Subgroups — C₂₃×#11637 #19159263

Order 5888.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2162	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2162	C₂	1	1	yes	maximal, characteristic
4	C₂₃×Q₈⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
6	C₈×Q₈×C₂₃	—	2	1	no	—
7	C₈×Q₈×C₂₃	—	2	1	no	—
8	C₈×Q₈×C₂₃	C₄	1	2	yes	—
9	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₂₃	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₁₆	—	2	1	no	—
15	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₄×C₁₆×C₂₃	—	2	1	no	—
19	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	—	2	1	no	—
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₂₃	—	2	1	no	—
25	C₄×Q₈×C₂₃	—	2	1	no	—
26	C₄×Q₈×C₂₃	C₈	1	2	yes	—
27	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	—	2	1	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₂₃	—	2	2	no	—
33	C₂×C₁₆×C₂₃	—	2	2	no	—
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×C₄⋊C₈	—	2	1	no	—
37	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₂₃	—	2	2	no	—
39	C₄×C₈×C₂₃	—	2	2	no	—
40	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂×Q₈×C₂₃	—	2	2	no	—
46	C₂×Q₈×C₂₃	C₁₆	1	4	yes	complemented
47	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	4	1	no	—
56	C₂×C₈×C₂₃	—	4	2	no	—
57	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
65	C₄²×C₂₃	—	2	2	no	—
66	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
70	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₁₆×C₂₃	—	4	2	no	—
72	C₁₆×C₂₃	—	4	2	no	—
73	#11637	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
75	Q₈×C₂₃	—	2	4	no	—
76	Q₈×C₂₃	—	2	2	no	—
77	Q₈×C₂₃	C₂×C₁₆	1	4	yes	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	4	1	no	—
84	C₂×C₄×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
89	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	2	1	no	—
95	C₈×C₂₃	—	2	1	no	—
96	C₈×C₂₃	—	4	2	no	—
97	C₈×C₂₃	#137	1	2	yes	—
98	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic
99	#2162	C₂×C₂₃	1	1	yes	characteristic
100	#2162	C₂×C₂₃	1	1	yes	characteristic
101	Q₈⋊C₁₆	C₂×C₂₃	1	4	yes	complemented
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	C₂²⋊C₁₆	1	2	yes	—
105	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
106	C₄×C₂₃	—	2	4	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	C₂²⋊C₁₆	1	2	yes	—
112	C₄×C₂₃	#419	1	1	yes	characteristic
113	C₄×C₂₃	#421	1	1	yes	characteristic
114	C₄×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
115	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
116	C₂×C₄×Q₈	C₄×C₂₃	1	1	yes	characteristic
117	C₈×Q₈	—	2	1	no	—
118	C₈×Q₈	—	2	1	no	—
119	C₈×Q₈	C₄×C₂₃	1	2	yes	—
120	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
121	C₂×C₄⋊C₈	—	2	1	no	—
122	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
123	C₂×C₄×C₈	—	2	1	no	—
124	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₄⋊C₁₆	—	2	1	no	—
126	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
127	C₂²⋊C₁₆	—	2	1	no	—
128	C₂²⋊C₁₆	—	2	1	no	—
129	C₄×C₁₆	—	2	1	no	—
130	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
131	C₂×C₂₃	—	2	2	no	—
132	C₂×C₂₃	#1479	1	1	yes	characteristic
133	C₂×C₂₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
134	C₂×C₂₃	#2117	1	1	yes	characteristic
135	C₂²×Q₈	C₈×C₂₃	1	1	yes	characteristic
136	C₂²×C₈	—	2	1	no	—
137	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
139	C₄×Q₈	—	2	1	no	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	C₈×C₂₃	1	2	yes	—
142	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	C₂×C₄⋊C₄	C₈×C₂₃	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
156	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
157	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
159	C₂₃	#11637	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	C₁₆×C₂₃	1	4	yes	complemented
163	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
184	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
185	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
186	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
187	C₁₆	—	4	2	no	—
188	C₁₆	—	4	2	no	—
189	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
190	Q₈	—	2	4	no	—
191	Q₈	—	2	2	no	—
192	Q₈	C₂×C₁₆×C₂₃	1	4	yes	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
201	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
202	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
204	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
207	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	C₂₃×#137	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂₃×C₂²⋊C₁₆	1	2	yes	—
216	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	4	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	C₂₃×C₂²⋊C₁₆	1	2	yes	—
223	C₄	C₂₃×#419	1	1	yes	characteristic
224	C₄	C₂₃×#421	1	1	yes	characteristic
225	C₄	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
226	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂₃×#1479	1	1	yes	characteristic
229	C₂	C₂×C₂₃×C₂²⋊C₁₆	1	1	yes	characteristic
230	C₂	C₂₃×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₂₃	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×Q₈⋊C₁₆	C₂	semidirect
46. C₂×Q₈×C₂₃	C₁₆	semidirect
73. #11637	C₂₃	direct (N × H)
101. Q₈⋊C₁₆	C₂×C₂₃	semidirect
159. C₂₃	#11637	direct (N × H)
162. C₂×Q₈	C₁₆×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×Q₈×C₂₃	2
C₂₃×#2162	2
C₂₃×Q₈⋊C₁₆	2
#11637	23

Subgroups — C23×#11637 #19159263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11637 #19159263