Subgroups — C₂₃×#12685 #19160311

Order 5888.

890 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3573	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2117	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃×#350	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#585	C₄	1	1	yes	characteristic
7	C₂₃×#568	—	2	4	no	—
8	C₂₃×#568	—	2	2	no	—
9	C₂₃×#568	C₄	1	1	yes	characteristic
10	C₂₃×#568	C₄	1	2	yes	—
11	C₂×C₂₃×#131	—	2	4	no	—
12	C₂×C₂₃×#131	—	2	2	no	—
13	C₂×C₂₃×#131	C₂²	1	2	yes	—
14	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₂₃×M₃₂	C₂²	1	2	yes	complemented
17	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×#350	—	2	4	no	—
19	C₂₃×#350	—	2	4	no	—
20	C₂₃×#349	—	2	2	no	—
21	C₂₃×C₂²⋊C₁₆	—	2	2	no	—
22	C₂₃×2₋¹⁺⁴	—	2	2	no	—
23	C₂₃×2₋¹⁺⁴	C₈	1	2	yes	—
24	C₂₃×H(5,𝔽₂)	—	2	2	no	—
25	C₂₃×H(5,𝔽₂)	—	2	2	no	—
26	C₂₃×H(5,𝔽₂)	C₈	1	2	yes	—
27	C₂×#41×C₂₃	—	2	4	no	—
28	C₂×#41×C₂₃	—	2	2	no	—
29	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂₃×#131	—	2	4	no	—
32	C₂₃×#131	—	2	8	no	—
33	C₂₃×#131	—	2	4	no	—
34	C₂₃×#131	—	2	4	no	—
35	C₂₃×#131	—	2	4	no	—
36	C₂₃×#131	—	2	4	no	—
37	C₂₃×#131	—	2	2	no	—
38	C₂₃×#131	—	2	2	no	—
39	C₂₃×#131	—	2	2	no	—
40	C₂₃×#131	—	2	2	no	—
41	C₂₃×#131	D₄	1	4	yes	—
42	C₂₃×#131	D₄	1	4	yes	—
43	C₂×M₁₆×C₂₃	—	2	8	no	—
44	C₂×M₁₆×C₂₃	—	2	4	no	—
45	C₂×M₁₆×C₂₃	—	2	2	no	—
46	C₂×M₁₆×C₂₃	—	2	4	no	—
47	C₂×M₁₆×C₂₃	—	2	2	no	—
48	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
49	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
51	C₂²×C₈×C₂₃	—	2	4	no	—
52	C₂²×C₈×C₂₃	—	2	2	no	—
53	C₂²×C₈×C₂₃	D₄	1	2	yes	—
54	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×M₃₂	—	2	2	no	—
56	C₂₃×M₃₂	—	2	2	no	—
57	C₂₃×M₃₂	—	2	2	no	—
58	C₂×C₁₆×C₂₃	—	2	1	no	—
59	C₂×C₁₆×C₂₃	—	2	1	no	—
60	C₂×C₁₆×C₂₃	—	2	2	no	—
61	C₂×C₁₆×C₂₃	—	2	2	no	—
62	#41×C₂₃	—	2	4	no	—
63	#41×C₂₃	—	2	8	no	—
64	#41×C₂₃	—	2	4	no	—
65	#41×C₂₃	—	2	4	no	—
66	#41×C₂₃	—	2	4	no	—
67	#41×C₂₃	—	2	4	no	—
68	#41×C₂₃	—	2	2	no	—
69	#41×C₂₃	—	2	2	no	—
70	#41×C₂₃	—	2	2	no	—
71	#41×C₂₃	—	2	2	no	—
72	#41×C₂₃	C₂²⋊C₄	1	4	yes	—
73	#41×C₂₃	C₂×C₈	1	4	yes	—
74	C₂×Q₈×C₂₃	—	2	2	no	—
75	C₂×Q₈×C₂₃	—	2	4	no	—
76	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
77	C₂×Q₈×C₂₃	M₁₆	1	2	yes	—
78	C₂×D₄×C₂₃	—	2	8	no	—
79	C₂×D₄×C₂₃	—	2	2	no	—
80	C₂×D₄×C₂₃	—	2	4	no	—
81	C₂×D₄×C₂₃	—	2	2	no	—
82	C₂×D₄×C₂₃	—	2	4	no	—
83	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
84	C₂×D₄×C₂₃	M₁₆	1	2	yes	—
85	C₂×D₄×C₂₃	M₁₆	1	2	yes	—
86	C₂²×C₄×C₂₃	—	2	4	no	—
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	M₁₆×C₂₃	—	2	4	no	—
91	M₁₆×C₂₃	—	2	8	no	—
92	M₁₆×C₂₃	—	2	2	no	—
93	M₁₆×C₂₃	—	2	2	no	—
94	M₁₆×C₂₃	—	2	4	no	—
95	M₁₆×C₂₃	—	2	4	no	—
96	M₁₆×C₂₃	—	2	4	no	—
97	M₁₆×C₂₃	C₂²⋊C₄	1	4	yes	—
98	C₂×C₈×C₂₃	—	2	4	no	—
99	C₂×C₈×C₂₃	—	2	8	no	—
100	C₂×C₈×C₂₃	—	2	2	no	—
101	C₂×C₈×C₂₃	—	2	4	no	—
102	C₂×C₈×C₂₃	—	2	2	no	—
103	C₂×C₈×C₂₃	—	2	4	no	—
104	C₂×C₈×C₂₃	—	2	2	no	—
105	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
110	C₁₆×C₂₃	—	4	1	no	—
111	C₁₆×C₂₃	—	4	2	no	—
112	C₁₆×C₂₃	—	4	1	no	—
113	#12685	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₂₃	—	2	4	no	—
115	C₂³×C₂₃	—	2	2	no	—
116	C₂³×C₂₃	C₂²⋊C₈	1	2	yes	—
117	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
118	Q₈×C₂₃	—	2	4	no	—
119	Q₈×C₂₃	—	2	2	no	—
120	Q₈×C₂₃	—	2	2	no	—
121	Q₈×C₂₃	C₂²⋊C₈	1	4	yes	—
122	D₄×C₂₃	—	2	8	no	—
123	D₄×C₂₃	—	2	4	no	—
124	D₄×C₂₃	—	2	4	no	—
125	D₄×C₂₃	—	2	4	no	—
126	D₄×C₂₃	—	2	4	no	—
127	D₄×C₂₃	—	2	2	no	—
128	D₄×C₂₃	—	2	2	no	—
129	D₄×C₂₃	C₂²⋊C₈	1	4	yes	—
130	C₂×C₄×C₂₃	—	2	4	no	—
131	C₂×C₄×C₂₃	—	2	8	no	—
132	C₂×C₄×C₂₃	—	2	2	no	—
133	C₂×C₄×C₂₃	—	2	4	no	—
134	C₂×C₄×C₂₃	—	2	2	no	—
135	C₂×C₄×C₂₃	—	2	4	no	—
136	C₂×C₄×C₂₃	—	2	2	no	—
137	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
138	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
142	C₈×C₂₃	—	2	4	no	—
143	C₈×C₂₃	—	2	2	no	—
144	C₈×C₂₃	#120	1	1	yes	characteristic
145	C₈×C₂₃	#120	1	2	yes	—
146	C₈×C₂₃	#120	1	1	yes	characteristic
147	#3573	C₂×C₂₃	1	1	yes	characteristic
148	#2117	C₂×C₂₃	1	2	yes	complemented
149	C₂×#350	C₂×C₂₃	1	2	yes	complemented
150	#2115	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#2113	C₂×C₂₃	1	1	yes	characteristic, complemented
152	C₂²×C₂₃	—	2	2	no	—
153	C₂²×C₂₃	—	2	4	no	—
154	C₂²×C₂₃	—	4	1	no	—
155	C₂²×C₂₃	—	4	2	no	—
156	C₂²×C₂₃	—	8	2	no	—
157	C₂²×C₂₃	—	8	4	no	—
158	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₂²×C₂₃	#407	1	2	yes	—
160	C₄×C₂₃	—	2	4	no	—
161	C₄×C₂₃	—	2	2	no	—
162	C₄×C₂₃	#379	1	1	yes	characteristic
163	C₄×C₂₃	C₂×C₂²⋊C₈	1	2	yes	—
164	C₄×C₂₃	#407	1	1	yes	characteristic
165	#585	C₄×C₂₃	1	1	yes	characteristic
166	#568	—	2	4	no	—
167	#568	—	2	2	no	—
168	#568	C₄×C₂₃	1	1	yes	characteristic
169	#568	C₄×C₂₃	1	2	yes	—
170	C₂×#131	—	2	4	no	—
171	C₂×#131	—	2	2	no	—
172	C₂×#131	C₂²×C₂₃	1	2	yes	—
173	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
174	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
175	C₂×M₃₂	C₂²×C₂₃	1	2	yes	complemented
176	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
177	#350	—	2	4	no	—
178	#350	—	2	4	no	—
179	#349	—	2	2	no	—
180	C₂²⋊C₁₆	—	2	2	no	—
181	C₂×C₂₃	—	2	1	no	—
182	C₂×C₂₃	—	2	2	no	—
183	C₂×C₂₃	—	4	2	no	—
184	C₂×C₂₃	—	4	4	no	—
185	C₂×C₂₃	#1779	1	1	yes	characteristic
186	2₋¹⁺⁴	—	2	2	no	—
187	2₋¹⁺⁴	C₈×C₂₃	1	2	yes	—
188	H(5,𝔽₂)	—	2	2	no	—
189	H(5,𝔽₂)	—	2	2	no	—
190	H(5,𝔽₂)	C₈×C₂₃	1	2	yes	—
191	C₂×#41	—	2	4	no	—
192	C₂×#41	—	2	2	no	—
193	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
194	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
195	#131	—	2	4	no	—
196	#131	—	2	8	no	—
197	#131	—	2	4	no	—
198	#131	—	2	4	no	—
199	#131	—	2	4	no	—
200	#131	—	2	4	no	—
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	#131	—	2	2	no	—
205	#131	D₄×C₂₃	1	4	yes	—
206	#131	D₄×C₂₃	1	4	yes	—
207	C₂×M₁₆	—	2	8	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	2	4	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
213	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
214	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
215	C₂²×C₈	—	2	4	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	D₄×C₂₃	1	2	yes	—
218	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
219	M₃₂	—	2	2	no	—
220	M₃₂	—	2	2	no	—
221	M₃₂	—	2	2	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	2	2	no	—
225	C₂×C₁₆	—	2	2	no	—
226	C₂₃	#12685	1	1	yes	characteristic, direct product factor, Sylow
227	#41	—	2	4	no	—
228	#41	—	2	8	no	—
229	#41	—	2	4	no	—
230	#41	—	2	4	no	—
231	#41	—	2	4	no	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	#41	—	2	4	no	—
237	#41	C₂²⋊C₄×C₂₃	1	4	yes	—
238	#41	C₂×C₈×C₂₃	1	4	yes	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	4	no	—
241	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
242	C₂×Q₈	M₁₆×C₂₃	1	2	yes	—
243	C₂×D₄	—	2	8	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	2	4	no	—
248	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
249	C₂×D₄	M₁₆×C₂₃	1	2	yes	—
250	C₂×D₄	M₁₆×C₂₃	1	2	yes	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	8	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	4	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	C₂²⋊C₄×C₂₃	1	4	yes	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	8	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
271	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
274	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
275	C₁₆	—	4	1	no	—
276	C₁₆	—	4	1	no	—
277	C₁₆	—	4	2	no	—
278	C₂³	—	2	4	no	—
279	C₂³	—	2	2	no	—
280	C₂³	C₂₃×C₂²⋊C₈	1	2	yes	—
281	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	2	2	no	—
284	Q₈	—	2	2	no	—
285	Q₈	C₂₃×C₂²⋊C₈	1	4	yes	—
286	D₄	—	2	8	no	—
287	D₄	—	2	4	no	—
288	D₄	—	2	4	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	2	2	no	—
292	D₄	—	2	2	no	—
293	D₄	C₂₃×C₂²⋊C₈	1	4	yes	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
302	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
306	C₈	—	2	4	no	—
307	C₈	—	2	2	no	—
308	C₈	C₂₃×#120	1	1	yes	characteristic
309	C₈	C₂₃×#120	1	1	yes	characteristic
310	C₈	C₂₃×#120	1	2	yes	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	4	no	—
317	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
318	C₂²	C₂₃×#407	1	2	yes	—
319	C₄	—	2	4	no	—
320	C₄	—	2	2	no	—
321	C₄	C₂₃×#379	1	1	yes	characteristic
322	C₄	C₂×C₂₃×C₂²⋊C₈	1	2	yes	—
323	C₄	C₂₃×#407	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	4	2	no	—
327	C₂	—	4	4	no	—
328	C₂	C₂₃×#1779	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#120
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1779
prime radical	C₂₃×#585	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2117	C₂	semidirect
3. C₂×C₂₃×#350	C₂	semidirect
4. C₂₃×#2115	C₂	semidirect
5. C₂₃×#2113	C₂	semidirect
15. C₂×C₂₃×M₃₂	C₂²	semidirect
16. C₂×C₂₃×M₃₂	C₂²	semidirect
17. C₂²×C₁₆×C₂₃	C₂²	semidirect
113. #12685	C₂₃	direct (N × H)
148. #2117	C₂×C₂₃	semidirect
149. C₂×#350	C₂×C₂₃	semidirect
150. #2115	C₂×C₂₃	semidirect
151. #2113	C₂×C₂₃	semidirect
174. C₂×M₃₂	C₂²×C₂₃	semidirect
175. C₂×M₃₂	C₂²×C₂₃	semidirect
176. C₂²×C₁₆	C₂²×C₂₃	semidirect
226. C₂₃	#12685	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×#3573	2
C₂₃×#2117	2
C₂×C₂₃×#350	2
C₂₃×#2115	2
C₂₃×#2113	2
#12685	23

Subgroups — C23×#12685 #19160311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12685 #19160311