Subgroups — C₂×C₇²×C₈×C₁₆ #72190897

Order 12544.

1990 conjugacy classes of subgroups.

The subgroup lattice has 159 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×C₈²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇²×C₁₆	C₂	1	2	yes	maximal
3	C₇²×C₈×C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₇²×C₈	C₄	1	2	yes	—
5	C₂×C₄×C₇²×C₈	C₂²	1	1	yes	characteristic
6	C₇²×C₈²	C₄	1	2	yes	—
7	C₇²×C₈²	C₂²	1	2	yes	—
8	C₂²×C₇²×C₁₆	C₄	1	4	yes	—
9	C₄×C₇²×C₁₆	C₄	1	4	yes	—
10	C₄×C₇²×C₁₆	C₂²	1	4	yes	—
11	C₂×C₇×C₈×C₁₆	C₇	1	8	yes	maximal, direct product factor
12	C₂²×C₇²×C₈	C₈	1	4	yes	—
13	C₂²×C₇²×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₇²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₇²×C₁₆	C₈	1	16	yes	direct product factor
16	C₂×C₇²×C₁₆	C₂×C₄	1	8	yes	—
17	C₄×C₇²×C₈	C₈	1	4	yes	—
18	C₄×C₇²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₄×C₇²×C₈	C₂×C₄	1	2	yes	—
20	C₄×C₇²×C₈	C₂×C₄	1	4	yes	—
21	C₄×C₇²×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₇×C₁₆	C₂×C₇	1	16	yes	—
23	C₂×C₇×C₈²	C₂×C₇	1	8	yes	—
24	C₇×C₈×C₁₆	C₂×C₇	1	32	yes	direct product factor
25	C₂²×C₄×C₇²	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₇²	C₂×C₈	1	2	yes	—
27	C₂×C₇²×C₈	C₁₆	1	16	yes	direct product factor
28	C₂×C₇²×C₈	C₄²	1	2	yes	—
29	C₂×C₇²×C₈	C₂×C₈	1	8	yes	—
30	C₂×C₇²×C₈	C₂×C₈	1	8	yes	—
31	C₂×C₇²×C₈	C₂²×C₄	1	2	yes	—
32	C₄²×C₇²	C₄²	1	1	yes	characteristic
33	C₄²×C₇²	C₂×C₈	1	2	yes	—
34	C₄²×C₇²	C₂²×C₄	1	1	yes	characteristic
35	C₇²×C₁₆	C₂×C₈	1	16	yes	direct product factor
36	C₂²×C₇×C₁₆	C₄×C₇	1	32	yes	—
37	C₂×C₄×C₇×C₈	C₄×C₇	1	16	yes	—
38	C₂×C₄×C₇×C₈	C₂²×C₇	1	8	yes	—
39	C₄×C₇×C₁₆	C₄×C₇	1	32	yes	—
40	C₄×C₇×C₁₆	C₂²×C₇	1	32	yes	—
41	C₇×C₈²	C₄×C₇	1	16	yes	—
42	C₇×C₈²	C₂²×C₇	1	16	yes	—
43	C₂³×C₇²	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₇²	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₇²	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₇²	C₄×C₈	1	4	yes	—
47	C₂×C₄×C₇²	C₂×C₁₆	1	8	yes	—
48	C₂×C₄×C₇²	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₇²	C₂²×C₈	1	2	yes	—
50	C₇²×C₈	C₄×C₈	1	4	yes	—
51	C₇²×C₈	C₂×C₁₆	1	16	yes	direct product factor
52	C₇²×C₈	C₂²×C₈	1	4	yes	—
53	C₂×C₈×C₁₆	C₇²	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₇×C₈	C₇×C₈	1	32	yes	—
55	C₂²×C₇×C₈	C₂×C₄×C₇	1	16	yes	—
56	C₂×C₄²×C₇	C₂×C₄×C₇	1	8	yes	—
57	C₂×C₇×C₁₆	C₇×C₈	1	128	yes	direct product factor
58	C₂×C₇×C₁₆	C₂×C₄×C₇	1	64	yes	—
59	C₄×C₇×C₈	C₇×C₈	1	32	yes	—
60	C₄×C₇×C₈	C₂×C₄×C₇	1	8	yes	—
61	C₄×C₇×C₈	C₂×C₄×C₇	1	16	yes	—
62	C₄×C₇×C₈	C₂×C₄×C₇	1	32	yes	—
63	C₄×C₇×C₈	C₂³×C₇	1	8	yes	—
64	C₂²×C₇²	C₈²	1	2	yes	—
65	C₂²×C₇²	C₄×C₁₆	1	4	yes	—
66	C₂²×C₇²	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₇²	C₈²	1	2	yes	—
68	C₄×C₇²	C₄×C₁₆	1	4	yes	—
69	C₄×C₇²	C₂×C₄×C₈	1	2	yes	—
70	C₄×C₇²	C₂²×C₁₆	1	4	yes	—
71	C₂×C₄×C₁₆	C₂×C₇²	1	2	yes	—
72	C₂×C₈²	C₂×C₇²	1	1	yes	characteristic
73	C₈×C₁₆	C₂×C₇²	1	4	yes	direct product factor
74	C₂²×C₄×C₇	C₄²×C₇	1	8	yes	—
75	C₂²×C₄×C₇	C₂×C₇×C₈	1	16	yes	—
76	C₂×C₇×C₈	C₇×C₁₆	1	128	yes	direct product factor
77	C₂×C₇×C₈	C₄²×C₇	1	16	yes	—
78	C₂×C₇×C₈	C₂×C₇×C₈	1	64	yes	—
79	C₂×C₇×C₈	C₂×C₇×C₈	1	64	yes	—
80	C₂×C₇×C₈	C₂²×C₄×C₇	1	16	yes	—
81	C₄²×C₇	C₄²×C₇	1	8	yes	—
82	C₄²×C₇	C₂×C₇×C₈	1	16	yes	—
83	C₄²×C₇	C₂²×C₄×C₇	1	8	yes	—
84	C₇×C₁₆	C₂×C₇×C₈	1	128	yes	direct product factor
85	C₂×C₇²	C₈×C₁₆	1	4	yes	direct product factor
86	C₂×C₇²	C₂×C₈²	1	1	yes	characteristic
87	C₂×C₇²	C₂×C₄×C₁₆	1	2	yes	—
88	C₂²×C₁₆	C₄×C₇²	1	4	yes	—
89	C₂×C₄×C₈	C₄×C₇²	1	2	yes	—
90	C₂×C₄×C₈	C₂²×C₇²	1	1	yes	characteristic
91	C₄×C₁₆	C₄×C₇²	1	4	yes	—
92	C₄×C₁₆	C₂²×C₇²	1	4	yes	—
93	C₈²	C₄×C₇²	1	2	yes	—
94	C₈²	C₂²×C₇²	1	2	yes	—
95	C₂³×C₇	C₄×C₇×C₈	1	8	yes	—
96	C₂×C₄×C₇	C₄×C₇×C₈	1	8	yes	—
97	C₂×C₄×C₇	C₄×C₇×C₈	1	16	yes	—
98	C₂×C₄×C₇	C₄×C₇×C₈	1	32	yes	—
99	C₂×C₄×C₇	C₂×C₇×C₁₆	1	64	yes	—
100	C₂×C₄×C₇	C₂×C₄²×C₇	1	8	yes	—
101	C₂×C₄×C₇	C₂²×C₇×C₈	1	16	yes	—
102	C₇×C₈	C₄×C₇×C₈	1	32	yes	—
103	C₇×C₈	C₂×C₇×C₁₆	1	128	yes	direct product factor
104	C₇×C₈	C₂²×C₇×C₈	1	32	yes	—
105	C₇²	C₂×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²×C₈	C₇²×C₈	1	4	yes	—
107	C₂²×C₈	C₂×C₄×C₇²	1	2	yes	—
108	C₂×C₄²	C₂×C₄×C₇²	1	1	yes	characteristic
109	C₂×C₁₆	C₇²×C₈	1	16	yes	direct product factor
110	C₂×C₁₆	C₂×C₄×C₇²	1	8	yes	—
111	C₄×C₈	C₇²×C₈	1	4	yes	—
112	C₄×C₈	C₂×C₄×C₇²	1	1	yes	characteristic
113	C₄×C₈	C₂×C₄×C₇²	1	2	yes	—
114	C₄×C₈	C₂×C₄×C₇²	1	4	yes	—
115	C₄×C₈	C₂³×C₇²	1	1	yes	characteristic
116	C₂²×C₇	C₇×C₈²	1	16	yes	—
117	C₂²×C₇	C₄×C₇×C₁₆	1	32	yes	—
118	C₂²×C₇	C₂×C₄×C₇×C₈	1	8	yes	—
119	C₄×C₇	C₇×C₈²	1	16	yes	—
120	C₄×C₇	C₄×C₇×C₁₆	1	32	yes	—
121	C₄×C₇	C₂×C₄×C₇×C₈	1	16	yes	—
122	C₄×C₇	C₂²×C₇×C₁₆	1	32	yes	—
123	C₂²×C₄	C₄²×C₇²	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₇²×C₈	1	2	yes	—
125	C₂×C₈	C₇²×C₁₆	1	16	yes	direct product factor
126	C₂×C₈	C₄²×C₇²	1	2	yes	—
127	C₂×C₈	C₂×C₇²×C₈	1	8	yes	—
128	C₂×C₈	C₂×C₇²×C₈	1	8	yes	—
129	C₂×C₈	C₂²×C₄×C₇²	1	2	yes	—
130	C₄²	C₄²×C₇²	1	1	yes	characteristic
131	C₄²	C₂×C₇²×C₈	1	2	yes	—
132	C₄²	C₂²×C₄×C₇²	1	1	yes	characteristic
133	C₁₆	C₂×C₇²×C₈	1	16	yes	direct product factor
134	C₂×C₇	C₇×C₈×C₁₆	1	32	yes	direct product factor
135	C₂×C₇	C₂×C₇×C₈²	1	8	yes	—
136	C₂×C₇	C₂×C₄×C₇×C₁₆	1	16	yes	—
137	C₂³	C₄×C₇²×C₈	1	1	yes	characteristic
138	C₂×C₄	C₄×C₇²×C₈	1	1	yes	characteristic
139	C₂×C₄	C₄×C₇²×C₈	1	2	yes	—
140	C₂×C₄	C₄×C₇²×C₈	1	4	yes	—
141	C₂×C₄	C₂×C₇²×C₁₆	1	8	yes	—
142	C₂×C₄	C₂×C₄²×C₇²	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₇²×C₈	1	2	yes	—
144	C₈	C₄×C₇²×C₈	1	4	yes	—
145	C₈	C₂×C₇²×C₁₆	1	16	yes	direct product factor
146	C₈	C₂²×C₇²×C₈	1	4	yes	—
147	C₇	C₂×C₇×C₈×C₁₆	1	8	yes	direct product factor
148	C₂²	C₄×C₇²×C₁₆	1	4	yes	—
149	C₂²	C₇²×C₈²	1	2	yes	—
150	C₂²	C₂×C₄×C₇²×C₈	1	1	yes	characteristic
151	C₄	C₄×C₇²×C₁₆	1	4	yes	—
152	C₄	C₂²×C₇²×C₁₆	1	4	yes	—
153	C₄	C₇²×C₈²	1	2	yes	—
154	C₄	C₂×C₄×C₇²×C₈	1	2	yes	—
155	C₂	C₇²×C₈×C₁₆	1	4	yes	direct product factor
156	C₂	C₂×C₄×C₇²×C₁₆	1	2	yes	—
157	C₂	C₂×C₇²×C₈²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈	C₂³×C₇²
socle, prime radical	C₂³×C₇²	C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇²×C₈×C₁₆	C₂	direct (N × H)
11. C₂×C₇×C₈×C₁₆	C₇	direct (N × H)
15. C₂×C₇²×C₁₆	C₈	direct (N × H)
24. C₇×C₈×C₁₆	C₂×C₇	direct (N × H)
27. C₂×C₇²×C₈	C₁₆	direct (N × H)
35. C₇²×C₁₆	C₂×C₈	direct (N × H)
51. C₇²×C₈	C₂×C₁₆	direct (N × H)
53. C₂×C₈×C₁₆	C₇²	direct (N × H)
57. C₂×C₇×C₁₆	C₇×C₈	direct (N × H)
73. C₈×C₁₆	C₂×C₇²	direct (N × H)
76. C₂×C₇×C₈	C₇×C₁₆	direct (N × H)
84. C₇×C₁₆	C₂×C₇×C₈	direct (N × H)
85. C₂×C₇²	C₈×C₁₆	direct (N × H)
103. C₇×C₈	C₂×C₇×C₁₆	direct (N × H)
105. C₇²	C₂×C₈×C₁₆	direct (N × H)
109. C₂×C₁₆	C₇²×C₈	direct (N × H)
125. C₂×C₈	C₇²×C₁₆	direct (N × H)
133. C₁₆	C₂×C₇²×C₈	direct (N × H)
134. C₂×C₇	C₇×C₈×C₁₆	direct (N × H)
145. C₈	C₂×C₇²×C₁₆	direct (N × H)
147. C₇	C₂×C₇×C₈×C₁₆	direct (N × H)
155. C₂	C₇²×C₈×C₁₆	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×C₇²×C₈²	2
C₂×C₄×C₇²×C₁₆	2
C₇²×C₈×C₁₆	2
C₂×C₇×C₈×C₁₆	7

Subgroups — C2×C72×C8×C16 #72190897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×C₈×C₁₆ #72190897