Group #38767996 — subgroups

Subgroups — #38767996

Order 25472.

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

normal subgroups only

This group has 218 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
2	#27164397	C₂	1	1	yes	maximal, characteristic, complemented
3	#27164309	C₂	1	2	yes	maximal, complemented
4	#27163745	C₂	1	2	yes	maximal, complemented
5	#333×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691876	—	2	2	no	—
7	#20263955	C₂²	1	1	yes	characteristic
8	#20263944	—	2	2	no	—
9	#20263930	—	2	2	no	—
10	C₄×D₇₉₆	C₂²	1	2	yes	—
11	#20263913	—	2	2	no	—
12	#20263887	—	2	2	no	—
13	#20263877	—	2	2	no	—
14	C₄×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₉₉	C₂²	1	2	yes	—
17	C₂×Dic₃₉₈	—	2	1	no	—
18	C₂×D₇₉₆	—	2	2	no	—
19	C₂×D₇₉₆	D₄	1	2	yes	—
20	C₂×C₄×D₁₉₉	—	2	2	no	—
21	#15691876	—	4	4	no	—
22	C₂×C₁₉₉⋊C₈	—	2	1	no	—
23	C₄⋊Dic₁₉₉	—	2	2	no	—
24	C₄⋊Dic₁₉₉	D₄	1	2	yes	—
25	(C₂×D₁₉₉)⋊C₄	—	2	4	no	—
26	(C₂×D₁₉₉)⋊C₄	—	2	2	no	—
27	C₄×Dic₁₉₉	—	2	2	no	—
28	C₂×Q₈×C₁₉₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₉₉	—	2	2	no	—
30	C₄⋊C₄×C₁₉₉	—	2	2	no	—
31	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₉₉	—	2	2	no	—
33	Dic₃₉₈	—	4	2	no	—
34	D₇₉₆	—	2	2	no	—
35	D₇₉₆	—	2	2	no	—
36	D₇₉₆	—	8	2	no	—
37	C₂×Dic₁₉₉	—	2	2	no	—
38	C₂×Dic₁₉₉	—	4	1	no	—
39	C₄×D₁₉₉	—	4	2	no	—
40	C₄×D₁₉₉	—	4	2	no	—
41	C₁₉₉⋊C₈	—	2	2	no	—
42	Q₈×C₁₉₉	—	4	2	no	—
43	C₂×C₄×C₁₉₉	—	2	2	no	—
44	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₉₉	—	4	2	no	—
47	C₂²×C₁₉₉	#120	1	1	yes	characteristic
48	C₂×D₁₉₉	—	4	2	no	—
49	C₂×D₁₉₉	—	4	2	no	—
50	C₂×D₁₉₉	—	4	2	no	—
51	C₄×C₁₉₉	—	2	1	no	—
52	C₄×C₁₉₉	—	4	2	no	—
53	C₄×C₁₉₉	#135	1	2	yes	—
54	C₄×C₁₉₉	#137	1	2	yes	—
55	Dic₁₉₉	—	4	2	no	—
56	Dic₁₉₉	—	8	1	no	—
57	C₂×C₁₉₉	#452	1	2	yes	—
58	C₂×C₁₉₉	#453	1	1	yes	characteristic
59	D₁₉₉	—	8	2	no	—
60	C₁₉₉	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	199	1	no	maximal, Sylow
62	#555	—	199	1	no	—
63	#494	—	199	1	no	—
64	#476	—	199	2	no	—
65	#333	D₁₉₉	1	1	yes	characteristic, complemented
66	D₄⋊C₈	—	199	2	no	—
67	C₂×Q₁₆	—	398	2	no	—
68	C₄⋊Q₈	—	199	1	no	—
69	C₄⋊Q₈	C₂×D₁₉₉	1	1	yes	characteristic
70	#124	—	398	2	no	—
71	#121	—	398	2	no	—
72	C₄×D₄	—	199	2	no	—
73	#117	—	398	2	no	—
74	#106	—	398	2	no	—
75	C₄⋊C₈	C₂×D₁₉₉	1	2	yes	—
76	#102	—	398	2	no	—
77	C₄×C₈	—	199	1	no	—
78	C₂×Q₈	—	398	1	no	—
79	C₂×Q₈	D₇₉₆	1	2	yes	—
80	C₂×D₄	—	398	2	no	—
81	C₂×D₄	—	199	2	no	—
82	C₂²×C₄	—	398	2	no	—
83	Q₁₆	—	796	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	398	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	398	2	no	—
88	C₄⋊C₄	—	199	2	no	—
89	C₂²⋊C₄	—	398	4	no	—
90	C₂²⋊C₄	—	398	2	no	—
91	C₄²	—	398	2	no	—
92	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
93	C₂³	—	398	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	796	2	no	—
96	D₄	—	1592	2	no	—
97	D₄	—	398	2	no	—
98	D₄	—	398	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	796	2	no	—
101	C₂×C₄	—	796	2	no	—
102	C₂×C₄	—	796	1	no	—
103	C₂×C₄	—	398	2	no	—
104	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
105	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
106	C₈	—	4	2	no	—
107	C₈	—	398	2	no	—
108	C₂²	—	796	2	no	—
109	C₂²	—	796	2	no	—
110	C₂²	—	796	2	no	—
111	C₂²	#20263925	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	796	2	no	—
115	C₄	—	1592	1	no	—
116	C₄	#20263976	1	2	yes	—
117	C₄	#20263987	1	2	yes	—
118	C₂	—	1592	2	no	—
119	C₂	#27164183	1	2	yes	—
120	C₂	#27164188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164734

C₂

yes

maximal, characteristic

#27164397

C₂

yes

maximal, characteristic, complemented

#27164309

C₂

yes

maximal, complemented

#27163745

C₂

yes

maximal, complemented

#333×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×#15691876

—

#20263955

C₂²

yes

characteristic

#20263944

—

#20263930

—

C₄×D₇₉₆

C₂²

yes

—

#20263913

—

#20263887

—

#20263877

—

C₄×C₁₉₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₉

C₂²

yes

—

C₂×Dic₃₉₈

—

C₂×D₇₉₆

—

C₂×D₇₉₆

D₄

yes

—

C₂×C₄×D₁₉₉

—

#15691876

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

D₄

yes

—

(C₂×D₁₉₉)⋊C₄

—

(C₂×D₁₉₉)⋊C₄

—

C₄×Dic₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

C₂²×D₁₉₉

—

Dic₃₉₈

—

D₇₉₆

—

D₇₉₆

—

D₇₉₆

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₁₉₉⋊C₈

—

Q₈×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₈×C₁₉₉

—

C₂²×C₁₉₉

#120

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#135

yes

—

C₄×C₁₉₉

#137

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

#452

yes

—

C₂×C₁₉₉

#453

yes

characteristic

D₁₉₉

—

C₁₉₉

#1634

yes

characteristic, complemented, Sylow

#1634

—

199

maximal, Sylow

#555

—

199

—

#494

—

199

—

#476

—

199

—

#333

D₁₉₉

yes

characteristic, complemented

D₄⋊C₈

—

199

—

C₂×Q₁₆

—

398

—

C₄⋊Q₈

—

199

—

C₄⋊Q₈

C₂×D₁₉₉

yes

characteristic

#124

—

398

—

#121

—

398

—

C₄×D₄

—

199

—

#117

—

398

—

#106

—

398

—

C₄⋊C₈

C₂×D₁₉₉

yes

—

#102

—

398

—

C₄×C₈

—

199

—

C₂×Q₈

—

398

—

C₂×Q₈

D₇₉₆

yes

—

C₂×D₄

—

398

—

C₂×D₄

—

199

—

C₂²×C₄

—

398

—

Q₁₆

—

796

—

C₂×C₈

—

C₂×C₈

—

398

—

C₄⋊C₄

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

199

—

C₂²⋊C₄

—

398

—

C₂²⋊C₄

—

398

—

C₄²

—

398

—

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

—

398

—

Q₈

—

Q₈

—

796

—

D₄

—

1592

—

D₄

—

398

—

D₄

—

398

—

C₂×C₄

—

100

C₂×C₄

—

796

—

101

C₂×C₄

—

796

—

102

C₂×C₄

—

796

—

103

C₂×C₄

—

398

—

104

C₂×C₄

D₄×D₁₉₉

yes

—

105

C₂×C₄

C₂×D₇₉₆

yes

characteristic

106

C₈

—

107

C₈

—

398

—

108

C₂²

—

796

—

109

C₂²

—

796

—

110

C₂²

—

796

—

111

C₂²

#20263925

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

796

—

115

C₄

—

1592

—

116

C₄

#20263976

yes

—

117

C₄

#20263987

yes

—

118

C₂

—

1592

—

119

C₂

#27164183

yes

—

120

C₂

#27164188

yes

characteristic

	subgroup	G/H
center	C₂²	#20263925
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#333×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#120
prime radical	#27164734	C₂
hypercenter	#333	D₁₉₉
hypocenter	C₁₉₉	#1634
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#20263925

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#333×C₁₉₉

C₂

C₂²×C₁₉₉

C₂

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #27164397	C₂	semidirect
3. #27164309	C₂	semidirect
4. #27163745	C₂	semidirect
5. #333×C₁₉₉	C₂	semidirect
60. C₁₉₉	#1634	semidirect
65. #333	D₁₉₉	semidirect

H ≅ G/N

type

2. #27164397

C₂

semidirect

3. #27164309

C₂

semidirect

4. #27163745

C₂

semidirect

5. #333×C₁₉₉

C₂

semidirect

60. C₁₉₉

#1634

semidirect

65. #333

D₁₉₉

semidirect

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

subgroup	index
#27164734	2
#27164397	2
#27164309	2
#27163745	2
#333×C₁₉₉	2
#1634	199

subgroup

index

#333×C₁₉₉

199

Subgroups — #38767996

Conjugacy classes of subgroups

Complements

Maximal subgroups