Subgroups — C₂²×#15690721 #27150140

Order 12416.

680 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11715895	C₂	1	2	yes	maximal
2	C₂×#15690721	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715895	C₄	1	16	yes	complemented
5	C₂²×#11715895	C₂²	1	12	yes	—
6	#15690721	C₂²	1	16	yes	direct product factor
7	C₂³×C₄×C₉₇	C₄	1	2	yes	—
8	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₉₇	C₄	1	6	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	6	yes	—
11	C₂×#11715895	C₂×C₄	1	48	yes	complemented
12	C₂×#11715895	C₂³	1	8	yes	—
13	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₉₇	—	2	8	no	—
15	C₂²×C₄×C₉₇	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₉₇	C₂×C₄	1	12	yes	—
17	C₂²×C₄×C₉₇	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
19	C₄²×C₉₇	C₂×C₄	1	12	yes	—
20	C₄²×C₉₇	C₂³	1	4	yes	—
21	C₂³×C₉₇	C₄²	1	4	yes	—
22	C₂³×C₉₇	M₁₆	1	8	yes	—
23	C₂³×C₉₇	C₂²×C₄	1	3	yes	—
24	C₂×C₄×C₉₇	—	2	24	no	—
25	C₂×C₄×C₉₇	C₄²	1	12	yes	—
26	C₂×C₄×C₉₇	C₂²×C₄	1	12	yes	—
27	C₂×C₄×C₉₇	C₂²×C₄	1	8	yes	—
28	C₂×C₄×C₉₇	C₂²×C₄	1	3	yes	—
29	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
30	#11715895	C₂²×C₄	1	16	yes	complemented
31	C₂²×C₉₇	#97	1	16	yes	complemented
32	C₂²×C₉₇	C₂×C₄²	1	6	yes	—
33	C₂²×C₉₇	C₂×M₁₆	1	12	yes	—
34	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₉₇	—	2	8	no	—
36	C₄×C₉₇	C₂×C₄²	1	6	yes	—
37	C₄×C₉₇	C₂³×C₄	1	2	yes	—
38	C₂×C₉₇	C₂×#97	1	12	yes	complemented
39	C₂×C₉₇	C₂²×C₄²	1	1	yes	characteristic
40	C₂×C₉₇	C₂²×M₁₆	1	2	yes	—
41	C₂²×#97	—	97	1	no	maximal, Sylow
42	C₉₇	C₂²×#97	1	1	yes	characteristic, complemented, Sylow
43	C₂³×C₈	—	97	2	no	—
44	C₂²×C₄²	D₉₇	1	1	yes	characteristic
45	C₂×#97	—	97	12	no	—
46	C₂³×C₄	Dic₉₇	1	2	yes	—
47	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
48	C₂²×C₈	—	97	16	no	—
49	C₂²×C₈	—	97	12	no	—
50	C₂×C₄²	Dic₉₇	1	6	yes	—
51	C₂×C₄²	C₂×D₉₇	1	6	yes	—
52	#97	—	97	16	no	—
53	C₂⁴	C₂×Dic₉₇	1	1	yes	characteristic
54	C₂²×C₄	—	2	8	no	—
55	C₂²×C₄	C₄×D₉₇	1	8	yes	—
56	C₂²×C₄	C₂×Dic₉₇	1	12	yes	—
57	C₂²×C₄	C₂×Dic₉₇	1	3	yes	—
58	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
59	C₂×C₈	—	97	48	no	—
60	C₂×C₈	—	97	8	no	—
61	C₄²	C₂×Dic₉₇	1	12	yes	—
62	C₄²	C₂²×D₉₇	1	4	yes	—
63	C₂³	C₄×Dic₉₇	1	4	yes	—
64	C₂³	#13095495	1	8	yes	—
65	C₂³	C₂²×Dic₉₇	1	3	yes	—
66	C₂×C₄	—	2	24	no	—
67	C₂×C₄	C₄×Dic₉₇	1	12	yes	—
68	C₂×C₄	C₂×C₄×D₉₇	1	12	yes	—
69	C₂×C₄	C₂²×Dic₉₇	1	8	yes	—
70	C₂×C₄	C₂²×Dic₉₇	1	3	yes	—
71	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
72	C₈	—	97	16	no	—
73	C₂²	#15690721	1	16	yes	direct product factor
74	C₂²	C₂×C₄×Dic₉₇	1	6	yes	—
75	C₂²	C₂×#13095495	1	12	yes	—
76	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
77	C₄	—	2	8	no	—
78	C₄	C₂×C₄×Dic₉₇	1	6	yes	—
79	C₄	C₂²×C₄×D₉₇	1	2	yes	—
80	C₂	C₂×#15690721	1	12	yes	direct product factor
81	C₂	C₂²×C₄×Dic₉₇	1	1	yes	characteristic
82	C₂	C₂²×#13095495	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×D₉₇
commutator	C₂×C₉₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂²×C₄²×C₉₇	C₂
socle, prime radical	C₂⁴×C₉₇	C₂×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂²×C₄²	D₉₇
hypocenter	C₉₇	C₂²×#97
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#15690721	C₂	direct (N × H)
4. C₂²×#11715895	C₄	semidirect
6. #15690721	C₂²	direct (N × H)
11. C₂×#11715895	C₂×C₄	semidirect
30. #11715895	C₂²×C₄	semidirect
31. C₂²×C₉₇	#97	semidirect
38. C₂×C₉₇	C₂×#97	semidirect
42. C₉₇	C₂²×#97	semidirect
73. C₂²	#15690721	direct (N × H)
80. C₂	C₂×#15690721	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11715895	2
C₂×#15690721	2
C₂²×C₄²×C₉₇	2
C₂²×#97	97

Subgroups — C22×#15690721 #27150140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15690721 #27150140