Subgroups — C₂₃×#11635 #19159261

Order 5888.

356 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1330	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#131	—	2	1	no	—
6	C₂₃×#532	C₄	1	1	yes	characteristic
7	C₂₃×#436	—	2	1	no	—
8	C₂₃×#436	C₂²	1	2	yes	—
9	C₈×D₄×C₂₃	—	2	1	no	—
10	C₈×D₄×C₂₃	C₄	1	2	yes	—
11	C₂₃×#423	C₄	1	1	yes	characteristic
12	C₂₃×#408	—	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	C₂₃×#346	—	2	1	no	—
19	C₂₃×#346	C₂²	1	2	yes	—
20	C₂×#41×C₂₃	—	2	1	no	—
21	C₂₃×#131	—	4	4	no	—
22	C₂×M₁₆×C₂₃	—	2	1	no	—
23	C₂×M₁₆×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
28	C₂₃×#127	C₈	1	1	yes	characteristic
29	C₂₃×#124	—	2	1	no	—
30	C₂₃×#121	—	2	1	no	—
31	C₂₃×#121	C₈	1	2	yes	—
32	C₄×D₄×C₂₃	—	2	1	no	—
33	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₂₃	—	2	2	no	—
36	C₂×C₁₆×C₂₃	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	1	no	—
38	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₂₃×C₂²⋊C₈	—	2	2	no	—
40	C₂₃×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	#41×C₂₃	—	4	2	no	—
46	#41×C₂₃	—	4	2	no	—
47	C₂×Q₈×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	—	2	2	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	2	1	no	—
51	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
56	M₁₆×C₂₃	—	4	2	no	—
57	M₁₆×C₂₃	—	4	4	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	4	1	no	—
61	C₂×C₈×C₂₃	—	4	4	no	—
62	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
68	C₂²⋊C₄×C₂₃	—	2	2	no	—
69	C₂²⋊C₄×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
73	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₁₆×C₂₃	—	4	2	no	—
75	C₁₆×C₂₃	—	4	2	no	—
76	#11635	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	—	2	2	no	—
78	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
79	Q₈×C₂₃	—	4	2	no	—
80	D₄×C₂₃	—	4	2	no	—
81	D₄×C₂₃	—	4	2	no	—
82	D₄×C₂₃	—	4	2	no	—
83	D₄×C₂₃	—	4	4	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	4	1	no	—
88	C₂×C₄×C₂₃	—	4	1	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	—	2	1	no	—
100	C₈×C₂₃	—	4	2	no	—
101	C₈×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
102	#2931	C₂×C₂₃	1	1	yes	characteristic
103	#2163	C₂×C₂₃	1	1	yes	characteristic, complemented
104	#2163	C₂×C₂₃	1	1	yes	characteristic, complemented
105	#1330	C₂×C₂₃	1	4	yes	complemented
106	C₂²×C₂₃	—	2	1	no	—
107	C₂²×C₂₃	—	2	1	no	—
108	C₂²×C₂₃	—	2	1	no	—
109	C₂²×C₂₃	—	4	2	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	#418	1	1	yes	characteristic
120	C₄×C₂₃	#421	1	1	yes	characteristic
121	C₂×#131	—	2	1	no	—
122	#532	C₄×C₂₃	1	1	yes	characteristic
123	#436	—	2	1	no	—
124	#436	C₂²×C₂₃	1	2	yes	—
125	C₈×D₄	—	2	1	no	—
126	C₈×D₄	C₄×C₂₃	1	2	yes	—
127	#423	C₄×C₂₃	1	1	yes	characteristic
128	#408	—	2	1	no	—
129	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
130	C₄⋊C₁₆	—	2	1	no	—
131	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
132	C₂²⋊C₁₆	—	2	1	no	—
133	C₂²⋊C₁₆	—	2	1	no	—
134	#346	—	2	1	no	—
135	#346	C₂²×C₂₃	1	2	yes	—
136	C₂×C₂₃	—	4	1	no	—
137	C₂×C₂₃	—	8	2	no	—
138	C₂×C₂₃	#1478	1	1	yes	characteristic
139	C₂×C₂₃	#2115	1	1	yes	characteristic
140	C₂×C₂₃	#2117	1	1	yes	characteristic
141	C₂×#41	—	2	1	no	—
142	#131	—	4	4	no	—
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
148	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
149	#127	C₈×C₂₃	1	1	yes	characteristic
150	#124	—	2	1	no	—
151	#121	—	2	1	no	—
152	#121	C₈×C₂₃	1	2	yes	—
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
164	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
165	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
166	C₂₃	#11635	1	1	yes	characteristic, direct product factor, Sylow
167	#41	—	4	2	no	—
168	#41	—	4	2	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
178	M₁₆	—	4	2	no	—
179	M₁₆	—	4	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
193	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
194	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
195	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
196	C₁₆	—	4	2	no	—
197	C₁₆	—	4	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
200	Q₈	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	4	4	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
213	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
214	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	2	no	—
236	C₄	C₂₃×#418	1	1	yes	characteristic
237	C₄	C₂₃×#421	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	8	2	no	—
240	C₂	C₂₃×#1478	1	1	yes	characteristic
241	C₂	C₂₃×#2115	1	1	yes	characteristic
242	C₂	C₂₃×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₂²⋊C₈
prime radical	C₂₃×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2163	C₂	semidirect
3. C₂₃×#2163	C₂	semidirect
4. C₂₃×#1330	C₂	semidirect
76. #11635	C₂₃	direct (N × H)
103. #2163	C₂×C₂₃	semidirect
104. #2163	C₂×C₂₃	semidirect
105. #1330	C₂×C₂₃	semidirect
166. C₂₃	#11635	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2931	2
C₂₃×#2163	2
C₂₃×#1330	2
#11635	23

Subgroups — C23×#11635 #19159261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11635 #19159261