Group #38767931 — subgroups

Subgroups — #38767931

Order 25472.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#27164734	C₂	1	1	yes	maximal, characteristic, complemented
2	#27164210	C₂	1	2	yes	maximal, complemented
3	#27163767	C₂	1	1	yes	maximal, characteristic, complemented
4	#27163747	C₂	1	2	yes	maximal, complemented
5	#492×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691871	—	2	2	no	—
7	#20263944	—	2	2	no	—
8	#20263930	—	2	2	no	—
9	C₄×D₇₉₆	C₂²	1	2	yes	—
10	#20263913	—	2	2	no	—
11	C₄⋊(C₁₉₉⋊C₈)	C₂²	1	2	yes	—
12	#20263876	—	2	2	no	—
13	C₂×SD₁₆×C₁₉₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	#127×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×D₇₉₆	—	2	2	no	—
18	C₂×D₇₉₆	D₄	1	2	yes	—
19	C₂×C₄×D₁₉₉	—	2	2	no	—
20	#15691871	—	4	4	no	—
21	C₂×C₁₉₉⋊C₈	—	2	2	no	—
22	C₄⋊Dic₁₉₉	D₄	1	2	yes	—
23	(C₂×D₁₉₉)⋊C₄	—	2	4	no	—
24	(C₂×D₁₉₉)⋊C₄	—	2	2	no	—
25	C₄×Dic₁₉₉	—	2	2	no	—
26	C₂×Q₈×C₁₉₉	D₄	1	2	yes	—
27	C₂×D₄×C₁₉₉	—	2	2	no	—
28	C₂×D₄×C₁₉₉	—	2	1	no	—
29	SD₁₆×C₁₉₉	—	4	4	no	—
30	C₂×C₈×C₁₉₉	—	2	1	no	—
31	C₄⋊C₄×C₁₉₉	—	2	2	no	—
32	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
33	C₂²×D₁₉₉	—	2	2	no	—
34	D₇₉₆	—	2	2	no	—
35	D₇₉₆	—	2	2	no	—
36	D₇₉₆	—	8	2	no	—
37	C₂×Dic₁₉₉	—	2	2	no	—
38	C₄×D₁₉₉	—	4	2	no	—
39	C₄×D₁₉₉	—	4	2	no	—
40	C₁₉₉⋊C₈	—	4	2	no	—
41	C₂³×C₁₉₉	—	4	1	no	—
42	Q₈×C₁₉₉	—	4	2	no	—
43	D₄×C₁₉₉	—	4	2	no	—
44	D₄×C₁₉₉	—	4	2	no	—
45	D₄×C₁₉₉	—	4	2	no	—
46	C₂×C₄×C₁₉₉	—	2	2	no	—
47	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₉₉	—	2	2	no	—
50	C₂²×C₁₉₉	—	8	2	no	—
51	C₂²×C₁₉₉	—	8	1	no	—
52	C₂²×C₁₉₉	#120	1	1	yes	characteristic
53	C₂×D₁₉₉	—	4	2	no	—
54	C₂×D₁₉₉	—	4	2	no	—
55	C₂×D₁₉₉	—	4	2	no	—
56	C₄×C₁₉₉	—	2	1	no	—
57	C₄×C₁₉₉	—	4	2	no	—
58	C₄×C₁₉₉	#135	1	2	yes	—
59	C₄×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₁₉₉	—	4	2	no	—
61	C₂×C₁₉₉	—	16	1	no	—
62	C₂×C₁₉₉	#449	1	2	yes	—
63	C₂×C₁₉₉	#453	1	1	yes	characteristic
64	D₁₉₉	—	8	2	no	—
65	C₁₉₉	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	199	1	no	maximal, Sylow
67	#555	—	199	1	no	—
68	#492	D₁₉₉	1	1	yes	characteristic, complemented
69	#459	—	199	2	no	—
70	#333	—	199	1	no	—
71	D₄⋊C₈	—	199	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	398	2	no	—
74	C₄⋊Q₈	C₂×D₁₉₉	1	1	yes	characteristic
75	#127	C₂×D₁₉₉	1	1	yes	characteristic
76	#124	—	398	2	no	—
77	#121	—	398	2	no	—
78	C₄×D₄	—	199	2	no	—
79	#117	—	398	2	no	—
80	C₄⋊C₈	—	199	2	no	—
81	#102	—	398	2	no	—
82	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₁₉₉	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	398	2	no	—
87	C₂×D₄	—	199	2	no	—
88	C₂²×C₄	—	398	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	796	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	398	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	199	2	no	—
95	C₂²⋊C₄	—	398	4	no	—
96	C₂²⋊C₄	—	398	2	no	—
97	C₄²	—	398	2	no	—
98	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	398	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	1592	2	no	—
106	D₄	—	398	2	no	—
107	D₄	—	398	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	796	2	no	—
110	C₂×C₄	—	796	2	no	—
111	C₂×C₄	—	398	2	no	—
112	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	796	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	796	2	no	—
119	C₂²	—	796	2	no	—
120	C₂²	—	796	2	no	—
121	C₂²	#20263923	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	796	2	no	—
125	C₄	#20263976	1	2	yes	—
126	C₄	#20263982	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	1592	2	no	—
129	C₂	#27164162	1	2	yes	—
130	C₂	#27164185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164734

C₂

yes

maximal, characteristic, complemented

#27164210

C₂

yes

maximal, complemented

#27163767

C₂

yes

maximal, characteristic, complemented

#27163747

C₂

yes

maximal, complemented

#492×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×#15691871

—

#20263944

—

#20263930

—

C₄×D₇₉₆

C₂²

yes

—

#20263913

—

C₄⋊(C₁₉₉⋊C₈)

C₂²

yes

—

#20263876

—

C₂×SD₁₆×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

#127×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×D₇₉₆

—

C₂×D₇₉₆

D₄

yes

—

C₂×C₄×D₁₉₉

—

#15691871

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

D₄

yes

—

(C₂×D₁₉₉)⋊C₄

—

(C₂×D₁₉₉)⋊C₄

—

C₄×Dic₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

—

SD₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

C₂²×D₁₉₉

—

D₇₉₆

—

D₇₉₆

—

D₇₉₆

—

C₂×Dic₁₉₉

—

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

Q₈×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#120

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#135

yes

—

C₄×C₁₉₉

AGL(1,ℤ/8)

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#449

yes

—

C₂×C₁₉₉

#453

yes

characteristic

D₁₉₉

—

C₁₉₉

#1624

yes

characteristic, complemented, Sylow

#1624

—

199

maximal, Sylow

#555

—

199

—

#492

D₁₉₉

yes

characteristic, complemented

#459

—

199

—

#333

—

199

—

D₄⋊C₈

—

199

—

C₂×SD₁₆

—

C₂×D₈

—

398

—

C₄⋊Q₈

C₂×D₁₉₉

yes

characteristic

#127

C₂×D₁₉₉

yes

characteristic

#124

—

398

—

#121

—

398

—

C₄×D₄

—

199

—

#117

—

398

—

C₄⋊C₈

—

199

—

#102

—

398

—

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

C₂²⋊D₁₉₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

398

—

C₂×D₄

—

199

—

C₂²×C₄

—

398

—

SD₁₆

—

D₈

—

796

—

C₂×C₈

—

C₂×C₈

—

398

—

C₄⋊C₄

—

C₄⋊C₄

—

199

—

C₂²⋊C₄

—

398

—

C₂²⋊C₄

—

398

—

C₄²

—

398

—

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

—

100

C₂³

—

398

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

1592

—

106

D₄

—

398

—

107

D₄

—

398

—

108

C₂×C₄

—

109

C₂×C₄

—

796

—

110

C₂×C₄

—

796

—

111

C₂×C₄

—

398

—

112

C₂×C₄

D₄×D₁₉₉

yes

—

113

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

114

C₈

—

115

C₈

—

796

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

796

—

119

C₂²

—

796

—

120

C₂²

—

796

—

121

C₂²

#20263923

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

796

—

125

C₄

#20263976

yes

—

126

C₄

#20263982

yes

—

127

C₂

—

128

C₂

—

1592

—

129

C₂

#27164162

yes

—

130

C₂

#27164185

yes

characteristic

	subgroup	G/H
center	C₂²	#20263923
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#492×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#120
hypercenter	#492	D₁₉₉
hypocenter	C₁₉₉	#1624
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20263923

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#492×C₁₉₉

C₂

C₂²×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #27164734	C₂	semidirect
2. #27164210	C₂	semidirect
3. #27163767	C₂	semidirect
4. #27163747	C₂	semidirect
5. #492×C₁₉₉	C₂	semidirect
65. C₁₉₉	#1624	semidirect
68. #492	D₁₉₉	semidirect

H ≅ G/N

type

1. #27164734

C₂

semidirect

2. #27164210

C₂

semidirect

3. #27163767

C₂

semidirect

4. #27163747

C₂

semidirect

5. #492×C₁₉₉

C₂

semidirect

65. C₁₉₉

#1624

semidirect

68. #492

D₁₉₉

semidirect

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

subgroup	index
#27164734	2
#27164210	2
#27163767	2
#27163747	2
#492×C₁₉₉	2
#1624	199

subgroup

index

#492×C₁₉₉

199

Subgroups — #38767931

Conjugacy classes of subgroups

Complements

Maximal subgroups