Finite Group Databasebeta

Subgroups — #36377956

Order 22592.

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	#26015038	C₂	1	1	yes	maximal, characteristic, complemented
2	#26015020	C₂	1	1	yes	maximal, characteristic, complemented
3	#26015017	C₂	1	1	yes	maximal, characteristic, complemented
4	#26014989	C₂	1	1	yes	maximal, characteristic, complemented
5	#26014920	C₂	1	1	yes	maximal, characteristic, complemented
6	#26014908	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#19150738	—	2	1	no	—
9	#19150737	—	2	1	no	—
10	#19150737	C₂²	1	1	yes	characteristic
11	C₂×D₁₄₁₂	—	2	1	no	—
12	C₂×D₁₄₁₂	—	2	2	no	—
13	C₂×D₁₄₁₂	C₂²	1	1	yes	characteristic
14	C₂×D₁₄₁₂	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₃₅₃	—	2	2	no	—
16	D₄×D₃₅₃	—	2	1	no	—
17	#19150726	—	2	1	no	—
18	#19150724	—	2	1	no	—
19	D₂₈₂₄	—	2	1	no	—
20	#19150722	—	2	1	no	—
21	#19150721	C₂²	1	1	yes	characteristic, complemented
22	#41×C₃₅₃	C₂²	1	1	yes	characteristic
23	M₁₆×C₃₅₃	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₃₅₃	—	2	2	no	—
26	C₂²×D₃₅₃	—	2	1	no	—
27	C₂²×D₃₅₃	D₄	1	2	yes	—
28	Dic₇₀₆	D₄	1	1	yes	characteristic, complemented
29	D₁₄₁₂	—	2	2	no	—
30	D₁₄₁₂	—	2	1	no	—
31	D₁₄₁₂	—	2	1	no	—
32	D₁₄₁₂	—	2	1	no	—
33	D₁₄₁₂	—	2	1	no	—
34	D₁₄₁₂	—	4	4	no	—
35	D₁₄₁₂	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₃₅₃	—	2	2	no	—
37	C₂²⋊D₃₅₃	—	2	1	no	—
38	C₄×D₃₅₃	—	2	2	no	—
39	C₄×D₃₅₃	—	2	1	no	—
40	#15667725	—	2	1	no	—
41	Q₈×C₃₅₃	D₄	1	1	yes	characteristic, complemented
42	D₄×C₃₅₃	—	2	1	no	—
43	D₄×C₃₅₃	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₃₅₃	—	2	1	no	—
45	C₂×C₄×C₃₅₃	—	2	1	no	—
46	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
47	C₈×C₃₅₃	—	2	1	no	—
48	C₂²×C₃₅₃	—	2	1	no	—
49	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×D₃₅₃	—	2	2	no	—
51	C₂×D₃₅₃	—	2	1	no	—
52	C₂×D₃₅₃	—	4	2	no	—
53	C₂×D₃₅₃	—	4	2	no	—
54	C₂×D₃₅₃	—	4	1	no	—
55	C₂×D₃₅₃	—	8	2	no	—
56	C₄×C₃₅₃	—	2	2	no	—
57	C₄×C₃₅₃	—	2	1	no	—
58	C₄×C₃₅₃	C₂×D₄	1	1	yes	characteristic
59	C₄×C₃₅₃	C₂×D₄	1	1	yes	characteristic
60	Dic₃₅₃	—	2	1	no	—
61	C₂×C₃₅₃	—	2	1	no	—
62	C₂×C₃₅₃	—	4	1	no	—
63	C₂×C₃₅₃	#120	1	1	yes	characteristic
64	D₃₅₃	—	4	2	no	—
65	D₃₅₃	—	4	1	no	—
66	D₃₅₃	—	8	1	no	—
67	C₃₅₃	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	353	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	353	1	no	—
70	AGL(1,ℤ/8)	—	353	1	no	—
71	AGL(1,ℤ/8)	—	353	1	no	—
72	#127	—	353	1	no	—
73	#104	—	353	1	no	—
74	#104	D₃₅₃	1	1	yes	characteristic, complemented
75	#100	—	353	1	no	—
76	#41	—	353	1	no	—
77	#41	—	706	1	no	—
78	#41	—	706	1	no	—
79	#41	C₂×D₃₅₃	1	1	yes	characteristic
80	C₂×D₄	—	353	1	no	—
81	C₂×D₄	—	353	1	no	—
82	C₂×D₄	—	706	1	no	—
83	C₂×D₄	—	706	2	no	—
84	C₂×D₄	—	706	2	no	—
85	C₂×D₄	—	706	1	no	—
86	SD₁₆	—	706	1	no	—
87	SD₁₆	—	706	1	no	—
88	D₈	—	706	1	no	—
89	D₈	—	706	1	no	—
90	M₁₆	—	353	1	no	—
91	M₁₆	C₂×D₃₅₃	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₃₅₃	1	1	yes	characteristic, complemented
93	C₂³	—	353	2	no	—
94	C₂³	—	706	2	no	—
95	C₂³	—	706	1	no	—
96	Q₈	—	353	1	no	—
97	Q₈	D₁₄₁₂	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	353	1	no	—
100	D₄	—	1412	4	no	—
101	D₄	—	706	2	no	—
102	D₄	—	706	2	no	—
103	D₄	—	706	1	no	—
104	D₄	—	706	1	no	—
105	D₄	—	706	1	no	—
106	D₄	—	706	1	no	—
107	D₄	—	706	1	no	—
108	D₄	D₁₄₁₂	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	706	2	no	—
112	C₂×C₄	—	706	1	no	—
113	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	706	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	2824	2	no	—
118	C₂²	—	1412	2	no	—
119	C₂²	—	1412	2	no	—
120	C₂²	—	1412	1	no	—
121	C₂²	—	706	1	no	—
122	C₂²	—	706	2	no	—
123	C₂²	D₄×D₃₅₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	706	1	no	—
127	C₄	D₄×D₃₅₃	1	1	yes	characteristic
128	C₄	C₂×D₁₄₁₂	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	2824	1	no	—
132	C₂	—	1412	1	no	—
133	C₂	—	1412	2	no	—
134	C₂	#26014962	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26014962
commutator, cosocle	C₂×C₄×C₃₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	#104×C₃₅₃	C₂
socle	C₂×C₃₅₃	#120
hypercenter	#104	D₃₅₃
hypocenter	C₃₅₃	#453
LCS	G ▷ C₂×C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #26015038	C₂	semidirect
2. #26015020	C₂	semidirect
3. #26015017	C₂	semidirect
4. #26014989	C₂	semidirect
5. #26014920	C₂	semidirect
6. #26014908	C₂	semidirect
7. #104×C₃₅₃	C₂	semidirect
14. C₂×D₁₄₁₂	C₂²	semidirect
21. #19150721	C₂²	semidirect
23. M₁₆×C₃₅₃	C₂²	semidirect
24. C₄²×C₃₅₃	C₂²	semidirect
28. Dic₇₀₆	D₄	semidirect
35. D₁₄₁₂	D₄	semidirect
41. Q₈×C₃₅₃	D₄	semidirect
43. D₄×C₃₅₃	D₄	semidirect
67. C₃₅₃	#453	semidirect
74. #104	D₃₅₃	semidirect
91. M₁₆	C₂×D₃₅₃	semidirect
92. C₄²	C₂×D₃₅₃	semidirect
97. Q₈	D₁₄₁₂	semidirect
108. D₄	D₁₄₁₂	semidirect

Maximal subgroups

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

subgroup	index
#26015038	2
#26015020	2
#26015017	2
#26014989	2
#26014920	2
#26014908	2
#104×C₃₅₃	2
#453	353