Group #27137792 — subgroups

Subgroups — C₉₇×#1834 #27137792

Order 12416.

248 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 152 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×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
2	#384×C₉₇	C₂	1	2	yes	maximal
3	#340×C₉₇	C₂	1	4	yes	maximal
4	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
5	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
6	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
7	C₂×C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
8	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₉₇	C₄	1	4	yes	complemented
10	C₂²×C₄×C₉₇	—	2	4	no	—
11	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
13	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
14	C₂×C₈×C₉₇	—	2	4	no	—
15	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
16	C₄⋊C₄×C₉₇	—	2	8	no	—
17	C₄⋊C₄×C₉₇	—	4	4	no	—
18	C₄⋊C₄×C₉₇	—	4	4	no	—
19	C₄²×C₉₇	C₂×C₄	1	2	yes	—
20	C₄²×C₉₇	Q₈	1	2	yes	—
21	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
22	C₂×C₄×C₉₇	—	4	4	no	—
23	C₂×C₄×C₉₇	—	4	4	no	—
24	C₂×C₄×C₉₇	—	4	4	no	—
25	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₉₇	D₈	1	2	yes	—
28	C₂×C₄×C₉₇	SD₁₆	1	4	yes	—
29	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
30	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₉₇	#41	1	4	yes	—
33	C₈×C₉₇	—	2	4	no	—
34	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₉₇	#124	1	2	yes	—
36	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
37	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
38	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
39	C₄×C₉₇	—	8	4	no	—
40	C₄×C₉₇	#106	1	4	yes	—
41	C₄×C₉₇	#107	1	4	yes	—
42	C₄×C₉₇	#117	1	2	yes	—
43	C₄×C₉₇	#125	1	2	yes	—
44	C₂×C₉₇	#382	1	1	yes	characteristic
45	C₂×C₉₇	C₂×#106	1	1	yes	characteristic
46	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
47	C₂×C₉₇	#486	1	2	yes	—
48	C₂×C₉₇	#487	1	2	yes	—
49	#1834	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₉₇	#1834	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₄×C₈	C₂×C₉₇	1	1	yes	characteristic
52	#384	C₂×C₉₇	1	2	yes	—
53	#340	C₂×C₉₇	1	4	yes	—
54	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
55	C₂×C₄⋊C₄	—	2	2	no	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄⋊C₄	C₂²×C₉₇	1	4	yes	—
58	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
59	C₄×C₈	C₄×C₉₇	1	4	yes	complemented
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
62	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
63	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	C₂×C₄×C₉₇	1	4	yes	—
66	C₄⋊C₄	—	2	8	no	—
67	C₄⋊C₄	—	4	4	no	—
68	C₄⋊C₄	—	4	4	no	—
69	C₄²	C₂×C₄×C₉₇	1	2	yes	—
70	C₄²	Q₈×C₉₇	1	2	yes	—
71	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
72	C₂×C₄	—	4	4	no	—
73	C₂×C₄	—	4	4	no	—
74	C₂×C₄	—	4	4	no	—
75	C₂×C₄	C₄⋊C₄×C₉₇	1	2	yes	—
76	C₂×C₄	C₄⋊C₄×C₉₇	1	2	yes	—
77	C₂×C₄	D₈×C₉₇	1	2	yes	—
78	C₂×C₄	SD₁₆×C₉₇	1	4	yes	—
79	C₂×C₄	Q₁₆×C₉₇	1	2	yes	—
80	C₂×C₄	C₂²×C₄×C₉₇	1	1	yes	characteristic
81	C₂×C₄	C₂×Q₈×C₉₇	1	1	yes	characteristic
82	C₂×C₄	#41×C₉₇	1	4	yes	—
83	C₈	—	2	4	no	—
84	C₂²	C₂×C₄⋊C₄×C₉₇	1	1	yes	characteristic
85	C₂²	#124×C₉₇	1	2	yes	—
86	C₂²	C₂×D₈×C₉₇	1	1	yes	characteristic
87	C₂²	C₂×SD₁₆×C₉₇	1	2	yes	—
88	C₂²	C₂×Q₁₆×C₉₇	1	1	yes	characteristic
89	C₄	—	8	4	no	—
90	C₄	#106×C₉₇	1	4	yes	—
91	C₄	#107×C₉₇	1	4	yes	—
92	C₄	#117×C₉₇	1	2	yes	—
93	C₄	#125×C₉₇	1	2	yes	—
94	C₂	#382×C₉₇	1	1	yes	characteristic
95	C₂	C₂×#106×C₉₇	1	1	yes	characteristic
96	C₂	C₂×#107×C₉₇	1	1	yes	characteristic
97	C₂	#486×C₉₇	1	2	yes	—
98	C₂	#487×C₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

#384×C₉₇

C₂

yes

maximal

#340×C₉₇

C₂

yes

maximal

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₄

yes

complemented

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

D₈

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₈×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#124

yes

—

C₂²×C₉₇

C₂×D₈

yes

characteristic

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₂²×C₉₇

C₂×Q₁₆

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

#106

yes

—

C₄×C₉₇

#107

yes

—

C₄×C₉₇

#117

yes

—

C₄×C₉₇

#125

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₂×C₉₇

#487

yes

—

#1834

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#1834

yes

characteristic, direct product factor, Sylow

C₂×C₄×C₈

C₂×C₉₇

yes

characteristic

#384

C₂×C₉₇

yes

—

#340

C₂×C₉₇

yes

—

C₂²×C₈

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₄²

C₂²×C₉₇

yes

characteristic

C₄×C₈

C₄×C₉₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

D₄×C₉₇

yes

characteristic

C₂²×C₄

D₄×C₉₇

yes

characteristic

C₂²×C₄

C₂³×C₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×C₄×C₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₉₇

yes

—

C₄²

Q₈×C₉₇

yes

—

C₂³

C₂×D₄×C₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₉₇

yes

—

C₂×C₄

C₄⋊C₄×C₉₇

yes

—

C₂×C₄

D₈×C₉₇

yes

—

C₂×C₄

SD₁₆×C₉₇

yes

—

C₂×C₄

Q₁₆×C₉₇

yes

—

C₂×C₄

C₂²×C₄×C₉₇

yes

characteristic

C₂×C₄

C₂×Q₈×C₉₇

yes

characteristic

C₂×C₄

#41×C₉₇

yes

—

C₈

—

C₂²

C₂×C₄⋊C₄×C₉₇

yes

characteristic

C₂²

#124×C₉₇

yes

—

C₂²

C₂×D₈×C₉₇

yes

characteristic

C₂²

C₂×SD₁₆×C₉₇

yes

—

C₂²

C₂×Q₁₆×C₉₇

yes

characteristic

C₄

—

C₄

#106×C₉₇

yes

—

C₄

#107×C₉₇

yes

—

C₄

#117×C₉₇

yes

—

C₄

#125×C₉₇

yes

—

C₂

#382×C₉₇

yes

characteristic

C₂

C₂×#106×C₉₇

yes

characteristic

C₂

C₂×#107×C₉₇

yes

characteristic

C₂

#486×C₉₇

yes

—

C₂

#487×C₉₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₉₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₉₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₉₇

Frattini, cosocle

C₂²×C₄

C₂³×C₉₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

N	H ≅ G/N	type
9. C₄×C₈×C₉₇	C₄	semidirect
49. #1834	C₉₇	direct (N × H)
50. C₉₇	#1834	direct (N × H)
59. C₄×C₈	C₄×C₉₇	semidirect

H ≅ G/N

type

9. C₄×C₈×C₉₇

C₄

semidirect

49. #1834

C₉₇

direct (N × H)

50. C₉₇

#1834

direct (N × H)

59. C₄×C₈

C₄×C₉₇

semidirect

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

subgroup	index
C₂×C₄×C₈×C₉₇	2
#384×C₉₇	2
#340×C₉₇	2
#1834	97

subgroup

index

C₂×C₄×C₈×C₉₇

#384×C₉₇

#340×C₉₇

#1834

Subgroups — C₉₇×#1834 #27137792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#1834 #27137792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1834 #27137792