D₄	—	148	2	no	—
118	C₂×D₄	C₂²×D₃₇	1	2	yes	—
119	Q₁₆	—	37	2	no	—
120	Q₁₆	—	74	4	no	—
121	SD₁₆	—	2	4	no	—
122	SD₁₆	—	2	4	no	—
123	SD₁₆	—	37	4	no	—
124	SD₁₆	—	74	4	no	—
125	SD₁₆	—	74	4	no	—
126	D₈	—	37	2	no	—
127	D₈	—	74	4	no	—
128	M₁₆	—	37	4	no	—
129	M₁₆	—	74	4	no	—
130	C₂×C₈	—	74	4	no	—
131	C₂×C₈	C₂²×D₃₇	1	2	yes	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄²	C₂²×D₃₇	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	—	148	2	no	—
136	Q₈	—	2	4	no	—
137	Q₈	—	37	4	no	—
138	Q₈	—	148	2	no	—
139	D₄	—	2	4	no	—
140	D₄	—	8	2	no	—
141	D₄	—	37	4	no	—
142	D₄	—	74	4	no	—
143	D₄	—	148	2	no	—
144	D₄	—	148	4	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	74	4	no	—
148	C₂×C₄	—	148	4	no	—
149	C₂×C₄	—	148	2	no	—
150	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
151	C₈	—	74	4	no	—
152	C₈	D₄×D₃₇	1	2	yes	—
153	C₈	D₄×D₃₇	1	2	yes	—
154	C₂²	—	4	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	74	4	no	—
157	C₂²	—	592	2	no	—
158	C₂²	#11924581	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	74	4	no	—
162	C₄	C₂×D₄×D₃₇	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	8	2	no	—
165	C₂	—	148	4	no	—
166	C₂	#14558168	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14558168
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#492	C₂
socle	C₂×C₃₇	D₄²
hypercenter	#492	D₃₇
hypocenter	C₃₇	#3291
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #14558339	C₂	semidirect
2. #14557793	C₂	semidirect
3. #14557715	C₂	semidirect
4. #14557700	C₂	semidirect
5. #14557651	C₂	semidirect
6. C₃₇×#492	C₂	semidirect
8. #11924581	C₂²	semidirect
9. #11924567	C₂²	semidirect
16. #11924542	C₂²	semidirect
17. #11924472	C₂²	semidirect
18. #11924464	C₂²	semidirect
19. #11924455	C₂²	semidirect
20. #11924453	C₂²	semidirect
21. C₂×SD₁₆×C₃₇	C₂²	semidirect
22. C₄⋊Q₈×C₃₇	C₂²	semidirect
24. C₄×C₈×C₃₇	C₂²	semidirect
45. Dic₁₄₈	D₄	semidirect
46. D₂₉₆	D₄	semidirect
47. #10596930	D₄	semidirect
85. #492	D₃₇	semidirect
90. C₃₇	#3291	semidirect
100. C₂×SD₁₆	C₂×D₃₇	semidirect
102. C₄⋊Q₈	C₂×D₃₇	semidirect
108. C₄×C₈	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#14558339	2
#14557793	2
#14557715	2
#14557700	2
#14557651	2
C₃₇×#492	2
#3291	37