Subgroups — C₂×#20261843 #27158966

Order 12416.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261843	C₂	1	4	yes	maximal, direct product factor
2	C₂×#15690900	C₂	1	2	yes	maximal, complemented
3	C₂²×#13095514	C₂	1	1	yes	maximal, characteristic
4	#15690900	C₂²	1	4	yes	complemented
5	C₂×#13095514	C₄	1	2	yes	—
6	C₂×#13095514	C₄	1	1	yes	characteristic
7	C₂×#13095514	C₂²	1	1	yes	characteristic
8	C₂×#13095514	C₂²	1	2	yes	—
9	C₂²×#11715895	C₄	1	1	yes	characteristic
10	#13095514	C₂×C₄	1	1	yes	characteristic
11	#13095514	C₂×C₄	1	2	yes	—
12	#13095514	C₂³	1	1	yes	characteristic
13	C₂×#11715895	C₈	1	2	yes	—
14	C₂×#11715895	C₈	1	1	yes	characteristic
15	C₂×#11715895	C₂×C₄	1	2	yes	—
16	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₉₇	C₈	1	1	yes	characteristic
18	C₂³×C₉₇	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₉₇	C₁₆	1	2	yes	—
20	C₂×C₄×C₉₇	C₁₆	1	1	yes	characteristic
21	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
22	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
23	#11715895	C₂×C₈	1	1	yes	characteristic
24	#11715895	C₂×C₈	1	2	yes	—
25	#11715895	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₉₇	—	2	2	no	—
27	C₂²×C₉₇	C₂×C₁₆	1	2	yes	—
28	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
29	C₄×C₉₇	C₂×C₁₆	1	2	yes	—
30	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₉₇	—	2	2	no	—
33	C₂×C₉₇	M₆₄	1	2	yes	complemented
34	C₂×C₉₇	C₂²×C₁₆	1	1	yes	characteristic
35	C₂×M₆₄	—	97	1	no	maximal, Sylow
36	C₉₇	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
37	C₂²×C₁₆	—	97	1	no	—
38	M₆₄	—	97	4	no	—
39	C₂×C₃₂	—	97	2	no	—
40	C₂²×C₈	—	97	1	no	—
41	C₂×C₁₆	—	97	1	no	—
42	C₂×C₁₆	—	97	2	no	—
43	C₂×C₁₆	—	97	2	no	—
44	C₂×C₁₆	—	97	1	no	—
45	C₃₂	—	97	4	no	—
46	C₂²×C₄	#11715906	1	1	yes	characteristic
47	C₂×C₈	—	97	2	no	—
48	C₂×C₈	—	97	2	no	—
49	C₂×C₈	—	97	1	no	—
50	C₂×C₈	—	97	1	no	—
51	C₁₆	—	97	1	no	—
52	C₁₆	—	97	2	no	—
53	C₁₆	—	97	1	no	—
54	C₂³	#13095523	1	1	yes	characteristic
55	C₂×C₄	#13095523	1	1	yes	characteristic
56	C₂×C₄	#13095523	1	2	yes	—
57	C₂×C₄	C₂×#11715906	1	2	yes	—
58	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
59	C₈	—	97	1	no	—
60	C₈	—	97	1	no	—
61	C₈	—	97	2	no	—
62	C₂²	—	2	2	no	—
63	C₂²	C₂×#13095523	1	2	yes	—
64	C₂²	C₂×#13095523	1	1	yes	characteristic
65	C₄	C₂×#13095523	1	2	yes	—
66	C₄	C₂×#13095523	1	1	yes	characteristic
67	C₄	C₂²×#11715906	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	#20261843	1	2	yes	direct product factor
70	C₂	C₂²×#13095523	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11715906
commutator	C₂×C₉₇	C₂²×C₁₆
Frattini	C₄	C₂²×#11715906
Fitting	C₂²×C₄×C₉₇	C₈
socle	C₂²×C₉₇	C₂×C₁₆
prime radical	C₂³×C₉₇	C₁₆
cosocle	#13095514	C₂³
hypercenter	C₂²×C₄	#11715906
hypocenter	C₉₇	C₂×M₆₄
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. #20261843	C₂	direct (N × H)
2. C₂×#15690900	C₂	semidirect
4. #15690900	C₂²	semidirect
33. C₂×C₉₇	M₆₄	semidirect
36. C₉₇	C₂×M₆₄	semidirect
69. C₂	#20261843	direct (N × H)

Maximal subgroups

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

subgroup	index
#20261843	2
C₂×#15690900	2
C₂²×#13095514	2
C₂×M₆₄	97

Subgroups — C2×#20261843 #27158966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20261843 #27158966