Subgroups — C₂×Q₈×C₁₆×C₆₇ #30712723

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 144 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, direct product factor
3	C₂×C₄⋊C₁₆×C₆₇	C₂	1	3	yes	maximal
4	C₂×C₄×C₁₆×C₆₇	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₆₇	C₂²	1	3	yes	—
7	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₆₇	C₄	1	4	yes	—
9	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₆₇	C₄	1	3	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
12	C₄⋊C₁₆×C₆₇	C₂²	1	12	yes	—
13	C₄×C₁₆×C₆₇	C₂²	1	12	yes	—
14	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
16	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
17	C₄×Q₈×C₆₇	C₈	1	4	yes	—
18	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
19	C₂×C₄⋊C₄×C₆₇	C₈	1	3	yes	—
20	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
21	C₂×C₁₆×C₆₇	—	2	6	no	—
22	C₂×C₁₆×C₆₇	Q₈	1	4	yes	direct product factor
23	C₂×C₁₆×C₆₇	C₂³	1	6	yes	—
24	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
25	C₄⋊C₈×C₆₇	C₂×C₄	1	12	yes	—
26	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
27	C₄×C₈×C₆₇	C₂³	1	6	yes	—
28	C₂×Q₈×C₆₇	C₁₆	1	8	yes	direct product factor
29	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
30	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
31	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
32	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	—	2	6	no	—
34	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
35	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
37	C₂×C₈×C₆₇	#41	1	2	yes	—
38	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
39	C₄⋊C₄×C₆₇	C₂×C₈	1	12	yes	—
40	C₄²×C₆₇	C₂×C₈	1	6	yes	—
41	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
42	C₁₆×C₆₇	—	2	6	no	—
43	C₁₆×C₆₇	C₂×Q₈	1	4	yes	direct product factor
44	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
45	Q₈×C₆₇	C₂×C₁₆	1	16	yes	direct product factor
46	C₂×C₄×C₆₇	—	2	6	no	—
47	C₂×C₄×C₆₇	C₂×C₁₆	1	12	yes	—
48	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
49	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
50	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂²×C₈	1	6	yes	—
52	C₂×C₄×C₆₇	#131	1	2	yes	—
53	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
54	C₈×C₆₇	—	2	6	no	—
55	C₈×C₆₇	C₄×Q₈	1	2	yes	—
56	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
57	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
58	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
59	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
60	C₂²×C₆₇	#504	1	2	yes	—
61	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
62	C₄×C₆₇	—	2	6	no	—
63	C₄×C₆₇	C₈×Q₈	1	2	yes	—
64	C₄×C₆₇	C₂²×C₁₆	1	12	yes	—
65	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
67	C₂×Q₈×C₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₆₇	Q₈×C₁₆	1	4	yes	direct product factor
69	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
70	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
71	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
72	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
73	Q₈×C₁₆	C₂×C₆₇	1	8	yes	direct product factor
74	C₂×C₄⋊C₁₆	C₂×C₆₇	1	3	yes	—
75	C₂×C₄×C₁₆	C₂×C₆₇	1	3	yes	—
76	C₆₇	C₂×Q₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
78	C₂²×C₁₆	C₂²×C₆₇	1	3	yes	—
79	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
80	C₈×Q₈	C₄×C₆₇	1	4	yes	—
81	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
82	C₂×C₄⋊C₈	C₄×C₆₇	1	3	yes	—
83	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
84	C₄⋊C₁₆	C₂²×C₆₇	1	12	yes	—
85	C₄×C₁₆	C₂²×C₆₇	1	12	yes	—
86	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
87	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
88	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
89	C₄×Q₈	C₈×C₆₇	1	4	yes	—
90	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
91	C₂×C₄⋊C₄	C₈×C₆₇	1	3	yes	—
92	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
93	C₂×C₁₆	—	2	6	no	—
94	C₂×C₁₆	Q₈×C₆₇	1	4	yes	direct product factor
95	C₂×C₁₆	C₂³×C₆₇	1	6	yes	—
96	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
97	C₄⋊C₈	C₂×C₄×C₆₇	1	12	yes	—
98	C₄×C₈	C₂×C₄×C₆₇	1	6	yes	—
99	C₄×C₈	C₂³×C₆₇	1	6	yes	—
100	C₂×Q₈	C₁₆×C₆₇	1	8	yes	direct product factor
101	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
102	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
103	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
104	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
105	C₂×C₈	—	2	6	no	—
106	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
107	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
108	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
109	C₂×C₈	#41×C₆₇	1	2	yes	—
110	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
111	C₄⋊C₄	C₂×C₈×C₆₇	1	12	yes	—
112	C₄²	C₂×C₈×C₆₇	1	6	yes	—
113	C₄²	C₂²×C₄×C₆₇	1	6	yes	—
114	C₁₆	—	2	6	no	—
115	C₁₆	C₂×Q₈×C₆₇	1	4	yes	direct product factor
116	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
117	Q₈	C₂×C₁₆×C₆₇	1	16	yes	direct product factor
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	C₂×C₁₆×C₆₇	1	12	yes	—
120	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
121	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
122	C₂×C₄	C₂²×C₈×C₆₇	1	6	yes	—
123	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
124	C₂×C₄	#131×C₆₇	1	2	yes	—
125	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
126	C₈	—	2	6	no	—
127	C₈	C₄×Q₈×C₆₇	1	2	yes	—
128	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
129	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
130	C₂²	C₈×Q₈×C₆₇	1	2	yes	—
131	C₂²	C₂²×C₁₆×C₆₇	1	2	yes	—
132	C₂²	#504×C₆₇	1	2	yes	—
133	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
134	C₄	—	2	6	no	—
135	C₄	C₈×Q₈×C₆₇	1	2	yes	—
136	C₄	C₂²×C₁₆×C₆₇	1	12	yes	—
137	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
138	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
139	C₂	Q₈×C₁₆×C₆₇	1	4	yes	direct product factor
140	C₂	C₂×C₈×Q₈×C₆₇	1	1	yes	characteristic
141	C₂	C₂³×C₁₆×C₆₇	1	1	yes	characteristic
142	C₂	C₂×#504×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. Q₈×C₁₆×C₆₇	C₂	direct (N × H)
22. C₂×C₁₆×C₆₇	Q₈	direct (N × H)
28. C₂×Q₈×C₆₇	C₁₆	direct (N × H)
43. C₁₆×C₆₇	C₂×Q₈	direct (N × H)
45. Q₈×C₆₇	C₂×C₁₆	direct (N × H)
67. C₂×Q₈×C₁₆	C₆₇	direct (N × H)
68. C₂×C₆₇	Q₈×C₁₆	direct (N × H)
73. Q₈×C₁₆	C₂×C₆₇	direct (N × H)
76. C₆₇	C₂×Q₈×C₁₆	direct (N × H)
94. C₂×C₁₆	Q₈×C₆₇	direct (N × H)
100. C₂×Q₈	C₁₆×C₆₇	direct (N × H)
115. C₁₆	C₂×Q₈×C₆₇	direct (N × H)
117. Q₈	C₂×C₁₆×C₆₇	direct (N × H)
139. C₂	Q₈×C₁₆×C₆₇	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×C₈×Q₈×C₆₇	2
Q₈×C₁₆×C₆₇	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×C₄×C₁₆×C₆₇	2
C₂×Q₈×C₁₆	67

Subgroups — C2×Q8×C16×C67 #30712723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₁₆×C₆₇ #30712723