Subgroups — C₂³×#13095521 #27158723

Order 12416.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13095521	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#11715897	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄×D₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095521	C₂²	1	112	yes	direct product factor
5	C₂²×#11715897	C₂²	1	28	yes	complemented
6	C₂⁴×D₉₇	C₄	1	1	yes	characteristic
7	C₂³×Dic₉₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×D₉₇	C₄	1	14	yes	—
9	C₂²×C₄×D₉₇	C₂²	1	14	yes	—
10	C₂³×C₄×C₉₇	C₄	1	1	yes	characteristic
11	#13095521	C₂³	1	64	yes	direct product factor
12	C₂×#11715897	C₂³	1	56	yes	complemented
13	C₂³×D₉₇	—	2	8	no	—
14	C₂³×D₉₇	C₂×C₄	1	14	yes	—
15	C₂²×Dic₉₇	C₂×C₄	1	7	yes	—
16	C₂²×Dic₉₇	C₂³	1	7	yes	—
17	C₂×C₄×D₉₇	C₂×C₄	1	84	yes	—
18	C₂×C₄×D₉₇	C₂³	1	28	yes	—
19	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	C₂×C₄	1	14	yes	—
21	C₂³×C₉₇	M₁₆	1	8	yes	complemented
22	C₂³×C₉₇	C₂²×C₄	1	7	yes	—
23	C₂²×D₉₇	—	2	56	no	—
24	C₂²×D₉₇	C₂²×C₄	1	28	yes	—
25	C₂×C₄×C₉₇	C₂²×C₄	1	28	yes	—
26	C₂×Dic₉₇	C₂²×C₄	1	21	yes	—
27	C₂×Dic₉₇	C₂⁴	1	7	yes	—
28	C₄×D₉₇	C₂²×C₄	1	56	yes	—
29	C₄×D₉₇	C₂⁴	1	8	yes	—
30	#11715897	C₂⁴	1	16	yes	complemented
31	C₂²×C₉₇	C₂×M₁₆	1	28	yes	complemented
32	C₂²×C₉₇	C₂³×C₄	1	7	yes	—
33	C₂×D₉₇	—	2	56	no	—
34	C₂×D₉₇	C₂³×C₄	1	8	yes	—
35	C₄×C₉₇	C₂³×C₄	1	8	yes	—
36	Dic₉₇	C₂³×C₄	1	7	yes	—
37	Dic₉₇	C₂⁵	1	1	yes	characteristic
38	C₂×C₉₇	C₂²×M₁₆	1	14	yes	complemented
39	C₂×C₉₇	C₂⁴×C₄	1	1	yes	characteristic
40	D₉₇	—	2	8	no	—
41	C₂³×M₁₆	—	97	1	no	maximal, Sylow
42	C₉₇	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
43	C₂⁴×C₄	—	97	1	no	—
44	C₂²×M₁₆	—	97	28	no	—
45	C₂³×C₈	—	97	2	no	—
46	C₂⁵	—	97	1	no	—
47	C₂³×C₄	—	97	1	no	—
48	C₂³×C₄	—	97	14	no	—
49	C₂³×C₄	—	97	14	no	—
50	C₂³×C₄	#87604	1	1	yes	characteristic
51	C₂×M₁₆	—	97	112	no	—
52	C₂²×C₈	—	97	28	no	—
53	C₂⁴	—	97	14	no	—
54	C₂⁴	—	194	8	no	—
55	C₂⁴	C₂×#87604	1	1	yes	characteristic
56	C₂²×C₄	—	97	7	no	—
57	C₂²×C₄	—	97	7	no	—
58	C₂²×C₄	—	97	84	no	—
59	C₂²×C₄	—	97	28	no	—
60	C₂²×C₄	C₂×#87604	1	14	yes	—
61	M₁₆	—	97	64	no	—
62	C₂×C₈	—	97	56	no	—
63	C₂³	—	97	28	no	—
64	C₂³	—	194	56	no	—
65	C₂³	#13095521	1	8	yes	direct product factor
66	C₂³	C₂²×#87604	1	7	yes	—
67	C₂×C₄	—	97	7	no	—
68	C₂×C₄	—	97	21	no	—
69	C₂×C₄	—	97	8	no	—
70	C₂×C₄	—	97	56	no	—
71	C₂×C₄	C₂²×#87604	1	28	yes	—
72	C₈	—	97	16	no	—
73	C₂²	—	97	8	no	—
74	C₂²	—	194	56	no	—
75	C₂²	C₂×#13095521	1	28	yes	direct product factor
76	C₂²	C₂³×#87604	1	7	yes	—
77	C₄	—	97	1	no	—
78	C₄	—	97	7	no	—
79	C₄	C₂³×#87604	1	8	yes	—
80	C₂	—	194	8	no	—
81	C₂	C₂²×#13095521	1	14	yes	direct product factor
82	C₂	C₂⁴×#87604	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×#87604
commutator	C₂×C₉₇	C₂⁴×C₄
Frattini	C₂	C₂⁴×#87604
Fitting	C₂³×C₄×C₉₇	C₄
socle	C₂⁴×C₉₇	C₂×C₄
prime radical	C₂⁴×D₉₇	C₄
cosocle	Dic₉₇	C₂⁵
hypercenter	C₂³×C₄	#87604
hypocenter	C₉₇	C₂³×M₁₆
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13095521	C₂	direct (N × H)
2. C₂³×#11715897	C₂	semidirect
4. C₂×#13095521	C₂²	direct (N × H)
5. C₂²×#11715897	C₂²	semidirect
11. #13095521	C₂³	direct (N × H)
12. C₂×#11715897	C₂³	semidirect
21. C₂³×C₉₇	M₁₆	semidirect
30. #11715897	C₂⁴	semidirect
31. C₂²×C₉₇	C₂×M₁₆	semidirect
38. C₂×C₉₇	C₂²×M₁₆	semidirect
42. C₉₇	C₂³×M₁₆	semidirect
65. C₂³	#13095521	direct (N × H)
75. C₂²	C₂×#13095521	direct (N × H)
81. C₂	C₂²×#13095521	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13095521	2
C₂³×#11715897	2
C₂³×C₄×D₉₇	2
C₂³×M₁₆	97

Subgroups — C23×#13095521 #27158723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13095521 #27158723