Subgroups — C₈×C₂₃×#127 #19169306

Order 5888.

1350 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₂₃	C₂	1	6	yes	maximal, complemented
2	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1954	C₂	1	6	yes	maximal, complemented
4	C₂₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₈×C₂₃	C₂²	1	4	yes	—
7	C₂×C₂₃×#127	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₂₃	C₂²	1	6	yes	—
9	C₈×D₄×C₂₃	—	2	24	no	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	6	yes	—
11	C₂×C₂₃×C₂²⋊C₈	C₂²	1	12	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
14	C₂₃×#390	C₄	1	6	yes	—
15	C₂₃×#384	C₄	1	1	yes	characteristic
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₂₃	C₂×C₄	1	6	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
19	C₂²×C₈×C₂₃	—	2	24	no	—
20	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
21	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
22	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×#127	C₈	1	8	yes	direct product factor
24	C₄×D₄×C₂₃	—	2	12	no	—
25	C₄×D₄×C₂₃	—	2	12	no	—
26	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	6	yes	—
27	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	12	yes	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
31	C₂₃×C₄⋊C₈	—	2	12	no	—
32	C₂₃×C₂²⋊C₈	—	2	24	no	—
33	C₄×C₈×C₂₃	—	2	6	no	—
34	C₄×C₈×C₂₃	D₄	1	12	yes	complemented
35	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
36	C₂×D₄×C₂₃	—	2	12	no	—
37	C₂×D₄×C₂₃	—	2	12	no	—
38	C₂×D₄×C₂₃	C₂×C₈	1	24	yes	complemented
39	C₂²×C₄×C₂₃	—	2	12	no	—
40	C₂²×C₄×C₂₃	—	2	12	no	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₂₃	—	2	6	no	—
46	C₂×C₈×C₂₃	—	2	12	no	—
47	C₂×C₈×C₂₃	—	4	16	no	—
48	C₂×C₈×C₂₃	C₂×D₄	1	6	yes	—
49	C₂×C₈×C₂₃	C₂×D₄	1	6	yes	—
50	C₄⋊C₄×C₂₃	—	2	12	no	—
51	C₂²⋊C₄×C₂₃	—	2	24	no	—
52	C₄²×C₂₃	—	2	6	no	—
53	C₄²×C₂₃	C₂×C₈	1	4	yes	complemented
54	C₄²×C₂₃	C₂×D₄	1	6	yes	—
55	C₄²×C₂₃	#41	1	6	yes	—
56	C₈×#127	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₂₃	—	2	12	no	—
58	C₂³×C₂₃	—	2	12	no	—
59	C₂³×C₂₃	C₂²×C₈	1	8	yes	—
60	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
61	D₄×C₂₃	—	2	48	no	—
62	C₂×C₄×C₂₃	—	2	6	no	—
63	C₂×C₄×C₂₃	—	2	12	no	—
64	C₂×C₄×C₂₃	—	4	4	no	—
65	C₂×C₄×C₂₃	—	4	12	no	—
66	C₂×C₄×C₂₃	C₄×D₄	1	6	yes	—
67	C₂×C₄×C₂₃	C₄×D₄	1	6	yes	—
68	C₂×C₄×C₂₃	C₂²×C₈	1	6	yes	—
69	C₂×C₄×C₂₃	#131	1	6	yes	—
70	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
71	C₂×C₄×C₂₃	C₂×#41	1	3	yes	—
72	C₈×C₂₃	—	2	6	no	—
73	C₈×C₂₃	—	4	4	no	—
74	C₈×C₂₃	#127	1	4	yes	direct product factor
75	C₂×C₈×D₄	C₂×C₂₃	1	6	yes	complemented
76	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
77	#1954	C₂×C₂₃	1	6	yes	complemented
78	#1842	C₂×C₂₃	1	1	yes	characteristic, complemented
79	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
80	C₂²×C₂₃	—	2	24	no	—
81	C₂²×C₂₃	—	4	4	no	—
82	C₂²×C₂₃	—	4	12	no	—
83	C₂²×C₂₃	C₂×C₄×D₄	1	3	yes	—
84	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
85	C₂²×C₂₃	C₂×#131	1	3	yes	—
86	C₄×C₂₃	—	2	6	no	—
87	C₄×C₂₃	—	4	4	no	—
88	C₄×C₂₃	C₈×D₄	1	12	yes	complemented
89	C₄×C₂₃	C₂×#127	1	1	yes	characteristic
90	C₄×C₂₃	#532	1	3	yes	—
91	C₂³×C₈	C₂²×C₂₃	1	4	yes	—
92	C₂×#127	C₄×C₂₃	1	1	yes	characteristic
93	C₂×C₄×D₄	C₂²×C₂₃	1	6	yes	—
94	C₈×D₄	—	2	24	no	—
95	C₂×C₄⋊C₈	C₂²×C₂₃	1	6	yes	—
96	C₂×C₂²⋊C₈	C₂²×C₂₃	1	12	yes	—
97	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
98	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
99	#390	C₄×C₂₃	1	6	yes	—
100	#384	C₄×C₂₃	1	1	yes	characteristic
101	C₄³	C₂²×C₂₃	1	1	yes	characteristic
102	C₂×C₂₃	—	4	8	no	—
103	C₂×C₂₃	C₄×#127	1	1	yes	characteristic
104	C₂×C₂₃	C₂×C₈×D₄	1	3	yes	—
105	C₂×C₂₃	#2931	1	3	yes	—
106	C₂²×D₄	C₂×C₄×C₂₃	1	6	yes	—
107	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
108	C₂²×C₈	—	2	24	no	—
109	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
110	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
111	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
112	#127	C₈×C₂₃	1	8	yes	direct product factor
113	C₄×D₄	—	2	12	no	—
114	C₄×D₄	—	2	12	no	—
115	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	6	yes	—
116	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	12	yes	—
117	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
118	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
119	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
120	C₄⋊C₈	—	2	12	no	—
121	C₂²⋊C₈	—	2	24	no	—
122	C₄×C₈	—	2	6	no	—
123	C₄×C₈	D₄×C₂₃	1	12	yes	complemented
124	C₂₃	C₈×#127	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
126	C₂×D₄	—	2	12	no	—
127	C₂×D₄	—	2	12	no	—
128	C₂×D₄	C₂×C₈×C₂₃	1	24	yes	complemented
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	—	2	12	no	—
131	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
132	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
133	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
134	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	—	2	12	no	—
137	C₂×C₈	—	4	16	no	—
138	C₂×C₈	C₂×D₄×C₂₃	1	6	yes	—
139	C₂×C₈	C₂×D₄×C₂₃	1	6	yes	—
140	C₄⋊C₄	—	2	12	no	—
141	C₂²⋊C₄	—	2	24	no	—
142	C₄²	—	2	6	no	—
143	C₄²	C₂×C₈×C₂₃	1	4	yes	complemented
144	C₄²	C₂×D₄×C₂₃	1	6	yes	—
145	C₄²	#41×C₂₃	1	6	yes	—
146	C₂³	—	2	12	no	—
147	C₂³	—	2	12	no	—
148	C₂³	C₂²×C₈×C₂₃	1	8	yes	—
149	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
150	D₄	—	2	48	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	12	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	12	no	—
155	C₂×C₄	C₄×D₄×C₂₃	1	6	yes	—
156	C₂×C₄	C₄×D₄×C₂₃	1	6	yes	—
157	C₂×C₄	C₂²×C₈×C₂₃	1	6	yes	—
158	C₂×C₄	C₂₃×#131	1	6	yes	—
159	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
160	C₂×C₄	C₂×#41×C₂₃	1	3	yes	—
161	C₈	—	2	6	no	—
162	C₈	—	4	4	no	—
163	C₈	C₂₃×#127	1	4	yes	direct product factor
164	C₂²	—	2	24	no	—
165	C₂²	—	4	12	no	—
166	C₂²	—	4	4	no	—
167	C₂²	C₂×C₄×D₄×C₂₃	1	3	yes	—
168	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
169	C₂²	C₂×C₂₃×#131	1	3	yes	—
170	C₄	—	2	6	no	—
171	C₄	—	4	4	no	—
172	C₄	C₈×D₄×C₂₃	1	12	yes	complemented
173	C₄	C₂×C₂₃×#127	1	1	yes	characteristic
174	C₄	C₂₃×#532	1	3	yes	—
175	C₂	—	4	8	no	—
176	C₂	C₄×C₂₃×#127	1	1	yes	characteristic
177	C₂	C₂×C₈×D₄×C₂₃	1	3	yes	—
178	C₂	C₂₃×#2931	1	3	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₃	C₂³
commutator	C₂²	C₂³×C₈×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂×C₂₃×#127	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₄×C₂₃	C₂	semidirect
3. C₂₃×#1954	C₂	semidirect
4. C₂₃×#1842	C₂	semidirect
5. C₄²×C₈×C₂₃	C₂	semidirect
23. C₂₃×#127	C₈	direct (N × H)
34. C₄×C₈×C₂₃	D₄	semidirect
38. C₂×D₄×C₂₃	C₂×C₈	semidirect
53. C₄²×C₂₃	C₂×C₈	semidirect
56. C₈×#127	C₂₃	direct (N × H)
74. C₈×C₂₃	#127	direct (N × H)
75. C₂×C₈×D₄	C₂×C₂₃	semidirect
77. #1954	C₂×C₂₃	semidirect
78. #1842	C₂×C₂₃	semidirect
79. C₄²×C₈	C₂×C₂₃	semidirect
88. C₄×C₂₃	C₈×D₄	semidirect
112. #127	C₈×C₂₃	direct (N × H)
123. C₄×C₈	D₄×C₂₃	semidirect
124. C₂₃	C₈×#127	direct (N × H)
128. C₂×D₄	C₂×C₈×C₂₃	semidirect
143. C₄²	C₂×C₈×C₂₃	semidirect
163. C₈	C₂₃×#127	direct (N × H)
172. C₄	C₈×D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₂₃	2
C₄×C₂₃×#127	2
C₂₃×#1954	2
C₂₃×#1842	2
C₄²×C₈×C₂₃	2
C₈×#127	23

Subgroups — C8×C23×#127 #19169306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂₃×#127 #19169306