Subgroups — C₆₇×#1373 #22555804

Order 8576.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#364×C₆₇	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	M₃₂×C₆₇	—	2	2	no	—
4	C₂×C₁₆×C₆₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₆₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
8	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
9	C₄²×C₆₇	Q₈	1	1	yes	characteristic
10	C₁₆×C₆₇	—	2	1	no	—
11	C₁₆×C₆₇	—	4	2	no	—
12	C₁₆×C₆₇	C₈	1	2	yes	—
13	C₂×C₄×C₆₇	—	2	1	no	—
14	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
15	C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
16	C₈×C₆₇	M₁₆	1	1	yes	characteristic
17	C₈×C₆₇	D₈	1	1	yes	characteristic
18	C₈×C₆₇	Q₁₆	1	1	yes	characteristic
19	C₂²×C₆₇	#108	1	1	yes	characteristic
20	C₄×C₆₇	—	2	2	no	—
21	C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₆₇	#107	1	1	yes	characteristic
23	C₂×C₆₇	—	2	1	no	—
24	C₂×C₆₇	#335	1	1	yes	characteristic
25	#1373	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₆₇	#1373	1	1	yes	characteristic, direct product factor, Sylow
27	#364	C₂×C₆₇	1	2	yes	—
28	C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
29	M₃₂	—	2	2	no	—
30	C₂×C₁₆	C₄×C₆₇	1	1	yes	characteristic
31	C₂×C₁₆	C₄×C₆₇	1	1	yes	characteristic
32	C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
33	C₂×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
34	C₂×C₈	D₄×C₆₇	1	1	yes	characteristic
35	C₄²	Q₈×C₆₇	1	1	yes	characteristic
36	C₁₆	—	2	1	no	—
37	C₁₆	—	4	2	no	—
38	C₁₆	C₈×C₆₇	1	2	yes	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
41	C₈	C₂×C₈×C₆₇	1	1	yes	characteristic
42	C₈	M₁₆×C₆₇	1	1	yes	characteristic
43	C₈	D₈×C₆₇	1	1	yes	characteristic
44	C₈	Q₁₆×C₆₇	1	1	yes	characteristic
45	C₂²	#108×C₆₇	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄⋊C₈×C₆₇	1	1	yes	characteristic
48	C₄	#107×C₆₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	#335×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₇	D₈
commutator	C₈	C₂×C₈×C₆₇
Frattini, cosocle	C₄×C₈	C₂²×C₆₇
socle	C₂×C₆₇	#335
prime radical	C₂²×C₆₇	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₇ ◁ C₂×C₈×C₆₇ ◁ C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
25. #1373	C₆₇	direct (N × H)
26. C₆₇	#1373	direct (N × H)

Maximal subgroups

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

subgroup	index
#364×C₆₇	2
C₄×C₁₆×C₆₇	2
#1373	67

Subgroups — C67×#1373 #22555804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#1373 #22555804