Subgroups — C₄²×#11715906 #27158868

Order 12416.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11715906	C₂	1	6	yes	maximal
2	C₄²×#87604	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715906	C₂²	1	4	yes	—
4	C₄×#11715906	C₄	1	24	yes	direct product factor
5	C₂×C₄×#87604	C₄	1	3	yes	—
6	C₂×C₄×#87604	C₂²	1	3	yes	—
7	C₄²×D₉₇	C₄	1	1	yes	characteristic
8	C₂×#11715906	C₂×C₄	1	24	yes	—
9	C₂²×#87604	C₂×C₄	1	3	yes	—
10	C₂²×#87604	C₂³	1	1	yes	characteristic
11	C₄×#87604	C₈	1	12	yes	complemented
12	C₄×#87604	C₂×C₄	1	12	yes	—
13	C₂×C₄×D₉₇	C₂×C₄	1	3	yes	—
14	C₄×Dic₉₇	C₈	1	3	yes	complemented
15	C₄²×C₉₇	C₈	1	1	yes	characteristic, complemented
16	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
17	C₂×#87604	C₄²	1	6	yes	—
18	C₂×#87604	C₂×C₈	1	12	yes	—
19	C₂×#87604	C₂²×C₄	1	6	yes	—
20	#11715906	C₄²	1	16	yes	direct product factor
21	C₂×C₄×C₉₇	C₂×C₈	1	3	yes	—
22	C₂×Dic₉₇	C₂×C₈	1	3	yes	—
23	C₄×D₉₇	C₄²	1	6	yes	—
24	C₄×D₉₇	C₂×C₈	1	6	yes	—
25	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
26	C₂×D₉₇	C₂×C₄²	1	3	yes	—
27	C₂×D₉₇	C₂²×C₈	1	3	yes	—
28	#87604	C₄×C₈	1	12	yes	complemented
29	#87604	C₂×C₄²	1	4	yes	—
30	C₄×C₉₇	C₄×C₈	1	6	yes	complemented
31	Dic₉₇	C₄×C₈	1	6	yes	complemented
32	C₂×C₉₇	C₂×C₄×C₈	1	3	yes	—
33	D₉₇	C₄³	1	1	yes	characteristic
34	D₉₇	C₂×C₄×C₈	1	3	yes	—
35	C₄²×C₈	—	97	1	no	maximal, Sylow
36	C₉₇	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₄×C₈	—	97	6	no	—
38	C₄³	—	97	1	no	—
39	C₂²×C₈	—	97	4	no	—
40	C₂×C₄²	—	97	3	no	—
41	C₂×C₄²	—	97	3	no	—
42	C₂×C₄²	—	97	1	no	—
43	C₄×C₈	—	97	24	no	—
44	C₂²×C₄	—	97	3	no	—
45	C₂²×C₄	—	97	1	no	—
46	C₂²×C₄	—	97	3	no	—
47	C₂×C₈	—	97	24	no	—
48	C₄²	—	97	12	no	—
49	C₄²	—	97	12	no	—
50	C₄²	—	97	3	no	—
51	C₄²	#11715906	1	1	yes	characteristic, direct product factor
52	C₂³	—	97	1	no	—
53	C₂×C₄	—	97	6	no	—
54	C₂×C₄	—	97	6	no	—
55	C₂×C₄	—	97	12	no	—
56	C₂×C₄	—	97	3	no	—
57	C₂×C₄	—	97	6	no	—
58	C₂×C₄	—	97	6	no	—
59	C₂×C₄	C₂×#11715906	1	3	yes	—
60	C₈	—	97	16	no	—
61	C₂²	—	97	3	no	—
62	C₂²	—	97	3	no	—
63	C₂²	C₂²×#11715906	1	1	yes	characteristic
64	C₄	—	97	4	no	—
65	C₄	—	97	12	no	—
66	C₄	—	97	6	no	—
67	C₄	C₄×#11715906	1	6	yes	direct product factor
68	C₂	—	97	1	no	—
69	C₂	—	97	3	no	—
70	C₂	C₂×C₄×#11715906	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²	#11715906
commutator, hypocenter	C₉₇	C₄²×C₈
Frattini	C₂²	C₂²×#11715906
Fitting	C₄²×C₉₇	C₈
socle	C₂²×C₉₇	C₂²×C₈
prime radical	C₂²×D₉₇	C₂²×C₄
cosocle	C₂²×#87604	C₂³
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. C₄×#11715906	C₄	direct (N × H)
11. C₄×#87604	C₈	semidirect
14. C₄×Dic₉₇	C₈	semidirect
15. C₄²×C₉₇	C₈	semidirect
20. #11715906	C₄²	direct (N × H)
28. #87604	C₄×C₈	semidirect
30. C₄×C₉₇	C₄×C₈	semidirect
31. Dic₉₇	C₄×C₈	semidirect
36. C₉₇	C₄²×C₈	semidirect
51. C₄²	#11715906	direct (N × H)
67. C₄	C₄×#11715906	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×#11715906	2
C₄²×#87604	2
C₄²×C₈	97

Subgroups — C42×#11715906 #27158868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#11715906 #27158868