Subgroups — C₂₃×#8808 #19156434

Order 5888.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1842	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#335	C₂	1	2	yes	maximal
5	C₂×C₂₃×#334	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#97	C₄	1	2	yes	—
11	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
12	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
14	C₄³×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#336	—	2	2	no	—
16	C₂₃×#335	—	2	4	no	—
17	C₂₃×#334	—	2	4	no	—
18	C₂²×C₈×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	—	2	1	no	—
20	C₂²×C₈×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	D₄	1	2	yes	—
26	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	—	2	2	no	—
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	—	2	2	no	—
31	C₂₃×C₄⋊C₈	—	2	2	no	—
32	C₂₃×C₄⋊C₈	—	4	4	no	—
33	C₂₃×#97	—	2	4	no	—
34	C₂₃×#97	—	2	2	no	—
35	C₂₃×#97	C₈	1	4	yes	complemented
36	C₄×C₈×C₂₃	D₄	1	4	yes	—
37	C₄×C₈×C₂₃	Q₈	1	4	yes	—
38	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	—	4	2	no	—
46	C₂×C₈×C₂₃	—	4	1	no	—
47	C₂×C₈×C₂₃	—	4	1	no	—
48	C₂×C₈×C₂₃	—	4	4	no	—
49	C₂×C₈×C₂₃	—	4	2	no	—
50	C₂×C₈×C₂₃	—	4	2	no	—
51	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
54	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
55	C₄²×C₂₃	—	2	2	no	—
56	C₄²×C₂₃	—	2	2	no	—
57	C₄²×C₂₃	—	2	2	no	—
58	C₄²×C₂₃	—	4	2	no	—
59	C₄²×C₂₃	C₂×C₈	1	2	yes	—
60	C₄²×C₂₃	M₁₆	1	2	yes	—
61	C₄²×C₂₃	C₂×D₄	1	2	yes	—
62	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
63	#8808	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	C₄⋊C₈	1	4	yes	—
72	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
75	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₂₃	#137	1	2	yes	—
81	C₈×C₂₃	—	2	4	no	—
82	C₈×C₂₃	—	8	2	no	—
83	C₈×C₂₃	—	8	1	no	—
84	C₈×C₂₃	—	8	1	no	—
85	C₈×C₂₃	C₄⋊C₈	1	8	yes	complemented
86	#1842	C₂×C₂₃	1	1	yes	characteristic
87	#1831	C₂×C₂₃	1	1	yes	characteristic
88	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
89	C₂×#335	C₂×C₂₃	1	2	yes	—
90	C₂×#334	C₂×C₂₃	1	2	yes	—
91	C₂²×C₂₃	#384	1	1	yes	characteristic
92	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₂₃	#423	1	1	yes	characteristic
94	C₂²×C₂₃	#428	1	2	yes	—
95	C₂²×C₂₃	#430	1	2	yes	—
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	4	2	no	—
99	C₄×C₂₃	C₂×C₄⋊C₈	1	2	yes	—
100	C₄×C₂₃	#423	1	2	yes	—
101	C₄×C₂₃	#496	1	1	yes	characteristic
102	C₄×C₂₃	#497	1	1	yes	characteristic
103	C₄×C₂₃	#501	1	2	yes	—
104	C₂×C₄⋊C₈	—	2	2	no	—
105	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
106	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
107	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
108	C₂×#97	C₄×C₂₃	1	2	yes	—
109	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
110	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
112	C₄³	C₂²×C₂₃	1	1	yes	characteristic
113	#336	—	2	2	no	—
114	#335	—	2	4	no	—
115	#334	—	2	4	no	—
116	C₂×C₂₃	#1565	1	2	yes	—
117	C₂×C₂₃	#1569	1	2	yes	—
118	C₂×C₂₃	#1842	1	1	yes	characteristic
119	C₂×C₂₃	#1847	1	1	yes	characteristic
120	C₂×C₂₃	#1849	1	1	yes	characteristic
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
125	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
126	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
127	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₄²	D₄×C₂₃	1	2	yes	—
129	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
130	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	4	4	no	—
136	#97	—	2	4	no	—
137	#97	—	2	2	no	—
138	#97	C₈×C₂₃	1	4	yes	complemented
139	C₄×C₈	D₄×C₂₃	1	4	yes	—
140	C₄×C₈	Q₈×C₂₃	1	4	yes	—
141	C₂₃	#8808	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
143	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
156	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
157	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
158	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂×C₈×C₂₃	1	2	yes	—
164	C₄²	M₁₆×C₂₃	1	2	yes	—
165	C₄²	C₂×D₄×C₂₃	1	2	yes	—
166	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
167	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₂₃×C₄⋊C₈	1	4	yes	—
175	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
176	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
177	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
178	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
181	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
183	C₂×C₄	C₂₃×#137	1	2	yes	—
184	C₈	—	2	4	no	—
185	C₈	—	8	2	no	—
186	C₈	—	8	1	no	—
187	C₈	—	8	1	no	—
188	C₈	C₂₃×C₄⋊C₈	1	8	yes	complemented
189	C₂²	C₂₃×#384	1	1	yes	characteristic
190	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
191	C₂²	C₂₃×#423	1	1	yes	characteristic
192	C₂²	C₂₃×#428	1	2	yes	—
193	C₂²	C₂₃×#430	1	2	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₂×C₂₃×C₄⋊C₈	1	2	yes	—
198	C₄	C₂₃×#423	1	2	yes	—
199	C₄	C₂₃×#496	1	1	yes	characteristic
200	C₄	C₂₃×#497	1	1	yes	characteristic
201	C₄	C₂₃×#501	1	2	yes	—
202	C₂	C₂₃×#1565	1	2	yes	—
203	C₂	C₂₃×#1569	1	2	yes	—
204	C₂	C₂₃×#1842	1	1	yes	characteristic
205	C₂	C₂₃×#1847	1	1	yes	characteristic
206	C₂	C₂₃×#1849	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
35. C₂₃×#97	C₈	semidirect
63. #8808	C₂₃	direct (N × H)
85. C₈×C₂₃	C₄⋊C₈	semidirect
138. #97	C₈×C₂₃	semidirect
141. C₂₃	#8808	direct (N × H)
188. C₈	C₂₃×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1842	2
C₂₃×#1831	2
C₄×C₂₃×#97	2
C₂×C₂₃×#335	2
C₂×C₂₃×#334	2
#8808	23

Subgroups — C23×#8808 #19156434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8808 #19156434