Finite Group Databasebeta

Subgroups — #23765416

Order 10112.

248 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 41 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	#19099305	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098736	C₂	1	2	yes	maximal, complemented
3	#19098360	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098359	C₂	1	2	yes	maximal, complemented
5	#454×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14579186	—	2	2	no	—
7	#14579150	C₂²	1	2	yes	—
8	#14579149	—	2	1	no	—
9	#14579147	—	2	2	no	—
10	#14579131	—	2	2	no	—
11	#14579129	—	2	1	no	—
12	#14579088	—	2	2	no	—
13	#14579080	C₂²	1	2	yes	complemented
14	2₋¹⁺⁴×C₇₉	C₂²	1	1	yes	characteristic, complemented
15	AGL(1,ℤ/8)×C₇₉	—	2	1	no	—
16	#124×C₇₉	C₂²	1	1	yes	characteristic
17	#104×C₇₉	—	2	1	no	—
18	#101×C₇₉	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₇₉	—	2	1	no	—
20	C₂×Dic₁₅₈	—	2	1	no	—
21	C₂×D₃₁₆	—	2	1	no	—
22	C₂×D₃₁₆	D₄	1	2	yes	—
23	C₂×C₂²⋊D₇₉	—	2	2	no	—
24	#11929611	—	4	2	no	—
25	#11929609	—	4	2	no	—
26	#11929608	—	2	2	no	—
27	C₂²⋊Dic₇₉	—	2	1	no	—
28	(C₂×D₇₉)⋊C₄	—	2	2	no	—
29	(C₂×D₇₉)⋊C₄	—	2	2	no	—
30	(C₂×D₇₉)⋊C₄	—	2	2	no	—
31	C₄×Dic₇₉	D₄	1	2	yes	—
32	#41×C₇₉	—	2	1	no	—
33	#41×C₇₉	—	2	2	no	—
34	#41×C₇₉	—	4	1	no	—
35	C₂×Q₈×C₇₉	—	4	1	no	—
36	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
38	SD₁₆×C₇₉	—	4	1	no	—
39	D₈×C₇₉	—	4	1	no	—
40	M₁₆×C₇₉	—	2	1	no	—
41	C₂²⋊C₄×C₇₉	—	2	2	no	—
42	C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂³×C₇₉	—	2	1	no	—
44	C₂²×D₇₉	—	2	2	no	—
45	C₂²×D₇₉	—	8	2	no	—
46	C₂²×D₇₉	—	8	1	no	—
47	Q₈×C₇₉	—	2	2	no	—
48	Q₈×C₇₉	—	2	1	no	—
49	Q₈×C₇₉	—	4	1	no	—
50	D₄×C₇₉	—	2	2	no	—
51	D₄×C₇₉	—	2	1	no	—
52	D₄×C₇₉	—	4	1	no	—
53	D₄×C₇₉	—	4	1	no	—
54	C₂×C₄×C₇₉	—	2	1	no	—
55	C₂×C₄×C₇₉	—	4	1	no	—
56	C₂×C₄×C₇₉	—	4	2	no	—
57	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₇₉	—	8	2	no	—
60	C₂×Dic₇₉	—	2	2	no	—
61	D₃₁₆	—	4	2	no	—
62	D₃₁₆	—	8	1	no	—
63	Dic₁₅₈	—	8	1	no	—
64	C₈×C₇₉	—	4	1	no	—
65	C₇₉⋊C₈	—	4	2	no	—
66	C₂²×C₇₉	—	4	2	no	—
67	C₂²×C₇₉	—	4	1	no	—
68	C₂²×C₇₉	—	4	1	no	—
69	C₂²×C₇₉	#120	1	1	yes	characteristic
70	C₂×D₇₉	—	4	2	no	—
71	C₂×D₇₉	—	4	2	no	—
72	C₂×D₇₉	—	4	2	no	—
73	C₂×D₇₉	—	8	4	no	—
74	C₄×C₇₉	—	2	1	no	—
75	C₄×C₇₉	—	2	2	no	—
76	C₄×C₇₉	—	4	1	no	—
77	C₄×C₇₉	—	4	1	no	—
78	Dic₇₉	—	4	2	no	—
79	C₂×C₇₉	—	2	1	no	—
80	C₂×C₇₉	—	8	1	no	—
81	C₂×C₇₉	—	8	1	no	—
82	C₂×C₇₉	B(4,𝔽₂)	1	1	yes	characteristic
83	D₇₉	—	8	2	no	—
84	#2202	—	79	1	no	maximal, Sylow
85	C₇₉	#2202	1	1	yes	characteristic, complemented, Sylow
86	H(3,𝔽₄)	—	79	1	no	—
87	#454	—	79	2	no	—
88	#454	D₇₉	1	1	yes	characteristic, complemented
89	#355	—	79	1	no	—
90	#355	—	79	2	no	—
91	2₋¹⁺⁴	C₂×D₇₉	1	1	yes	characteristic, complemented
92	AGL(1,ℤ/8)	—	2	1	no	—
93	AGL(1,ℤ/8)	—	158	2	no	—
94	#124	—	79	2	no	—
95	#124	—	158	2	no	—
96	#124	—	158	1	no	—
97	#124	C₂×D₇₉	1	1	yes	characteristic
98	#120	—	158	2	no	—
99	#120	—	158	1	no	—
100	#104	—	2	1	no	—
101	#104	—	158	2	no	—
102	#101	—	79	2	no	—
103	#101	C₂×D₇₉	1	1	yes	characteristic, complemented
104	C₂⁴	—	158	1	no	—
105	#41	—	2	2	no	—
106	#41	—	2	1	no	—
107	#41	—	4	1	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	158	1	no	—
110	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
111	C₂×D₄	—	79	2	no	—
112	C₂×D₄	—	158	1	no	—
113	C₂×D₄	—	158	2	no	—
114	C₂×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
115	SD₁₆	—	4	1	no	—
116	SD₁₆	—	316	2	no	—
117	D₈	—	4	1	no	—
118	D₈	—	316	2	no	—
119	M₁₆	—	2	1	no	—
120	M₁₆	—	158	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	158	2	no	—
123	C₂²⋊C₄	—	158	1	no	—
124	C₂²⋊C₄	—	158	2	no	—
125	C₂²⋊C₄	—	158	2	no	—
126	C₄²	—	79	2	no	—
127	C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
128	C₂³	—	2	1	no	—
129	C₂³	—	632	2	no	—
130	C₂³	—	632	1	no	—
131	C₂³	—	158	2	no	—
132	Q₈	—	2	1	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	632	1	no	—
136	D₄	—	2	1	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	316	2	no	—
141	D₄	—	632	1	no	—
142	D₄	—	632	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	158	2	no	—
147	C₂×C₄	D₄×D₇₉	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	316	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	316	2	no	—
155	C₂²	—	316	2	no	—
156	C₂²	—	316	2	no	—
157	C₂²	—	632	4	no	—
158	C₂²	#14579129	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	1	no	—
163	C₄	—	316	2	no	—
164	C₂	—	2	1	no	—
165	C₂	—	8	1	no	—
166	C₂	—	8	1	no	—
167	C₂	—	632	2	no	—
168	C₂	#19098749	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#19098749
commutator, cosocle	C₂×Q₈×C₇₉	C₂³
Frattini	C₂×Q₈	C₂²×D₇₉
Fitting	#454×C₇₉	C₂
socle	C₂×C₇₉	B(4,𝔽₂)
hypercenter	#454	D₇₉
hypocenter	C₇₉	#2202
LCS	G ▷ C₂×Q₈×C₇₉ ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #454

Complements

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

N	H ≅ G/N	type
1. #19099305	C₂	semidirect
2. #19098736	C₂	semidirect
3. #19098360	C₂	semidirect
4. #19098359	C₂	semidirect
5. #454×C₇₉	C₂	semidirect
13. #14579080	C₂²	semidirect
14. 2₋¹⁺⁴×C₇₉	C₂²	semidirect
18. #101×C₇₉	C₂²	semidirect
85. C₇₉	#2202	semidirect
88. #454	D₇₉	semidirect
91. 2₋¹⁺⁴	C₂×D₇₉	semidirect
103. #101	C₂×D₇₉	semidirect