Subgroups — C₆₇×#8803 #30688054

Order 17152.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#1455	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1291	C₂	1	3	yes	maximal
3	C₆₇×#1288	C₂	1	3	yes	maximal
4	#432×C₆₇	C₂²	1	3	yes	—
5	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
6	#383×C₆₇	C₂²	1	1	yes	characteristic
7	#322×C₆₇	C₄	1	3	yes	complemented
8	C₈²×C₆₇	C₄	1	1	yes	characteristic, complemented
9	C₂×M₁₆×C₆₇	—	2	3	no	—
10	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
11	C₄⋊C₈×C₆₇	—	2	3	no	—
12	C₂²⋊C₈×C₆₇	—	2	3	no	—
13	#97×C₆₇	—	2	3	no	—
14	#97×C₆₇	—	2	3	no	—
15	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
16	C₄×C₈×C₆₇	C₂×C₄	1	3	yes	—
17	#95×C₆₇	—	2	3	no	—
18	C₂²×C₄×C₆₇	—	4	1	no	—
19	C₂²×C₄×C₆₇	#41	1	3	yes	—
20	M₁₆×C₆₇	—	4	6	no	—
21	C₂×C₈×C₆₇	—	2	3	no	—
22	C₂×C₈×C₆₇	—	2	3	no	—
23	C₂×C₈×C₆₇	—	4	3	no	—
24	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
25	C₄²×C₆₇	#41	1	3	yes	—
26	C₂³×C₆₇	#126	1	1	yes	characteristic
27	C₂×C₄×C₆₇	—	4	3	no	—
28	C₂×C₄×C₆₇	—	8	3	no	—
29	C₂×C₄×C₆₇	#117	1	3	yes	—
30	C₂×C₄×C₆₇	#126	1	3	yes	—
31	C₈×C₆₇	—	2	6	no	—
32	C₈×C₆₇	—	4	3	no	—
33	C₈×C₆₇	—	8	3	no	—
34	C₂²×C₆₇	—	2	3	no	—
35	C₂²×C₆₇	#383	1	1	yes	characteristic
36	C₄×C₆₇	—	2	3	no	—
37	C₄×C₆₇	—	2	3	no	—
38	C₄×C₆₇	—	16	1	no	—
39	#8803	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₆₇	—	4	1	no	—
41	C₂×C₆₇	#1841	1	3	yes	—
42	#1455	C₂×C₆₇	1	1	yes	characteristic
43	#1291	C₂×C₆₇	1	3	yes	—
44	#1288	C₂×C₆₇	1	3	yes	—
45	C₆₇	#8803	1	1	yes	characteristic, direct product factor, Sylow
46	#432	C₂²×C₆₇	1	3	yes	—
47	C₄×M₁₆	C₂²×C₆₇	1	3	yes	—
48	#383	C₂²×C₆₇	1	1	yes	characteristic
49	#322	C₄×C₆₇	1	3	yes	complemented
50	C₈²	C₄×C₆₇	1	1	yes	characteristic, complemented
51	C₂×M₁₆	—	2	3	no	—
52	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
53	C₄⋊C₈	—	2	3	no	—
54	C₂²⋊C₈	—	2	3	no	—
55	#97	—	2	3	no	—
56	#97	—	2	3	no	—
57	C₄×C₈	C₂×C₄×C₆₇	1	3	yes	—
58	C₄×C₈	C₂×C₄×C₆₇	1	3	yes	—
59	#95	—	2	3	no	—
60	C₂²×C₄	—	4	1	no	—
61	C₂²×C₄	#41×C₆₇	1	3	yes	—
62	M₁₆	—	4	6	no	—
63	C₂×C₈	—	2	3	no	—
64	C₂×C₈	—	2	3	no	—
65	C₂×C₈	—	4	3	no	—
66	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
67	C₄²	#41×C₆₇	1	3	yes	—
68	C₂³	#126×C₆₇	1	1	yes	characteristic
69	C₂×C₄	—	4	3	no	—
70	C₂×C₄	—	8	3	no	—
71	C₂×C₄	#117×C₆₇	1	3	yes	—
72	C₂×C₄	#126×C₆₇	1	3	yes	—
73	C₈	—	2	6	no	—
74	C₈	—	4	3	no	—
75	C₈	—	8	3	no	—
76	C₂²	—	2	3	no	—
77	C₂²	#383×C₆₇	1	1	yes	characteristic
78	C₄	—	2	3	no	—
79	C₄	—	2	3	no	—
80	C₄	—	16	1	no	—
81	C₂	—	4	1	no	—
82	C₂	C₆₇×#1841	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	#383
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂³×C₆₇	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
7. #322×C₆₇	C₄	semidirect
8. C₈²×C₆₇	C₄	semidirect
39. #8803	C₆₇	direct (N × H)
45. C₆₇	#8803	direct (N × H)
49. #322	C₄×C₆₇	semidirect
50. C₈²	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#1455	2
C₆₇×#1291	2
C₆₇×#1288	2
#8803	67

Subgroups — C67×#8803 #30688054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#8803 #30688054