Subgroups — #30787033

Order 17152.

1278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558436	C₂	1	14	yes	maximal
2	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₇⋊#95	C₂²	1	28	yes	—
4	C₂²×C₄²×C₆₇	C₂²	1	7	yes	—
5	C₄³×C₆₇	C₄	1	8	yes	complemented
6	C₂³×Dic₆₇	C₂³	1	8	yes	—
7	C₆₇⋊#95	—	2	56	no	—
8	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
9	C₂×C₄²×C₆₇	—	2	28	no	—
10	C₂×C₄²×C₆₇	C₂×C₄	1	28	yes	—
11	C₂²×Dic₆₇	—	2	56	no	—
12	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
13	C₂²×C₄×C₆₇	—	2	28	no	—
14	C₂²×C₄×C₆₇	C₂²×C₄	1	14	yes	—
15	C₂²×C₄×C₆₇	#41	1	28	yes	—
16	C₄²×C₆₇	—	2	56	no	—
17	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
18	C₂³×C₆₇	C₂×#41	1	14	yes	—
19	C₂×C₄×C₆₇	—	2	28	no	—
20	C₂×C₄×C₆₇	—	2	56	no	—
21	C₂×C₄×C₆₇	#117	1	28	yes	—
22	C₂×Dic₆₇	—	4	56	no	—
23	C₂²×C₆₇	C₂×#117	1	7	yes	—
24	C₂²×C₆₇	#529	1	28	yes	—
25	C₄×C₆₇	—	2	28	no	—
26	Dic₆₇	—	8	8	no	—
27	#14690	—	67	1	no	maximal, Sylow
28	C₂×C₆₇	#2283	1	7	yes	—
29	C₂×C₆₇	#2865	1	8	yes	—
30	C₂×C₄³	D₆₇	1	1	yes	characteristic
31	#1434	—	67	14	no	—
32	C₆₇	#14690	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₄²	C₂×D₆₇	1	7	yes	—
34	C₂×#95	—	67	28	no	—
35	C₄³	Dic₆₇	1	8	yes	complemented
36	C₂³×C₄	—	67	8	no	—
37	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
38	C₂×C₄²	—	2	28	no	—
39	C₂×C₄²	C₂×Dic₆₇	1	28	yes	—
40	#95	—	134	56	no	—
41	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
42	C₂²×C₄	—	2	28	no	—
43	C₂²×C₄	—	134	56	no	—
44	C₂²×C₄	C₂²×Dic₆₇	1	14	yes	—
45	C₂²×C₄	#11761628	1	28	yes	—
46	C₄²	—	2	56	no	—
47	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
48	C₂³	C₂×#11761628	1	14	yes	—
49	C₂×C₄	—	2	28	no	—
50	C₂×C₄	—	2	56	no	—
51	C₂×C₄	—	268	56	no	—
52	C₂×C₄	#14524002	1	28	yes	—
53	C₂²	C₂×#14524002	1	7	yes	—
54	C₂²	#16863930	1	28	yes	—
55	C₄	—	2	28	no	—
56	C₄	—	536	8	no	—
57	C₂	#22562607	1	7	yes	—
58	C₂	#22568598	1	8	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₆₇
commutator	C₂³×C₆₇	C₂³×C₄
Fitting	C₂×C₄³×C₆₇	C₂
socle, prime radical, cosocle	C₂⁴×C₆₇	C₂⁴
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	#14690
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
5. C₄³×C₆₇	C₄	semidirect
32. C₆₇	#14690	semidirect
35. C₄³	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22558436	2
C₂×C₄³×C₆₇	2
#14690	67