Subgroups — C₂×C₄×Dic₇₀₆ #36378252

Order 22592.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₀₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₇₀₆	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₃₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#19150716	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₃₅₃	C₂	1	2	yes	maximal
6	C₂×C₄²×C₃₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₇₀₆	C₄	1	8	yes	direct product factor
8	C₂×Dic₇₀₆	C₂²	1	4	yes	—
9	C₂²×Dic₃₅₃	C₂²	1	2	yes	—
10	C₂²×Dic₃₅₃	C₂²	1	2	yes	—
11	C₄⋊Dic₃₅₃	C₂²	1	4	yes	—
12	#19150716	C₂²	1	8	yes	—
13	C₄×Dic₃₅₃	C₂²	1	8	yes	—
14	C₂²×C₄×C₃₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₃₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₃₅₃	C₂²	1	1	yes	characteristic
17	C₄²×C₃₅₃	C₂²	1	4	yes	—
18	Dic₇₀₆	C₂×C₄	1	16	yes	direct product factor
19	C₂×Dic₃₅₃	—	2	4	no	—
20	C₂×Dic₃₅₃	C₂×C₄	1	8	yes	—
21	C₂×Dic₃₅₃	C₂³	1	4	yes	—
22	C₂×Dic₃₅₃	C₂³	1	4	yes	—
23	C₂³×C₃₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₅₃	—	2	2	no	—
25	C₂×C₄×C₃₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄×C₃₅₃	Q₈	1	4	yes	complemented
27	C₂×C₄×C₃₅₃	C₂³	1	2	yes	—
28	C₂×C₄×C₃₅₃	C₂³	1	2	yes	—
29	C₂×C₄×C₃₅₃	C₂³	1	2	yes	—
30	C₂²×C₃₅₃	C₂²×C₄	1	2	yes	—
31	C₂²×C₃₅₃	C₂×Q₈	1	2	yes	—
32	C₂²×C₃₅₃	#41	1	2	yes	—
33	C₂²×C₃₅₃	C₂⁴	1	1	yes	characteristic
34	C₄×C₃₅₃	—	2	2	no	—
35	C₄×C₃₅₃	C₂²×C₄	1	4	yes	—
36	C₄×C₃₅₃	C₂×Q₈	1	4	yes	complemented
37	Dic₃₅₃	—	2	4	no	—
38	Dic₃₅₃	C₂²×C₄	1	8	yes	—
39	C₂×C₃₅₃	C₄×Q₈	1	4	yes	complemented
40	C₂×C₃₅₃	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₃₅₃	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₃₅₃	C₂×#41	1	1	yes	characteristic
43	C₃₅₃	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄×Q₈	—	353	1	no	maximal, Sylow
45	C₂²×Q₈	—	353	1	no	—
46	C₄×Q₈	—	353	8	no	—
47	C₂×C₄⋊C₄	—	353	2	no	—
48	C₂×C₄⋊C₄	—	353	1	no	—
49	C₂×C₄²	—	353	2	no	—
50	C₂×C₄²	D₃₅₃	1	1	yes	characteristic
51	C₂×Q₈	—	353	8	no	—
52	C₂×Q₈	—	353	4	no	—
53	C₂²×C₄	—	353	2	no	—
54	C₂²×C₄	—	353	2	no	—
55	C₂²×C₄	C₂×D₃₅₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₃₅₃	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₃₅₃	1	1	yes	characteristic
58	C₄⋊C₄	—	353	8	no	—
59	C₄⋊C₄	—	353	4	no	—
60	C₄²	—	353	8	no	—
61	C₄²	C₂×D₃₅₃	1	4	yes	—
62	C₂³	C₂²×D₃₅₃	1	1	yes	characteristic
63	Q₈	—	353	16	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	353	8	no	—
66	C₂×C₄	—	353	4	no	—
67	C₂×C₄	—	353	4	no	—
68	C₂×C₄	—	706	4	no	—
69	C₂×C₄	C₄×D₃₅₃	1	4	yes	—
70	C₂×C₄	Dic₇₀₆	1	4	yes	direct product factor
71	C₂×C₄	C₂²×D₃₅₃	1	2	yes	—
72	C₂×C₄	C₂²×D₃₅₃	1	2	yes	—
73	C₂×C₄	C₂²×D₃₅₃	1	2	yes	—
74	C₂²	C₂×C₄×D₃₅₃	1	2	yes	—
75	C₂²	C₂×Dic₇₀₆	1	2	yes	—
76	C₂²	#19150737	1	2	yes	—
77	C₂²	C₂³×D₃₅₃	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	353	8	no	—
80	C₄	—	706	4	no	—
81	C₄	C₂×C₄×D₃₅₃	1	4	yes	—
82	C₄	C₂×Dic₇₀₆	1	4	yes	direct product factor
83	C₂	C₄×Dic₇₀₆	1	4	yes	direct product factor
84	C₂	C₂²×C₄×D₃₅₃	1	1	yes	characteristic
85	C₂	C₂²×Dic₇₀₆	1	1	yes	characteristic
86	C₂	C₂×#19150737	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₃
commutator	C₂×C₃₅₃	C₂³×C₄
Frattini	C₂²	C₂³×D₃₅₃
Fitting	C₂×C₄²×C₃₅₃	C₂
socle, prime radical	C₂³×C₃₅₃	C₂³
cosocle	C₂²×C₃₅₃	C₂⁴
hypercenter	C₂×C₄²	D₃₅₃
hypocenter	C₃₅₃	C₂×C₄×Q₈
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₇₀₆	C₂	direct (N × H)
7. C₂×Dic₇₀₆	C₄	direct (N × H)
18. Dic₇₀₆	C₂×C₄	direct (N × H)
26. C₂×C₄×C₃₅₃	Q₈	semidirect
36. C₄×C₃₅₃	C₂×Q₈	semidirect
39. C₂×C₃₅₃	C₄×Q₈	semidirect
43. C₃₅₃	C₂×C₄×Q₈	semidirect
70. C₂×C₄	Dic₇₀₆	direct (N × H)
82. C₄	C₂×Dic₇₀₆	direct (N × H)
83. C₂	C₄×Dic₇₀₆	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×Dic₇₀₆	2
C₄×Dic₇₀₆	2
C₂×C₄⋊Dic₃₅₃	2
C₂×#19150716	2
C₂×C₄×Dic₃₅₃	2
C₂×C₄²×C₃₅₃	2
C₂×C₄×Q₈	353

Subgroups — C2×C4×Dic706 #36378252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₇₀₆ #36378252