Group #38738608 — subgroups

Subgroups — #1674×C₁₉₇ #38738608

Order 25216.

156 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 72 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	#500×C₁₉₇	C₂	1	1	yes	maximal, characteristic
2	#462×C₁₉₇	C₂	1	2	yes	maximal
3	C₈×Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#335×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	#332×C₁₉₇	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₁₉₇	—	2	2	no	—
7	C₄⋊Q₈×C₁₉₇	C₂²	1	2	yes	—
8	C₄×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
9	#107×C₁₉₇	—	2	1	no	—
10	#107×C₁₉₇	—	2	1	no	—
11	C₄⋊C₈×C₁₉₇	—	2	1	no	—
12	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₉₇	C₂²	1	2	yes	—
14	#103×C₁₉₇	—	2	2	no	—
15	C₄×C₈×C₁₉₇	—	2	1	no	—
16	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₉₇	—	2	2	no	—
18	C₂×Q₈×C₁₉₇	D₄	1	1	yes	characteristic
19	Q₁₆×C₁₉₇	—	4	4	no	—
20	C₂×C₈×C₁₉₇	—	2	1	no	—
21	C₂×C₈×C₁₉₇	—	2	2	no	—
22	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₁₉₇	—	2	1	no	—
25	C₄⋊C₄×C₁₉₇	—	2	2	no	—
26	C₄⋊C₄×C₁₉₇	—	2	2	no	—
27	C₄⋊C₄×C₁₉₇	D₄	1	1	yes	characteristic
28	C₄²×C₁₉₇	—	2	1	no	—
29	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
30	Q₈×C₁₉₇	—	2	1	no	—
31	Q₈×C₁₉₇	—	4	2	no	—
32	Q₈×C₁₉₇	—	4	2	no	—
33	Q₈×C₁₉₇	D₈	1	2	yes	—
34	C₂×C₄×C₁₉₇	—	2	1	no	—
35	C₂×C₄×C₁₉₇	—	2	1	no	—
36	C₂×C₄×C₁₉₇	—	4	2	no	—
37	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
40	C₈×C₁₉₇	—	2	1	no	—
41	C₈×C₁₉₇	—	4	2	no	—
42	C₈×C₁₉₇	—	4	1	no	—
43	C₈×C₁₉₇	Q₁₆	1	2	yes	complemented
44	C₂²×C₁₉₇	#121	1	1	yes	characteristic
45	C₄×C₁₉₇	—	2	2	no	—
46	C₄×C₁₉₇	—	2	1	no	—
47	C₄×C₁₉₇	—	4	1	no	—
48	C₄×C₁₉₇	—	8	2	no	—
49	C₄×C₁₉₇	C₂×D₈	1	1	yes	characteristic
50	C₄×C₁₉₇	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₁₉₇	#135	1	1	yes	characteristic
52	C₄×C₁₉₇	#137	1	1	yes	characteristic
53	C₂×C₁₉₇	#462	1	1	yes	characteristic
54	C₂×C₁₉₇	#466	1	1	yes	characteristic
55	C₂×C₁₉₇	#473	1	1	yes	characteristic
56	C₁₉₇	#1674	1	1	yes	characteristic, direct product factor, Sylow
57	#1674	C₁₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	#500	C₂×C₁₉₇	1	1	yes	characteristic
59	#462	C₂×C₁₉₇	1	2	yes	—
60	C₈×Q₈	C₂×C₁₉₇	1	1	yes	characteristic
61	#335	C₂×C₁₉₇	1	1	yes	characteristic
62	#332	C₂×C₁₉₇	1	2	yes	—
63	C₂×Q₁₆	—	2	2	no	—
64	C₄⋊Q₈	C₂²×C₁₉₇	1	2	yes	—
65	C₄×Q₈	C₂²×C₁₉₇	1	1	yes	characteristic
66	#107	—	2	1	no	—
67	#107	—	2	1	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	C₂²×C₁₉₇	1	1	yes	characteristic
70	C₄⋊C₈	C₂²×C₁₉₇	1	2	yes	—
71	#103	—	2	2	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	C₂²×C₁₉₇	1	1	yes	characteristic
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	D₄×C₁₉₇	1	1	yes	characteristic
76	Q₁₆	—	4	4	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	D₄×C₁₉₇	1	1	yes	characteristic
80	C₂×C₈	D₄×C₁₉₇	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	D₄×C₁₉₇	1	1	yes	characteristic
85	C₄²	—	2	1	no	—
86	C₄²	C₂³×C₁₉₇	1	1	yes	characteristic
87	Q₈	—	2	1	no	—
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	D₈×C₁₉₇	1	2	yes	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	C₂×D₄×C₁₉₇	1	1	yes	characteristic
95	C₂×C₄	C₂×D₄×C₁₉₇	1	1	yes	characteristic
96	C₂×C₄	#41×C₁₉₇	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	4	2	no	—
99	C₈	—	4	1	no	—
100	C₈	Q₁₆×C₁₉₇	1	2	yes	complemented
101	C₂²	#121×C₁₉₇	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	8	2	no	—
106	C₄	C₂×D₈×C₁₉₇	1	1	yes	characteristic
107	C₄	C₂×Q₁₆×C₁₉₇	1	1	yes	characteristic
108	C₄	#135×C₁₉₇	1	1	yes	characteristic
109	C₄	#137×C₁₉₇	1	1	yes	characteristic
110	C₂	#462×C₁₉₇	1	1	yes	characteristic
111	C₂	#466×C₁₉₇	1	1	yes	characteristic
112	C₂	#473×C₁₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₁₉₇

C₂

yes

maximal, characteristic

#462×C₁₉₇

C₂

yes

maximal

C₈×Q₈×C₁₉₇

C₂

yes

maximal, characteristic

#335×C₁₉₇

C₂

yes

maximal, characteristic

#332×C₁₉₇

C₂

yes

maximal

C₂×Q₁₆×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

C₄×Q₈×C₁₉₇

C₂²

yes

characteristic

#107×C₁₉₇

—

#107×C₁₉₇

—

C₄⋊C₈×C₁₉₇

—

C₄⋊C₈×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

C₂²

yes

—

#103×C₁₉₇

—

C₄×C₈×C₁₉₇

—

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

characteristic

Q₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

characteristic

C₂×C₈×C₁₉₇

D₄

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

D₈

yes

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

characteristic

C₈×C₁₉₇

—

C₈×C₁₉₇

—

C₈×C₁₉₇

—

C₈×C₁₉₇

Q₁₆

yes

complemented

C₂²×C₁₉₇

#121

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×D₈

yes

characteristic

C₄×C₁₉₇

C₂×Q₁₆

yes

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, direct product factor, Sylow

#1674

C₁₉₇

yes

maximal, characteristic, direct product factor, Sylow

#500

C₂×C₁₉₇

yes

characteristic

#462

C₂×C₁₉₇

yes

—

C₈×Q₈

C₂×C₁₉₇

yes

characteristic

#335

C₂×C₁₉₇

yes

characteristic

#332

C₂×C₁₉₇

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₁₉₇

yes

—

C₄×Q₈

C₂²×C₁₉₇

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₁₉₇

yes

characteristic

C₄⋊C₈

C₂²×C₁₉₇

yes

—

#103

—

C₄×C₈

—

C₄×C₈

C₂²×C₁₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₁₉₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₁₉₇

yes

characteristic

C₂×C₈

D₄×C₁₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₉₇

yes

characteristic

C₄²

—

C₄²

C₂³×C₁₉₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

D₈×C₁₉₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₉₇

yes

characteristic

C₂×C₄

C₂×D₄×C₁₉₇

yes

characteristic

C₂×C₄

#41×C₁₉₇

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

Q₁₆×C₁₉₇

yes

complemented

101

C₂²

#121×C₁₉₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₂×D₈×C₁₉₇

yes

characteristic

107

C₄

C₂×Q₁₆×C₁₉₇

yes

characteristic

108

C₄

#135×C₁₉₇

yes

characteristic

109

C₄

#137×C₁₉₇

yes

characteristic

110

C₂

#462×C₁₉₇

yes

characteristic

111

C₂

#466×C₁₉₇

yes

characteristic

112

C₂

#473×C₁₉₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₁₉₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₇ ◁ C₄²×C₁₉₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₉₇

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₁₉₇

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉₇ ◁ C₄²×C₁₉₇ ◁ G

N	H ≅ G/N	type
43. C₈×C₁₉₇	Q₁₆	semidirect
56. C₁₉₇	#1674	direct (N × H)
57. #1674	C₁₉₇	direct (N × H)
100. C₈	Q₁₆×C₁₉₇	semidirect

H ≅ G/N

type

43. C₈×C₁₉₇

Q₁₆

semidirect

56. C₁₉₇

#1674

direct (N × H)

57. #1674

C₁₉₇

direct (N × H)

100. C₈

Q₁₆×C₁₉₇

semidirect

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

subgroup	index
#500×C₁₉₇	2
#462×C₁₉₇	2
C₈×Q₈×C₁₉₇	2
#335×C₁₉₇	2
#332×C₁₉₇	2
#1674	197

subgroup

index

#500×C₁₉₇

#462×C₁₉₇

C₈×Q₈×C₁₉₇

#335×C₁₉₇

#332×C₁₉₇

#1674

197

Subgroups — #1674×C₁₉₇ #38738608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1674×C197 #38738608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1674×C₁₉₇ #38738608