Subgroups — C₆₇×#33414 #30712665

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2916	C₂	1	2	yes	maximal
3	C₆₇×#2161	C₂	1	6	yes	maximal, complemented
4	C₂×#368×C₆₇	C₂	1	6	yes	maximal, complemented
5	#568×C₆₇	—	2	2	no	—
6	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
7	#504×C₆₇	C₄	1	8	yes	—
8	C₂×M₃₂×C₆₇	C₂²	1	3	yes	—
9	C₂²×C₁₆×C₆₇	C₂²	1	3	yes	—
10	#430×C₆₇	—	2	6	no	—
11	C₂×#108×C₆₇	C₂²	1	12	yes	—
12	#368×C₆₇	C₂²	1	4	yes	—
13	#368×C₆₇	C₂²	1	12	yes	—
14	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
15	#131×C₆₇	—	4	8	no	—
16	#131×C₆₇	C₂×C₄	1	4	yes	—
17	#131×C₆₇	Q₈	1	4	yes	—
18	C₂×M₁₆×C₆₇	—	2	12	no	—
19	C₂×M₁₆×C₆₇	D₄	1	3	yes	—
20	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
21	M₃₂×C₆₇	C₂×C₄	1	12	yes	—
22	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	—
23	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
24	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
25	#108×C₆₇	—	2	12	no	—
26	#108×C₆₇	C₂³	1	8	yes	—
27	#41×C₆₇	C₄⋊C₄	1	4	yes	—
28	#41×C₆₇	D₈	1	4	yes	—
29	C₂×Q₈×C₆₇	Q₁₆	1	1	yes	characteristic
30	C₂×D₄×C₆₇	Q₁₆	1	3	yes	—
31	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
32	M₁₆×C₆₇	—	2	8	no	—
33	M₁₆×C₆₇	—	4	12	no	—
34	M₁₆×C₆₇	C₄⋊C₄	1	12	yes	—
35	C₂×C₈×C₆₇	—	4	12	no	—
36	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
37	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
38	C₂×C₈×C₆₇	C₂×Q₈	1	6	yes	—
39	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
40	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
41	C₁₆×C₆₇	C₂²×C₄	1	6	yes	—
42	C₂³×C₆₇	C₂×Q₁₆	1	3	yes	—
43	Q₈×C₆₇	#107	1	4	yes	—
44	D₄×C₆₇	#107	1	12	yes	—
45	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
46	C₂×C₄×C₆₇	C₂×D₈	1	6	yes	—
47	C₂×C₄×C₆₇	C₂×Q₁₆	1	3	yes	—
48	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
49	C₈×C₆₇	—	4	8	no	—
50	C₈×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
51	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
52	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₆₇	—	4	3	no	—
54	C₂²×C₆₇	C₂×#107	1	6	yes	—
55	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
56	C₄×C₆₇	C₂×#107	1	6	yes	—
57	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
59	#33414	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₆₇	—	2	1	no	—
61	C₂×C₆₇	—	2	6	no	—
62	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
63	C₂×#504	C₂×C₆₇	1	1	yes	characteristic
64	#2916	C₂×C₆₇	1	2	yes	—
65	#2161	C₂×C₆₇	1	6	yes	complemented
66	C₂×#368	C₂×C₆₇	1	6	yes	complemented
67	C₆₇	#33414	1	1	yes	characteristic, direct product factor, Sylow
68	#568	—	2	2	no	—
69	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
70	#504	C₄×C₆₇	1	8	yes	—
71	C₂×M₃₂	C₂²×C₆₇	1	3	yes	—
72	C₂²×C₁₆	C₂²×C₆₇	1	3	yes	—
73	#430	—	2	6	no	—
74	C₂×#108	C₂²×C₆₇	1	12	yes	—
75	#368	C₂²×C₆₇	1	4	yes	—
76	#368	C₂²×C₆₇	1	12	yes	—
77	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
78	#131	—	4	8	no	—
79	#131	C₂×C₄×C₆₇	1	4	yes	—
80	#131	Q₈×C₆₇	1	4	yes	—
81	C₂×M₁₆	—	2	12	no	—
82	C₂×M₁₆	D₄×C₆₇	1	3	yes	—
83	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
84	M₃₂	C₂×C₄×C₆₇	1	12	yes	—
85	C₂×C₁₆	C₂×C₄×C₆₇	1	12	yes	—
86	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
87	C₂×C₁₆	C₂³×C₆₇	1	3	yes	—
88	#108	—	2	12	no	—
89	#108	C₂³×C₆₇	1	8	yes	—
90	#41	C₄⋊C₄×C₆₇	1	4	yes	—
91	#41	D₈×C₆₇	1	4	yes	—
92	C₂×Q₈	Q₁₆×C₆₇	1	1	yes	characteristic
93	C₂×D₄	Q₁₆×C₆₇	1	3	yes	—
94	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
95	M₁₆	—	2	8	no	—
96	M₁₆	—	4	12	no	—
97	M₁₆	C₄⋊C₄×C₆₇	1	12	yes	—
98	C₂×C₈	—	4	12	no	—
99	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
100	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
101	C₂×C₈	C₂×Q₈×C₆₇	1	6	yes	—
102	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
103	C₁₆	C₂²×C₄×C₆₇	1	2	yes	—
104	C₁₆	C₂²×C₄×C₆₇	1	6	yes	—
105	C₂³	C₂×Q₁₆×C₆₇	1	3	yes	—
106	Q₈	#107×C₆₇	1	4	yes	—
107	D₄	#107×C₆₇	1	12	yes	—
108	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
109	C₂×C₄	C₂×D₈×C₆₇	1	6	yes	—
110	C₂×C₄	C₂×Q₁₆×C₆₇	1	3	yes	—
111	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
112	C₈	—	4	8	no	—
113	C₈	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
114	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
115	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
116	C₂²	—	4	3	no	—
117	C₂²	C₂×#107×C₆₇	1	6	yes	—
118	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
119	C₄	C₂×#107×C₆₇	1	6	yes	—
120	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
121	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	2	6	no	—
124	C₂	C₂²×#107×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂²×D₈
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂²×#107
prime radical	C₂×#41×C₆₇	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂×#41×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2161	C₂	semidirect
4. C₂×#368×C₆₇	C₂	semidirect
59. #33414	C₆₇	direct (N × H)
65. #2161	C₂×C₆₇	semidirect
66. C₂×#368	C₂×C₆₇	semidirect
67. C₆₇	#33414	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×#504×C₆₇	2
C₆₇×#2916	2
C₆₇×#2161	2
C₂×#368×C₆₇	2
#33414	67

Subgroups — C67×#33414 #30712665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33414 #30712665