Subgroups — C₄×C₈×Dic₉₇ #27141417

Order 12416.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₉₇	C₂	1	4	yes	maximal
3	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄×Dic₉₇	C₄	1	4	yes	—
7	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
8	C₄×#11715895	C₄	1	4	yes	complemented
9	C₈×Dic₉₇	C₄	1	16	yes	direct product factor
10	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₉₇	C₄	1	4	yes	complemented
13	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
14	C₂×#11715895	C₂×C₄	1	8	yes	—
15	C₂×#11715895	C₂×C₄	1	4	yes	—
16	C₄×Dic₉₇	C₈	1	16	yes	direct product factor
17	C₄×Dic₉₇	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₉₇	C₂×C₄	1	8	yes	—
21	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
22	C₄²×C₉₇	C₂×C₄	1	2	yes	—
23	C₄²×C₉₇	C₂×C₄	1	2	yes	—
24	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₄²	1	4	yes	—
26	C₂×C₄×C₉₇	C₂×C₈	1	8	yes	—
27	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₉₇	C₄²	1	8	yes	—
31	C₂×Dic₉₇	C₂×C₈	1	16	yes	—
32	C₈×C₉₇	C₄²	1	8	yes	complemented
33	#11715895	C₄²	1	8	yes	complemented
34	C₂²×C₉₇	C₂×C₄²	1	2	yes	—
35	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
36	C₂²×C₉₇	C₂²×C₈	1	4	yes	—
37	C₄×C₉₇	C₄×C₈	1	8	yes	complemented
38	C₄×C₉₇	C₂×C₄²	1	2	yes	—
39	C₄×C₉₇	C₂×C₄²	1	2	yes	—
40	Dic₉₇	C₄×C₈	1	16	yes	direct product factor
41	C₂×C₉₇	C₄³	1	1	yes	characteristic
42	C₂×C₉₇	C₂×C₄×C₈	1	4	yes	—
43	C₂×C₉₇	C₂×C₄×C₈	1	2	yes	—
44	C₄²×C₈	—	97	1	no	maximal, Sylow
45	C₉₇	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄×C₈	—	97	4	no	—
47	C₂×C₄×C₈	—	97	1	no	—
48	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
49	C₄³	—	97	1	no	—
50	C₂²×C₈	—	97	2	no	—
51	C₂²×C₈	C₂×D₉₇	1	2	yes	—
52	C₂×C₄²	—	97	4	no	—
53	C₂×C₄²	—	97	2	no	—
54	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
55	C₄×C₈	—	97	16	no	—
56	C₄×C₈	—	97	4	no	—
57	C₄×C₈	Dic₉₇	1	4	yes	direct product factor
58	C₂²×C₄	—	97	4	no	—
59	C₂²×C₄	C₄×D₉₇	1	2	yes	—
60	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
61	C₂×C₈	—	97	4	no	—
62	C₂×C₈	—	97	8	no	—
63	C₂×C₈	C₄×D₉₇	1	8	yes	—
64	C₂×C₈	C₂×Dic₉₇	1	4	yes	—
65	C₄²	—	97	16	no	—
66	C₄²	—	97	8	no	—
67	C₄²	C₄×D₉₇	1	2	yes	—
68	C₄²	C₂×Dic₉₇	1	2	yes	—
69	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
70	C₂×C₄	—	97	8	no	—
71	C₂×C₄	—	97	16	no	—
72	C₂×C₄	C₄×Dic₉₇	1	4	yes	—
73	C₂×C₄	C₈×D₉₇	1	8	yes	—
74	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
75	C₂×C₄	C₂×C₄×D₉₇	1	4	yes	—
76	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
77	C₈	—	97	8	no	—
78	C₈	C₄×Dic₉₇	1	8	yes	direct product factor
79	C₂²	C₄²×D₉₇	1	2	yes	—
80	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₉₇	1	4	yes	—
82	C₄	—	97	16	no	—
83	C₄	C₈×Dic₉₇	1	8	yes	direct product factor
84	C₄	C₄²×D₉₇	1	2	yes	—
85	C₄	C₂×C₄×Dic₉₇	1	2	yes	—
86	C₂	C₄²×Dic₉₇	1	1	yes	characteristic
87	C₂	C₄×C₈×D₉₇	1	4	yes	—
88	C₂	C₂×C₈×Dic₉₇	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	D₉₇
commutator, hypocenter	C₉₇	C₄²×C₈
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₄×#11715895	C₄	semidirect
9. C₈×Dic₉₇	C₄	direct (N × H)
12. C₄×C₈×C₉₇	C₄	semidirect
16. C₄×Dic₉₇	C₈	direct (N × H)
32. C₈×C₉₇	C₄²	semidirect
33. #11715895	C₄²	semidirect
37. C₄×C₉₇	C₄×C₈	semidirect
40. Dic₉₇	C₄×C₈	direct (N × H)
45. C₉₇	C₄²×C₈	semidirect
57. C₄×C₈	Dic₉₇	direct (N × H)
78. C₈	C₄×Dic₉₇	direct (N × H)
83. C₄	C₈×Dic₉₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×#11715895	2
C₂×C₈×Dic₉₇	2
C₄²×Dic₉₇	2
C₂×C₄×C₈×C₉₇	2
C₄²×C₈	97

Subgroups — C4×C8×Dic97 #27141417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₈×Dic₉₇ #27141417