Subgroups — #328×C₃₀₇ #34087347

Order 19648.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊Q₈×C₃₀₇	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₈×C₃₀₇	C₂	1	1	yes	maximal, characteristic
3	#95×C₃₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₃₀₇	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₄×C₃₀₇	—	2	1	no	—
6	C₂²×C₄×C₃₀₇	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₃₀₇	—	2	1	no	—
8	C₄⋊C₄×C₃₀₇	—	2	1	no	—
9	C₄⋊C₄×C₃₀₇	C₄	1	1	yes	characteristic, complemented
10	C₂²⋊C₄×C₃₀₇	—	2	1	no	—
11	C₂³×C₃₀₇	D₄	1	1	yes	characteristic
12	Q₈×C₃₀₇	—	4	1	no	—
13	C₂×C₄×C₃₀₇	—	2	2	no	—
14	C₂×C₄×C₃₀₇	—	2	1	no	—
15	C₂×C₄×C₃₀₇	—	2	1	no	—
16	C₂×C₄×C₃₀₇	—	4	1	no	—
17	C₂×C₄×C₃₀₇	—	4	1	no	—
18	C₂×C₄×C₃₀₇	—	4	1	no	—
19	C₂×C₄×C₃₀₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₀₇	D₄	1	1	yes	characteristic
21	C₈×C₃₀₇	—	4	1	no	—
22	C₂²×C₃₀₇	—	2	1	no	—
23	C₂²×C₃₀₇	—	2	1	no	—
24	C₂²×C₃₀₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₃₀₇	SD₁₆	1	1	yes	characteristic
26	C₂²×C₃₀₇	Q₁₆	1	1	yes	characteristic
27	C₄×C₃₀₇	—	2	1	no	—
28	C₄×C₃₀₇	—	2	1	no	—
29	C₄×C₃₀₇	—	4	2	no	—
30	C₄×C₃₀₇	—	4	1	no	—
31	C₄×C₃₀₇	—	4	1	no	—
32	C₂×C₃₀₇	—	2	1	no	—
33	C₂×C₃₀₇	—	2	1	no	—
34	C₂×C₃₀₇	#99	1	1	yes	characteristic
35	C₂×C₃₀₇	#103	1	1	yes	characteristic
36	C₂×C₃₀₇	#104	1	1	yes	characteristic
37	C₃₀₇	#328	1	1	yes	characteristic, direct product factor, Sylow
38	#328	C₃₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²⋊Q₈	C₂×C₃₀₇	1	1	yes	characteristic
40	C₂²⋊C₈	C₂×C₃₀₇	1	1	yes	characteristic
41	#95	C₂×C₃₀₇	1	1	yes	characteristic
42	C₂×Q₈	C₄×C₃₀₇	1	1	yes	characteristic, complemented
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	C₂²×C₃₀₇	1	1	yes	characteristic
45	C₂×C₈	—	2	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₄×C₃₀₇	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	2	1	no	—
49	C₂³	D₄×C₃₀₇	1	1	yes	characteristic
50	Q₈	—	4	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	C₂×C₄×C₃₀₇	1	1	yes	characteristic
58	C₂×C₄	D₄×C₃₀₇	1	1	yes	characteristic
59	C₈	—	4	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	—	2	1	no	—
62	C₂²	C₂²⋊C₄×C₃₀₇	1	1	yes	characteristic
63	C₂²	SD₁₆×C₃₀₇	1	1	yes	characteristic
64	C₂²	Q₁₆×C₃₀₇	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	—	4	2	no	—
68	C₄	—	4	1	no	—
69	C₄	—	4	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	#99×C₃₀₇	1	1	yes	characteristic
73	C₂	#103×C₃₀₇	1	1	yes	characteristic
74	C₂	#104×C₃₀₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₀₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₃₀₇
Frattini, cosocle	C₂²×C₄	C₂²×C₃₀₇
prime radical	C₂³×C₃₀₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₀₇ ◁ C₂²×C₄×C₃₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×Q₈×C₃₀₇	C₄	semidirect
9. C₄⋊C₄×C₃₀₇	C₄	semidirect
37. C₃₀₇	#328	direct (N × H)
38. #328	C₃₀₇	direct (N × H)
42. C₂×Q₈	C₄×C₃₀₇	semidirect
47. C₄⋊C₄	C₄×C₃₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊Q₈×C₃₀₇	2
C₂²⋊C₈×C₃₀₇	2
#95×C₃₀₇	2
#328	307

Subgroups — #328×C307 #34087347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #328×C₃₀₇ #34087347