Finite Group Databasebeta

Subgroups — #23776548

Order 10112.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#19099380	C₂	1	1	yes	maximal, characteristic, complemented
2	#19099368	C₂	1	1	yes	maximal, characteristic, complemented
3	#19099006	C₂	1	3	yes	maximal, complemented
4	#19099001	C₂	1	3	yes	maximal, complemented
5	#19098995	C₂	1	3	yes	maximal, complemented
6	C₂×#14579106	C₂	1	3	yes	maximal, complemented
7	#504×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14579208	—	2	1	no	—
9	#14579207	—	2	1	no	—
10	#14579194	—	2	3	no	—
11	#14579189	—	2	3	no	—
12	#14579183	C₂²	1	3	yes	—
13	#14579183	C₂²	1	3	yes	complemented
14	C₂×Dic₃₁₆	C₂²	1	3	yes	complemented
15	C₂×D₆₃₂	C₂²	1	3	yes	—
16	Dic₆₃₂	C₂²	1	3	yes	complemented
17	#14579106	C₂²	1	3	yes	complemented
18	#14579106	C₂²	1	6	yes	complemented
19	#14579106	C₂²	1	1	yes	characteristic, complemented
20	D₁₂₆₄	C₂²	1	3	yes	complemented
21	#131×C₇₉	C₂²	1	1	yes	characteristic
22	M₃₂×C₇₉	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₇₉	C₂²	1	3	yes	complemented
24	#11929625	—	4	1	no	—
25	#11929624	—	2	3	no	—
26	#11929624	—	2	3	no	—
27	#11929623	—	4	3	no	—
28	C₂×Dic₁₅₈	—	2	3	no	—
29	Q₈×D₇₉	—	4	1	no	—
30	C₂×D₃₁₆	—	2	3	no	—
31	D₄×D₇₉	—	4	3	no	—
32	Dic₃₁₆	—	2	3	no	—
33	Dic₃₁₆	C₂³	1	3	yes	—
34	Dic₃₁₆	C₂³	1	1	yes	characteristic
35	#11929613	—	2	3	no	—
36	#11929613	—	2	3	no	—
37	D₆₃₂	—	2	3	no	—
38	D₆₃₂	C₂³	1	1	yes	characteristic
39	D₆₃₂	C₂³	1	3	yes	—
40	#41×C₇₉	D₄	1	1	yes	characteristic
41	M₁₆×C₇₉	D₄	1	3	yes	—
42	C₂×C₈×C₇₉	C₂³	1	3	yes	—
43	C₁₆×C₇₉	C₂³	1	1	yes	characteristic
44	C₁₆×C₇₉	C₂³	1	3	yes	—
45	C₂²×D₇₉	—	4	3	no	—
46	Q₈×C₇₉	D₈	1	1	yes	characteristic
47	D₄×C₇₉	D₈	1	3	yes	—
48	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
49	C₂²⋊D₇₉	—	4	3	no	—
50	C₂²⋊D₇₉	—	4	3	no	—
51	C₂×Dic₇₉	—	4	3	no	—
52	D₃₁₆	—	2	3	no	—
53	D₃₁₆	—	2	1	no	—
54	D₃₁₆	—	4	3	no	—
55	C₄×D₇₉	—	4	3	no	—
56	C₄×D₇₉	—	4	3	no	—
57	Dic₁₅₈	—	2	1	no	—
58	Dic₁₅₈	—	2	3	no	—
59	Dic₁₅₈	—	4	3	no	—
60	C₈×C₇₉	C₂×D₄	1	3	yes	—
61	C₈×C₇₉	C₂⁴	1	1	yes	characteristic
62	C₂²×C₇₉	C₂×D₈	1	3	yes	—
63	C₂×D₇₉	—	4	1	no	—
64	C₂×D₇₉	—	4	3	no	—
65	C₂×D₇₉	—	16	3	no	—
66	C₄×C₇₉	C₂×D₈	1	3	yes	—
67	C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
68	Dic₇₉	—	4	1	no	—
69	Dic₇₉	—	4	3	no	—
70	C₂×C₇₉	—	2	3	no	—
71	C₂×C₇₉	C₂²×D₈	1	1	yes	characteristic
72	D₇₉	—	8	1	no	—
73	D₇₉	—	8	3	no	—
74	#3416	—	79	1	no	maximal, Sylow
75	C₇₉	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	—	79	1	no	—
77	#576	—	79	1	no	—
78	#510	—	79	3	no	—
79	#509	—	79	3	no	—
80	#508	—	79	3	no	—
81	C₂×SD₃₂	—	79	3	no	—
82	#504	D₇₉	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	158	1	no	—
84	H(5,𝔽₂)	—	158	1	no	—
85	#137	—	158	3	no	—
86	AGL(1,ℤ/8)	—	158	3	no	—
87	#135	—	79	3	no	—
88	#135	—	79	3	no	—
89	C₂×Q₁₆	—	79	3	no	—
90	C₂×D₈	—	79	3	no	—
91	#131	C₂×D₇₉	1	1	yes	characteristic
92	Q₃₂	—	79	3	no	—
93	SD₃₂	—	79	3	no	—
94	SD₃₂	—	79	6	no	—
95	SD₃₂	—	79	1	no	—
96	D₁₆	—	79	3	no	—
97	M₃₂	C₂×D₇₉	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₇₉	1	3	yes	complemented
99	#41	—	316	1	no	—
100	#41	—	316	3	no	—
101	#41	—	158	3	no	—
102	#41	—	158	3	no	—
103	#41	D₃₁₆	1	1	yes	characteristic
104	C₂×Q₈	—	316	1	no	—
105	C₂×Q₈	—	158	3	no	—
106	C₂×D₄	—	316	3	no	—
107	C₂×D₄	—	158	3	no	—
108	Q₁₆	—	79	3	no	—
109	Q₁₆	—	79	1	no	—
110	Q₁₆	—	158	3	no	—
111	SD₁₆	—	158	3	no	—
112	SD₁₆	—	158	3	no	—
113	D₈	—	79	1	no	—
114	D₈	—	79	3	no	—
115	D₈	—	158	3	no	—
116	M₁₆	D₃₁₆	1	3	yes	—
117	C₂×C₈	C₂²×D₇₉	1	3	yes	—
118	C₁₆	C₂²×D₇₉	1	1	yes	characteristic
119	C₁₆	C₂²×D₇₉	1	3	yes	—
120	C₂³	—	316	3	no	—
121	Q₈	—	316	3	no	—
122	Q₈	—	158	1	no	—
123	Q₈	—	158	3	no	—
124	Q₈	D₆₃₂	1	1	yes	characteristic
125	D₄	—	316	3	no	—
126	D₄	—	316	3	no	—
127	D₄	—	316	3	no	—
128	D₄	—	158	3	no	—
129	D₄	—	158	1	no	—
130	D₄	D₆₃₂	1	3	yes	—
131	C₂×C₄	—	316	3	no	—
132	C₂×C₄	—	316	3	no	—
133	C₂×C₄	—	316	3	no	—
134	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
135	C₈	C₂×D₃₁₆	1	3	yes	—
136	C₈	C₂³×D₇₉	1	1	yes	characteristic
137	C₂²	—	316	1	no	—
138	C₂²	—	316	3	no	—
139	C₂²	—	1264	3	no	—
140	C₂²	C₂×D₆₃₂	1	3	yes	—
141	C₄	—	316	1	no	—
142	C₄	—	316	3	no	—
143	C₄	C₂×D₆₃₂	1	3	yes	—
144	C₄	C₂²×D₃₁₆	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	632	3	no	—
147	C₂	—	632	1	no	—
148	C₂	C₂²×D₆₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₆₃₂
commutator, cosocle	C₈×C₇₉	C₂⁴
Frattini	C₈	C₂³×D₇₉
Fitting	#504×C₇₉	C₂
socle	C₂×C₇₉	C₂²×D₈
hypercenter	#504	D₇₉
hypocenter	C₇₉	#3416
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #19099380	C₂	semidirect
2. #19099368	C₂	semidirect
3. #19099006	C₂	semidirect
4. #19099001	C₂	semidirect
5. #19098995	C₂	semidirect
6. C₂×#14579106	C₂	semidirect
7. #504×C₇₉	C₂	semidirect
13. #14579183	C₂²	semidirect
14. C₂×Dic₃₁₆	C₂²	semidirect
16. Dic₆₃₂	C₂²	semidirect
17. #14579106	C₂²	semidirect
18. #14579106	C₂²	semidirect
19. #14579106	C₂²	semidirect
20. D₁₂₆₄	C₂²	semidirect
22. M₃₂×C₇₉	C₂²	semidirect
23. C₂×C₁₆×C₇₉	C₂²	semidirect
75. C₇₉	#3416	semidirect
82. #504	D₇₉	semidirect
97. M₃₂	C₂×D₇₉	semidirect
98. C₂×C₁₆	C₂×D₇₉	semidirect