Group #38767143 — subgroups

Subgroups — #38767143

Order 25472.

196 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 117 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	#27164071	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#27163740	C₂	1	2	yes	maximal, complemented
4	#27163739	C₂	1	2	yes	maximal, complemented
5	#431×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₉⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₁₉₉⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₁₉₉⋊C₈)	C₄	1	2	yes	—
9	C₄×C₁₉₉⋊C₈	C₄	1	2	yes	—
10	C₄×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₁₉₉⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₉₉	C₄	1	2	yes	—
15	C₂²⋊C₈×C₁₉₉	C₄	1	2	yes	—
16	C₄×C₈×C₁₉₉	C₂²	1	2	yes	—
17	C₂×C₁₉₉⋊C₈	—	2	4	no	—
18	C₂×C₁₉₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₁₉₉⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₉₉⋊C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₉₉	C₂×C₄	1	4	yes	—
25	C₄²×C₁₉₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₉₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₉₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₁₉₉⋊C₈	—	2	2	no	—
30	C₁₉₉⋊C₈	—	2	2	no	—
31	C₁₉₉⋊C₈	—	2	4	no	—
32	C₂³×C₁₉₉	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₉₉	—	2	2	no	—
34	C₂×C₄×C₁₉₉	—	2	2	no	—
35	C₂×C₄×C₁₉₉	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₁₉₉	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₁₉₉	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₉₉	M₁₆	1	4	yes	—
39	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
42	C₈×C₁₉₉	—	2	4	no	—
43	C₂²×C₁₉₉	—	2	1	no	—
44	C₂²×C₁₉₉	—	2	1	no	—
45	C₂²×C₁₉₉	#97	1	2	yes	—
46	C₂²×C₁₉₉	C₂×C₄²	1	1	yes	characteristic
47	C₄×C₁₉₉	—	2	1	no	—
48	C₄×C₁₉₉	—	2	1	no	—
49	C₄×C₁₉₉	#97	1	2	yes	—
50	C₄×C₁₉₉	C₂×M₁₆	1	2	yes	—
51	C₄×C₁₉₉	#131	1	2	yes	—
52	C₄×C₁₉₉	#131	1	2	yes	—
53	C₂×C₁₉₉	—	2	2	no	—
54	C₂×C₁₉₉	C₂×#97	1	1	yes	characteristic
55	C₂×C₁₉₉	#405	1	1	yes	characteristic
56	C₂×C₁₉₉	#405	1	1	yes	characteristic
57	C₁₉₉	#1452	1	1	yes	characteristic, complemented, Sylow
58	#1452	—	199	1	no	maximal, Sylow
59	#431	—	199	1	no	—
60	#431	D₁₉₉	1	1	yes	characteristic
61	C₂×C₄×C₈	—	199	1	no	—
62	#322	—	199	2	no	—
63	#322	—	199	2	no	—
64	C₂²×C₈	—	199	2	no	—
65	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
66	C₄⋊C₈	—	199	2	no	—
67	C₄⋊C₈	Dic₁₉₉	1	2	yes	—
68	C₂²⋊C₈	—	199	2	no	—
69	C₂²⋊C₈	Dic₁₉₉	1	2	yes	—
70	C₄×C₈	—	199	2	no	—
71	C₄×C₈	—	199	1	no	—
72	C₄×C₈	—	199	1	no	—
73	C₄×C₈	—	199	2	no	—
74	C₄×C₈	C₂×D₁₉₉	1	2	yes	—
75	C₂²×C₄	C₄×D₁₉₉	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₁₉₉	1	1	yes	characteristic
77	C₂²×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
78	C₂×C₈	—	398	4	no	—
79	C₂×C₈	—	199	2	no	—
80	C₂×C₈	—	199	2	no	—
81	C₂×C₈	—	199	4	no	—
82	C₂×C₈	C₂×Dic₁₉₉	1	4	yes	—
83	C₄²	C₄×D₁₉₉	1	1	yes	characteristic
84	C₄²	C₄×D₁₉₉	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₉₉	1	1	yes	characteristic
86	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
87	C₂³	C₄×Dic₁₉₉	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
91	C₂×C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
93	C₂×C₄	#15691869	1	4	yes	—
94	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
96	C₂×C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
97	C₈	—	2	4	no	—
98	C₈	—	398	2	no	—
99	C₈	—	398	2	no	—
100	C₈	—	398	4	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	#20263865	1	2	yes	—
104	C₂²	C₂×C₄×Dic₁₉₉	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#20263865	1	2	yes	—
108	C₄	C₂×#15691869	1	2	yes	—
109	C₄	#20263962	1	2	yes	—
110	C₄	#20263963	1	2	yes	—
111	C₂	—	2	2	no	—
112	C₂	C₂×#20263865	1	1	yes	characteristic
113	C₂	#27163974	1	1	yes	characteristic
114	C₂	#27163976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164071

C₂

yes

maximal, characteristic

C₂×C₄×C₁₉₉⋊C₈

C₂

yes

maximal, characteristic

#27163740

C₂

yes

maximal, complemented

#27163739

C₂

yes

maximal, complemented

#431×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₄

yes

—

C₄⋊(C₁₉₉⋊C₈)

C₄

yes

—

C₂²⋊(C₁₉₉⋊C₈)

C₄

yes

—

C₄×C₁₉₉⋊C₈

C₄

yes

—

C₄×C₁₉₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₉₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₉

C₄

yes

—

C₂²⋊C₈×C₁₉₉

C₄

yes

—

C₄×C₈×C₁₉₉

C₂²

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

C₂×C₄

yes

characteristic

C₄²×C₁₉₉

C₂×C₄

yes

characteristic

C₄²×C₁₉₉

C₂×C₄

yes

characteristic

C₄²×C₁₉₉

C₂³

yes

characteristic

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₄²

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₄²

yes

characteristic

C₂×C₄×C₁₉₉

C₄²

yes

characteristic

C₂×C₄×C₁₉₉

C₄²

yes

characteristic

C₂×C₄×C₁₉₉

M₁₆

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#97

yes

—

C₂²×C₁₉₉

C₂×C₄²

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#97

yes

—

C₄×C₁₉₉

C₂×M₁₆

yes

—

C₄×C₁₉₉

#131

yes

—

C₄×C₁₉₉

#131

yes

—

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂×#97

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#1452

—

199

maximal, Sylow

#431

—

199

—

#431

D₁₉₉

yes

characteristic

C₂×C₄×C₈

—

199

—

#322

—

199

—

#322

—

199

—

C₂²×C₈

—

199

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₄⋊C₈

—

199

—

C₄⋊C₈

Dic₁₉₉

yes

—

C₂²⋊C₈

—

199

—

C₂²⋊C₈

Dic₁₉₉

yes

—

C₄×C₈

—

199

—

C₄×C₈

—

199

—

C₄×C₈

—

199

—

C₄×C₈

—

199

—

C₄×C₈

C₂×D₁₉₉

yes

—

C₂²×C₄

C₄×D₁₉₉

yes

characteristic

C₂²×C₄

C₄×D₁₉₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₉

yes

characteristic

C₂×C₈

—

398

—

C₂×C₈

—

199

—

C₂×C₈

—

199

—

C₂×C₈

—

199

—

C₂×C₈

C₂×Dic₁₉₉

yes

—

C₄²

C₄×D₁₉₉

yes

characteristic

C₄²

C₄×D₁₉₉

yes

characteristic

C₄²

C₂×Dic₁₉₉

yes

characteristic

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

C₄×Dic₁₉₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₉₉

yes

characteristic

C₂×C₄

C₄×Dic₁₉₉

yes

characteristic

C₂×C₄

C₄×Dic₁₉₉

yes

characteristic

C₂×C₄

#15691869

yes

—

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

C₂×C₄

C₂²×Dic₁₉₉

yes

characteristic

C₈

—

C₈

—

398

—

C₈

—

398

—

100

C₈

—

398

—

101

C₂²

—

102

C₂²

—

103

C₂²

#20263865

yes

—

104

C₂²

C₂×C₄×Dic₁₉₉

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

#20263865

yes

—

108

C₄

C₂×#15691869

yes

—

109

C₄

#20263962

yes

—

110

C₄

#20263963

yes

—

111

C₂

—

112

C₂

C₂×#20263865

yes

characteristic

113

C₂

#27163974

yes

characteristic

114

C₂

#27163976

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₉₉
commutator, socle	C₂²×C₁₉₉	C₂×C₄²
Fitting	#431×C₁₉₉	C₂
prime radical	C₂³×C₁₉₉	C₄²
cosocle	C₄²×C₁₉₉	C₂³
hypercenter	#431	D₁₉₉
hypocenter	C₁₉₉	#1452
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₉₉

commutator, socle

C₂²×C₁₉₉

C₂×C₄²

Fitting

#431×C₁₉₉

C₂

prime radical

C₂³×C₁₉₉

C₄²

cosocle

C₄²×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂²×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #27163740	C₂	semidirect
4. #27163739	C₂	semidirect
57. C₁₉₉	#1452	semidirect

H ≅ G/N

type

3. #27163740

C₂

semidirect

4. #27163739

C₂

semidirect

57. C₁₉₉

#1452

semidirect

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

subgroup	index
#27164071	2
C₂×C₄×C₁₉₉⋊C₈	2
#27163740	2
#27163739	2
#431×C₁₉₉	2
#1452	199

subgroup

index

C₂×C₄×C₁₉₉⋊C₈

#431×C₁₉₉

199

Subgroups — #38767143

Conjugacy classes of subgroups

Complements

Maximal subgroups