Subgroups — C₂₃×#12829 #19160455

Order 5888.

378 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2957	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2148	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2138	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
9	C₂₃×#568	—	2	1	no	—
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#444	C₄	1	4	yes	complemented
13	C₂₃×#443	—	2	2	no	—
14	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
16	C₂₃×#427	C₂²	1	1	yes	characteristic
17	C₂₃×#421	—	2	1	no	—
18	C₂₃×#405	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊C₁₆	—	2	2	no	—
20	C₂₃×#362	—	2	1	no	—
21	C₂₃×#361	—	2	1	no	—
22	C₂₃×#358	—	2	1	no	—
23	C₂₃×#357	—	2	1	no	—
24	C₂₃×#344	—	2	1	no	—
25	C₂₃×H(3,ℤ/4)	—	2	1	no	—
26	C₂×#41×C₂₃	—	2	1	no	—
27	C₂₃×#135	—	2	1	no	—
28	C₂₃×#135	—	2	1	no	—
29	C₂₃×#135	C₂×C₄	1	2	yes	—
30	C₂₃×#135	Q₈	1	2	yes	—
31	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
32	C₂×SD₁₆×C₂₃	—	2	1	no	—
33	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂₃×#131	—	2	4	no	—
35	C₂₃×#131	—	4	2	no	—
36	C₂×M₁₆×C₂₃	—	2	1	no	—
37	C₂×M₁₆×C₂₃	—	2	1	no	—
38	C₂×M₁₆×C₂₃	—	4	2	no	—
39	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×#117	—	2	1	no	—
42	C₂₃×#117	D₄	1	1	yes	characteristic
43	C₂₃×M₃₂	—	2	1	no	—
44	C₂₃×M₃₂	—	2	1	no	—
45	C₂×C₁₆×C₂₃	—	2	1	no	—
46	C₂×C₁₆×C₂₃	—	2	1	no	—
47	C₂×C₁₆×C₂₃	—	2	1	no	—
48	C₂×C₁₆×C₂₃	—	2	1	no	—
49	C₂₃×#108	—	2	1	no	—
50	C₂₃×#108	—	2	1	no	—
51	C₂₃×#108	—	2	1	no	—
52	C₂₃×#108	C₂×C₄	1	2	yes	—
53	C₂₃×#107	—	2	1	no	—
54	C₂₃×#106	—	2	1	no	—
55	C₂₃×#104	—	2	4	no	—
56	C₂₃×#97	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₂₃	D₄	1	2	yes	—
58	#41×C₂₃	—	2	2	no	—
59	#41×C₂₃	—	2	2	no	—
60	#41×C₂₃	—	4	1	no	—
61	#41×C₂₃	—	4	1	no	—
62	C₂×Q₈×C₂₃	—	2	1	no	—
63	C₂×D₄×C₂₃	—	2	1	no	—
64	C₂×D₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	—	4	1	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	2	1	no	—
68	Q₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
69	SD₁₆×C₂₃	—	2	2	no	—
70	SD₁₆×C₂₃	—	2	2	no	—
71	D₈×C₂₃	—	2	1	no	—
72	D₈×C₂₃	C₄⋊C₄	1	2	yes	—
73	M₁₆×C₂₃	—	2	1	no	—
74	M₁₆×C₂₃	—	2	1	no	—
75	M₁₆×C₂₃	—	2	2	no	—
76	M₁₆×C₂₃	—	4	4	no	—
77	M₁₆×C₂₃	—	4	1	no	—
78	M₁₆×C₂₃	—	4	1	no	—
79	M₁₆×C₂₃	—	4	1	no	—
80	C₂×C₈×C₂₃	—	4	4	no	—
81	C₂×C₈×C₂₃	—	4	1	no	—
82	C₂×C₈×C₂₃	—	4	1	no	—
83	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
90	C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₄⋊C₄×C₂₃	—	4	1	no	—
92	C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₂²⋊C₄×C₂₃	—	4	1	no	—
94	C₄²×C₂₃	—	4	1	no	—
95	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
96	C₁₆×C₂₃	—	4	1	no	—
97	C₁₆×C₂₃	—	4	1	no	—
98	#12829	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₂₃	—	4	1	no	—
100	C₂³×C₂₃	#135	1	1	yes	characteristic
101	Q₈×C₂₃	—	2	2	no	—
102	Q₈×C₂₃	—	4	1	no	—
103	D₄×C₂₃	—	2	2	no	—
104	D₄×C₂₃	—	4	2	no	—
105	D₄×C₂₃	—	4	2	no	—
106	D₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	8	1	no	—
112	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
116	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₈×C₂₃	—	2	2	no	—
119	C₈×C₂₃	—	4	2	no	—
120	C₈×C₂₃	—	8	1	no	—
121	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₂₃	#117	1	1	yes	characteristic
123	C₈×C₂₃	#121	1	1	yes	characteristic
124	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
125	#2957	C₂×C₂₃	1	1	yes	characteristic
126	#2151	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#2148	C₂×C₂₃	1	1	yes	characteristic
128	#2138	C₂×C₂₃	1	1	yes	characteristic
129	#1833	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#1389	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#1381	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂²×C₂₃	—	4	1	no	—
133	C₂²×C₂₃	—	4	2	no	—
134	C₂²×C₂₃	—	16	1	no	—
135	C₂²×C₂₃	#416	1	1	yes	characteristic
136	C₂²×C₂₃	#463	1	1	yes	characteristic
137	C₂²×C₂₃	#478	1	1	yes	characteristic
138	C₄×C₂₃	—	4	2	no	—
139	C₄×C₂₃	—	4	2	no	—
140	C₄×C₂₃	—	16	1	no	—
141	C₄×C₂₃	#380	1	1	yes	characteristic
142	C₄×C₂₃	#418	1	1	yes	characteristic
143	C₄×C₂₃	#463	1	1	yes	characteristic
144	C₄×C₂₃	#478	1	1	yes	characteristic
145	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
146	#568	—	2	1	no	—
147	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
148	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
149	#444	C₄×C₂₃	1	4	yes	complemented
150	#443	—	2	2	no	—
151	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
153	#427	C₂²×C₂₃	1	1	yes	characteristic
154	#421	—	2	1	no	—
155	#405	C₂²×C₂₃	1	1	yes	characteristic
156	C₄⋊C₁₆	—	2	2	no	—
157	#362	—	2	1	no	—
158	#361	—	2	1	no	—
159	#358	—	2	1	no	—
160	#357	—	2	1	no	—
161	#344	—	2	1	no	—
162	H(3,ℤ/4)	—	2	1	no	—
163	C₂×C₂₃	—	2	1	no	—
164	C₂×C₂₃	—	2	1	no	—
165	C₂×C₂₃	—	2	1	no	—
166	C₂×C₂₃	—	8	2	no	—
167	C₂×C₂₃	#1804	1	1	yes	characteristic
168	C₂×#41	—	2	1	no	—
169	#135	—	2	1	no	—
170	#135	—	2	1	no	—
171	#135	C₂×C₄×C₂₃	1	2	yes	—
172	#135	Q₈×C₂₃	1	2	yes	—
173	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
174	C₂×SD₁₆	—	2	1	no	—
175	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
176	#131	—	2	4	no	—
177	#131	—	4	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	4	2	no	—
181	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	#117	—	2	1	no	—
184	#117	D₄×C₂₃	1	1	yes	characteristic
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	#108	—	2	1	no	—
192	#108	—	2	1	no	—
193	#108	—	2	1	no	—
194	#108	C₂×C₄×C₂₃	1	2	yes	—
195	#107	—	2	1	no	—
196	#106	—	2	1	no	—
197	#104	—	2	4	no	—
198	#97	C₂×C₄×C₂₃	1	2	yes	—
199	C₄×C₈	D₄×C₂₃	1	2	yes	—
200	C₂₃	#12829	1	1	yes	characteristic, direct product factor, Sylow
201	#41	—	2	2	no	—
202	#41	—	2	2	no	—
203	#41	—	4	1	no	—
204	#41	—	4	1	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	Q₁₆	—	2	1	no	—
211	Q₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	2	no	—
214	D₈	—	2	1	no	—
215	D₈	C₄⋊C₄×C₂₃	1	2	yes	—
216	M₁₆	—	2	1	no	—
217	M₁₆	—	2	1	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	4	4	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	C₂×C₈	—	4	4	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
231	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
232	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₂²⋊C₄	—	2	1	no	—
236	C₂²⋊C₄	—	4	1	no	—
237	C₄²	—	4	1	no	—
238	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
239	C₁₆	—	4	1	no	—
240	C₁₆	—	4	1	no	—
241	C₂³	—	4	1	no	—
242	C₂³	C₂₃×#135	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	4	1	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
256	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
258	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	8	1	no	—
263	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
264	C₈	C₂₃×#117	1	1	yes	characteristic
265	C₈	C₂₃×#121	1	1	yes	characteristic
266	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	16	1	no	—
270	C₂²	C₂₃×#416	1	1	yes	characteristic
271	C₂²	C₂₃×#463	1	1	yes	characteristic
272	C₂²	C₂₃×#478	1	1	yes	characteristic
273	C₄	—	4	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	16	1	no	—
276	C₄	C₂₃×#380	1	1	yes	characteristic
277	C₄	C₂₃×#418	1	1	yes	characteristic
278	C₄	C₂₃×#463	1	1	yes	characteristic
279	C₄	C₂₃×#478	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	2	no	—
284	C₂	C₂₃×#1804	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#418
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1804
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2151	C₂	semidirect
5. C₂₃×#1833	C₂	semidirect
6. C₂₃×#1389	C₂	semidirect
7. C₂₃×#1381	C₂	semidirect
12. C₂₃×#444	C₄	semidirect
98. #12829	C₂₃	direct (N × H)
126. #2151	C₂×C₂₃	semidirect
129. #1833	C₂×C₂₃	semidirect
130. #1389	C₂×C₂₃	semidirect
131. #1381	C₂×C₂₃	semidirect
149. #444	C₄×C₂₃	semidirect
200. C₂₃	#12829	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×#2957	2
C₂₃×#2151	2
C₂₃×#2148	2
C₂₃×#2138	2
C₂₃×#1833	2
C₂₃×#1389	2
C₂₃×#1381	2
#12829	23

Subgroups — C23×#12829 #19160455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12829 #19160455