Finite Group Databasebeta

Subgroups — #35203392

Order 20288.

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	#23778826	C₂	1	1	yes	maximal, characteristic, complemented
2	#23778808	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778805	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778777	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778708	C₂	1	1	yes	maximal, characteristic, complemented
6	#23778696	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#19099575	—	2	1	no	—
9	#19099574	—	2	1	no	—
10	#19099574	C₂²	1	1	yes	characteristic
11	C₂×D₁₂₆₈	—	2	1	no	—
12	C₂×D₁₂₆₈	—	2	2	no	—
13	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
14	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₃₁₇	—	2	2	no	—
16	D₄×D₃₁₇	—	2	1	no	—
17	#19099563	—	2	1	no	—
18	#19099561	—	2	1	no	—
19	D₂₅₃₆	—	2	1	no	—
20	#19099559	—	2	1	no	—
21	#19099558	C₂²	1	1	yes	characteristic, complemented
22	#41×C₃₁₇	C₂²	1	1	yes	characteristic
23	M₁₆×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₃₁₇	—	2	2	no	—
26	C₂²×D₃₁₇	—	2	1	no	—
27	C₂²×D₃₁₇	D₄	1	2	yes	—
28	Dic₆₃₄	D₄	1	1	yes	characteristic, complemented
29	D₁₂₆₈	—	2	2	no	—
30	D₁₂₆₈	—	2	1	no	—
31	D₁₂₆₈	—	2	1	no	—
32	D₁₂₆₈	—	2	1	no	—
33	D₁₂₆₈	—	2	1	no	—
34	D₁₂₆₈	—	4	4	no	—
35	D₁₂₆₈	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₃₁₇	—	2	2	no	—
37	C₂²⋊D₃₁₇	—	2	1	no	—
38	C₄×D₃₁₇	—	2	2	no	—
39	C₄×D₃₁₇	—	2	1	no	—
40	#14579261	—	2	1	no	—
41	Q₈×C₃₁₇	D₄	1	1	yes	characteristic, complemented
42	D₄×C₃₁₇	—	2	1	no	—
43	D₄×C₃₁₇	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₃₁₇	—	2	1	no	—
45	C₂×C₄×C₃₁₇	—	2	1	no	—
46	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
47	C₈×C₃₁₇	—	2	1	no	—
48	C₂²×C₃₁₇	—	2	1	no	—
49	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×D₃₁₇	—	2	2	no	—
51	C₂×D₃₁₇	—	2	1	no	—
52	C₂×D₃₁₇	—	4	2	no	—
53	C₂×D₃₁₇	—	4	2	no	—
54	C₂×D₃₁₇	—	4	1	no	—
55	C₂×D₃₁₇	—	8	2	no	—
56	C₄×C₃₁₇	—	2	2	no	—
57	C₄×C₃₁₇	—	2	1	no	—
58	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
59	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
60	Dic₃₁₇	—	2	1	no	—
61	C₂×C₃₁₇	—	2	1	no	—
62	C₂×C₃₁₇	—	4	1	no	—
63	C₂×C₃₁₇	#120	1	1	yes	characteristic
64	D₃₁₇	—	4	2	no	—
65	D₃₁₇	—	4	1	no	—
66	D₃₁₇	—	8	1	no	—
67	C₃₁₇	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	317	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	317	1	no	—
70	AGL(1,ℤ/8)	—	317	1	no	—
71	AGL(1,ℤ/8)	—	317	1	no	—
72	#127	—	317	1	no	—
73	#104	—	317	1	no	—
74	#104	D₃₁₇	1	1	yes	characteristic, complemented
75	#100	—	317	1	no	—
76	#41	—	317	1	no	—
77	#41	—	634	1	no	—
78	#41	—	634	1	no	—
79	#41	C₂×D₃₁₇	1	1	yes	characteristic
80	C₂×D₄	—	317	1	no	—
81	C₂×D₄	—	317	1	no	—
82	C₂×D₄	—	634	1	no	—
83	C₂×D₄	—	634	2	no	—
84	C₂×D₄	—	634	2	no	—
85	C₂×D₄	—	634	1	no	—
86	SD₁₆	—	634	1	no	—
87	SD₁₆	—	634	1	no	—
88	D₈	—	634	1	no	—
89	D₈	—	634	1	no	—
90	M₁₆	—	317	1	no	—
91	M₁₆	C₂×D₃₁₇	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₃₁₇	1	1	yes	characteristic, complemented
93	C₂³	—	317	2	no	—
94	C₂³	—	634	2	no	—
95	C₂³	—	634	1	no	—
96	Q₈	—	317	1	no	—
97	Q₈	D₁₂₆₈	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	317	1	no	—
100	D₄	—	634	2	no	—
101	D₄	—	634	2	no	—
102	D₄	—	634	1	no	—
103	D₄	—	634	1	no	—
104	D₄	—	634	1	no	—
105	D₄	—	634	1	no	—
106	D₄	—	634	1	no	—
107	D₄	—	1268	4	no	—
108	D₄	D₁₂₆₈	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	634	2	no	—
112	C₂×C₄	—	634	1	no	—
113	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	634	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	634	1	no	—
118	C₂²	—	634	2	no	—
119	C₂²	—	2536	2	no	—
120	C₂²	—	1268	2	no	—
121	C₂²	—	1268	2	no	—
122	C₂²	—	1268	1	no	—
123	C₂²	D₄×D₃₁₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	634	1	no	—
127	C₄	D₄×D₃₁₇	1	1	yes	characteristic
128	C₄	C₂×D₁₂₆₈	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	2536	1	no	—
132	C₂	—	1268	1	no	—
133	C₂	—	1268	2	no	—
134	C₂	#23778750	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23778750
commutator, cosocle	C₂×C₄×C₃₁₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	#104×C₃₁₇	C₂
socle	C₂×C₃₁₇	#120
hypercenter	#104	D₃₁₇
hypocenter	C₃₁₇	#453
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #23778826	C₂	semidirect
2. #23778808	C₂	semidirect
3. #23778805	C₂	semidirect
4. #23778777	C₂	semidirect
5. #23778708	C₂	semidirect
6. #23778696	C₂	semidirect
7. #104×C₃₁₇	C₂	semidirect
14. C₂×D₁₂₆₈	C₂²	semidirect
21. #19099558	C₂²	semidirect
23. M₁₆×C₃₁₇	C₂²	semidirect
24. C₄²×C₃₁₇	C₂²	semidirect
28. Dic₆₃₄	D₄	semidirect
35. D₁₂₆₈	D₄	semidirect
41. Q₈×C₃₁₇	D₄	semidirect
43. D₄×C₃₁₇	D₄	semidirect
67. C₃₁₇	#453	semidirect
74. #104	D₃₁₇	semidirect
91. M₁₆	C₂×D₃₁₇	semidirect
92. C₄²	C₂×D₃₁₇	semidirect
97. Q₈	D₁₂₆₈	semidirect
108. D₄	D₁₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
#23778826	2
#23778808	2
#23778805	2
#23778777	2
#23778708	2
#23778696	2
#104×C₃₁₇	2
#453	317