Group #23776375 — subgroups

Subgroups — #23776375

Order 10112.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19099235	C₂	1	2	yes	maximal, complemented
2	#19098940	C₂	1	3	yes	maximal, complemented
3	#19098759	C₂	1	6	yes	maximal, complemented
4	C₄×D₆₃₂	C₂	1	3	yes	maximal, complemented
5	C₈×Q₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11929625	—	2	2	no	—
7	C₂×D₆₃₂	C₂²	1	3	yes	—
8	C₂×D₆₃₂	C₂²	1	6	yes	—
9	#14579158	C₂²	1	6	yes	—
10	#14579136	—	2	6	no	—
11	C₄×D₃₁₆	C₂²	1	6	yes	—
12	#14579095	C₂²	1	1	yes	characteristic
13	#14579083	C₂²	1	6	yes	—
14	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₉	C₂²	1	3	yes	—
16	C₄×C₈×C₇₉	C₂²	1	3	yes	—
17	#11929625	—	4	8	no	—
18	C₂×D₃₁₆	—	2	6	no	—
19	C₂×D₃₁₆	—	2	6	no	—
20	C₂×D₃₁₆	C₂³	1	6	yes	—
21	C₂×C₄×D₇₉	—	2	6	no	—
22	D₆₃₂	—	2	6	no	—
23	D₆₃₂	—	4	6	no	—
24	C₄⋊Dic₇₉	C₂³	1	2	yes	—
25	(C₂×D₇₉)⋊C₄	—	2	6	no	—
26	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₉	C₂³	1	3	yes	—
28	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₉	D₄	1	3	yes	—
30	C₄²×C₇₉	C₂³	1	3	yes	—
31	C₂²×D₇₉	—	2	6	no	—
32	Q₈×C₇₉	D₈	1	4	yes	complemented
33	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₇₉	C₂×D₄	1	3	yes	—
35	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
36	C₂×Dic₇₉	—	2	2	no	—
37	D₃₁₆	—	2	6	no	—
38	D₃₁₆	—	2	6	no	—
39	D₃₁₆	—	4	12	no	—
40	D₃₁₆	—	8	6	no	—
41	C₄×D₇₉	—	4	12	no	—
42	C₈×C₇₉	—	2	3	no	—
43	C₈×C₇₉	#41	1	2	yes	complemented
44	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
45	C₂×D₇₉	—	4	6	no	—
46	C₂×D₇₉	—	4	6	no	—
47	C₂×D₇₉	—	4	6	no	—
48	C₄×C₇₉	—	2	3	no	—
49	C₄×C₇₉	C₂×D₈	1	6	yes	—
50	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
51	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
52	Dic₇₉	—	4	2	no	—
53	C₂×C₇₉	#550	1	1	yes	characteristic
54	C₂×C₇₉	C₂²×D₈	1	1	yes	characteristic
55	C₂×C₇₉	#576	1	1	yes	characteristic
56	D₇₉	—	8	6	no	—
57	#3391	—	79	1	no	maximal, Sylow
58	C₇₉	#3391	1	1	yes	characteristic, complemented, Sylow
59	#550	—	79	2	no	—
60	#493	—	79	3	no	—
61	#459	—	79	6	no	—
62	C₈×Q₈	D₇₉	1	1	yes	characteristic, complemented
63	C₄×D₈	—	79	3	no	—
64	C₂×#41	—	158	2	no	—
65	C₂×D₈	—	79	6	no	—
66	C₂×D₈	—	79	3	no	—
67	#127	—	79	6	no	—
68	#121	—	158	6	no	—
69	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
70	C₄×D₄	—	79	6	no	—
71	#107	—	79	1	no	—
72	C₄⋊C₈	C₂×D₇₉	1	3	yes	—
73	#102	—	79	6	no	—
74	C₄×C₈	C₂×D₇₉	1	3	yes	—
75	#41	—	316	8	no	—
76	C₂×Q₈	D₃₁₆	1	1	yes	characteristic
77	C₂×D₄	—	79	6	no	—
78	C₂×D₄	—	158	6	no	—
79	C₂×D₄	—	158	6	no	—
80	C₂²×C₄	—	158	6	no	—
81	D₈	—	316	6	no	—
82	D₈	—	158	6	no	—
83	C₂×C₈	C₂²×D₇₉	1	3	yes	—
84	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
85	C₄⋊C₄	—	79	2	no	—
86	C₄⋊C₄	D₃₁₆	1	3	yes	—
87	C₂²⋊C₄	—	158	6	no	—
88	C₄²	C₂²×D₇₉	1	3	yes	—
89	C₂³	—	158	6	no	—
90	Q₈	D₆₃₂	1	4	yes	complemented
91	D₄	—	316	12	no	—
92	D₄	—	632	6	no	—
93	D₄	—	158	6	no	—
94	D₄	—	158	6	no	—
95	C₂×C₄	—	316	12	no	—
96	C₂×C₄	—	158	2	no	—
97	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
98	C₂×C₄	C₂×D₃₁₆	1	3	yes	—
99	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
100	C₈	—	2	3	no	—
101	C₈	#11929625	1	2	yes	complemented
102	C₂²	—	316	6	no	—
103	C₂²	—	316	6	no	—
104	C₂²	—	316	6	no	—
105	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
106	C₄	—	2	3	no	—
107	C₄	—	316	2	no	—
108	C₄	C₂×D₆₃₂	1	6	yes	—
109	C₄	C₂×#11929625	1	1	yes	characteristic
110	C₄	#14579207	1	1	yes	characteristic
111	C₂	—	632	6	no	—
112	C₂	#19099235	1	1	yes	characteristic
113	C₂	C₂²×D₆₃₂	1	1	yes	characteristic
114	C₂	#19099368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099235

C₂

yes

maximal, complemented

#19098940

C₂

yes

maximal, complemented

#19098759

C₂

yes

maximal, complemented

C₄×D₆₃₂

C₂

yes

maximal, complemented

C₈×Q₈×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929625

—

C₂×D₆₃₂

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

—

#14579158

C₂²

yes

—

#14579136

—

C₄×D₃₁₆

C₂²

yes

—

#14579095

C₂²

yes

characteristic

#14579083

C₂²

yes

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

—

#11929625

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₄×D₇₉

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

C₂³

yes

—

(C₂×D₇₉)⋊C₄

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂²×D₇₉

—

Q₈×C₇₉

D₈

yes

complemented

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

complemented

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

—

C₄×C₇₉

C₂×#41

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

Dic₇₉

—

C₂×C₇₉

#550

yes

characteristic

C₂×C₇₉

C₂²×D₈

yes

characteristic

C₂×C₇₉

#576

yes

characteristic

D₇₉

—

#3391

—

maximal, Sylow

C₇₉

#3391

yes

characteristic, complemented, Sylow

#550

—

#493

—

#459

—

C₈×Q₈

D₇₉

yes

characteristic, complemented

C₄×D₈

—

C₂×#41

—

158

—

C₂×D₈

—

C₂×D₈

—

#127

—

#121

—

158

—

C₄×Q₈

C₂×D₇₉

yes

characteristic

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₇₉

yes

—

#102

—

C₄×C₈

C₂×D₇₉

yes

—

#41

—

316

—

C₂×Q₈

D₃₁₆

yes

characteristic

C₂×D₄

—

C₂×D₄

—

158

—

C₂×D₄

—

158

—

C₂²×C₄

—

158

—

D₈

—

316

—

D₈

—

158

—

C₂×C₈

C₂²×D₇₉

yes

—

C₂×C₈

C₂²×D₇₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₃₁₆

yes

—

C₂²⋊C₄

—

158

—

C₄²

C₂²×D₇₉

yes

—

C₂³

—

158

—

Q₈

D₆₃₂

yes

complemented

D₄

—

316

—

D₄

—

632

—

D₄

—

158

—

D₄

—

158

—

C₂×C₄

—

316

—

C₂×C₄

—

158

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₂×C₄

C₂×D₃₁₆

yes

—

C₂×C₄

C₂³×D₇₉

yes

characteristic

100

C₈

—

101

C₈

#11929625

yes

complemented

102

C₂²

—

316

—

103

C₂²

—

316

—

104

C₂²

—

316

—

105

C₂²

C₂²×D₃₁₆

yes

characteristic

106

C₄

—

107

C₄

—

316

—

108

C₄

C₂×D₆₃₂

yes

—

109

C₄

C₂×#11929625

yes

characteristic

110

C₄

#14579207

yes

characteristic

111

C₂

—

632

—

112

C₂

#19099235

yes

characteristic

113

C₂

C₂²×D₆₃₂

yes

characteristic

114

C₂

#19099368

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	C₈×Q₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	C₈×Q₈	D₇₉
hypocenter	C₇₉	#3391
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

C₈×Q₈×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

hypercenter

C₈×Q₈

D₇₉

hypocenter

C₇₉

#3391

LCS

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #19099235	C₂	semidirect
2. #19098940	C₂	semidirect
3. #19098759	C₂	semidirect
4. C₄×D₆₃₂	C₂	semidirect
5. C₈×Q₈×C₇₉	C₂	semidirect
32. Q₈×C₇₉	D₈	semidirect
43. C₈×C₇₉	#41	semidirect
58. C₇₉	#3391	semidirect
62. C₈×Q₈	D₇₉	semidirect
90. Q₈	D₆₃₂	semidirect
101. C₈	#11929625	semidirect

H ≅ G/N

type

1. #19099235

C₂

semidirect

2. #19098940

C₂

semidirect

3. #19098759

C₂

semidirect

4. C₄×D₆₃₂

C₂

semidirect

5. C₈×Q₈×C₇₉

C₂

semidirect

32. Q₈×C₇₉

D₈

semidirect

43. C₈×C₇₉

#41

semidirect

58. C₇₉

#3391

semidirect

62. C₈×Q₈

D₇₉

semidirect

90. Q₈

D₆₃₂

semidirect

101. C₈

#11929625

semidirect

Subgroups — #23776375

Conjugacy classes of subgroups

Complements