Subgroups — C₆₇×#1849 #22556280

Order 8576.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×C₆₇	C₂	1	4	yes	maximal
2	#423×C₆₇	C₂	1	2	yes	maximal
3	C₂×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×C₆₇	C₂²	1	4	yes	—
5	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
6	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
7	#108×C₆₇	—	2	8	no	—
8	C₄⋊C₈×C₆₇	—	2	4	no	—
9	#97×C₆₇	C₄	1	4	yes	—
10	C₂²×C₄×C₆₇	Q₈	1	2	yes	—
11	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
12	M₁₆×C₆₇	—	2	8	no	—
13	C₂×C₈×C₆₇	—	2	4	no	—
14	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
15	C₂×C₈×C₆₇	D₄	1	4	yes	—
16	C₂×C₈×C₆₇	Q₈	1	4	yes	—
17	C₄²×C₆₇	C₂×C₄	1	2	yes	—
18	C₄²×C₆₇	D₄	1	2	yes	—
19	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₄×C₆₇	—	4	2	no	—
21	C₂×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
24	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
26	C₈×C₆₇	—	4	4	no	—
27	C₈×C₆₇	C₄⋊C₄	1	8	yes	—
28	C₂²×C₆₇	—	2	2	no	—
29	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₆₇	C₄⋊Q₈	1	2	yes	—
31	C₄×C₆₇	—	2	4	no	—
32	C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
33	C₄×C₆₇	#127	1	1	yes	characteristic
34	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
35	C₂×C₆₇	—	2	2	no	—
36	C₂×C₆₇	#384	1	1	yes	characteristic
37	C₂×C₆₇	#430	1	2	yes	—
38	#1849	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₆₇	#1849	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×#108	C₂×C₆₇	1	4	yes	—
41	#423	C₂×C₆₇	1	2	yes	—
42	C₂×#97	C₂×C₆₇	1	1	yes	characteristic
43	C₂×M₁₆	C₂²×C₆₇	1	4	yes	—
44	C₂²×C₈	C₂²×C₆₇	1	2	yes	—
45	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
46	#108	—	2	8	no	—
47	C₄⋊C₈	—	2	4	no	—
48	#97	C₄×C₆₇	1	4	yes	—
49	C₂²×C₄	Q₈×C₆₇	1	2	yes	—
50	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
51	M₁₆	—	2	8	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	C₂×C₄×C₆₇	1	4	yes	—
54	C₂×C₈	D₄×C₆₇	1	4	yes	—
55	C₂×C₈	Q₈×C₆₇	1	4	yes	—
56	C₄²	C₂×C₄×C₆₇	1	2	yes	—
57	C₄²	D₄×C₆₇	1	2	yes	—
58	C₂³	C₂×Q₈×C₆₇	1	1	yes	characteristic
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
61	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
62	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
63	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
64	C₂×C₄	C₂×Q₈×C₆₇	1	2	yes	—
65	C₈	—	4	4	no	—
66	C₈	C₄⋊C₄×C₆₇	1	8	yes	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
69	C₂²	C₄⋊Q₈×C₆₇	1	2	yes	—
70	C₄	—	2	4	no	—
71	C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
72	C₄	#127×C₆₇	1	1	yes	characteristic
73	C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
74	C₂	—	2	2	no	—
75	C₂	#384×C₆₇	1	1	yes	characteristic
76	C₂	#430×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×C₄⋊C₄
prime radical	C₂³×C₆₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
38. #1849	C₆₇	direct (N × H)
39. C₆₇	#1849	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#108×C₆₇	2
#423×C₆₇	2
C₂×#97×C₆₇	2
#1849	67

Subgroups — C67×#1849 #22556280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#1849 #22556280