Finite Group Databasebeta

Subgroups — #37559591

Order 23872.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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	#26045838	C₂	1	1	yes	maximal, characteristic, complemented
2	#26045820	C₂	1	1	yes	maximal, characteristic, complemented
3	#26045817	C₂	1	1	yes	maximal, characteristic, complemented
4	#26045789	C₂	1	1	yes	maximal, characteristic, complemented
5	#26045720	C₂	1	1	yes	maximal, characteristic, complemented
6	#26045708	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#20238611	—	2	1	no	—
9	#20238610	—	2	1	no	—
10	#20238610	C₂²	1	1	yes	characteristic
11	C₂×D₁₄₉₂	—	2	1	no	—
12	C₂×D₁₄₉₂	—	2	2	no	—
13	C₂×D₁₄₉₂	C₂²	1	1	yes	characteristic
14	C₂×D₁₄₉₂	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₃₇₃	—	2	2	no	—
16	D₄×D₃₇₃	—	2	1	no	—
17	#20238599	—	2	1	no	—
18	#20238597	—	2	1	no	—
19	D₂₉₈₄	—	2	1	no	—
20	#20238595	—	2	1	no	—
21	#20238594	C₂²	1	1	yes	characteristic, complemented
22	#41×C₃₇₃	C₂²	1	1	yes	characteristic
23	M₁₆×C₃₇₃	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₃₇₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₃₇₃	—	2	2	no	—
26	C₂²×D₃₇₃	—	2	1	no	—
27	C₂²×D₃₇₃	D₄	1	2	yes	—
28	Dic₇₄₆	D₄	1	1	yes	characteristic, complemented
29	D₁₄₉₂	—	2	2	no	—
30	D₁₄₉₂	—	2	1	no	—
31	D₁₄₉₂	—	2	1	no	—
32	D₁₄₉₂	—	2	1	no	—
33	D₁₄₉₂	—	2	1	no	—
34	D₁₄₉₂	—	4	4	no	—
35	D₁₄₉₂	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₃₇₃	—	2	2	no	—
37	C₂²⋊D₃₇₃	—	2	1	no	—
38	C₄×D₃₇₃	—	2	2	no	—
39	C₄×D₃₇₃	—	2	1	no	—
40	#15688412	—	2	1	no	—
41	Q₈×C₃₇₃	D₄	1	1	yes	characteristic, complemented
42	D₄×C₃₇₃	—	2	1	no	—
43	D₄×C₃₇₃	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₃₇₃	—	2	1	no	—
45	C₂×C₄×C₃₇₃	—	2	1	no	—
46	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
47	C₈×C₃₇₃	—	2	1	no	—
48	C₂²×C₃₇₃	—	2	1	no	—
49	C₂²×C₃₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×D₃₇₃	—	2	2	no	—
51	C₂×D₃₇₃	—	2	1	no	—
52	C₂×D₃₇₃	—	4	2	no	—
53	C₂×D₃₇₃	—	4	2	no	—
54	C₂×D₃₇₃	—	4	1	no	—
55	C₂×D₃₇₃	—	8	2	no	—
56	C₄×C₃₇₃	—	2	2	no	—
57	C₄×C₃₇₃	—	2	1	no	—
58	C₄×C₃₇₃	C₂×D₄	1	1	yes	characteristic
59	C₄×C₃₇₃	C₂×D₄	1	1	yes	characteristic
60	Dic₃₇₃	—	2	1	no	—
61	C₂×C₃₇₃	—	2	1	no	—
62	C₂×C₃₇₃	—	4	1	no	—
63	C₂×C₃₇₃	#120	1	1	yes	characteristic
64	D₃₇₃	—	4	2	no	—
65	D₃₇₃	—	4	1	no	—
66	D₃₇₃	—	8	1	no	—
67	C₃₇₃	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	373	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	373	1	no	—
70	AGL(1,ℤ/8)	—	373	1	no	—
71	AGL(1,ℤ/8)	—	373	1	no	—
72	#127	—	373	1	no	—
73	#104	—	373	1	no	—
74	#104	D₃₇₃	1	1	yes	characteristic, complemented
75	#100	—	373	1	no	—
76	#41	—	373	1	no	—
77	#41	—	746	1	no	—
78	#41	—	746	1	no	—
79	#41	C₂×D₃₇₃	1	1	yes	characteristic
80	C₂×D₄	—	373	1	no	—
81	C₂×D₄	—	373	1	no	—
82	C₂×D₄	—	746	1	no	—
83	C₂×D₄	—	746	2	no	—
84	C₂×D₄	—	746	2	no	—
85	C₂×D₄	—	746	1	no	—
86	SD₁₆	—	746	1	no	—
87	SD₁₆	—	746	1	no	—
88	D₈	—	746	1	no	—
89	D₈	—	746	1	no	—
90	M₁₆	—	373	1	no	—
91	M₁₆	C₂×D₃₇₃	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₃₇₃	1	1	yes	characteristic, complemented
93	C₂³	—	373	2	no	—
94	C₂³	—	746	2	no	—
95	C₂³	—	746	1	no	—
96	Q₈	—	373	1	no	—
97	Q₈	D₁₄₉₂	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	373	1	no	—
100	D₄	—	1492	4	no	—
101	D₄	—	746	2	no	—
102	D₄	—	746	2	no	—
103	D₄	—	746	1	no	—
104	D₄	—	746	1	no	—
105	D₄	—	746	1	no	—
106	D₄	—	746	1	no	—
107	D₄	—	746	1	no	—
108	D₄	D₁₄₉₂	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	746	2	no	—
112	C₂×C₄	—	746	1	no	—
113	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	746	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2984	2	no	—
118	C₂²	—	1492	2	no	—
119	C₂²	—	1492	2	no	—
120	C₂²	—	1492	1	no	—
121	C₂²	—	746	1	no	—
122	C₂²	—	746	2	no	—
123	C₂²	D₄×D₃₇₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	746	1	no	—
127	C₄	D₄×D₃₇₃	1	1	yes	characteristic
128	C₄	C₂×D₁₄₉₂	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	2984	1	no	—
132	C₂	—	1492	1	no	—
133	C₂	—	1492	2	no	—
134	C₂	#26045762	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26045762
commutator, cosocle	C₂×C₄×C₃₇₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₇₃
Fitting	#104×C₃₇₃	C₂
socle	C₂×C₃₇₃	#120
hypercenter	#104	D₃₇₃
hypocenter	C₃₇₃	#453
LCS	G ▷ C₂×C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #26045838	C₂	semidirect
2. #26045820	C₂	semidirect
3. #26045817	C₂	semidirect
4. #26045789	C₂	semidirect
5. #26045720	C₂	semidirect
6. #26045708	C₂	semidirect
7. #104×C₃₇₃	C₂	semidirect
14. C₂×D₁₄₉₂	C₂²	semidirect
21. #20238594	C₂²	semidirect
23. M₁₆×C₃₇₃	C₂²	semidirect
24. C₄²×C₃₇₃	C₂²	semidirect
28. Dic₇₄₆	D₄	semidirect
35. D₁₄₉₂	D₄	semidirect
41. Q₈×C₃₇₃	D₄	semidirect
43. D₄×C₃₇₃	D₄	semidirect
67. C₃₇₃	#453	semidirect
74. #104	D₃₇₃	semidirect
91. M₁₆	C₂×D₃₇₃	semidirect
92. C₄²	C₂×D₃₇₃	semidirect
97. Q₈	D₁₄₉₂	semidirect
108. D₄	D₁₄₉₂	semidirect

Maximal subgroups

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

subgroup	index
#26045838	2
#26045820	2
#26045817	2
#26045789	2
#26045720	2
#26045708	2
#104×C₃₇₃	2
#453	373