Subgroups — C₂²×C₄⋊C₁₆×C₆₇ #30712645

Order 17152.

862 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₆₇	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆×C₆₇	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₆₇	—	2	8	no	—
8	C₂²×C₁₆×C₆₇	C₂²	1	12	yes	—
9	C₂×C₄×C₈×C₆₇	C₄	1	6	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	6	yes	—
11	C₄⋊C₁₆×C₆₇	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₆₇	C₈	1	1	yes	characteristic
13	C₂³×C₄×C₆₇	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	—	2	4	no	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
18	C₂²×C₈×C₆₇	D₄	1	4	yes	—
19	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
20	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
21	C₂×C₄²×C₆₇	C₈	1	6	yes	—
22	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
23	C₂×C₁₆×C₆₇	—	2	24	no	—
24	C₂×C₁₆×C₆₇	C₂³	1	8	yes	—
25	C₄×C₈×C₆₇	C₂×C₄	1	12	yes	—
26	C₄×C₈×C₆₇	C₂³	1	4	yes	—
27	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	—	2	4	no	—
29	C₂²×C₄×C₆₇	C₁₆	1	8	yes	complemented
30	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
31	C₂²×C₄×C₆₇	C₂×C₈	1	6	yes	—
32	C₂²×C₄×C₆₇	C₂×C₈	1	3	yes	—
33	C₂²×C₄×C₆₇	C₂×C₈	1	6	yes	—
34	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
36	C₂×C₈×C₆₇	—	2	12	no	—
37	C₂×C₈×C₆₇	C₄⋊C₄	1	12	yes	—
38	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
39	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
40	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
41	C₂×C₈×C₆₇	C₂×Q₈	1	6	yes	—
42	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
43	C₄²×C₆₇	C₂×C₈	1	12	yes	—
44	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
45	C₁₆×C₆₇	—	2	8	no	—
46	C₂³×C₆₇	C₄⋊C₈	1	4	yes	—
47	C₂³×C₆₇	C₂×C₁₆	1	4	yes	—
48	C₂³×C₆₇	M₃₂	1	4	yes	—
49	C₂³×C₆₇	C₂²×C₈	1	3	yes	—
50	C₂×C₄×C₆₇	—	2	12	no	—
51	C₂×C₄×C₆₇	C₄⋊C₈	1	12	yes	—
52	C₂×C₄×C₆₇	C₂×C₁₆	1	24	yes	complemented
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
54	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
55	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
57	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
58	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
59	C₈×C₆₇	—	2	4	no	—
60	C₈×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
61	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
62	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
63	C₂²×C₆₇	C₄⋊C₁₆	1	16	yes	direct product factor
64	C₂²×C₆₇	C₂×C₄⋊C₈	1	6	yes	—
65	C₂²×C₆₇	C₂²×C₁₆	1	6	yes	—
66	C₂²×C₆₇	C₂×M₃₂	1	6	yes	—
67	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
68	C₄×C₆₇	—	2	4	no	—
69	C₄×C₆₇	C₂×C₄⋊C₈	1	6	yes	—
70	C₄×C₆₇	C₂²×C₁₆	1	8	yes	complemented
71	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
73	C₂²×C₄⋊C₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₆₇	C₂×C₄⋊C₁₆	1	12	yes	direct product factor
75	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
77	C₂×C₆₇	C₂²×M₃₂	1	1	yes	characteristic
78	C₂³×C₁₆	C₂×C₆₇	1	2	yes	—
79	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
80	C₂×C₄⋊C₁₆	C₂×C₆₇	1	12	yes	direct product factor
81	C₆₇	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
82	C₂³×C₈	C₄×C₆₇	1	1	yes	characteristic
83	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
84	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
85	C₂²×C₁₆	—	2	8	no	—
86	C₂²×C₁₆	C₂²×C₆₇	1	12	yes	—
87	C₂×C₄×C₈	C₄×C₆₇	1	6	yes	—
88	C₂×C₄×C₈	C₂²×C₆₇	1	6	yes	—
89	C₄⋊C₁₆	C₂²×C₆₇	1	16	yes	direct product factor
90	C₂³×C₄	C₈×C₆₇	1	1	yes	characteristic
91	C₂³×C₄	C₈×C₆₇	1	1	yes	characteristic
92	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
93	C₂²×C₈	—	2	4	no	—
94	C₂²×C₈	C₂×C₄×C₆₇	1	6	yes	—
95	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
96	C₂²×C₈	D₄×C₆₇	1	4	yes	—
97	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
98	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
99	C₂×C₄²	C₈×C₆₇	1	6	yes	—
100	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
101	C₂×C₁₆	—	2	24	no	—
102	C₂×C₁₆	C₂³×C₆₇	1	8	yes	—
103	C₄×C₈	C₂×C₄×C₆₇	1	12	yes	—
104	C₄×C₈	C₂³×C₆₇	1	4	yes	—
105	C₂⁴	C₂×C₈×C₆₇	1	1	yes	characteristic
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	C₁₆×C₆₇	1	8	yes	complemented
108	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
109	C₂²×C₄	C₂×C₈×C₆₇	1	6	yes	—
110	C₂²×C₄	C₂×C₈×C₆₇	1	3	yes	—
111	C₂²×C₄	C₂×C₈×C₆₇	1	6	yes	—
112	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
113	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
114	C₂×C₈	—	2	12	no	—
115	C₂×C₈	C₄⋊C₄×C₆₇	1	12	yes	—
116	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
117	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
118	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
119	C₂×C₈	C₂×Q₈×C₆₇	1	6	yes	—
120	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
121	C₄²	C₂×C₈×C₆₇	1	12	yes	—
122	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
123	C₁₆	—	2	8	no	—
124	C₂³	C₄⋊C₈×C₆₇	1	4	yes	—
125	C₂³	C₂×C₁₆×C₆₇	1	4	yes	—
126	C₂³	M₃₂×C₆₇	1	4	yes	—
127	C₂³	C₂²×C₈×C₆₇	1	3	yes	—
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	C₄⋊C₈×C₆₇	1	12	yes	—
130	C₂×C₄	C₂×C₁₆×C₆₇	1	24	yes	complemented
131	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
132	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
133	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
134	C₂×C₄	C₂²×C₈×C₆₇	1	3	yes	—
135	C₂×C₄	C₂×M₁₆×C₆₇	1	6	yes	—
136	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
139	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
140	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
141	C₂²	C₄⋊C₁₆×C₆₇	1	16	yes	direct product factor
142	C₂²	C₂×C₄⋊C₈×C₆₇	1	6	yes	—
143	C₂²	C₂²×C₁₆×C₆₇	1	6	yes	—
144	C₂²	C₂×M₃₂×C₆₇	1	6	yes	—
145	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	C₂×C₄⋊C₈×C₆₇	1	6	yes	—
148	C₄	C₂²×C₁₆×C₆₇	1	8	yes	complemented
149	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
150	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
151	C₂	C₂×C₄⋊C₁₆×C₆₇	1	12	yes	direct product factor
152	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
153	C₂	C₂³×C₁₆×C₆₇	1	1	yes	characteristic
154	C₂	C₂²×M₃₂×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊C₁₆×C₆₇	C₂	direct (N × H)
11. C₄⋊C₁₆×C₆₇	C₂²	direct (N × H)
29. C₂²×C₄×C₆₇	C₁₆	semidirect
52. C₂×C₄×C₆₇	C₂×C₁₆	semidirect
63. C₂²×C₆₇	C₄⋊C₁₆	direct (N × H)
70. C₄×C₆₇	C₂²×C₁₆	semidirect
73. C₂²×C₄⋊C₁₆	C₆₇	direct (N × H)
74. C₂×C₆₇	C₂×C₄⋊C₁₆	direct (N × H)
80. C₂×C₄⋊C₁₆	C₂×C₆₇	direct (N × H)
81. C₆₇	C₂²×C₄⋊C₁₆	direct (N × H)
89. C₄⋊C₁₆	C₂²×C₆₇	direct (N × H)
107. C₂²×C₄	C₁₆×C₆₇	semidirect
130. C₂×C₄	C₂×C₁₆×C₆₇	semidirect
141. C₂²	C₄⋊C₁₆×C₆₇	direct (N × H)
148. C₄	C₂²×C₁₆×C₆₇	semidirect
151. C₂	C₂×C₄⋊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
C₂³×C₁₆×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂²×C₄⋊C₁₆	67

Subgroups — C22×C4⋊C16×C67 #30712645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊C₁₆×C₆₇ #30712645