Group #38769598 — subgroups

Subgroups — #38769598

Order 25472.

208 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20263891	C₂	1	1	yes	maximal, characteristic
2	#27164056	C₂	1	1	yes	maximal, characteristic
3	#27164044	C₂	1	2	yes	maximal
4	#27163778	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691870	C₂²	1	2	yes	—
7	#20263914	C₂²	1	2	yes	—
8	#20263891	—	2	2	no	—
9	#20263889	C₄	1	2	yes	complemented
10	#20263887	C₄	1	2	yes	complemented
11	Dic₁₉₉⋊C₈	—	2	4	no	—
12	C₂²×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₉₉	—	2	2	no	—
16	#15691870	—	2	2	no	—
17	#15691870	—	2	2	no	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₉₉	C₂×C₄	1	4	yes	—
20	C₂²⋊Dic₁₉₉	—	2	2	no	—
21	C₄×Dic₁₉₉	—	2	2	no	—
22	C₂²×C₄×C₁₉₉	—	2	1	no	—
23	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₉₉	—	2	1	no	—
25	C₂×C₈×C₁₉₉	—	2	1	no	—
26	C₂×C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
30	C₂×C₈×C₁₉₉	Q₈	1	2	yes	—
31	C₄²×C₁₉₉	—	2	1	no	—
32	C₄²×C₁₉₉	Q₈	1	2	yes	—
33	C₂×Dic₁₉₉	—	2	4	no	—
34	C₁₉₉⋊C₈	—	4	2	no	—
35	C₂³×C₁₉₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₉₉	—	2	4	no	—
37	C₂×C₄×C₁₉₉	—	2	2	no	—
38	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
44	C₈×C₁₉₉	—	2	2	no	—
45	C₈×C₁₉₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₉₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₉₉	—	2	2	no	—
48	C₂²×C₁₉₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₉₉	#125	1	1	yes	characteristic
51	C₄×C₁₉₉	—	2	4	no	—
52	C₄×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₉₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₉₉	#121	1	1	yes	characteristic
55	C₄×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₉₉	—	4	4	no	—
57	C₂×C₁₉₉	—	2	1	no	—
58	C₂×C₁₉₉	—	2	1	no	—
59	C₂×C₁₉₉	#389	1	1	yes	characteristic
60	C₂×C₁₉₉	#443	1	2	yes	—
61	C₁₉₉	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	199	1	no	maximal, Sylow
63	C₂×#108	—	199	1	no	—
64	#427	—	199	1	no	—
65	#424	—	199	2	no	—
66	C₂×C₄×C₈	D₁₉₉	1	1	yes	characteristic
67	#339	—	199	2	no	—
68	C₂×M₁₆	—	199	2	no	—
69	C₂²×C₈	C₂×D₁₉₉	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₁₉₉	1	1	yes	characteristic
71	#117	—	199	2	no	—
72	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
73	#108	—	398	2	no	—
74	#107	—	199	2	no	—
75	#106	—	199	2	no	—
76	C₄⋊C₈	—	398	4	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
80	M₁₆	—	398	2	no	—
81	M₁₆	—	398	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	398	2	no	—
85	C₂×C₈	C₄×D₁₉₉	1	1	yes	characteristic
86	C₂×C₈	C₄×D₁₉₉	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊D₁₉₉	1	2	yes	—
88	C₂×C₈	D₇₉₆	1	2	yes	—
89	C₂×C₈	Dic₃₉₈	1	2	yes	—
90	C₄⋊C₄	—	199	4	no	—
91	C₂²⋊C₄	—	398	2	no	—
92	C₄²	—	2	1	no	—
93	C₄²	—	398	2	no	—
94	C₄²	Dic₃₉₈	1	2	yes	—
95	C₂³	#15691886	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	398	4	no	—
99	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
102	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
104	C₂×C₄	#15691886	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	796	2	no	—
107	C₈	Dic₁₉₉⋊C₄	1	2	yes	—
108	C₈	Dic₁₉₉⋊C₄	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	C₄×D₇₉₆	1	1	yes	characteristic
111	C₂²	#20263934	1	1	yes	characteristic
112	C₂²	#20263947	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	796	4	no	—
115	C₄	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
116	C₄	C₄×Dic₃₉₈	1	1	yes	characteristic
117	C₄	#20263929	1	1	yes	characteristic
118	C₄	#20263955	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#27163916	1	1	yes	characteristic
122	C₂	#27164136	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263891

C₂

yes

maximal, characteristic

#27164056

C₂

yes

maximal, characteristic

#27164044

C₂

yes

maximal

#27163778

C₂

yes

maximal

C₂×C₄×C₈×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691870

C₂²

yes

—

#20263914

C₂²

yes

—

#20263891

—

#20263889

C₄

yes

complemented

#20263887

C₄

yes

complemented

Dic₁₉₉⋊C₈

—

C₂²×C₈×C₁₉₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

—

#15691870

—

#15691870

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

C₂×C₄

yes

—

C₂²⋊Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₂×C₈×C₁₉₉

Q₈

yes

—

C₄²×C₁₉₉

—

C₄²×C₁₉₉

Q₈

yes

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

#41

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₈×C₁₉₉

—

C₈×C₁₉₉

C₄⋊C₄

yes

—

C₈×C₁₉₉

C₄⋊C₄

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₄×D₄

yes

characteristic

C₂²×C₁₉₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₉

#125

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₉

C₄×Q₈

yes

characteristic

C₄×C₁₉₉

#121

yes

characteristic

C₄×C₁₉₉

C₄⋊Q₈

yes

characteristic

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#389

yes

characteristic

C₂×C₁₉₉

#443

yes

—

C₁₉₉

#1945

yes

characteristic, complemented, Sylow

#1945

—

199

maximal, Sylow

C₂×#108

—

199

—

#427

—

199

—

#424

—

199

—

C₂×C₄×C₈

D₁₉₉

yes

characteristic

#339

—

199

—

C₂×M₁₆

—

199

—

C₂²×C₈

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

C₂×D₁₉₉

yes

characteristic

#117

—

199

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#108

—

398

—

#107

—

199

—

#106

—

199

—

C₄⋊C₈

—

398

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

M₁₆

—

398

—

M₁₆

—

398

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

C₄×D₁₉₉

yes

characteristic

C₂×C₈

C₄×D₁₉₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₉₉

yes

—

C₂×C₈

D₇₉₆

yes

—

C₂×C₈

Dic₃₉₈

yes

—

C₄⋊C₄

—

199

—

C₂²⋊C₄

—

398

—

C₄²

—

C₄²

—

398

—

C₄²

Dic₃₉₈

yes

—

C₂³

#15691886

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

398

—

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

101

C₂×C₄

C₂×D₇₉₆

yes

characteristic

102

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

103

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

104

C₂×C₄

#15691886

yes

characteristic

105

C₈

—

106

C₈

—

796

—

107

C₈

Dic₁₉₉⋊C₄

yes

—

108

C₈

Dic₁₉₉⋊C₄

yes

—

109

C₂²

—

110

C₂²

C₄×D₇₉₆

yes

characteristic

111

C₂²

#20263934

yes

characteristic

112

C₂²

#20263947

yes

characteristic

113

C₄

—

114

C₄

—

796

—

115

C₄

C₂×Dic₁₉₉⋊C₄

yes

characteristic

116

C₄

C₄×Dic₃₉₈

yes

characteristic

117

C₄

#20263929

yes

characteristic

118

C₄

#20263955

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#27163916

yes

characteristic

122

C₂

#27164136

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₉₆
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₂×C₄×C₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₄×D₄
prime radical	C₂³×C₁₉₉	#41
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₉₉
hypocenter	C₁₉₉	#1945
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₇₉₆

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₂×C₄×C₈×C₁₉₉

C₂

socle

C₂²×C₁₉₉

C₄×D₄

prime radical

C₂³×C₁₉₉

#41

cosocle

C₂²×C₄×C₁₉₉

C₂³

hypercenter

C₂×C₄×C₈

D₁₉₉

hypocenter

C₁₉₉

#1945

LCS

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. #20263889	C₄	semidirect
10. #20263887	C₄	semidirect
61. C₁₉₉	#1945	semidirect

H ≅ G/N

type

9. #20263889

C₄

semidirect

10. #20263887

C₄

semidirect

61. C₁₉₉

#1945

semidirect

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

subgroup	index
C₂×#20263891	2
#27164056	2
#27164044	2
#27163778	2
C₂×C₄×C₈×C₁₉₉	2
#1945	199

subgroup

index

C₂×C₄×C₈×C₁₉₉

199

Subgroups — #38769598

Conjugacy classes of subgroups

Complements

Maximal subgroups