Subgroups — C₆₇×#33496 #30712747

Order 17152.

526 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₆₇	C₂	1	8	yes	maximal, complemented
3	C₆₇×#2162	C₂	1	6	yes	maximal
4	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
5	C₈×Q₈×C₆₇	C₄	1	4	yes	—
6	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
7	C₂×C₄⋊C₈×C₆₇	C₄	1	3	yes	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	C₄⋊C₁₆×C₆₇	C₂²	1	12	yes	—
10	C₂²⋊C₁₆×C₆₇	C₂²	1	4	yes	—
11	C₄×C₁₆×C₆₇	C₂²	1	12	yes	—
12	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
14	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
15	C₄×Q₈×C₆₇	C₈	1	4	yes	—
16	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
17	C₂×C₄⋊C₄×C₆₇	C₈	1	3	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
19	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
20	C₄⋊C₈×C₆₇	C₂×C₄	1	12	yes	—
21	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
22	C₄×C₈×C₆₇	C₂³	1	6	yes	—
23	C₂×Q₈×C₆₇	C₁₆	1	8	yes	complemented
24	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
26	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
27	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	—	4	3	no	—
30	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
31	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
33	C₄⋊C₄×C₆₇	C₂×C₈	1	12	yes	—
34	C₄²×C₆₇	C₂×C₈	1	6	yes	—
35	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
36	C₁₆×C₆₇	—	2	8	no	—
37	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
38	Q₈×C₆₇	C₂×C₁₆	1	16	yes	—
39	C₂×C₄×C₆₇	—	2	1	no	—
40	C₂×C₄×C₆₇	—	2	1	no	—
41	C₂×C₄×C₆₇	—	4	3	no	—
42	C₂×C₄×C₆₇	C₂×C₁₆	1	12	yes	—
43	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
45	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
46	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
47	C₈×C₆₇	—	2	1	no	—
48	C₈×C₆₇	—	2	6	no	—
49	C₈×C₆₇	2₋¹⁺⁴	1	2	yes	—
50	C₂²×C₆₇	—	2	1	no	—
51	C₂²×C₆₇	—	2	1	no	—
52	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
53	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
54	C₄×C₆₇	—	2	6	no	—
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	C₂²×C₁₆	1	12	yes	—
57	C₄×C₆₇	#519	1	1	yes	characteristic
58	C₄×C₆₇	#568	1	1	yes	characteristic
59	#33496	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₆₇	—	2	2	no	—
61	C₂×C₆₇	#2982	1	1	yes	characteristic
62	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
63	C₂×C₆₇	#3407	1	1	yes	characteristic
64	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
65	Q₈×C₁₆	C₂×C₆₇	1	8	yes	complemented
66	#2162	C₂×C₆₇	1	6	yes	—
67	C₆₇	#33496	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
69	C₈×Q₈	C₄×C₆₇	1	4	yes	—
70	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
71	C₂×C₄⋊C₈	C₄×C₆₇	1	3	yes	—
72	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
73	C₄⋊C₁₆	C₂²×C₆₇	1	12	yes	—
74	C₂²⋊C₁₆	C₂²×C₆₇	1	4	yes	—
75	C₄×C₁₆	C₂²×C₆₇	1	12	yes	—
76	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
78	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
79	C₄×Q₈	C₈×C₆₇	1	4	yes	—
80	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
81	C₂×C₄⋊C₄	C₈×C₆₇	1	3	yes	—
82	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
83	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
84	C₄⋊C₈	C₂×C₄×C₆₇	1	12	yes	—
85	C₄×C₈	C₂×C₄×C₆₇	1	6	yes	—
86	C₄×C₈	C₂³×C₆₇	1	6	yes	—
87	C₂×Q₈	C₁₆×C₆₇	1	8	yes	complemented
88	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
89	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
90	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
91	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	3	no	—
94	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
95	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
96	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
97	C₄⋊C₄	C₂×C₈×C₆₇	1	12	yes	—
98	C₄²	C₂×C₈×C₆₇	1	6	yes	—
99	C₄²	C₂²×C₄×C₆₇	1	6	yes	—
100	C₁₆	—	2	8	no	—
101	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
102	Q₈	C₂×C₁₆×C₆₇	1	16	yes	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	4	3	no	—
106	C₂×C₄	C₂×C₁₆×C₆₇	1	12	yes	—
107	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
108	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
109	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
110	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
111	C₈	—	2	1	no	—
112	C₈	—	2	6	no	—
113	C₈	2₋¹⁺⁴×C₆₇	1	2	yes	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	C₂²×C₁₆×C₆₇	1	2	yes	—
117	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
118	C₄	—	2	6	no	—
119	C₄	—	2	1	no	—
120	C₄	C₂²×C₁₆×C₆₇	1	12	yes	—
121	C₄	#519×C₆₇	1	1	yes	characteristic
122	C₄	#568×C₆₇	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	C₆₇×#2982	1	1	yes	characteristic
125	C₂	C₂³×C₁₆×C₆₇	1	1	yes	characteristic
126	C₂	C₆₇×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂⁴
commutator	C₂	C₂³×C₁₆×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂³×C₆₇	C₂²×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×C₁₆×C₆₇	C₂	semidirect
23. C₂×Q₈×C₆₇	C₁₆	semidirect
59. #33496	C₆₇	direct (N × H)
65. Q₈×C₁₆	C₂×C₆₇	semidirect
67. C₆₇	#33496	direct (N × H)
87. C₂×Q₈	C₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₆₇	2
Q₈×C₁₆×C₆₇	2
C₆₇×#2162	2
#33496	67

Subgroups — C67×#33496 #30712747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33496 #30712747