Group #23765299 — subgroups

Subgroups — #23765299

Order 10112.

154 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 59 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	#19098685	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098675	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098386	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098376	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098373	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098340	C₂	1	1	yes	maximal, characteristic, complemented
7	#509×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#14579168	—	2	1	no	—
9	D₄×Dic₇₉	—	2	1	no	—
10	#14579101	C₂²	1	1	yes	characteristic
11	#14579097	C₂²	1	1	yes	characteristic
12	#14579093	C₂²	1	1	yes	characteristic
13	#14579088	—	2	1	no	—
14	#14579082	—	2	1	no	—
15	#14579071	C₂²	1	1	yes	characteristic
16	#135×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
18	SD₃₂×C₇₉	C₄	1	2	yes	complemented
19	D₁₆×C₇₉	C₄	1	2	yes	complemented
20	M₃₂×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×Dic₇₉	—	4	1	no	—
22	#11929608	—	2	1	no	—
23	#11929608	—	4	1	no	—
24	C₂×C₇₉⋊C₈	—	4	1	no	—
25	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
26	C₄⋊Dic₇₉	—	2	1	no	—
27	C₂²⋊Dic₇₉	—	4	1	no	—
28	C₄×Dic₇₉	D₄	1	1	yes	characteristic
29	C₇₉⋊C₁₆	—	2	1	no	—
30	#41×C₇₉	—	2	1	no	—
31	C₂×D₄×C₇₉	—	2	1	no	—
32	Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic, complemented
33	SD₁₆×C₇₉	—	2	1	no	—
34	D₈×C₇₉	—	2	1	no	—
35	D₈×C₇₉	C₂×C₄	1	2	yes	—
36	D₈×C₇₉	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
38	C₁₆×C₇₉	C₂×C₄	1	2	yes	complemented
39	C₂³×C₇₉	—	4	1	no	—
40	Q₈×C₇₉	—	2	1	no	—
41	D₄×C₇₉	—	2	2	no	—
42	D₄×C₇₉	—	2	1	no	—
43	D₄×C₇₉	—	4	1	no	—
44	D₄×C₇₉	—	4	1	no	—
45	C₂×C₄×C₇₉	—	4	1	no	—
46	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₇₉	—	2	1	no	—
48	C₂×Dic₇₉	—	4	1	no	—
49	C₂×Dic₇₉	—	8	2	no	—
50	C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₈×C₇₉	#41	1	1	yes	characteristic
52	C₇₉⋊C₈	—	2	1	no	—
53	C₇₉⋊C₈	—	4	1	no	—
54	C₂²×C₇₉	—	4	1	no	—
55	C₂²×C₇₉	—	4	2	no	—
56	C₂²×C₇₉	—	8	1	no	—
57	C₂²×C₇₉	—	8	1	no	—
58	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₄×C₇₉	—	4	1	no	—
60	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
61	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₇₉	—	4	1	no	—
63	Dic₇₉	—	8	1	no	—
64	C₂×C₇₉	—	2	1	no	—
65	C₂×C₇₉	—	8	1	no	—
66	C₂×C₇₉	—	8	2	no	—
67	C₂×C₇₉	#442	1	1	yes	characteristic
68	AGL(1,ℤ/16)	—	79	1	no	maximal, Sylow
69	C₇₉	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
70	#509	D₇₉	1	1	yes	characteristic
71	#444	—	79	1	no	—
72	#442	—	79	1	no	—
73	#365	—	79	1	no	—
74	#361	—	79	1	no	—
75	#360	—	79	1	no	—
76	#347	—	79	1	no	—
77	#135	C₂×D₇₉	1	1	yes	characteristic
78	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
79	#131	—	158	1	no	—
80	C₄×D₄	—	158	1	no	—
81	SD₃₂	Dic₇₉	1	2	yes	complemented
82	D₁₆	Dic₇₉	1	2	yes	complemented
83	M₃₂	—	79	1	no	—
84	M₃₂	C₂×D₇₉	1	1	yes	characteristic
85	#108	—	79	1	no	—
86	#106	—	79	1	no	—
87	#104	—	158	1	no	—
88	#102	—	158	1	no	—
89	#97	—	79	1	no	—
90	#41	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂²×C₄	—	316	1	no	—
93	Q₁₆	C₂×Dic₇₉	1	1	yes	characteristic, complemented
94	SD₁₆	—	2	1	no	—
95	D₈	—	2	1	no	—
96	D₈	C₂×Dic₇₉	1	2	yes	—
97	D₈	C₂×Dic₇₉	1	1	yes	characteristic, complemented
98	M₁₆	—	316	1	no	—
99	M₁₆	—	158	1	no	—
100	C₂×C₈	—	316	1	no	—
101	C₂×C₈	—	79	1	no	—
102	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
103	C₄⋊C₄	—	158	1	no	—
104	C₂²⋊C₄	—	316	1	no	—
105	C₄²	—	79	1	no	—
106	C₁₆	—	158	1	no	—
107	C₁₆	C₂×Dic₇₉	1	2	yes	complemented
108	C₂³	—	4	1	no	—
109	Q₈	—	2	1	no	—
110	D₄	—	2	2	no	—
111	D₄	—	2	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	316	1	no	—
116	C₂×C₄	—	632	2	no	—
117	C₂×C₄	—	158	1	no	—
118	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
119	C₈	—	316	1	no	—
120	C₈	—	158	1	no	—
121	C₈	C₂²×Dic₇₉	1	1	yes	characteristic
122	C₈	#11929623	1	1	yes	characteristic
123	C₂²	—	4	1	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	#14579185	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	316	1	no	—
130	C₄	—	632	1	no	—
131	C₄	D₄×Dic₇₉	1	1	yes	characteristic
132	C₄	#14579185	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	8	2	no	—
135	C₂	—	8	1	no	—
136	C₂	#19098675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19098685

C₂

yes

maximal, characteristic, complemented

#19098675

C₂

yes

maximal, characteristic, complemented

#19098386

C₂

yes

maximal, characteristic, complemented

#19098376

C₂

yes

maximal, characteristic, complemented

#19098373

C₂

yes

maximal, characteristic, complemented

#19098340

C₂

yes

maximal, characteristic, complemented

#509×C₇₉

C₂

yes

maximal, characteristic

#14579168

—

D₄×Dic₇₉

—

#14579101

C₂²

yes

characteristic

#14579097

C₂²

yes

characteristic

#14579093

C₂²

yes

—

—

C₂²

yes

characteristic

#135×C₇₉

C₂²

yes

characteristic

C₂×D₈×C₇₉

C₂²

yes

characteristic

SD₃₂×C₇₉

C₄

yes

complemented

D₁₆×C₇₉

C₄

yes

complemented

M₃₂×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

D₄

yes

characteristic

C₇₉⋊C₁₆

—

#41×C₇₉

—

C₂×D₄×C₇₉

—

Q₁₆×C₇₉

C₂×C₄

yes

characteristic, complemented

SD₁₆×C₇₉

—

D₈×C₇₉

—

D₈×C₇₉

C₂×C₄

yes

—

D₈×C₇₉

C₂×C₄

yes

characteristic, complemented

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₁₆×C₇₉

C₂×C₄

yes

complemented

C₂³×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

C₂²×C₄

yes

characteristic

C₈×C₇₉

#41

yes

characteristic

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

AGL(1,ℤ/8)

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

AGL(1,ℤ/8)

yes

characteristic

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

yes

—

C₇₉

yes

characteristic, complemented, Sylow

#509

D₇₉

yes

—

—

—

—

—

—

C₂×D₇₉

yes

characteristic

C₂×D₈

C₂×D₇₉

yes

characteristic

#131

—

158

—

C₄×D₄

—

158

—

SD₃₂

Dic₇₉

yes

complemented

D₁₆

Dic₇₉

yes

complemented

M₃₂

—

M₃₂

C₂×D₇₉

yes

—

—

—

158

—

#102

—

158

—

#97

—

#41

—

C₂×D₄

—

C₂²×C₄

—

316

—

Q₁₆

C₂×Dic₇₉

yes

characteristic, complemented

SD₁₆

—

D₈

—

D₈

C₂×Dic₇₉

yes

—

D₈

C₂×Dic₇₉

yes

characteristic, complemented

M₁₆

—

316

—

M₁₆

—

158

—

100

C₂×C₈

—

316

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₇₉

yes

characteristic

103

C₄⋊C₄

—

158

—

104

C₂²⋊C₄

—

316

—

105

C₄²

—

106

C₁₆

—

158

—

107

C₁₆

C₂×Dic₇₉

yes

complemented

108

C₂³

—

109

Q₈

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

316

—

116

C₂×C₄

—

632

—

117

C₂×C₄

—

158

—

118

C₂×C₄

D₄×D₇₉

yes

characteristic

119

C₈

—

316

—

120

C₈

—

158

—

121

C₈

C₂²×Dic₇₉

yes

characteristic

122

C₈

#11929623

yes

characteristic

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#14579185

yes

characteristic

128

C₄

—

129

C₄

—

316

—

130

C₄

—

632

—

131

C₄

D₄×Dic₇₉

yes

characteristic

132

C₄

#14579185

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

—

136

C₂

#19098675

yes

characteristic

	subgroup	G/H
center	C₂	#19098675
commutator	C₈×C₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₉
Fitting, prime radical	#509×C₇₉	C₂
socle	C₂×C₇₉	#442
cosocle	C₂×C₈×C₇₉	C₂³
hypercenter	#509	D₇₉
hypocenter	C₇₉	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

subgroup

G/H

center

C₂

#19098675

commutator

C₈×C₇₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₉

Fitting, prime radical

#509×C₇₉

C₂

socle

C₂×C₇₉

#442

cosocle

C₂×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

N	H ≅ G/N	type
1. #19098685	C₂	semidirect
2. #19098675	C₂	semidirect
3. #19098386	C₂	semidirect
4. #19098376	C₂	semidirect
5. #19098373	C₂	semidirect
6. #19098340	C₂	semidirect
18. SD₃₂×C₇₉	C₄	semidirect
19. D₁₆×C₇₉	C₄	semidirect
32. Q₁₆×C₇₉	C₂×C₄	semidirect
36. D₈×C₇₉	C₂×C₄	semidirect
38. C₁₆×C₇₉	C₂×C₄	semidirect
69. C₇₉	AGL(1,ℤ/16)	semidirect
81. SD₃₂	Dic₇₉	semidirect
82. D₁₆	Dic₇₉	semidirect
93. Q₁₆	C₂×Dic₇₉	semidirect
97. D₈	C₂×Dic₇₉	semidirect
107. C₁₆	C₂×Dic₇₉	semidirect

H ≅ G/N

type

1. #19098685

C₂

semidirect

2. #19098675

C₂

semidirect

3. #19098386

C₂

semidirect

4. #19098376

C₂

semidirect

5. #19098373

C₂

semidirect

6. #19098340

C₂

semidirect

18. SD₃₂×C₇₉

C₄

semidirect

19. D₁₆×C₇₉

C₄

semidirect

32. Q₁₆×C₇₉

C₂×C₄

semidirect

36. D₈×C₇₉

C₂×C₄

semidirect

38. C₁₆×C₇₉

C₂×C₄

semidirect

69. C₇₉

AGL(1,ℤ/16)

semidirect

81. SD₃₂

Dic₇₉

semidirect

82. D₁₆

Dic₇₉

semidirect

93. Q₁₆

C₂×Dic₇₉

semidirect

97. D₈

C₂×Dic₇₉

semidirect

107. C₁₆

C₂×Dic₇₉

semidirect

Subgroups — #23765299

Conjugacy classes of subgroups

Complements