Group #38769844 — subgroups

Subgroups — #38769844

Order 25472.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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	#27164042	C₂	1	1	yes	maximal, characteristic
2	C₄×#15691870	C₂	1	1	yes	maximal, characteristic
3	#27163787	C₂	1	4	yes	maximal
4	C₂×#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691870	C₂²	1	2	yes	—
6	C₂×#15691870	C₂²	1	2	yes	—
7	C₄⋊(C₁₉₉⋊C₈)	—	2	2	no	—
8	#20263866	—	2	1	no	—
9	C₄×C₁₉₉⋊C₈	—	2	1	no	—
10	#125×C₁₉₉	—	2	1	no	—
11	#125×C₁₉₉	—	2	1	no	—
12	#125×C₁₉₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₉₉	C₄	1	4	yes	—
14	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	#15691870	—	2	2	no	—
17	#15691870	—	2	4	no	—
18	#15691870	Q₈	1	4	yes	—
19	C₂×C₁₉₉⋊C₈	—	2	2	no	—
20	C₂×C₁₉₉⋊C₈	—	2	2	no	—
21	C₂²×C₄×C₁₉₉	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₉₉	—	2	4	no	—
25	C₄⋊C₄×C₁₉₉	—	2	4	no	—
26	C₄⋊C₄×C₁₉₉	—	2	2	no	—
27	C₄⋊C₄×C₁₉₉	—	2	2	no	—
28	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
29	C₄²×C₁₉₉	D₄	1	1	yes	characteristic
30	C₄²×C₁₉₉	D₄	1	2	yes	—
31	C₁₉₉⋊C₈	—	2	4	no	—
32	C₁₉₉⋊C₈	—	4	2	no	—
33	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₉	—	2	4	no	—
35	C₂×C₄×C₁₉₉	—	4	2	no	—
36	C₂×C₄×C₁₉₉	—	4	4	no	—
37	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
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	2	yes	—
42	C₂×C₄×C₁₉₉	#41	1	2	yes	—
43	C₂²×C₁₉₉	—	2	2	no	—
44	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₉₉	C₄×Q₈	1	2	yes	—
46	C₄×C₁₉₉	—	2	2	no	—
47	C₄×C₁₉₉	—	2	2	no	—
48	C₄×C₁₉₉	—	4	4	no	—
49	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₁₉₉	#124	1	1	yes	characteristic
51	C₄×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₁₉₉	—	2	2	no	—
53	C₂×C₁₉₉	#391	1	1	yes	characteristic
54	C₂×C₁₉₉	#413	1	2	yes	—
55	C₁₉₉	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	199	1	no	maximal, Sylow
57	C₂×#125	D₁₉₉	1	1	yes	characteristic
58	#423	—	199	1	no	—
59	C₄×M₁₆	—	199	1	no	—
60	#343	—	199	4	no	—
61	C₂×M₁₆	—	199	2	no	—
62	C₂×M₁₆	—	199	2	no	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	Dic₁₉₉	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
68	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
69	C₄⋊C₈	—	398	2	no	—
70	#97	—	398	1	no	—
71	C₄×C₈	—	398	1	no	—
72	C₂²×C₄	C₂×Dic₁₉₉	1	4	yes	—
73	C₂²×C₄	C₂×Dic₁₉₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
75	M₁₆	—	398	4	no	—
76	M₁₆	—	398	2	no	—
77	M₁₆	—	199	4	no	—
78	C₂×C₈	—	398	2	no	—
79	C₂×C₈	—	398	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄²	C₂²⋊D₁₉₉	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₉₉	1	1	yes	characteristic
86	C₄²	C₂²⋊D₁₉₉	1	2	yes	—
87	C₂³	C₂²×Dic₁₉₉	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₁₉₉	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₁₉₉	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₁₉₉	1	2	yes	—
96	C₂×C₄	#15691887	1	2	yes	—
97	C₈	—	796	2	no	—
98	C₈	—	398	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊Dic₁₉₉	1	1	yes	characteristic
101	C₂²	Q₈×Dic₁₉₉	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#20263931	1	2	yes	—
106	C₄	#20263944	1	1	yes	characteristic
107	C₄	#20263953	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#27163925	1	1	yes	characteristic
110	C₂	#27164002	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27164042

C₂

yes

maximal, characteristic

C₄×#15691870

C₂

yes

maximal, characteristic

#27163787

C₂

yes

maximal

C₂×#125×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691870

C₂²

yes

—

C₂×#15691870

C₂²

yes

—

C₄⋊(C₁₉₉⋊C₈)

—

#20263866

—

C₄×C₁₉₉⋊C₈

—

#125×C₁₉₉

—

#125×C₁₉₉

—

#125×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

C₄

yes

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

—

—

Q₈

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

D₄

yes

characteristic

C₄²×C₁₉₉

D₄

yes

characteristic

C₄²×C₁₉₉

D₄

yes

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₉

C₄×Q₈

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

C₄×C₁₉₉

#124

yes

characteristic

C₄×C₁₉₉

C₄⋊Q₈

yes

characteristic

C₂×C₁₉₉

—

C₂×C₁₉₉

#391

yes

characteristic

C₂×C₁₉₉

#413

yes

—

C₁₉₉

#1998

yes

characteristic, complemented, Sylow

#1998

—

199

maximal, Sylow

C₂×#125

D₁₉₉

yes

characteristic

#423

—

199

—

C₄×M₁₆

—

199

—

#343

—

199

—

C₂×M₁₆

—

199

—

C₂×M₁₆

—

199

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₁₉₉

yes

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₄⋊C₈

—

398

—

#97

—

398

—

C₄×C₈

—

398

—

C₂²×C₄

C₂×Dic₁₉₉

yes

—

C₂²×C₄

C₂×Dic₁₉₉

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

M₁₆

—

398

—

M₁₆

—

398

—

M₁₆

—

199

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₉₉

yes

characteristic

C₄²

C₂²⋊D₁₉₉

yes

characteristic

C₄²

C₂²⋊D₁₉₉

yes

—

C₂³

C₂²×Dic₁₉₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₉₉

yes

—

C₂×C₄

C₂²⋊Dic₁₉₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

C₂×C₄

Q₈×D₁₉₉

yes

—

C₂×C₄

#15691887

yes

—

C₈

—

796

—

C₈

—

398

—

C₂²

—

100

C₂²

C₂×C₂²⋊Dic₁₉₉

yes

characteristic

101

C₂²

Q₈×Dic₁₉₉

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#20263931

yes

—

106

C₄

#20263944

yes

characteristic

107

C₄

#20263953

yes

characteristic

108

C₂

—

109

C₂

#27163925

yes

characteristic

110

C₂

#27164002

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₉₉
commutator, prime radical	C₂³×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₂×#125×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂×#125	D₁₉₉
hypocenter	C₁₉₉	#1998
LCS	G ▷ C₂³×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₉₉

commutator, prime radical

C₂³×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₂×#125×C₁₉₉

C₂

socle

C₂²×C₁₉₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂³×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
55. C₁₉₉	#1998	semidirect

H ≅ G/N

type

55. C₁₉₉

#1998

semidirect

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

subgroup	index
#27164042	2
C₄×#15691870	2
#27163787	2
C₂×#125×C₁₉₉	2
#1998	199

subgroup

index

C₂×#125×C₁₉₉

199

Subgroups — #38769844

Conjugacy classes of subgroups

Complements

Maximal subgroups