Subgroups — C₂×#20261623 #27157936

Order 12416.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13095514	C₂	1	2	yes	maximal
2	#20261623	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	C₄	1	8	yes	complemented
5	C₂×#13095514	C₂²	1	4	yes	—
6	C₂²×#11715895	C₄	1	1	yes	characteristic
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	C₄×#11715895	C₄	1	2	yes	—
9	C₄×#11715895	C₂²	1	2	yes	—
10	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic
11	#13095514	C₂×C₄	1	8	yes	complemented
12	C₂×#11715895	C₈	1	4	yes	—
13	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
14	C₂×#11715895	C₂×C₄	1	4	yes	—
15	C₂×#11715895	C₂×C₄	1	2	yes	—
16	C₂×#11715895	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₉₇	C₈	1	2	yes	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₉₇	C₈	1	2	yes	—
20	C₄²×C₉₇	C₂×C₄	1	2	yes	—
21	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
22	C₂×C₄×C₉₇	—	2	4	no	—
23	C₂×C₄×C₉₇	C₄²	1	2	yes	—
24	C₂×C₄×C₉₇	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
28	#11715895	C₄²	1	2	yes	—
29	#11715895	C₂×C₈	1	4	yes	—
30	#11715895	C₂²×C₄	1	2	yes	—
31	C₂²×C₉₇	C₄×C₈	1	2	yes	—
32	C₂²×C₉₇	M₃₂	1	4	yes	—
33	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
34	C₄×C₉₇	—	2	4	no	—
35	C₄×C₉₇	C₄×C₈	1	2	yes	—
36	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
37	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
38	C₂×C₉₇	#346	1	4	yes	complemented
39	C₂×C₉₇	C₂×C₄×C₈	1	1	yes	characteristic
40	C₂×C₉₇	C₂×M₃₂	1	2	yes	—
41	C₂×#346	—	97	1	no	maximal, Sylow
42	C₉₇	C₂×#346	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₁₆	—	97	2	no	—
44	C₂×C₄×C₈	—	97	1	no	—
45	#346	—	97	4	no	—
46	C₂²×C₈	—	97	1	no	—
47	C₂²×C₈	—	97	1	no	—
48	C₂×C₄²	#87604	1	1	yes	characteristic
49	C₂×C₁₆	—	97	4	no	—
50	C₂×C₁₆	—	97	8	no	—
51	C₄×C₈	—	97	2	no	—
52	C₄×C₈	—	97	2	no	—
53	C₂²×C₄	#11715897	1	2	yes	—
54	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
55	C₂×C₈	—	97	4	no	—
56	C₂×C₈	—	97	4	no	—
57	C₂×C₈	—	97	1	no	—
58	C₂×C₈	—	97	2	no	—
59	C₂×C₈	—	97	1	no	—
60	C₄²	#11715897	1	2	yes	—
61	C₄²	C₂×#87604	1	2	yes	—
62	C₁₆	—	97	8	no	—
63	C₂³	C₂×#11715897	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	C₄×#87604	1	2	yes	—
66	C₂×C₄	C₂×#11715897	1	4	yes	—
67	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
68	C₂×C₄	#13095519	1	2	yes	—
69	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
70	C₈	—	97	2	no	—
71	C₈	—	97	2	no	—
72	C₈	—	97	4	no	—
73	C₂²	C₄×#11715897	1	2	yes	—
74	C₂²	#15690884	1	4	yes	—
75	C₂²	C₂²×#11715897	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	C₄×#11715897	1	2	yes	—
78	C₄	C₂×C₄×#87604	1	1	yes	characteristic
79	C₄	C₂×#13095519	1	1	yes	characteristic
80	C₂	#20261623	1	4	yes	direct product factor
81	C₂	C₂×C₄×#11715897	1	1	yes	characteristic
82	C₂	C₂×#15690884	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×#87604
commutator	C₂×C₉₇	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₂×C₄²×C₉₇	C₄
socle, prime radical	C₂³×C₉₇	C₂×C₈
cosocle	C₂×#11715895	C₂³
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	C₂×#346
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #20261623	C₂	direct (N × H)
4. C₂×#13095514	C₄	semidirect
11. #13095514	C₂×C₄	semidirect
38. C₂×C₉₇	#346	semidirect
42. C₉₇	C₂×#346	semidirect
80. C₂	#20261623	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×#13095514	2
#20261623	2
C₂×C₄×#11715895	2
C₂×#346	97

Subgroups — C2×#20261623 #27157936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261623 #27157936