Subgroups — C₉₇×#1829 #27137787

Order 12416.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#518×C₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#100×C₉₇	C₂	1	2	yes	maximal
3	#344×C₉₇	C₂	1	4	yes	maximal, complemented
4	C₂²×D₄×C₉₇	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₉₇	C₂²	1	4	yes	—
6	C₄×D₄×C₉₇	—	2	4	no	—
7	#117×C₉₇	C₂²	1	2	yes	—
8	C₂×C₂²⋊C₄×C₉₇	C₄	1	4	yes	—
9	#100×C₉₇	—	2	4	no	—
10	C₂⁴×C₉₇	C₂×C₄	1	2	yes	—
11	C₂×D₄×C₉₇	—	2	4	no	—
12	C₂×D₄×C₉₇	D₄	1	4	yes	—
13	C₂²×C₄×C₉₇	—	2	4	no	—
14	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
15	M₁₆×C₉₇	—	2	4	no	—
16	M₁₆×C₉₇	—	2	4	no	—
17	C₂×C₈×C₉₇	—	2	4	no	—
18	C₄⋊C₄×C₉₇	—	2	2	no	—
19	C₂²⋊C₄×C₉₇	—	2	4	no	—
20	C₂²⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
21	C₄²×C₉₇	—	2	2	no	—
22	C₂³×C₉₇	—	2	4	no	—
23	C₂³×C₉₇	—	8	2	no	—
24	C₂³×C₉₇	C₂²⋊C₄	1	4	yes	—
25	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
26	D₄×C₉₇	—	2	8	no	—
27	C₂×C₄×C₉₇	—	2	4	no	—
28	C₂×C₄×C₉₇	—	8	4	no	—
29	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
30	C₂×C₄×C₉₇	#41	1	4	yes	—
31	C₈×C₉₇	—	4	4	no	—
32	C₂²×C₉₇	—	2	4	no	—
33	C₂²×C₉₇	—	4	1	no	—
34	C₂²×C₉₇	—	4	4	no	—
35	C₂²×C₉₇	—	8	4	no	—
36	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₉₇	#117	1	2	yes	—
38	C₄×C₉₇	—	4	4	no	—
39	C₄×C₉₇	#123	1	4	yes	—
40	C₂×C₉₇	—	2	1	no	—
41	C₂×C₉₇	—	2	2	no	—
42	C₂×C₉₇	—	4	4	no	—
43	C₂×C₉₇	#381	1	1	yes	characteristic
44	#1829	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₉₇	#1829	1	1	yes	characteristic, direct product factor, Sylow
46	#518	C₂×C₉₇	1	1	yes	characteristic
47	C₂×#100	C₂×C₉₇	1	2	yes	—
48	#344	C₂×C₉₇	1	4	yes	complemented
49	C₂²×D₄	C₂²×C₉₇	1	1	yes	characteristic
50	C₂×M₁₆	C₂²×C₉₇	1	4	yes	—
51	C₄×D₄	—	2	4	no	—
52	#117	C₂²×C₉₇	1	2	yes	—
53	C₂×C₂²⋊C₄	C₄×C₉₇	1	4	yes	—
54	#100	—	2	4	no	—
55	C₂⁴	C₂×C₄×C₉₇	1	2	yes	—
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	D₄×C₉₇	1	4	yes	—
58	C₂²×C₄	—	2	4	no	—
59	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
60	M₁₆	—	2	4	no	—
61	M₁₆	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₂²⋊C₄	—	2	4	no	—
65	C₂²⋊C₄	C₂×C₄×C₉₇	1	4	yes	—
66	C₄²	—	2	2	no	—
67	C₂³	—	2	4	no	—
68	C₂³	—	8	2	no	—
69	C₂³	C₂²⋊C₄×C₉₇	1	4	yes	—
70	C₂³	C₂²×C₄×C₉₇	1	1	yes	characteristic
71	D₄	—	2	8	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	8	4	no	—
74	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
75	C₂×C₄	#41×C₉₇	1	4	yes	—
76	C₈	—	4	4	no	—
77	C₂²	—	2	4	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	4	no	—
80	C₂²	—	8	4	no	—
81	C₂²	C₂×C₂²⋊C₄×C₉₇	1	1	yes	characteristic
82	C₂²	#117×C₉₇	1	2	yes	—
83	C₄	—	4	4	no	—
84	C₄	#123×C₉₇	1	4	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	2	2	no	—
87	C₂	—	4	4	no	—
88	C₂	#381×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₉₇	#381
commutator	C₂³	C₂²×C₄×C₉₇
Frattini, cosocle	C₂²×C₄	C₂³×C₉₇
prime radical	C₂²×D₄×C₉₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉₇ ◁ C₂²×C₄×C₉₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #344×C₉₇	C₂	semidirect
44. #1829	C₉₇	direct (N × H)
45. C₉₇	#1829	direct (N × H)
48. #344	C₂×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
#518×C₉₇	2
C₂×#100×C₉₇	2
#344×C₉₇	2
#1829	97

Subgroups — C97×#1829 #27137787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1829 #27137787