Group #38768527 — subgroups

Subgroups — #38768527

Order 25472.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#27164423	C₂	1	1	yes	maximal, characteristic
2	#27164423	C₂	1	1	yes	maximal, characteristic
3	#27164416	C₂	1	1	yes	maximal, characteristic
4	#27164398	C₂	1	1	yes	maximal, characteristic
5	#27164369	C₂	1	1	yes	maximal, characteristic
6	#27164369	C₂	1	1	yes	maximal, characteristic
7	#322×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₉₆	—	2	2	no	—
9	#20263955	C₂²	1	1	yes	characteristic
10	#20263955	C₂²	1	1	yes	characteristic
11	#20263955	C₂²	1	1	yes	characteristic
12	#20263955	C₂²	1	1	yes	characteristic
13	#20263889	—	2	2	no	—
14	#20263889	—	2	2	no	—
15	#20263887	—	2	2	no	—
16	#20263880	—	2	2	no	—
17	#20263880	—	2	2	no	—
18	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₉₈	—	2	1	no	—
22	C₂×Dic₃₉₈	—	2	1	no	—
23	C₂×Dic₃₉₈	—	2	1	no	—
24	C₂×Dic₃₉₈	—	2	1	no	—
25	Dic₇₉₆	—	4	4	no	—
26	C₄⋊Dic₁₉₉	—	2	1	no	—
27	C₄⋊Dic₁₉₉	—	2	1	no	—
28	C₄⋊Dic₁₉₉	—	2	1	no	—
29	C₄⋊Dic₁₉₉	—	2	1	no	—
30	C₄⋊Dic₁₉₉	—	2	1	no	—
31	C₄⋊Dic₁₉₉	—	2	1	no	—
32	C₄⋊Dic₁₉₉	—	2	1	no	—
33	C₄⋊Dic₁₉₉	—	2	1	no	—
34	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
35	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
36	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
37	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
38	Dic₃₉₈	—	4	2	no	—
39	Dic₃₉₈	—	4	2	no	—
40	Dic₃₉₈	—	4	2	no	—
41	Dic₃₉₈	—	4	2	no	—
42	C₂×Dic₁₉₉	—	4	1	no	—
43	C₂×Dic₁₉₉	—	4	1	no	—
44	C₂×Dic₁₉₉	—	4	1	no	—
45	C₂×Dic₁₉₉	—	4	1	no	—
46	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₉₉	—	2	2	no	—
50	C₈×C₁₉₉	—	2	2	no	—
51	C₈×C₁₉₉	Q₁₆	1	4	yes	complemented
52	C₂²×C₁₉₉	#127	1	1	yes	characteristic
53	C₄×C₁₉₉	C₂×Q₁₆	1	2	yes	—
54	C₄×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₉₉	#137	1	2	yes	—
56	Dic₁₉₉	—	8	1	no	—
57	Dic₁₉₉	—	8	1	no	—
58	Dic₁₉₉	—	8	1	no	—
59	Dic₁₉₉	—	8	1	no	—
60	C₂×C₁₉₉	#494	1	1	yes	characteristic
61	C₂×C₁₉₉	#496	1	1	yes	characteristic
62	C₂×C₁₉₉	#497	1	1	yes	characteristic
63	C₁₉₉	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	199	1	no	maximal, Sylow
65	#500	—	199	1	no	—
66	#500	—	199	1	no	—
67	#498	—	199	1	no	—
68	#494	—	199	1	no	—
69	#487	—	199	1	no	—
70	#487	—	199	1	no	—
71	#322	D₁₉₉	1	1	yes	characteristic
72	C₂×Q₁₆	—	398	2	no	—
73	C₄⋊Q₈	—	199	1	no	—
74	C₄⋊Q₈	—	199	1	no	—
75	C₄⋊Q₈	—	199	1	no	—
76	C₄⋊Q₈	—	199	1	no	—
77	#107	—	398	2	no	—
78	#107	—	398	2	no	—
79	#106	—	398	2	no	—
80	#103	—	398	2	no	—
81	#103	—	398	2	no	—
82	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
83	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
84	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
85	C₂×Q₈	—	398	1	no	—
86	C₂×Q₈	—	398	1	no	—
87	C₂×Q₈	—	398	1	no	—
88	C₂×Q₈	—	398	1	no	—
89	Q₁₆	—	796	4	no	—
90	C₂×C₈	D₇₉₆	1	2	yes	—
91	C₂×C₈	D₇₉₆	1	2	yes	—
92	C₂×C₈	D₇₉₆	1	2	yes	—
93	C₄⋊C₄	—	398	1	no	—
94	C₄⋊C₄	—	398	1	no	—
95	C₄⋊C₄	—	398	1	no	—
96	C₄⋊C₄	—	398	1	no	—
97	C₄⋊C₄	—	398	1	no	—
98	C₄⋊C₄	—	398	1	no	—
99	C₄⋊C₄	—	398	1	no	—
100	C₄⋊C₄	—	398	1	no	—
101	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
102	Q₈	—	796	2	no	—
103	Q₈	—	796	2	no	—
104	Q₈	—	796	2	no	—
105	Q₈	—	796	2	no	—
106	C₂×C₄	—	796	1	no	—
107	C₂×C₄	—	796	1	no	—
108	C₂×C₄	—	796	1	no	—
109	C₂×C₄	—	796	1	no	—
110	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
111	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
112	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	Dic₇₉₆	1	4	yes	complemented
116	C₂²	#20263952	1	1	yes	characteristic
117	C₄	—	1592	1	no	—
118	C₄	—	1592	1	no	—
119	C₄	—	1592	1	no	—
120	C₄	—	1592	1	no	—
121	C₄	C₂×Dic₇₉₆	1	2	yes	—
122	C₄	#20263983	1	2	yes	—
123	C₄	#20263988	1	2	yes	—
124	C₂	#27164398	1	1	yes	characteristic
125	C₂	#27164407	1	1	yes	characteristic
126	C₂	#27164412	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164423

C₂

yes

maximal, characteristic

#27164423

C₂

yes

maximal, characteristic

#27164416

C₂

yes

maximal, characteristic

#27164398

C₂

yes

maximal, characteristic

#27164369

C₂

yes

maximal, characteristic

#27164369

C₂

yes

maximal, characteristic

#322×C₁₉₉

C₂

yes

maximal, characteristic

C₂×Dic₇₉₆

—

#20263955

C₂²

yes

characteristic

#20263955

C₂²

yes

characteristic

#20263955

C₂²

yes

characteristic

#20263955

C₂²

yes

—

—

—

—

—

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

Dic₇₉₆

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₄²×C₁₉₉

C₂³

yes

characteristic

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₈×C₁₉₉

—

C₈×C₁₉₉

—

C₈×C₁₉₉

Q₁₆

yes

complemented

C₂²×C₁₉₉

#127

yes

characteristic

C₄×C₁₉₉

C₂×Q₁₆

yes

—

C₄×C₁₉₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₉

#137

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#1723

—

199

maximal, Sylow

#500

—

199

—

#500

—

199

—

#498

—

199

—

#494

—

199

—

#487

—

199

—

#487

—

199

—

#322

D₁₉₉

yes

characteristic

C₂×Q₁₆

—

398

—

C₄⋊Q₈

—

199

—

C₄⋊Q₈

—

199

—

C₄⋊Q₈

—

199

—

C₄⋊Q₈

—

199

—

#107

—

398

—

#107

—

398

—

#106

—

398

—

#103

—

398

—

#103

—

398

—

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

—

398

—

C₂×Q₈

—

398

—

C₂×Q₈

—

398

—

C₂×Q₈

—

398

—

Q₁₆

—

796

—

C₂×C₈

D₇₉₆

yes

—

C₂×C₈

D₇₉₆

yes

—

C₂×C₈

D₇₉₆

yes

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

100

C₄⋊C₄

—

398

—

101

C₄²

C₂²×D₁₉₉

yes

characteristic

102

Q₈

—

796

—

103

Q₈

—

796

—

104

Q₈

—

796

—

105

Q₈

—

796

—

106

C₂×C₄

—

796

—

107

C₂×C₄

—

796

—

108

C₂×C₄

—

796

—

109

C₂×C₄

—

796

—

110

C₂×C₄

C₂×D₇₉₆

yes

characteristic

111

C₂×C₄

C₂×D₇₉₆

yes

characteristic

112

C₂×C₄

C₂×D₇₉₆

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

Dic₇₉₆

yes

complemented

116

C₂²

#20263952

yes

characteristic

117

C₄

—

1592

—

118

C₄

—

1592

—

119

C₄

—

1592

—

120

C₄

—

1592

—

121

C₄

C₂×Dic₇₉₆

yes

—

122

C₄

#20263983

yes

—

123

C₄

#20263988

yes

—

124

C₂

#27164398

yes

characteristic

125

C₂

#27164407

yes

characteristic

126

C₂

#27164412

yes

characteristic

	subgroup	G/H
center	C₂²	#20263952
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#322×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	#127
hypercenter	#322	D₁₉₉
hypocenter	C₁₉₉	#1723
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#20263952

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#322×C₁₉₉

C₂

C₂²×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
51. C₈×C₁₉₉	Q₁₆	semidirect
63. C₁₉₉	#1723	semidirect
115. C₈	Dic₇₉₆	semidirect

H ≅ G/N

type

51. C₈×C₁₉₉

Q₁₆

semidirect

63. C₁₉₉

#1723

semidirect

115. C₈

Dic₇₉₆

semidirect

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

subgroup	index
#27164423	2
#27164416	2
#27164398	2
#27164369	2
#322×C₁₉₉	2
#1723	199

subgroup

index

#322×C₁₉₉

199

Subgroups — #38768527

Conjugacy classes of subgroups

Complements

Maximal subgroups