Subgroups — C₂³×C₂₃×C₂²⋊C₈ #19207685

Order 5888.

7646 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₈×C₂₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂₃×C₂²⋊C₈	C₂	1	28	yes	maximal, direct product factor
4	C₂⁶×C₂₃	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₂₃	—	2	16	no	—
6	C₂⁴×C₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₂₃	C₄	1	14	yes	—
8	C₂⁴×C₄×C₂₃	C₂²	1	14	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₂₃	—	2	16	no	—
11	C₂³×C₈×C₂₃	C₂²	1	28	yes	complemented
12	C₂×C₂₃×C₂²⋊C₈	C₂²	1	112	yes	direct product factor
13	C₂⁵×C₂₃	—	2	8	no	—
14	C₂⁵×C₂₃	—	2	8	no	—
15	C₂⁵×C₂₃	C₈	1	16	yes	complemented
16	C₂⁵×C₂₃	C₂×C₄	1	14	yes	—
17	C₂⁵×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	—	2	8	no	—
19	C₂³×C₄×C₂₃	—	2	112	no	—
20	C₂³×C₄×C₂₃	—	2	112	no	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	14	yes	—
22	C₂³×C₄×C₂₃	C₂×C₄	1	84	yes	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	7	yes	—
24	C₂³×C₄×C₂₃	D₄	1	16	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	28	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	7	yes	—
27	C₂²×C₈×C₂₃	—	2	112	no	—
28	C₂²×C₈×C₂₃	C₂³	1	56	yes	complemented
29	C₂₃×C₂²⋊C₈	C₂³	1	64	yes	direct product factor
30	C₂⁴×C₂₃	—	2	128	no	—
31	C₂⁴×C₂₃	—	2	56	no	—
32	C₂⁴×C₂₃	—	2	56	no	—
33	C₂⁴×C₂₃	C₂²⋊C₄	1	8	yes	—
34	C₂⁴×C₂₃	C₂×C₈	1	8	yes	—
35	C₂⁴×C₂₃	C₂×C₈	1	112	yes	complemented
36	C₂⁴×C₂₃	M₁₆	1	8	yes	—
37	C₂⁴×C₂₃	C₂²×C₄	1	28	yes	—
38	C₂⁴×C₂₃	C₂²×C₄	1	7	yes	—
39	C₂²×C₄×C₂₃	—	2	56	no	—
40	C₂²×C₄×C₂₃	—	2	336	no	—
41	C₂²×C₄×C₂₃	—	2	112	no	—
42	C₂²×C₄×C₂₃	C₂²⋊C₄	1	56	yes	—
43	C₂²×C₄×C₂₃	C₂²×C₄	1	28	yes	—
44	C₂²×C₄×C₂₃	C₂²×C₄	1	56	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	21	yes	—
46	C₂²×C₄×C₂₃	C₂×D₄	1	56	yes	—
47	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
48	C₂²×C₄×C₂₃	C₂⁴	1	7	yes	—
49	C₂×C₈×C₂₃	—	2	112	no	—
50	C₂×C₈×C₂₃	C₂⁴	1	16	yes	complemented
51	C₂³×C₂²⋊C₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	—	2	56	no	—
53	C₂³×C₂₃	—	2	56	no	—
54	C₂³×C₂₃	—	2	448	no	—
55	C₂³×C₂₃	C₂²⋊C₈	1	64	yes	direct product factor
56	C₂³×C₂₃	C₂×C₂²⋊C₄	1	28	yes	—
57	C₂³×C₂₃	C₂²×C₈	1	28	yes	—
58	C₂³×C₂₃	C₂²×C₈	1	112	yes	complemented
59	C₂³×C₂₃	C₂×M₁₆	1	28	yes	—
60	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
61	C₂³×C₂₃	C₂³×C₄	1	7	yes	—
62	C₂×C₄×C₂₃	—	2	56	no	—
63	C₂×C₄×C₂₃	—	2	112	no	—
64	C₂×C₄×C₂₃	—	2	16	no	—
65	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	84	yes	—
66	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
67	C₂×C₄×C₂₃	C₂³×C₄	1	7	yes	—
68	C₂×C₄×C₂₃	C₂²×D₄	1	28	yes	—
69	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
70	C₈×C₂₃	—	2	16	no	—
71	C₂⁵×C₄	C₂×C₂₃	1	1	yes	characteristic
72	C₂⁴×C₈	C₂×C₂₃	1	2	yes	complemented
73	C₂²×C₂²⋊C₈	C₂×C₂₃	1	28	yes	direct product factor
74	C₂²×C₂₃	—	2	224	no	—
75	C₂²×C₂₃	—	2	8	no	—
76	C₂²×C₂₃	—	2	8	no	—
77	C₂²×C₂₃	C₂×C₂²⋊C₈	1	112	yes	direct product factor
78	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	14	yes	—
79	C₂²×C₂₃	C₂³×C₈	1	14	yes	—
80	C₂²×C₂₃	C₂³×C₈	1	16	yes	complemented
81	C₂²×C₂₃	C₂²×M₁₆	1	14	yes	—
82	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
83	C₄×C₂₃	—	2	8	no	—
84	C₄×C₂₃	C₂²×C₂²⋊C₄	1	14	yes	—
85	C₄×C₂₃	C₂³×D₄	1	2	yes	—
86	C₂⁶	C₄×C₂₃	1	1	yes	characteristic
87	C₂⁴×C₄	—	2	16	no	—
88	C₂⁴×C₄	C₄×C₂₃	1	1	yes	characteristic
89	C₂⁴×C₄	C₄×C₂₃	1	14	yes	—
90	C₂⁴×C₄	C₂²×C₂₃	1	14	yes	—
91	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
92	C₂³×C₈	—	2	16	no	—
93	C₂³×C₈	C₂²×C₂₃	1	28	yes	complemented
94	C₂×C₂²⋊C₈	C₂²×C₂₃	1	112	yes	direct product factor
95	C₂×C₂₃	—	2	16	no	—
96	C₂×C₂₃	C₂²×C₂²⋊C₈	1	28	yes	direct product factor
97	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₂₃	C₂⁴×C₈	1	1	yes	characteristic
99	C₂×C₂₃	C₂³×M₁₆	1	1	yes	characteristic
100	C₂⁵	—	2	8	no	—
101	C₂⁵	—	2	8	no	—
102	C₂⁵	C₈×C₂₃	1	16	yes	complemented
103	C₂⁵	C₂×C₄×C₂₃	1	14	yes	—
104	C₂⁵	C₂×C₄×C₂₃	1	1	yes	characteristic
105	C₂³×C₄	—	2	8	no	—
106	C₂³×C₄	—	2	112	no	—
107	C₂³×C₄	—	2	112	no	—
108	C₂³×C₄	C₂×C₄×C₂₃	1	14	yes	—
109	C₂³×C₄	C₂×C₄×C₂₃	1	84	yes	—
110	C₂³×C₄	C₂×C₄×C₂₃	1	7	yes	—
111	C₂³×C₄	D₄×C₂₃	1	16	yes	—
112	C₂³×C₄	C₂³×C₂₃	1	28	yes	—
113	C₂³×C₄	C₂³×C₂₃	1	7	yes	—
114	C₂²×C₈	—	2	112	no	—
115	C₂²×C₈	C₂³×C₂₃	1	56	yes	complemented
116	C₂²⋊C₈	C₂³×C₂₃	1	64	yes	direct product factor
117	C₂₃	C₂³×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
118	C₂⁴	—	2	56	no	—
119	C₂⁴	—	2	56	no	—
120	C₂⁴	—	2	128	no	—
121	C₂⁴	C₂²⋊C₄×C₂₃	1	8	yes	—
122	C₂⁴	C₂×C₈×C₂₃	1	8	yes	—
123	C₂⁴	C₂×C₈×C₂₃	1	112	yes	complemented
124	C₂⁴	M₁₆×C₂₃	1	8	yes	—
125	C₂⁴	C₂²×C₄×C₂₃	1	28	yes	—
126	C₂⁴	C₂²×C₄×C₂₃	1	7	yes	—
127	C₂²×C₄	—	2	56	no	—
128	C₂²×C₄	—	2	336	no	—
129	C₂²×C₄	—	2	112	no	—
130	C₂²×C₄	C₂²⋊C₄×C₂₃	1	56	yes	—
131	C₂²×C₄	C₂²×C₄×C₂₃	1	28	yes	—
132	C₂²×C₄	C₂²×C₄×C₂₃	1	56	yes	—
133	C₂²×C₄	C₂²×C₄×C₂₃	1	21	yes	—
134	C₂²×C₄	C₂×D₄×C₂₃	1	56	yes	—
135	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
136	C₂²×C₄	C₂⁴×C₂₃	1	7	yes	—
137	C₂×C₈	—	2	112	no	—
138	C₂×C₈	C₂⁴×C₂₃	1	16	yes	complemented
139	C₂³	—	2	56	no	—
140	C₂³	—	2	448	no	—
141	C₂³	—	2	56	no	—
142	C₂³	C₂₃×C₂²⋊C₈	1	64	yes	direct product factor
143	C₂³	C₂×C₂²⋊C₄×C₂₃	1	28	yes	—
144	C₂³	C₂²×C₈×C₂₃	1	28	yes	—
145	C₂³	C₂²×C₈×C₂₃	1	112	yes	complemented
146	C₂³	C₂×M₁₆×C₂₃	1	28	yes	—
147	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
148	C₂³	C₂³×C₄×C₂₃	1	7	yes	—
149	C₂×C₄	—	2	56	no	—
150	C₂×C₄	—	2	16	no	—
151	C₂×C₄	—	2	112	no	—
152	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	84	yes	—
153	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
154	C₂×C₄	C₂³×C₄×C₂₃	1	7	yes	—
155	C₂×C₄	C₂²×D₄×C₂₃	1	28	yes	—
156	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
157	C₈	—	2	16	no	—
158	C₂²	—	2	8	no	—
159	C₂²	—	2	224	no	—
160	C₂²	—	2	8	no	—
161	C₂²	C₂×C₂₃×C₂²⋊C₈	1	112	yes	direct product factor
162	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	14	yes	—
163	C₂²	C₂³×C₈×C₂₃	1	14	yes	—
164	C₂²	C₂³×C₈×C₂₃	1	16	yes	complemented
165	C₂²	C₂²×M₁₆×C₂₃	1	14	yes	—
166	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
167	C₄	—	2	8	no	—
168	C₄	C₂²×C₂²⋊C₄×C₂₃	1	14	yes	—
169	C₄	C₂³×D₄×C₂₃	1	2	yes	—
170	C₂	—	2	16	no	—
171	C₂	C₂²×C₂₃×C₂²⋊C₈	1	28	yes	direct product factor
172	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
173	C₂	C₂⁴×C₈×C₂₃	1	1	yes	characteristic
174	C₂	C₂³×M₁₆×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄×C₂₃	C₂²
commutator	C₂	C₂⁴×C₈×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂⁵×C₂₃	C₂×C₄
prime radical	C₂⁶×C₂₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₈×C₂₃	C₂	semidirect
3. C₂²×C₂₃×C₂²⋊C₈	C₂	direct (N × H)
11. C₂³×C₈×C₂₃	C₂²	semidirect
12. C₂×C₂₃×C₂²⋊C₈	C₂²	direct (N × H)
15. C₂⁵×C₂₃	C₈	semidirect
28. C₂²×C₈×C₂₃	C₂³	semidirect
29. C₂₃×C₂²⋊C₈	C₂³	direct (N × H)
35. C₂⁴×C₂₃	C₂×C₈	semidirect
50. C₂×C₈×C₂₃	C₂⁴	semidirect
51. C₂³×C₂²⋊C₈	C₂₃	direct (N × H)
55. C₂³×C₂₃	C₂²⋊C₈	direct (N × H)
58. C₂³×C₂₃	C₂²×C₈	semidirect
72. C₂⁴×C₈	C₂×C₂₃	semidirect
73. C₂²×C₂²⋊C₈	C₂×C₂₃	direct (N × H)
77. C₂²×C₂₃	C₂×C₂²⋊C₈	direct (N × H)
80. C₂²×C₂₃	C₂³×C₈	semidirect
93. C₂³×C₈	C₂²×C₂₃	semidirect
94. C₂×C₂²⋊C₈	C₂²×C₂₃	direct (N × H)
96. C₂×C₂₃	C₂²×C₂²⋊C₈	direct (N × H)
102. C₂⁵	C₈×C₂₃	semidirect
115. C₂²×C₈	C₂³×C₂₃	semidirect
116. C₂²⋊C₈	C₂³×C₂₃	direct (N × H)
117. C₂₃	C₂³×C₂²⋊C₈	direct (N × H)
123. C₂⁴	C₂×C₈×C₂₃	semidirect
138. C₂×C₈	C₂⁴×C₂₃	semidirect
142. C₂³	C₂₃×C₂²⋊C₈	direct (N × H)
145. C₂³	C₂²×C₈×C₂₃	semidirect
161. C₂²	C₂×C₂₃×C₂²⋊C₈	direct (N × H)
164. C₂²	C₂³×C₈×C₂₃	semidirect
171. C₂	C₂²×C₂₃×C₂²⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₂₃	2
C₂⁴×C₈×C₂₃	2
C₂²×C₂₃×C₂²⋊C₈	2
C₂³×C₂²⋊C₈	23

Subgroups — C23×C23×C22⋊C8 #19207685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×C₂²⋊C₈ #19207685