Subgroups — C₂×#1356 #11910

Order 256.

157 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#491	C₂	1	1	yes	maximal, characteristic
2	C₂×#347	C₂	1	2	yes	maximal
3	#1356	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
5	C₂×#125	C₄	1	1	yes	characteristic
6	C₂×M₃₂	—	2	2	no	—
7	#491	—	2	2	no	—
8	#491	C₄	1	2	yes	—
9	#491	C₂²	1	2	yes	—
10	C₂×#103	—	2	2	no	—
11	C₂×#97	C₂²	1	1	yes	characteristic
12	#347	—	2	2	no	—
13	#347	C₂²	1	4	yes	—
14	C₂²×Q₈	—	2	1	no	—
15	C₂²×C₈	D₄	1	2	yes	—
16	C₄⋊Q₈	—	2	1	no	—
17	C₄⋊Q₈	—	4	1	no	—
18	C₄⋊Q₈	C₂×C₄	1	2	yes	—
19	#125	—	2	1	no	—
20	#125	—	4	1	no	—
21	#125	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊C₄	—	2	1	no	—
23	C₂×C₄⋊C₄	—	4	1	no	—
24	C₂×C₄⋊C₄	—	4	1	no	—
25	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
26	M₃₂	—	2	4	no	—
27	M₃₂	—	2	4	no	—
28	C₂×C₁₆	—	2	4	no	—
29	#103	—	2	4	no	—
30	#103	—	2	4	no	—
31	#97	C₂×C₄	1	1	yes	characteristic
32	#97	D₄	1	2	yes	—
33	#97	C₂³	1	1	yes	characteristic
34	C₂×Q₈	—	2	2	no	—
35	C₂×Q₈	—	4	1	no	—
36	C₂×Q₈	—	4	2	no	—
37	C₂×Q₈	—	4	2	no	—
38	C₂²×C₄	—	2	1	no	—
39	C₂²×C₄	—	4	1	no	—
40	C₂²×C₄	—	4	1	no	—
41	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈	C₂×D₄	1	2	yes	—
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	—	4	2	no	—
48	C₄⋊C₄	—	4	2	no	—
49	C₄⋊C₄	—	4	2	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄²	—	2	1	no	—
53	C₄²	C₂²⋊C₄	1	1	yes	characteristic
54	C₄²	C₂²×C₄	1	1	yes	characteristic
55	C₁₆	—	2	8	no	—
56	C₂³	#100	1	1	yes	characteristic
57	Q₈	—	4	4	no	—
58	Q₈	—	8	2	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	—	8	2	no	—
65	C₂×C₄	—	8	1	no	—
66	C₂×C₄	—	8	1	no	—
67	C₂×C₄	#100	1	1	yes	characteristic
68	C₂×C₄	#104	1	2	yes	—
69	C₂×C₄	#104	1	2	yes	—
70	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	2	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	#329	1	1	yes	characteristic
75	C₂²	#329	1	1	yes	characteristic
76	C₂²	C₂×#100	1	1	yes	characteristic
77	C₄	—	4	1	no	—
78	C₄	—	4	1	no	—
79	C₄	—	8	2	no	—
80	C₄	—	8	2	no	—
81	C₄	#329	1	2	yes	—
82	C₄	C₂×#104	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	2	1	no	—
85	C₂	#1356	1	2	yes	direct product factor
86	C₂	C₂×#329	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#329
commutator	C₄²	C₂²×C₄
Frattini, cosocle	#97	C₂³
prime radical	C₂³	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1356	C₂	direct (N × H)
85. C₂	#1356	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×#491	2
C₂×#347	2
#1356	2

Subgroups — C2×#1356 #11910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1356 #11910