Group #38767509 — subgroups

Subgroups — #38767509

Order 25472.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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	#27164566	C₂	1	1	yes	maximal, characteristic
2	#27164070	C₂	1	2	yes	maximal, complemented
3	#27163766	C₂	1	2	yes	maximal, complemented
4	#27163762	C₂	1	1	yes	maximal, characteristic, complemented
5	#331×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691885	C₄	1	2	yes	—
7	#20263955	C₂²	1	1	yes	characteristic
8	#20263942	C₄	1	2	yes	—
9	#20263926	—	2	2	no	—
10	D₄×Dic₁₉₉	—	2	2	no	—
11	C₄²×D₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₉₉⋊C₈)	C₂²	1	2	yes	—
13	(C₂×D₁₉₉)⋊C₈	—	2	2	no	—
14	C₈×Dic₁₉₉	—	2	2	no	—
15	#127×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₉₉	C₂²	1	2	yes	—
17	#15691885	—	4	4	no	—
18	C₂×Dic₃₉₈	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊D₁₉₉	—	2	2	no	—
20	C₂²×Dic₁₉₉	—	2	2	no	—
21	C₂×C₄×D₁₉₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₉₉	D₄	1	2	yes	—
23	C₂×C₁₉₉⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₉₉	—	2	2	no	—
25	C₂²⋊Dic₁₉₉	—	2	4	no	—
26	C₄×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₉₉	D₄	1	2	yes	—
28	C₂×D₄×C₁₉₉	—	2	2	no	—
29	C₂×D₄×C₁₉₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₉₉	—	2	2	no	—
31	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₃₉₈	—	4	2	no	—
34	C₂²⋊D₁₉₉	—	4	4	no	—
35	C₂×Dic₁₉₉	—	2	2	no	—
36	C₂×Dic₁₉₉	—	4	4	no	—
37	C₂×Dic₁₉₉	C₂²⋊C₄	1	2	yes	—
38	C₂×Dic₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×D₁₉₉	—	2	2	no	—
40	C₄×D₁₉₉	—	2	2	no	—
41	C₄×D₁₉₉	—	2	2	no	—
42	C₁₉₉⋊C₈	—	4	2	no	—
43	C₂³×C₁₉₉	—	2	2	no	—
44	D₄×C₁₉₉	—	4	2	no	—
45	D₄×C₁₉₉	—	4	2	no	—
46	D₄×C₁₉₉	—	4	2	no	—
47	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
49	C₈×C₁₉₉	—	4	2	no	—
50	C₂²×C₁₉₉	—	4	2	no	—
51	C₂²×C₁₉₉	—	4	4	no	—
52	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂×D₁₉₉	—	2	1	no	—
54	C₂×D₁₉₉	—	2	2	no	—
55	C₄×C₁₉₉	—	2	1	no	—
56	C₄×C₁₉₉	#135	1	2	yes	—
57	C₄×C₁₉₉	#135	1	2	yes	—
58	Dic₁₉₉	—	2	2	no	—
59	Dic₁₉₉	—	4	2	no	—
60	Dic₁₉₉	#100	1	2	yes	—
61	C₂×C₁₉₉	—	8	2	no	—
62	C₂×C₁₉₉	C₂×#100	1	1	yes	characteristic
63	C₂×C₁₉₉	#416	1	2	yes	—
64	D₁₉₉	—	4	1	no	—
65	C₁₉₉	#1543	1	1	yes	characteristic, complemented, Sylow
66	#1543	—	199	1	no	maximal, Sylow
67	#532	—	199	1	no	—
68	#431	—	199	2	no	—
69	#333	—	199	2	no	—
70	#331	—	199	1	no	—
71	#331	D₁₉₉	1	1	yes	characteristic, complemented
72	C₂×#41	—	199	2	no	—
73	C₄⋊Q₈	—	199	1	no	—
74	#127	C₂×D₁₉₉	1	1	yes	characteristic
75	#124	—	199	2	no	—
76	#121	—	398	2	no	—
77	C₄×D₄	—	398	2	no	—
78	C₂×C₄²	—	199	1	no	—
79	C₄⋊C₈	—	199	2	no	—
80	C₄⋊C₈	C₂×D₁₉₉	1	2	yes	—
81	C₂²⋊C₈	—	398	2	no	—
82	C₄×C₈	—	398	2	no	—
83	#41	—	796	4	no	—
84	C₂×Q₈	—	199	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	398	2	no	—
87	C₂×D₄	C₄×D₁₉₉	1	2	yes	—
88	C₂²×C₄	—	398	2	no	—
89	C₂²×C₄	—	199	1	no	—
90	C₂²×C₄	—	199	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	398	2	no	—
93	C₄⋊C₄	—	398	2	no	—
94	C₂²⋊C₄	—	398	4	no	—
95	C₄²	—	199	1	no	—
96	C₄²	—	199	2	no	—
97	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	199	1	no	—
100	Q₈	—	796	2	no	—
101	D₄	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	796	4	no	—
105	C₂×C₄	—	796	4	no	—
106	C₂×C₄	—	398	2	no	—
107	C₂×C₄	—	398	2	no	—
108	C₂×C₄	—	398	2	no	—
109	C₂×C₄	—	398	2	no	—
110	C₂×C₄	—	199	2	no	—
111	C₂×C₄	—	199	1	no	—
112	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
113	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	796	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	4	no	—
118	C₂²	—	398	1	no	—
119	C₂²	—	398	2	no	—
120	C₂²	C₂²⋊C₄×D₁₉₉	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	796	2	no	—
123	C₄	—	398	2	no	—
124	C₄	—	199	2	no	—
125	C₄	#20263974	1	2	yes	—
126	C₄	#20263976	1	2	yes	—
127	C₂	—	8	2	no	—
128	C₂	—	796	1	no	—
129	C₂	#100×D₁₉₉	1	1	yes	characteristic
130	C₂	#27164011	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27164566

C₂

yes

maximal, characteristic

#27164070

C₂

yes

maximal, complemented

#27163766

C₂

yes

maximal, complemented

#27163762

C₂

yes

maximal, characteristic, complemented

#331×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×#15691885

C₄

yes

—

#20263955

C₂²

yes

characteristic

#20263942

C₄

yes

—

#20263926

—

D₄×Dic₁₉₉

—

C₄²×D₁₉₉

C₂²

yes

characteristic

C₄⋊(C₁₉₉⋊C₈)

C₂²

yes

—

(C₂×D₁₉₉)⋊C₈

—

C₈×Dic₁₉₉

—

#127×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₉

C₂²

yes

—

#15691885

—

C₂×Dic₃₉₈

C₂×C₄

yes

—

C₂×C₂²⋊D₁₉₉

—

C₂²×Dic₁₉₉

—

C₂×C₄×D₁₉₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₉₉

D₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

—

C₄×Dic₁₉₉

C₂×C₄

yes

characteristic

C₄×Dic₁₉₉

D₄

yes

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

C₂×C₄

yes

—

C₂×C₈×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

C₂²×D₁₉₉

C₂²⋊C₄

yes

characteristic

Dic₃₉₈

—

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

C₂²⋊C₄

yes

—

C₂×Dic₁₉₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#135

yes

—

C₄×C₁₉₉

#135

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

#100

yes

—

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂×#100

yes

characteristic

C₂×C₁₉₉

#416

yes

—

D₁₉₉

—

C₁₉₉

#1543

yes

characteristic, complemented, Sylow

#1543

—

199

maximal, Sylow

#532

—

199

—

#431

—

199

—

#333

—

199

—

#331

—

199

—

#331

D₁₉₉

yes

characteristic, complemented

C₂×#41

—

199

—

C₄⋊Q₈

—

199

—

#127

C₂×D₁₉₉

yes

characteristic

#124

—

199

—

#121

—

398

—

C₄×D₄

—

398

—

C₂×C₄²

—

199

—

C₄⋊C₈

—

199

—

C₄⋊C₈

C₂×D₁₉₉

yes

—

C₂²⋊C₈

—

398

—

C₄×C₈

—

398

—

#41

—

796

—

C₂×Q₈

—

199

—

C₂×D₄

—

C₂×D₄

—

398

—

C₂×D₄

C₄×D₁₉₉

yes

—

C₂²×C₄

—

398

—

C₂²×C₄

—

199

—

C₂²×C₄

—

199

—

C₂×C₈

—

C₂×C₈

—

398

—

C₄⋊C₄

—

398

—

C₂²⋊C₄

—

398

—

C₄²

—

199

—

C₄²

—

199

—

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

—

C₂³

—

199

—

100

Q₈

—

796

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

—

796

—

105

C₂×C₄

—

796

—

106

C₂×C₄

—

398

—

107

C₂×C₄

—

398

—

108

C₂×C₄

—

398

—

109

C₂×C₄

—

398

—

110

C₂×C₄

—

199

—

111

C₂×C₄

—

199

—

112

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

113

C₂×C₄

D₄×D₁₉₉

yes

—

114

C₈

—

115

C₈

—

796

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

398

—

119

C₂²

—

398

—

120

C₂²

C₂²⋊C₄×D₁₉₉

yes

characteristic

121

C₄

—

122

C₄

—

796

—

123

C₄

—

398

—

124

C₄

—

199

—

125

C₄

#20263974

yes

—

126

C₄

#20263976

yes

—

127

C₂

—

128

C₂

—

796

—

129

C₂

#100×D₁₉₉

yes

characteristic

130

C₂

#27164011

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₉
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#331×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×C₂²⋊C₄
prime radical	#27164566	C₂
cosocle	C₄²×C₁₉₉	C₂³
hypercenter	#331	D₁₉₉
hypocenter	C₁₉₉	#1543
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₉

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#331×C₁₉₉

C₂

socle

C₂²×C₁₉₉

C₂×C₂²⋊C₄

prime radical

#27164566

C₂

cosocle

C₄²×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #27164070	C₂	semidirect
3. #27163766	C₂	semidirect
4. #27163762	C₂	semidirect
5. #331×C₁₉₉	C₂	semidirect
65. C₁₉₉	#1543	semidirect
71. #331	D₁₉₉	semidirect

H ≅ G/N

type

2. #27164070

C₂

semidirect

3. #27163766

C₂

semidirect

4. #27163762

C₂

semidirect

5. #331×C₁₉₉

C₂

semidirect

65. C₁₉₉

#1543

semidirect

71. #331

D₁₉₉

semidirect

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

subgroup	index
#27164566	2
#27164070	2
#27163766	2
#27163762	2
#331×C₁₉₉	2
#1543	199

subgroup

index

#331×C₁₉₉

199

Subgroups — #38767509

Conjugacy classes of subgroups

Complements

Maximal subgroups