Group #27164588 — subgroups

Subgroups — #27164588

Order 12736.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂×#15691885	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263955	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263953	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263942	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263937	C₂	1	4	yes	maximal, complemented
7	#20263932	C₂	1	4	yes	maximal, complemented
8	#20263912	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#15691885	—	2	4	no	—
11	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₉₈	C₂²	1	2	yes	—
14	Q₈×D₁₉₉	—	2	4	no	—
15	C₂×C₂²⋊D₁₉₉	C₂²	1	2	yes	—
16	C₂²×Dic₁₉₉	C₂²	1	2	yes	—
17	C₂×C₄×D₁₉₉	C₂²	1	2	yes	—
18	C₂×C₄×D₁₉₉	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₉₉	C₂²	1	4	yes	—
20	Dic₁₉₉⋊C₄	C₂²	1	4	yes	—
21	Dic₁₉₉⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₉₉	C₂²	1	4	yes	—
23	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
24	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₁₉₉	C₂²	1	4	yes	—
28	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
29	C₂²×D₁₉₉	C₂³	1	1	yes	characteristic
30	Dic₃₉₈	—	2	2	no	—
31	Dic₃₉₈	—	2	2	no	—
32	Dic₃₉₈	—	2	4	no	—
33	C₂²⋊D₁₉₉	—	2	4	no	—
34	C₂×Dic₁₉₉	—	2	4	no	—
35	C₂×Dic₁₉₉	C₂³	1	2	yes	—
36	C₂×Dic₁₉₉	C₂³	1	2	yes	—
37	C₂×Dic₁₉₉	C₂³	1	2	yes	—
38	C₂×Dic₁₉₉	C₂³	1	1	yes	characteristic
39	C₄×D₁₉₉	—	2	4	no	—
40	C₄×D₁₉₉	D₄	1	4	yes	—
41	C₂³×C₁₉₉	C₂³	1	2	yes	—
42	Q₈×C₁₉₉	—	2	2	no	—
43	D₄×C₁₉₉	—	2	2	no	—
44	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
45	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
46	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
47	C₂²×C₁₉₉	—	2	2	no	—
48	C₂²×C₁₉₉	—	2	4	no	—
49	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
50	C₂×D₁₉₉	—	2	2	no	—
51	C₂×D₁₉₉	C₂×D₄	1	2	yes	—
52	C₄×C₁₉₉	—	2	2	no	—
53	C₄×C₁₉₉	—	2	2	no	—
54	C₄×C₁₉₉	C₂×D₄	1	2	yes	—
55	Dic₁₉₉	—	2	2	no	—
56	Dic₁₉₉	—	2	2	no	—
57	Dic₁₉₉	—	2	2	no	—
58	Dic₁₉₉	C₂×D₄	1	2	yes	—
59	C₂×C₁₉₉	—	4	2	no	—
60	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₁₉₉	2₋¹⁺⁴	1	2	yes	—
62	D₁₉₉	—	2	2	no	—
63	C₁₉₉	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	199	1	no	maximal, Sylow
65	C₂×#41	—	199	1	no	—
66	C₂²×Q₈	—	199	1	no	—
67	C₄⋊Q₈	—	199	1	no	—
68	C₄⋊Q₈	—	199	1	no	—
69	#124	—	199	1	no	—
70	#124	D₁₉₉	1	1	yes	characteristic, complemented
71	#123	—	199	4	no	—
72	C₂²⋊Q₈	—	199	4	no	—
73	#117	—	199	1	no	—
74	#41	—	398	4	no	—
75	C₂×Q₈	—	398	4	no	—
76	C₂×Q₈	—	199	2	no	—
77	C₂×Q₈	—	199	1	no	—
78	C₂×Q₈	—	199	1	no	—
79	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
80	C₂×D₄	—	199	2	no	—
81	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
82	C₂²×C₄	—	199	2	no	—
83	C₂²×C₄	—	199	2	no	—
84	C₂²×C₄	—	199	1	no	—
85	C₄⋊C₄	—	199	2	no	—
86	C₄⋊C₄	—	199	4	no	—
87	C₄⋊C₄	—	199	4	no	—
88	C₂²⋊C₄	—	199	4	no	—
89	C₂²⋊C₄	—	199	2	no	—
90	C₂²⋊C₄	C₂×D₁₉₉	1	4	yes	—
91	C₄²	—	199	1	no	—
92	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
93	C₂³	—	199	1	no	—
94	C₂³	C₂²×D₁₉₉	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	398	4	no	—
97	Q₈	—	398	2	no	—
98	Q₈	—	398	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	398	4	no	—
101	C₂×C₄	—	398	4	no	—
102	C₂×C₄	—	398	4	no	—
103	C₂×C₄	—	199	2	no	—
104	C₂×C₄	—	199	2	no	—
105	C₂×C₄	—	199	2	no	—
106	C₂×C₄	—	199	1	no	—
107	C₂×C₄	—	199	4	no	—
108	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
109	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
110	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	398	2	no	—
114	C₂²	—	199	2	no	—
115	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	398	2	no	—
119	C₄	—	398	2	no	—
120	C₄	—	398	2	no	—
121	C₄	—	199	2	no	—
122	C₄	D₄×D₁₉₉	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	398	2	no	—
125	C₂	C₂×D₄×D₁₉₉	1	1	yes	characteristic
126	C₂	#20264002	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15691885

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₉₉

C₂

yes

maximal, characteristic, complemented

#20263955

C₂

yes

maximal, characteristic, complemented

#20263953

C₂

yes

maximal, characteristic, complemented

#20263942

C₂

yes

maximal, characteristic, complemented

#20263937

C₂

yes

maximal, complemented

#20263932

C₂

yes

maximal, complemented

#20263912

C₂

yes

maximal, characteristic, complemented

#124×C₁₉₉

C₂

yes

maximal, characteristic, complemented

#15691885

—

C₂×Dic₃₉₈

C₂²

yes

characteristic

C₂×Dic₃₉₈

C₂²

yes

characteristic

C₂×Dic₃₉₈

C₂²

yes

—

Q₈×D₁₉₉

—

C₂×C₂²⋊D₁₉₉

C₂²

yes

—

C₂²×Dic₁₉₉

C₂²

yes

—

C₂×C₄×D₁₉₉

C₂²

yes

—

C₂×C₄×D₁₉₉

C₂²

yes

characteristic

C₄⋊Dic₁₉₉

C₂²

yes

—

Dic₁₉₉⋊C₄

C₂²

yes

—

Dic₁₉₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₉₉

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

C₄×Dic₁₉₉

C₂²

yes

characteristic

C₂×Q₈×C₁₉₉

C₂²

yes

characteristic

C₂×D₄×C₁₉₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₉₉

C₂²

yes

—

C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×D₁₉₉

C₂³

yes

characteristic

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

—

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

characteristic

C₄×D₁₉₉

—

C₄×D₁₉₉

D₄

yes

—

C₂³×C₁₉₉

C₂³

yes

—

Q₈×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂⁴

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

C₂×D₄

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×D₄

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

C₂×D₄

yes

—

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂²×D₄

yes

characteristic

C₂×C₁₉₉

2₋¹⁺⁴

yes

—

D₁₉₉

—

C₁₉₉

#536

yes

characteristic, complemented, Sylow

#536

—

199

maximal, Sylow

C₂×#41

—

199

—

C₂²×Q₈

—

199

—

C₄⋊Q₈

—

199

—

C₄⋊Q₈

—

199

—

#124

—

199

—

#124

D₁₉₉

yes

characteristic, complemented

#123

—

199

—

C₂²⋊Q₈

—

199

—

#117

—

199

—

#41

—

398

—

C₂×Q₈

—

398

—

C₂×Q₈

—

199

—

C₂×Q₈

—

199

—

C₂×Q₈

—

199

—

C₂×Q₈

C₂×D₁₉₉

yes

characteristic

C₂×D₄

—

199

—

C₂×D₄

C₂×D₁₉₉

yes

characteristic

C₂²×C₄

—

199

—

C₂²×C₄

—

199

—

C₂²×C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

C₂×D₁₉₉

yes

—

C₄²

—

199

—

C₄²

C₂×D₁₉₉

yes

characteristic

C₂³

—

199

—

C₂³

C₂²×D₁₉₉

yes

—

Q₈

—

Q₈

—

398

—

Q₈

—

398

—

Q₈

—

398

—

D₄

—

100

D₄

—

398

—

101

C₂×C₄

—

398

—

102

C₂×C₄

—

398

—

103

C₂×C₄

—

199

—

104

C₂×C₄

—

199

—

105

C₂×C₄

—

199

—

106

C₂×C₄

—

199

—

107

C₂×C₄

—

199

—

108

C₂×C₄

C₂²×D₁₉₉

yes

—

109

C₂×C₄

C₂²×D₁₉₉

yes

—

110

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

398

—

114

C₂²

—

199

—

115

C₂²

C₂³×D₁₉₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

398

—

119

C₄

—

398

—

120

C₄

—

398

—

121

C₄

—

199

—

122

C₄

D₄×D₁₉₉

yes

—

123

C₂

—

124

C₂

—

398

—

125

C₂

C₂×D₄×D₁₉₉

yes

characteristic

126

C₂

#20264002

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Fitting	#124×C₁₉₉	C₂
prime radical	C₂×#15691885	C₂
hypercenter	#124	D₁₉₉
hypocenter	C₁₉₉	#536
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉₉

commutator, socle, cosocle

C₂²×C₁₉₉

C₂⁴

Fitting

#124×C₁₉₉

C₂

C₂

D₁₉₉

C₁₉₉

G ▷ C₂²×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₁₉₉	C₂	semidirect
3. #20263955	C₂	semidirect
4. #20263953	C₂	semidirect
5. #20263942	C₂	semidirect
6. #20263937	C₂	semidirect
7. #20263932	C₂	semidirect
8. #20263912	C₂	semidirect
9. #124×C₁₉₉	C₂	semidirect
63. C₁₉₉	#536	semidirect
70. #124	D₁₉₉	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₁₉₉

C₂

semidirect

3. #20263955

C₂

semidirect

4. #20263953

C₂

semidirect

5. #20263942

C₂

semidirect

6. #20263937

C₂

semidirect

7. #20263932

C₂

semidirect

8. #20263912

C₂

semidirect

9. #124×C₁₉₉

C₂

semidirect

63. C₁₉₉

#536

semidirect

70. #124

D₁₉₉

semidirect

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

subgroup	index
C₂×#15691885	2
C₂×Q₈×D₁₉₉	2
#20263955	2
#20263953	2
#20263942	2
#20263937	2
#20263932	2
#20263912	2
#124×C₁₉₉	2
#536	199

subgroup

index

C₂×Q₈×D₁₉₉

#124×C₁₉₉

199

Subgroups — #27164588

Conjugacy classes of subgroups

Complements

Maximal subgroups