Subgroups — C₂₃×#12825 #19160451

Order 5888.

400 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2148	C₂	1	2	yes	maximal
4	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1385	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#131	—	2	1	no	—
8	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
9	C₂₃×#443	—	2	1	no	—
10	C₂₃×#443	—	2	1	no	—
11	C₂₃×#443	C₄	1	2	yes	—
12	C₂₃×#443	C₄	1	2	yes	—
13	C₂×C₂₃×#108	—	2	1	no	—
14	C₂×C₂₃×#108	C₂²	1	2	yes	—
15	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
16	C₂₃×#423	—	2	1	no	—
17	C₂×C₂₃×#104	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#364	—	2	1	no	—
20	C₂₃×#364	—	2	1	no	—
21	C₂₃×#362	—	2	2	no	—
22	C₂₃×#361	—	2	2	no	—
23	C₂×#41×C₂₃	—	2	1	no	—
24	C₂₃×#135	—	2	1	no	—
25	C₂₃×#135	—	2	1	no	—
26	C₂₃×#135	C₂×C₄	1	2	yes	—
27	C₂₃×#135	Q₈	1	2	yes	—
28	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
29	C₂×SD₁₆×C₂₃	—	2	1	no	—
30	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
31	C₂₃×#131	—	2	2	no	—
32	C₂₃×#131	—	2	2	no	—
33	C₂₃×#131	—	4	1	no	—
34	C₂₃×#131	—	4	1	no	—
35	C₂×M₁₆×C₂₃	—	2	1	no	—
36	C₂×M₁₆×C₂₃	—	2	2	no	—
37	C₂×M₁₆×C₂₃	—	4	1	no	—
38	C₂²×C₈×C₂₃	—	4	1	no	—
39	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
42	C₂₃×M₃₂	—	2	2	no	—
43	C₂₃×M₃₂	—	2	2	no	—
44	C₂×C₁₆×C₂₃	—	2	2	no	—
45	C₂₃×#108	—	2	1	no	—
46	C₂₃×#108	—	2	2	no	—
47	C₂₃×#108	—	2	2	no	—
48	C₂₃×#108	—	4	1	no	—
49	C₂₃×#108	—	4	1	no	—
50	C₂₃×#108	C₂×C₄	1	2	yes	—
51	C₂₃×C₄⋊C₈	—	4	2	no	—
52	C₂₃×#104	—	2	2	no	—
53	C₂₃×#104	—	2	2	no	—
54	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
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	—	2	1	no	—
66	C₂²×C₄×C₂₃	—	4	1	no	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₂₃	—	2	1	no	—
69	Q₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
70	SD₁₆×C₂₃	—	2	2	no	—
71	SD₁₆×C₂₃	—	2	2	no	—
72	D₈×C₂₃	—	2	1	no	—
73	D₈×C₂₃	C₄⋊C₄	1	2	yes	—
74	M₁₆×C₂₃	—	2	2	no	—
75	M₁₆×C₂₃	—	4	1	no	—
76	M₁₆×C₂₃	—	4	2	no	—
77	M₁₆×C₂₃	—	4	2	no	—
78	M₁₆×C₂₃	—	4	2	no	—
79	M₁₆×C₂₃	—	4	2	no	—
80	C₂×C₈×C₂₃	—	4	1	no	—
81	C₂×C₈×C₂₃	—	4	2	no	—
82	C₂×C₈×C₂₃	—	4	2	no	—
83	C₂×C₈×C₂₃	—	4	2	no	—
84	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
87	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
88	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
89	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
90	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
96	C₄²×C₂₃	—	2	1	no	—
97	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₁₆×C₂₃	—	4	2	no	—
100	#12825	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₃	—	4	1	no	—
102	C₂³×C₂₃	#137	1	1	yes	characteristic
103	Q₈×C₂₃	—	2	2	no	—
104	Q₈×C₂₃	—	4	1	no	—
105	D₄×C₂₃	—	2	2	no	—
106	D₄×C₂₃	—	4	2	no	—
107	D₄×C₂₃	—	4	2	no	—
108	D₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₈×C₂₃	—	4	2	no	—
122	C₈×C₂₃	—	8	2	no	—
123	C₈×C₂₃	#102	1	1	yes	characteristic
124	C₈×C₂₃	#102	1	1	yes	characteristic
125	C₈×C₂₃	#102	1	1	yes	characteristic
126	C₈×C₂₃	#102	1	1	yes	characteristic
127	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₈×C₂₃	#117	1	1	yes	characteristic
129	C₈×C₂₃	#121	1	1	yes	characteristic
130	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×#443	C₂×C₂₃	1	1	yes	characteristic
132	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#2148	C₂×C₂₃	1	2	yes	—
134	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#1385	C₂×C₂₃	1	2	yes	complemented
136	C₂²×C₂₃	—	4	1	no	—
137	C₂²×C₂₃	—	4	2	no	—
138	C₂²×C₂₃	—	8	2	no	—
139	C₂²×C₂₃	#417	1	1	yes	characteristic
140	C₂²×C₂₃	#461	1	1	yes	characteristic
141	C₂²×C₂₃	#474	1	1	yes	characteristic
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	4	1	no	—
144	C₄×C₂₃	—	4	2	no	—
145	C₄×C₂₃	#380	1	1	yes	characteristic
146	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
147	C₄×C₂₃	#459	1	1	yes	characteristic
148	C₄×C₂₃	#476	1	1	yes	characteristic
149	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×#131	—	2	1	no	—
151	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
152	#443	—	2	1	no	—
153	#443	—	2	1	no	—
154	#443	C₄×C₂₃	1	2	yes	—
155	#443	C₄×C₂₃	1	2	yes	—
156	C₂×#108	—	2	1	no	—
157	C₂×#108	C₂²×C₂₃	1	2	yes	—
158	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
159	#423	—	2	1	no	—
160	C₂×#104	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
162	#364	—	2	1	no	—
163	#364	—	2	1	no	—
164	#362	—	2	2	no	—
165	#361	—	2	2	no	—
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	—	2	1	no	—
168	C₂×C₂₃	—	2	1	no	—
169	C₂×C₂₃	—	8	2	no	—
170	C₂×C₂₃	#1802	1	1	yes	characteristic
171	C₂×#41	—	2	1	no	—
172	#135	—	2	1	no	—
173	#135	—	2	1	no	—
174	#135	C₂×C₄×C₂₃	1	2	yes	—
175	#135	Q₈×C₂₃	1	2	yes	—
176	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
177	C₂×SD₁₆	—	2	1	no	—
178	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
179	#131	—	2	2	no	—
180	#131	—	2	2	no	—
181	#131	—	4	1	no	—
182	#131	—	4	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	4	1	no	—
186	C₂²×C₈	—	4	1	no	—
187	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
190	M₃₂	—	2	2	no	—
191	M₃₂	—	2	2	no	—
192	C₂×C₁₆	—	2	2	no	—
193	#108	—	2	1	no	—
194	#108	—	2	2	no	—
195	#108	—	2	2	no	—
196	#108	—	4	1	no	—
197	#108	—	4	1	no	—
198	#108	C₂×C₄×C₂₃	1	2	yes	—
199	C₄⋊C₈	—	4	2	no	—
200	#104	—	2	2	no	—
201	#104	—	2	2	no	—
202	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
203	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
204	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
205	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
206	C₂₃	#12825	1	1	yes	characteristic, direct product factor, Sylow
207	#41	—	2	2	no	—
208	#41	—	2	2	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	Q₁₆	—	2	1	no	—
218	Q₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
219	SD₁₆	—	2	2	no	—
220	SD₁₆	—	2	2	no	—
221	D₈	—	2	1	no	—
222	D₈	C₄⋊C₄×C₂₃	1	2	yes	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
235	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
236	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
237	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
238	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
239	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
243	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
244	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
245	C₄²	—	2	1	no	—
246	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
247	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
248	C₁₆	—	4	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	C₂₃×#137	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	Q₈	—	4	1	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
265	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
266	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	8	2	no	—
271	C₈	C₂₃×#102	1	1	yes	characteristic
272	C₈	C₂₃×#102	1	1	yes	characteristic
273	C₈	C₂₃×#102	1	1	yes	characteristic
274	C₈	C₂₃×#102	1	1	yes	characteristic
275	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
276	C₈	C₂₃×#117	1	1	yes	characteristic
277	C₈	C₂₃×#121	1	1	yes	characteristic
278	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₂₃×#417	1	1	yes	characteristic
283	C₂²	C₂₃×#461	1	1	yes	characteristic
284	C₂²	C₂₃×#474	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₄	C₂₃×#380	1	1	yes	characteristic
289	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
290	C₄	C₂₃×#459	1	1	yes	characteristic
291	C₄	C₂₃×#476	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	C₂₃×#1802	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#476
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1802
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×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×#364	C₂	semidirect
4. C₂₃×#1848	C₂	semidirect
5. C₂₃×#1385	C₂	semidirect
100. #12825	C₂₃	direct (N × H)
132. C₂×#364	C₂×C₂₃	semidirect
134. #1848	C₂×C₂₃	semidirect
135. #1385	C₂×C₂₃	semidirect
206. C₂₃	#12825	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#443	2
C₂×C₂₃×#364	2
C₂₃×#2148	2
C₂₃×#1848	2
C₂₃×#1385	2
#12825	23

Subgroups — C23×#12825 #19160451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12825 #19160451