Group #27139340 — subgroups

Subgroups — Q₈×Q₁₆×C₉₇ #27139340

Order 12416.

336 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 208 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	Q₈²×C₉₇	C₂	1	2	yes	maximal
2	#500×C₉₇	C₂	1	3	yes	maximal
3	#477×C₉₇	C₂	1	6	yes	maximal
4	C₈×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₉₇	C₂	1	3	yes	maximal
6	C₂×Q₁₆×C₉₇	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₉₇	—	2	6	no	—
8	C₄⋊Q₈×C₉₇	C₂²	1	6	yes	—
9	C₄×Q₈×C₉₇	—	2	2	no	—
10	C₄×Q₈×C₉₇	C₂²	1	6	yes	—
11	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
12	#107×C₉₇	C₂²	1	3	yes	—
13	#107×C₉₇	C₂²	1	6	yes	—
14	C₄⋊C₈×C₉₇	C₂²	1	3	yes	—
15	#103×C₉₇	C₂²	1	6	yes	—
16	C₄×C₈×C₉₇	C₂²	1	3	yes	—
17	C₂×Q₈×C₉₇	—	2	6	no	—
18	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
19	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
20	Q₁₆×C₉₇	Q₈	1	4	yes	direct product factor
21	C₂×C₈×C₉₇	C₂³	1	3	yes	—
22	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₉₇	—	2	6	no	—
24	C₄⋊C₄×C₉₇	—	2	6	no	—
25	C₄⋊C₄×C₉₇	D₄	1	3	yes	—
26	C₄⋊C₄×C₉₇	C₂³	1	6	yes	—
27	C₄²×C₉₇	—	2	6	no	—
28	C₄²×C₉₇	C₂³	1	3	yes	—
29	Q₈×C₉₇	—	2	2	no	—
30	Q₈×C₉₇	—	8	6	no	—
31	Q₈×C₉₇	Q₁₆	1	4	yes	direct product factor
32	Q₈×C₉₇	C₂×Q₈	1	4	yes	—
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	2	6	no	—
35	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
36	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
37	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
38	C₈×C₉₇	—	2	3	no	—
39	C₈×C₉₇	C₂×Q₈	1	2	yes	—
40	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₉₇	—	2	4	no	—
42	C₄×C₉₇	—	2	3	no	—
43	C₄×C₉₇	—	4	6	no	—
44	C₄×C₉₇	C₂×Q₁₆	1	6	yes	—
45	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
47	C₂×C₉₇	D₄×Q₈	1	1	yes	characteristic
48	C₂×C₉₇	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂×C₉₇	#576	1	1	yes	characteristic
50	Q₈×Q₁₆	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₉₇	Q₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
52	Q₈²	C₂×C₉₇	1	2	yes	—
53	#500	C₂×C₉₇	1	3	yes	—
54	#477	C₂×C₉₇	1	6	yes	—
55	C₈×Q₈	C₂×C₉₇	1	1	yes	characteristic
56	C₄×Q₁₆	C₂×C₉₇	1	3	yes	—
57	C₂×Q₁₆	C₂²×C₉₇	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	6	no	—
59	C₄⋊Q₈	C₂²×C₉₇	1	6	yes	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×Q₈	C₂²×C₉₇	1	6	yes	—
62	C₄×Q₈	C₂²×C₉₇	1	1	yes	characteristic
63	#107	C₂²×C₉₇	1	3	yes	—
64	#107	C₂²×C₉₇	1	6	yes	—
65	C₄⋊C₈	C₂²×C₉₇	1	3	yes	—
66	#103	C₂²×C₉₇	1	6	yes	—
67	C₄×C₈	C₂²×C₉₇	1	3	yes	—
68	C₂×Q₈	—	2	6	no	—
69	C₂×Q₈	D₄×C₉₇	1	1	yes	characteristic
70	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
71	Q₁₆	Q₈×C₉₇	1	4	yes	direct product factor
72	C₂×C₈	C₂³×C₉₇	1	3	yes	—
73	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
74	C₄⋊C₄	—	2	6	no	—
75	C₄⋊C₄	—	2	6	no	—
76	C₄⋊C₄	D₄×C₉₇	1	3	yes	—
77	C₄⋊C₄	C₂³×C₉₇	1	6	yes	—
78	C₄²	—	2	6	no	—
79	C₄²	C₂³×C₉₇	1	3	yes	—
80	Q₈	—	2	2	no	—
81	Q₈	—	8	6	no	—
82	Q₈	Q₁₆×C₉₇	1	4	yes	direct product factor
83	Q₈	C₂×Q₈×C₉₇	1	4	yes	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	6	no	—
86	C₂×C₄	C₂×D₄×C₉₇	1	3	yes	—
87	C₂×C₄	C₂×D₄×C₉₇	1	3	yes	—
88	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	C₂×Q₈×C₉₇	1	2	yes	—
91	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
92	C₄	—	2	4	no	—
93	C₄	—	2	3	no	—
94	C₄	—	4	6	no	—
95	C₄	C₂×Q₁₆×C₉₇	1	6	yes	—
96	C₄	C₂²×Q₈×C₉₇	1	1	yes	characteristic
97	C₄	2₋¹⁺⁴×C₉₇	1	1	yes	characteristic
98	C₂	D₄×Q₈×C₉₇	1	1	yes	characteristic
99	C₂	C₂²×Q₁₆×C₉₇	1	1	yes	characteristic
100	C₂	#576×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₉₇

C₂

yes

maximal

#500×C₉₇

C₂

yes

maximal

#477×C₉₇

C₂

yes

maximal

C₈×Q₈×C₉₇

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₉₇

C₂

yes

maximal

C₂×Q₁₆×C₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₉₇

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

C₄×Q₈×C₉₇

—

C₄×Q₈×C₉₇

C₂²

yes

—

C₄×Q₈×C₉₇

C₂²

yes

characteristic

#107×C₉₇

C₂²

yes

—

#107×C₉₇

C₂²

yes

—

C₄⋊C₈×C₉₇

C₂²

yes

—

#103×C₉₇

C₂²

yes

—

C₄×C₈×C₉₇

C₂²

yes

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

D₄

yes

characteristic

C₂×Q₈×C₉₇

C₂³

yes

—

Q₁₆×C₉₇

Q₈

yes

direct product factor

C₂×C₈×C₉₇

C₂³

yes

—

C₂×C₈×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

C₂³

yes

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂³

yes

—

Q₈×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

Q₁₆

yes

direct product factor

Q₈×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂⁴

yes

characteristic

C₈×C₉₇

—

C₈×C₉₇

C₂×Q₈

yes

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×Q₁₆

yes

—

C₄×C₉₇

C₂²×Q₈

yes

characteristic

C₄×C₉₇

2₋¹⁺⁴

yes

characteristic

C₂×C₉₇

D₄×Q₈

yes

characteristic

C₂×C₉₇

C₂²×Q₁₆

yes

characteristic

C₂×C₉₇

#576

yes

characteristic

Q₈×Q₁₆

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

Q₈×Q₁₆

yes

characteristic, direct product factor, Sylow

Q₈²

C₂×C₉₇

yes

—

#500

C₂×C₉₇

yes

—

#477

C₂×C₉₇

yes

—

C₈×Q₈

C₂×C₉₇

yes

characteristic

C₄×Q₁₆

C₂×C₉₇

yes

—

C₂×Q₁₆

C₂²×C₉₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₉₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₉₇

yes

—

C₄×Q₈

C₂²×C₉₇

yes

characteristic

#107

C₂²×C₉₇

yes

—

#107

C₂²×C₉₇

yes

—

C₄⋊C₈

C₂²×C₉₇

yes

—

#103

C₂²×C₉₇

yes

—

C₄×C₈

C₂²×C₉₇

yes

—

C₂×Q₈

—

C₂×Q₈

D₄×C₉₇

yes

characteristic

C₂×Q₈

C₂³×C₉₇

yes

—

Q₁₆

Q₈×C₉₇

yes

direct product factor

C₂×C₈

C₂³×C₉₇

yes

—

C₂×C₈

C₂³×C₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₉₇

yes

—

C₄⋊C₄

C₂³×C₉₇

yes

—

C₄²

—

C₄²

C₂³×C₉₇

yes

—

Q₈

—

Q₈

—

Q₈

Q₁₆×C₉₇

yes

direct product factor

Q₈

C₂×Q₈×C₉₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₉₇

yes

—

C₂×C₄

C₂×D₄×C₉₇

yes

—

C₂×C₄

C₂⁴×C₉₇

yes

characteristic

C₈

—

C₈

C₂×Q₈×C₉₇

yes

—

C₂²

C₂²×D₄×C₉₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×Q₁₆×C₉₇

yes

—

C₄

C₂²×Q₈×C₉₇

yes

characteristic

C₄

2₋¹⁺⁴×C₉₇

yes

characteristic

C₂

D₄×Q₈×C₉₇

yes

characteristic

C₂

C₂²×Q₁₆×C₉₇

yes

characteristic

100

C₂

#576×C₉₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₉₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₄×Q₈×C₉₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₉₇

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₉₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₉₇ ◁ C₄×Q₈×C₉₇ ◁ G

N	H ≅ G/N	type
20. Q₁₆×C₉₇	Q₈	direct (N × H)
31. Q₈×C₉₇	Q₁₆	direct (N × H)
50. Q₈×Q₁₆	C₉₇	direct (N × H)
51. C₉₇	Q₈×Q₁₆	direct (N × H)
71. Q₁₆	Q₈×C₉₇	direct (N × H)
82. Q₈	Q₁₆×C₉₇	direct (N × H)

H ≅ G/N

type

20. Q₁₆×C₉₇

Q₈

direct (N × H)

31. Q₈×C₉₇

Q₁₆

direct (N × H)

50. Q₈×Q₁₆

C₉₇

direct (N × H)

51. C₉₇

Q₈×Q₁₆

direct (N × H)

71. Q₁₆

Q₈×C₉₇

direct (N × H)

82. Q₈

Q₁₆×C₉₇

direct (N × H)

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

subgroup	index
Q₈²×C₉₇	2
#500×C₉₇	2
#477×C₉₇	2
C₈×Q₈×C₉₇	2
C₄×Q₁₆×C₉₇	2
Q₈×Q₁₆	97

subgroup

index

Q₈²×C₉₇

#500×C₉₇

#477×C₉₇

C₈×Q₈×C₉₇

C₄×Q₁₆×C₉₇

Q₈×Q₁₆

Subgroups — Q₈×Q₁₆×C₉₇ #27139340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×Q16×C97 #27139340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Q₁₆×C₉₇ #27139340