Subgroups — C₆₇×#2845 #22557276

Order 8576.

340 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#398×C₆₇	C₂	1	12	yes	maximal
2	#382×C₆₇	C₂	1	3	yes	maximal
3	C₂×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
4	C₂×C₄⋊C₄×C₆₇	C₂²	1	12	yes	—
5	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
6	#95×C₆₇	C₂²	1	12	yes	—
7	#95×C₆₇	C₂²	1	4	yes	—
8	C₂²×C₄×C₆₇	C₂³	1	12	yes	—
9	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
10	C₄⋊C₄×C₆₇	—	4	6	no	—
11	C₄⋊C₄×C₆₇	—	4	12	no	—
12	C₄²×C₆₇	Q₈	1	4	yes	—
13	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
14	C₂×C₄×C₆₇	—	2	3	no	—
15	C₂×C₄×C₆₇	—	2	3	no	—
16	C₂×C₄×C₆₇	—	2	12	no	—
17	C₂×C₄×C₆₇	—	2	12	no	—
18	C₂×C₄×C₆₇	—	2	12	no	—
19	C₂×C₄×C₆₇	C₂×Q₈	1	6	yes	—
20	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
21	C₂²×C₆₇	H(5,𝔽₂)	1	3	yes	—
22	C₂²×C₆₇	2₋¹⁺⁴	1	3	yes	—
23	C₄×C₆₇	—	2	6	no	—
24	C₄×C₆₇	—	4	12	no	—
25	C₂×C₆₇	#544	1	3	yes	—
26	C₂×C₆₇	#562	1	3	yes	—
27	C₂×C₆₇	#564	1	1	yes	characteristic
28	#2845	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₆₇	#2845	1	1	yes	characteristic, direct product factor, Sylow
30	#398	C₂×C₆₇	1	12	yes	—
31	#382	C₂×C₆₇	1	3	yes	—
32	C₂×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
33	C₂×C₄⋊C₄	C₂²×C₆₇	1	12	yes	—
34	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
35	#95	C₂²×C₆₇	1	12	yes	—
36	#95	C₂²×C₆₇	1	4	yes	—
37	C₂²×C₄	C₂³×C₆₇	1	12	yes	—
38	C₂²×C₄	C₂³×C₆₇	1	3	yes	—
39	C₄⋊C₄	—	4	6	no	—
40	C₄⋊C₄	—	4	12	no	—
41	C₄²	Q₈×C₆₇	1	4	yes	—
42	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
43	C₂×C₄	—	2	3	no	—
44	C₂×C₄	—	2	3	no	—
45	C₂×C₄	—	2	12	no	—
46	C₂×C₄	—	2	12	no	—
47	C₂×C₄	—	2	12	no	—
48	C₂×C₄	C₂×Q₈×C₆₇	1	6	yes	—
49	C₂²	C₂²×Q₈×C₆₇	1	1	yes	characteristic
50	C₂²	H(5,𝔽₂)×C₆₇	1	3	yes	—
51	C₂²	2₋¹⁺⁴×C₆₇	1	3	yes	—
52	C₄	—	2	6	no	—
53	C₄	—	4	12	no	—
54	C₂	#544×C₆₇	1	3	yes	—
55	C₂	#562×C₆₇	1	3	yes	—
56	C₂	#564×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
28. #2845	C₆₇	direct (N × H)
29. C₆₇	#2845	direct (N × H)

Maximal subgroups

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

subgroup	index
#398×C₆₇	2
#382×C₆₇	2
#2845	67

Subgroups — C67×#2845 #22557276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2845 #22557276