Group #38740329 — subgroups

Subgroups — #3395×C₁₉₇ #38740329

Order 25216.

348 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 192 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	#557×C₁₉₇	C₂	1	2	yes	maximal
2	#494×C₁₉₇	C₂	1	3	yes	maximal
3	#462×C₁₉₇	C₂	1	6	yes	maximal
4	C₈×Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₁₉₇	C₂	1	3	yes	maximal
6	C₂×Q₁₆×C₁₉₇	C₂²	1	3	yes	—
7	C₂×Q₁₆×C₁₉₇	C₂²	1	6	yes	—
8	C₄⋊Q₈×C₁₉₇	C₂²	1	6	yes	—
9	#125×C₁₉₇	—	2	6	no	—
10	C₄×Q₈×C₁₉₇	—	2	2	no	—
11	C₄×Q₈×C₁₉₇	C₂²	1	6	yes	—
12	C₄×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
13	#107×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₉₇	C₂²	1	3	yes	—
15	#103×C₁₉₇	C₂²	1	6	yes	—
16	C₄×C₈×C₁₉₇	C₂²	1	3	yes	—
17	C₂×Q₈×C₁₉₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₉₇	C₂³	1	6	yes	—
19	Q₁₆×C₁₉₇	—	2	6	no	—
20	Q₁₆×C₁₉₇	—	4	6	no	—
21	C₂×C₈×C₁₉₇	C₂³	1	3	yes	—
22	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₉₇	—	2	6	no	—
24	C₄⋊C₄×C₁₉₇	—	2	6	no	—
25	C₄⋊C₄×C₁₉₇	—	2	6	no	—
26	C₄⋊C₄×C₁₉₇	D₄	1	3	yes	—
27	C₄⋊C₄×C₁₉₇	C₂³	1	2	yes	—
28	C₄²×C₁₉₇	—	2	6	no	—
29	C₄²×C₁₉₇	C₂³	1	3	yes	—
30	Q₈×C₁₉₇	—	2	6	no	—
31	Q₈×C₁₉₇	—	2	6	no	—
32	Q₈×C₁₉₇	Q₁₆	1	4	yes	complemented
33	C₂×C₄×C₁₉₇	—	2	2	no	—
34	C₂×C₄×C₁₉₇	—	2	6	no	—
35	C₂×C₄×C₁₉₇	C₂×D₄	1	3	yes	—
36	C₂×C₄×C₁₉₇	C₂×D₄	1	3	yes	—
37	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
38	C₈×C₁₉₇	—	2	3	no	—
39	C₈×C₁₉₇	#41	1	2	yes	—
40	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₁₉₇	—	2	3	no	—
42	C₄×C₁₉₇	—	4	6	no	—
43	C₄×C₁₉₇	—	4	2	no	—
44	C₄×C₁₉₇	C₂×Q₁₆	1	6	yes	—
45	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
46	C₄×C₁₉₇	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂×C₁₉₇	#550	1	1	yes	characteristic
48	C₂×C₁₉₇	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂×C₁₉₇	#578	1	1	yes	characteristic
50	C₁₉₇	#3395	1	1	yes	characteristic, direct product factor, Sylow
51	#3395	C₁₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	#557	C₂×C₁₉₇	1	2	yes	—
53	#494	C₂×C₁₉₇	1	3	yes	—
54	#462	C₂×C₁₉₇	1	6	yes	—
55	C₈×Q₈	C₂×C₁₉₇	1	1	yes	characteristic
56	C₄×Q₁₆	C₂×C₁₉₇	1	3	yes	—
57	C₂×Q₁₆	C₂²×C₁₉₇	1	3	yes	—
58	C₂×Q₁₆	C₂²×C₁₉₇	1	6	yes	—
59	C₄⋊Q₈	C₂²×C₁₉₇	1	6	yes	—
60	#125	—	2	6	no	—
61	C₄×Q₈	—	2	2	no	—
62	C₄×Q₈	C₂²×C₁₉₇	1	6	yes	—
63	C₄×Q₈	C₂²×C₁₉₇	1	1	yes	characteristic
64	#107	C₂²×C₁₉₇	1	1	yes	characteristic
65	C₄⋊C₈	C₂²×C₁₉₇	1	3	yes	—
66	#103	C₂²×C₁₉₇	1	6	yes	—
67	C₄×C₈	C₂²×C₁₉₇	1	3	yes	—
68	C₂×Q₈	D₄×C₁₉₇	1	1	yes	characteristic
69	C₂×Q₈	C₂³×C₁₉₇	1	6	yes	—
70	Q₁₆	—	2	6	no	—
71	Q₁₆	—	4	6	no	—
72	C₂×C₈	C₂³×C₁₉₇	1	3	yes	—
73	C₂×C₈	C₂³×C₁₉₇	1	1	yes	characteristic
74	C₄⋊C₄	—	2	6	no	—
75	C₄⋊C₄	—	2	6	no	—
76	C₄⋊C₄	—	2	6	no	—
77	C₄⋊C₄	D₄×C₁₉₇	1	3	yes	—
78	C₄⋊C₄	C₂³×C₁₉₇	1	2	yes	—
79	C₄²	—	2	6	no	—
80	C₄²	C₂³×C₁₉₇	1	3	yes	—
81	Q₈	—	2	6	no	—
82	Q₈	—	2	6	no	—
83	Q₈	Q₁₆×C₁₉₇	1	4	yes	complemented
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	6	no	—
86	C₂×C₄	C₂×D₄×C₁₉₇	1	3	yes	—
87	C₂×C₄	C₂×D₄×C₁₉₇	1	3	yes	—
88	C₂×C₄	C₂⁴×C₁₉₇	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	#41×C₁₉₇	1	2	yes	—
91	C₂²	C₂²×D₄×C₁₉₇	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	4	6	no	—
94	C₄	—	4	2	no	—
95	C₄	C₂×Q₁₆×C₁₉₇	1	6	yes	—
96	C₄	C₂×#41×C₁₉₇	1	1	yes	characteristic
97	C₄	H(5,𝔽₂)×C₁₉₇	1	1	yes	characteristic
98	C₂	#550×C₁₉₇	1	1	yes	characteristic
99	C₂	C₂²×Q₁₆×C₁₉₇	1	1	yes	characteristic
100	C₂	#578×C₁₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#557×C₁₉₇

C₂

yes

maximal

#494×C₁₉₇

C₂

yes

maximal

#462×C₁₉₇

C₂

yes

maximal

C₈×Q₈×C₁₉₇

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₁₉₇

C₂

yes

maximal

C₂×Q₁₆×C₁₉₇

C₂²

yes

—

C₂×Q₁₆×C₁₉₇

C₂²

yes

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

#125×C₁₉₇

—

C₄×Q₈×C₁₉₇

—

C₄×Q₈×C₁₉₇

C₂²

yes

—

C₄×Q₈×C₁₉₇

C₂²

yes

characteristic

#107×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

C₂²

yes

—

#103×C₁₉₇

C₂²

yes

—

C₄×C₈×C₁₉₇

C₂²

yes

—

C₂×Q₈×C₁₉₇

D₄

yes

characteristic

C₂×Q₈×C₁₉₇

C₂³

yes

—

Q₁₆×C₁₉₇

—

Q₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂³

yes

—

C₂×C₈×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

D₄

yes

—

C₄⋊C₄×C₁₉₇

C₂³

yes

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

Q₁₆

yes

complemented

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₈×C₁₉₇

—

C₈×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×Q₁₆

yes

—

C₄×C₁₉₇

C₂×#41

yes

C₄×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

C₂²×Q₁₆

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, direct product factor, Sylow

#3395

C₁₉₇

yes

maximal, characteristic, direct product factor, Sylow

#557

C₂×C₁₉₇

yes

—

#494

C₂×C₁₉₇

yes

—

#462

C₂×C₁₉₇

yes

—

C₈×Q₈

C₂×C₁₉₇

yes

characteristic

C₄×Q₁₆

C₂×C₁₉₇

yes

—

C₂×Q₁₆

C₂²×C₁₉₇

yes

—

C₂×Q₁₆

C₂²×C₁₉₇

yes

—

C₄⋊Q₈

C₂²×C₁₉₇

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₉₇

yes

—

C₄×Q₈

C₂²×C₁₉₇

yes

characteristic

#107

C₂²×C₁₉₇

yes

characteristic

C₄⋊C₈

C₂²×C₁₉₇

yes

—

#103

C₂²×C₁₉₇

yes

—

C₄×C₈

C₂²×C₁₉₇

yes

—

C₂×Q₈

D₄×C₁₉₇

yes

characteristic

C₂×Q₈

C₂³×C₁₉₇

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

C₂³×C₁₉₇

yes

—

C₂×C₈

C₂³×C₁₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₉₇

yes

—

C₄⋊C₄

C₂³×C₁₉₇

yes

—

C₄²

—

C₄²

C₂³×C₁₉₇

yes

—

Q₈

—

Q₈

—

Q₈

Q₁₆×C₁₉₇

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₉₇

yes

—

C₂×C₄

C₂×D₄×C₁₉₇

yes

—

C₂×C₄

C₂⁴×C₁₉₇

yes

characteristic

C₈

—

C₈

#41×C₁₉₇

yes

—

C₂²

C₂²×D₄×C₁₉₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×Q₁₆×C₁₉₇

yes

—

C₄

C₂×#41×C₁₉₇

yes

characteristic

C₄

H(5,𝔽₂)×C₁₉₇

yes

characteristic

C₂

#550×C₁₉₇

yes

characteristic

C₂

C₂²×Q₁₆×C₁₉₇

yes

characteristic

100

C₂

#578×C₁₉₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₇ ◁ C₄×Q₈×C₁₉₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₉₇

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₉₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉₇ ◁ C₄×Q₈×C₁₉₇ ◁ G

N	H ≅ G/N	type
32. Q₈×C₁₉₇	Q₁₆	semidirect
50. C₁₉₇	#3395	direct (N × H)
51. #3395	C₁₉₇	direct (N × H)
83. Q₈	Q₁₆×C₁₉₇	semidirect

H ≅ G/N

type

32. Q₈×C₁₉₇

Q₁₆

semidirect

50. C₁₉₇

#3395

direct (N × H)

51. #3395

C₁₉₇

direct (N × H)

83. Q₈

Q₁₆×C₁₉₇

semidirect

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

subgroup	index
#557×C₁₉₇	2
#494×C₁₉₇	2
#462×C₁₉₇	2
C₈×Q₈×C₁₉₇	2
C₄×Q₁₆×C₁₉₇	2
#3395	197

subgroup

index

#557×C₁₉₇

#494×C₁₉₇

#462×C₁₉₇

C₈×Q₈×C₁₉₇

C₄×Q₁₆×C₁₉₇

#3395

197

Subgroups — #3395×C₁₉₇ #38740329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3395×C197 #38740329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3395×C₁₉₇ #38740329