Group #38767926 — subgroups

Subgroups — #38767926

Order 25472.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27164748	C₂	1	1	yes	maximal, characteristic, complemented
2	#27164215	C₂	1	2	yes	maximal
3	#27163767	C₂	1	1	yes	maximal, characteristic, complemented
4	#27163750	C₂	1	2	yes	maximal, complemented
5	#492×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691874	—	2	2	no	—
7	#20263954	—	2	2	no	—
8	#20263953	—	2	2	no	—
9	C₄×Dic₃₉₈	C₂²	1	2	yes	—
10	#20263920	—	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×Dic₃₉₈	—	2	2	no	—
18	C₂×Dic₃₉₈	D₄	1	2	yes	—
19	#15691874	—	4	4	no	—
20	C₂×C₁₉₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₉₉	D₄	1	2	yes	—
22	Dic₁₉₉⋊C₄	—	2	4	no	—
23	Dic₁₉₉⋊C₄	—	2	2	no	—
24	C₄×Dic₁₉₉	—	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	Dic₃₉₈	—	2	2	no	—
34	Dic₃₉₈	—	8	2	no	—
35	Dic₃₉₈	SD₁₆	1	4	yes	complemented
36	C₂×Dic₁₉₉	—	2	2	no	—
37	C₂×Dic₁₉₉	—	2	2	no	—
38	C₁₉₉⋊C₈	—	4	2	no	—
39	C₂³×C₁₉₉	—	4	1	no	—
40	Q₈×C₁₉₉	—	4	2	no	—
41	D₄×C₁₉₉	—	4	2	no	—
42	D₄×C₁₉₉	—	4	2	no	—
43	D₄×C₁₉₉	—	4	2	no	—
44	C₂×C₄×C₁₉₉	—	2	2	no	—
45	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₉₉	—	2	2	no	—
48	C₂²×C₁₉₉	—	8	2	no	—
49	C₂²×C₁₉₉	—	8	1	no	—
50	C₂²×C₁₉₉	#120	1	1	yes	characteristic
51	C₄×C₁₉₉	—	2	1	no	—
52	C₄×C₁₉₉	—	4	2	no	—
53	C₄×C₁₉₉	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₉₉	#137	1	2	yes	—
55	Dic₁₉₉	—	2	4	no	—
56	Dic₁₉₉	—	4	2	no	—
57	C₂×C₁₉₉	—	16	1	no	—
58	C₂×C₁₉₉	#448	1	2	yes	—
59	C₂×C₁₉₉	#453	1	1	yes	characteristic
60	C₁₉₉	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	199	1	no	maximal, Sylow
62	Q₈²	—	199	1	no	—
63	#492	D₁₉₉	1	1	yes	characteristic
64	#460	—	199	2	no	—
65	#333	—	199	1	no	—
66	Q₈⋊C₈	—	199	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	398	2	no	—
69	C₄⋊Q₈	—	398	2	no	—
70	C₄⋊Q₈	—	398	2	no	—
71	C₄⋊Q₈	C₂×D₁₉₉	1	1	yes	characteristic
72	#127	C₂×D₁₉₉	1	1	yes	characteristic
73	C₄×Q₈	—	398	2	no	—
74	C₄×Q₈	—	199	2	no	—
75	C₄⋊C₈	—	199	2	no	—
76	#102	—	398	2	no	—
77	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
78	C₂×Q₈	—	398	2	no	—
79	C₂×Q₈	—	199	2	no	—
80	C₂×Q₈	C₂²⋊D₁₉₉	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	796	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	398	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	398	4	no	—
89	C₄⋊C₄	—	398	2	no	—
90	C₄⋊C₄	—	199	2	no	—
91	C₄²	—	398	2	no	—
92	C₄²	—	398	2	no	—
93	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	1592	2	no	—
97	Q₈	—	398	2	no	—
98	Q₈	—	199	4	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	398	2	no	—
104	C₂×C₄	—	398	2	no	—
105	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	796	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	2	no	—
111	C₂²	#20263923	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	796	2	no	—
115	C₄	—	398	4	no	—
116	C₄	SD₁₆×D₁₉₉	1	2	yes	—
117	C₄	#20263984	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#27164155	1	2	yes	—
120	C₂	#27164185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164748

C₂

yes

maximal, characteristic, complemented

#27164215

C₂

yes

maximal

#27163767

C₂

yes

maximal, characteristic, complemented

#27163750

C₂

yes

maximal, complemented

#492×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691874

—

#20263954

—

#20263953

—

C₄×Dic₃₉₈

C₂²

yes

—

#20263920

—

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×Dic₃₉₈

—

C₂×Dic₃₉₈

D₄

yes

—

#15691874

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

D₄

yes

—

Dic₁₉₉⋊C₄

—

Dic₁₉₉⋊C₄

—

C₄×Dic₁₉₉

—

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

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

SD₁₆

yes

complemented

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

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

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×SD₁₆

yes

—

C₄×C₁₉₉

#137

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#448

yes

—

C₂×C₁₉₉

#453

yes

characteristic

C₁₉₉

#1623

yes

characteristic, complemented, Sylow

#1623

—

199

maximal, Sylow

Q₈²

—

199

—

#492

D₁₉₉

yes

characteristic

#460

—

199

—

#333

—

199

—

Q₈⋊C₈

—

199

—

C₂×SD₁₆

—

C₂×SD₁₆

—

398

—

C₄⋊Q₈

—

398

—

C₄⋊Q₈

—

398

—

C₄⋊Q₈

C₂×D₁₉₉

yes

characteristic

#127

C₂×D₁₉₉

yes

characteristic

C₄×Q₈

—

398

—

C₄×Q₈

—

199

—

C₄⋊C₈

—

199

—

#102

—

398

—

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

—

398

—

C₂×Q₈

—

199

—

C₂×Q₈

C₂²⋊D₁₉₉

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

796

—

C₂×C₈

—

C₂×C₈

—

398

—

C₄⋊C₄

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

199

—

C₄²

—

398

—

C₄²

—

398

—

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

1592

—

Q₈

—

398

—

Q₈

—

199

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

398

—

104

C₂×C₄

—

398

—

105

C₂×C₄

D₄×D₁₉₉

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

107

C₈

—

108

C₈

—

796

—

109

C₂²

—

110

C₂²

—

111

C₂²

#20263923

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

796

—

115

C₄

—

398

—

116

C₄

SD₁₆×D₁₉₉

yes

—

117

C₄

#20263984

yes

—

118

C₂

—

119

C₂

#27164155

yes

—

120

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
prime radical	#127×C₁₉₉	C₂²
hypercenter	#492	D₁₉₉
hypocenter	C₁₉₉	#1623
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

socle

C₂²×C₁₉₉

#120

prime radical

#127×C₁₉₉

C₂²

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #27164748	C₂	semidirect
3. #27163767	C₂	semidirect
4. #27163750	C₂	semidirect
35. Dic₃₉₈	SD₁₆	semidirect
60. C₁₉₉	#1623	semidirect

H ≅ G/N

type

1. #27164748

C₂

semidirect

3. #27163767

C₂

semidirect

4. #27163750

C₂

semidirect

35. Dic₃₉₈

SD₁₆

semidirect

60. C₁₉₉

#1623

semidirect

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

subgroup	index
#27164748	2
#27164215	2
#27163767	2
#27163750	2
#492×C₁₉₉	2
#1623	199

subgroup

index

#492×C₁₉₉

199

Subgroups — #38767926

Conjugacy classes of subgroups

Complements

Maximal subgroups