Subgroups — C₂₃×#11929 #19159555

Order 5888.

414 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1362	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#135	—	2	1	no	—
7	C₂×C₂₃×#135	C₄	1	2	yes	—
8	C₂₃×#532	—	2	1	no	—
9	C₂²×C₁₆×C₂₃	—	2	1	no	—
10	C₂₃×#495	—	2	1	no	—
11	C₂₃×#495	C₄	1	2	yes	—
12	C₂₃×#494	C₂²	1	1	yes	characteristic
13	C₂₃×#493	C₂²	1	1	yes	characteristic
14	C₂₃×#492	—	2	1	no	—
15	C₂₃×#431	—	2	1	no	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊C₁₆	—	2	1	no	—
18	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂₃×#335	C₂²	1	2	yes	—
20	C₂₃×#334	—	2	1	no	—
21	C₂×#41×C₂₃	—	2	2	no	—
22	C₂₃×#135	—	4	4	no	—
23	C₂₃×#135	—	4	2	no	—
24	C₂₃×#135	—	4	2	no	—
25	C₂×Q₁₆×C₂₃	—	2	1	no	—
26	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₂₃	—	2	2	no	—
28	C₂×SD₁₆×C₂₃	—	2	2	no	—
29	C₂×D₈×C₂₃	—	2	1	no	—
30	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂₃×C₄⋊Q₈	—	2	1	no	—
34	C₂₃×#127	—	2	1	no	—
35	C₂₃×#124	—	2	2	no	—
36	C₂₃×#121	—	4	1	no	—
37	C₂₃×#121	—	4	1	no	—
38	C₄×D₄×C₂₃	—	4	1	no	—
39	C₄×D₄×C₂₃	—	4	1	no	—
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₂₃	—	2	2	no	—
42	C₂×C₁₆×C₂₃	—	4	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₂²⋊C₈	—	4	1	no	—
45	C₄×C₈×C₂₃	—	4	1	no	—
46	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	#41×C₂₃	—	4	4	no	—
51	#41×C₂₃	—	4	4	no	—
52	C₂×Q₈×C₂₃	—	2	2	no	—
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	—	4	2	no	—
55	C₂×D₄×C₂₃	—	4	1	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
61	Q₁₆×C₂₃	—	4	2	no	—
62	Q₁₆×C₂₃	—	4	1	no	—
63	Q₁₆×C₂₃	—	4	1	no	—
64	SD₁₆×C₂₃	—	4	4	no	—
65	SD₁₆×C₂₃	—	4	4	no	—
66	D₈×C₂₃	—	4	2	no	—
67	D₈×C₂₃	—	4	1	no	—
68	D₈×C₂₃	—	4	1	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	D₈	1	2	yes	—
74	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
75	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₂₃	—	4	1	no	—
78	C₄⋊C₄×C₂₃	—	4	1	no	—
79	C₂²⋊C₄×C₂₃	—	4	2	no	—
80	C₂²⋊C₄×C₂₃	—	4	2	no	—
81	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₂₃	D₈	1	1	yes	characteristic
83	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
84	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₁₆×C₂₃	—	4	2	no	—
86	#11929	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₃	—	4	2	no	—
88	C₂³×C₂₃	#102	1	1	yes	characteristic
89	Q₈×C₂₃	—	4	2	no	—
90	Q₈×C₂₃	—	4	2	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	2	no	—
93	D₄×C₂₃	—	8	2	no	—
94	D₄×C₂₃	—	8	4	no	—
95	D₄×C₂₃	—	8	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	4	1	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	8	4	no	—
100	C₂×C₄×C₂₃	#100	1	2	yes	—
101	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
103	C₂×C₄×C₂₃	#102	1	2	yes	—
104	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₈×C₂₃	—	2	1	no	—
109	C₈×C₂₃	—	2	1	no	—
110	C₈×C₂₃	—	8	2	no	—
111	C₈×C₂₃	#102	1	2	yes	—
112	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
113	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
114	#3147	C₂×C₂₃	1	1	yes	characteristic
115	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
116	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#1362	C₂×C₂₃	1	2	yes	complemented
118	#1361	C₂×C₂₃	1	2	yes	complemented
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	8	2	no	—
122	C₂²×C₂₃	—	8	2	no	—
123	C₂²×C₂₃	—	8	2	no	—
124	C₂²×C₂₃	#331	1	2	yes	—
125	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	2	1	no	—
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	—	8	2	no	—
130	C₄×C₂₃	#331	1	2	yes	—
131	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
132	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
133	C₄×C₂₃	#508	1	2	yes	—
134	C₂×#135	—	2	1	no	—
135	C₂×#135	C₄×C₂₃	1	2	yes	—
136	#532	—	2	1	no	—
137	C₂²×C₁₆	—	2	1	no	—
138	#495	—	2	1	no	—
139	#495	C₄×C₂₃	1	2	yes	—
140	#494	C₂²×C₂₃	1	1	yes	characteristic
141	#493	C₂²×C₂₃	1	1	yes	characteristic
142	#492	—	2	1	no	—
143	#431	—	2	1	no	—
144	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₄⋊C₁₆	—	2	1	no	—
146	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
147	#335	C₂²×C₂₃	1	2	yes	—
148	#334	—	2	1	no	—
149	C₂×C₂₃	—	2	2	no	—
150	C₂×C₂₃	—	16	2	no	—
151	C₂×C₂₃	C₂×#331	1	1	yes	characteristic
152	C₂×C₂₃	#2138	1	1	yes	characteristic
153	C₂×C₂₃	#2148	1	1	yes	characteristic
154	C₂×#41	—	2	2	no	—
155	#135	—	4	4	no	—
156	#135	—	4	2	no	—
157	#135	—	4	2	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
160	C₂×SD₁₆	—	2	2	no	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
164	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
166	C₄⋊Q₈	—	2	1	no	—
167	#127	—	2	1	no	—
168	#124	—	2	2	no	—
169	#121	—	4	1	no	—
170	#121	—	4	1	no	—
171	C₄×D₄	—	4	1	no	—
172	C₄×D₄	—	4	1	no	—
173	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	4	2	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	4	1	no	—
178	C₄×C₈	—	4	1	no	—
179	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
181	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
182	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	C₂₃	#11929	1	1	yes	characteristic, direct product factor, Sylow
184	#41	—	4	4	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
195	Q₁₆	—	4	2	no	—
196	Q₁₆	—	4	1	no	—
197	Q₁₆	—	4	1	no	—
198	SD₁₆	—	4	4	no	—
199	SD₁₆	—	4	4	no	—
200	D₈	—	4	2	no	—
201	D₈	—	4	1	no	—
202	D₈	—	4	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₈	D₈×C₂₃	1	2	yes	—
208	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
209	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₂²⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
216	C₄²	D₈×C₂₃	1	1	yes	characteristic
217	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₁₆	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂₃×#102	1	1	yes	characteristic
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	4	no	—
228	D₄	—	8	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	C₂₃×#100	1	2	yes	—
234	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
235	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
236	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
237	C₂×C₄	C₂₃×#102	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	8	2	no	—
244	C₈	C₂₃×#102	1	2	yes	—
245	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
246	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	C₂₃×#331	1	2	yes	—
253	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	2	no	—
258	C₄	C₂₃×#331	1	2	yes	—
259	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
260	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
261	C₄	C₂₃×#508	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	16	2	no	—
264	C₂	C₂×C₂₃×#331	1	1	yes	characteristic
265	C₂	C₂₃×#2138	1	1	yes	characteristic
266	C₂	C₂₃×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#102
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×C₄⋊C₁₆	C₂	semidirect
3. C₂₃×#1564	C₂	semidirect
4. C₂₃×#1362	C₂	semidirect
5. C₂₃×#1361	C₂	semidirect
86. #11929	C₂₃	direct (N × H)
115. C₂×C₄⋊C₁₆	C₂×C₂₃	semidirect
116. #1564	C₂×C₂₃	semidirect
117. #1362	C₂×C₂₃	semidirect
118. #1361	C₂×C₂₃	semidirect
183. C₂₃	#11929	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×#3147	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂₃×#1564	2
C₂₃×#1362	2
C₂₃×#1361	2
#11929	23

Subgroups — C23×#11929 #19159555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11929 #19159555