Subgroups — C₆₇×#2110 #22556541

Order 8576.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×C₆₇	C₂	1	2	yes	maximal
2	#405×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#347×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₆₇	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	#117×C₆₇	C₄	1	1	yes	characteristic
7	M₃₂×C₆₇	C₄	1	4	yes	complemented
8	M₃₂×C₆₇	C₂²	1	4	yes	—
9	C₂×C₁₆×C₆₇	C₄	1	4	yes	complemented
10	#97×C₆₇	C₂²	1	2	yes	—
11	C₄×C₈×C₆₇	C₄	1	2	yes	—
12	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
13	M₁₆×C₆₇	—	2	2	no	—
14	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
15	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₂²⋊C₄×C₆₇	—	2	1	no	—
21	C₄²×C₆₇	C₂×C₄	1	2	yes	—
22	C₁₆×C₆₇	C₂×C₄	1	8	yes	—
23	C₂³×C₆₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₆₇	—	2	2	no	—
25	C₂×C₄×C₆₇	C₄²	1	2	yes	—
26	C₂×C₄×C₆₇	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₆₇	M₁₆	1	2	yes	—
28	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₆₇	—	2	2	no	—
30	C₈×C₆₇	C₄²	1	1	yes	characteristic
31	C₈×C₆₇	C₄²	1	1	yes	characteristic
32	C₈×C₆₇	C₂²×C₄	1	2	yes	—
33	C₂²×C₆₇	—	4	1	no	—
34	C₂²×C₆₇	#97	1	2	yes	—
35	C₂²×C₆₇	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₆₇	—	4	2	no	—
37	C₄×C₆₇	#97	1	2	yes	—
38	C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
40	C₂×C₆₇	—	2	1	no	—
41	C₂×C₆₇	—	2	2	no	—
42	C₂×C₆₇	C₂×#97	1	1	yes	characteristic
43	#2110	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₆₇	#2110	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×M₃₂	C₂×C₆₇	1	2	yes	—
46	#405	C₂×C₆₇	1	1	yes	characteristic
47	#347	C₂×C₆₇	1	4	yes	complemented
48	C₂×M₁₆	C₄×C₆₇	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₆₇	1	1	yes	characteristic
50	#117	C₄×C₆₇	1	1	yes	characteristic
51	M₃₂	C₄×C₆₇	1	4	yes	complemented
52	M₃₂	C₂²×C₆₇	1	4	yes	—
53	C₂×C₁₆	C₄×C₆₇	1	4	yes	complemented
54	#97	C₂²×C₆₇	1	2	yes	—
55	C₄×C₈	C₄×C₆₇	1	2	yes	—
56	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
57	M₁₆	—	2	2	no	—
58	C₂×C₈	C₂×C₄×C₆₇	1	2	yes	—
59	C₂×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
60	C₂×C₈	C₂×C₄×C₆₇	1	2	yes	—
61	C₂×C₈	C₂×C₄×C₆₇	1	2	yes	—
62	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
63	C₄⋊C₄	—	2	1	no	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×C₄×C₆₇	1	2	yes	—
66	C₁₆	C₂×C₄×C₆₇	1	8	yes	—
67	C₂³	M₁₆×C₆₇	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	C₄²×C₆₇	1	2	yes	—
70	C₂×C₄	M₁₆×C₆₇	1	2	yes	—
71	C₂×C₄	M₁₆×C₆₇	1	1	yes	characteristic
72	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	C₄²×C₆₇	1	1	yes	characteristic
75	C₈	C₄²×C₆₇	1	1	yes	characteristic
76	C₈	C₂²×C₄×C₆₇	1	2	yes	—
77	C₂²	—	4	1	no	—
78	C₂²	#97×C₆₇	1	2	yes	—
79	C₂²	C₂×M₁₆×C₆₇	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	#97×C₆₇	1	2	yes	—
82	C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
83	C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#97×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₇	C₄²
commutator	C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂×C₈	C₂³×C₆₇
socle	C₂×C₆₇	C₂×#97
prime radical	C₂³×C₆₇	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #347×C₆₇	C₂	semidirect
7. M₃₂×C₆₇	C₄	semidirect
9. C₂×C₁₆×C₆₇	C₄	semidirect
43. #2110	C₆₇	direct (N × H)
44. C₆₇	#2110	direct (N × H)
47. #347	C₂×C₆₇	semidirect
51. M₃₂	C₄×C₆₇	semidirect
53. C₂×C₁₆	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×C₆₇	2
#405×C₆₇	2
#347×C₆₇	2
#2110	67

Subgroups — C67×#2110 #22556541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2110 #22556541