Subgroups — C₂₃×#32851 #19180477

Order 5888.

532 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2864	C₂	1	2	yes	maximal
2	C₂₃×#2101	C₂	1	2	yes	maximal
3	C₂₃×#2101	C₂	1	2	yes	maximal
4	C₂₃×#2088	C₂	1	2	yes	maximal
5	C₂₃×#2087	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#2086	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#2085	C₂	1	2	yes	maximal
8	C₂₃×#2085	C₂	1	2	yes	maximal
9	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#106	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₃×#389	C₂²	1	2	yes	—
16	C₂₃×#389	C₂²	1	2	yes	—
17	C₂₃×#389	C₂²	1	2	yes	—
18	C₂₃×#389	C₂²	1	2	yes	—
19	C₂₃×#387	—	2	2	no	—
20	C₂₃×#387	—	2	2	no	—
21	C₂₃×#387	—	2	2	no	—
22	C₂₃×#387	—	2	2	no	—
23	C₂₃×#382	C₂²	1	2	yes	—
24	C₂₃×#382	C₂²	1	2	yes	—
25	C₄³×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#340	C₂²	1	2	yes	—
27	C₂₃×#340	C₂²	1	2	yes	—
28	C₂₃×#340	C₂²	1	2	yes	—
29	C₂₃×#340	C₂²	1	2	yes	—
30	C₂₃×#340	C₂²	1	1	yes	characteristic
31	C₂₃×#340	C₂²	1	1	yes	characteristic
32	C₂₃×#340	C₂²	1	2	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	D₄	1	2	yes	—
48	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×#107	—	4	1	no	—
52	C₂₃×#107	—	4	1	no	—
53	C₂₃×#106	—	4	2	no	—
54	C₂₃×C₄⋊C₈	—	4	1	no	—
55	C₂₃×C₄⋊C₈	—	4	1	no	—
56	C₂₃×#95	—	2	2	no	—
57	C₂₃×#95	—	2	2	no	—
58	C₂₃×#95	—	2	2	no	—
59	C₂₃×#95	—	2	2	no	—
60	C₂₃×#95	—	2	2	no	—
61	C₂₃×#95	—	2	2	no	—
62	C₂₃×#95	—	2	2	no	—
63	C₂₃×#95	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	—	2	2	no	—
66	C₂²×C₄×C₂₃	—	2	2	no	—
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	2	1	no	—
76	C₂×C₈×C₂₃	—	2	4	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	4	4	no	—
88	C₄⋊C₄×C₂₃	—	4	4	no	—
89	C₄⋊C₄×C₂₃	—	4	4	no	—
90	C₄⋊C₄×C₂₃	—	4	4	no	—
91	C₄²×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	—	2	1	no	—
93	C₄²×C₂₃	—	2	1	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	4	1	no	—
97	C₄²×C₂₃	—	4	2	no	—
98	#32851	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	#135	1	2	yes	—
123	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
125	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
126	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₈×C₂₃	—	4	2	no	—
128	C₈×C₂₃	—	4	1	no	—
129	C₈×C₂₃	—	4	1	no	—
130	#2864	C₂×C₂₃	1	2	yes	—
131	#2101	C₂×C₂₃	1	2	yes	—
132	#2101	C₂×C₂₃	1	2	yes	—
133	#2088	C₂×C₂₃	1	2	yes	—
134	#2087	C₂×C₂₃	1	1	yes	characteristic
135	#2086	C₂×C₂₃	1	1	yes	characteristic
136	#2085	C₂×C₂₃	1	2	yes	—
137	#2085	C₂×C₂₃	1	2	yes	—
138	#1831	C₂×C₂₃	1	1	yes	characteristic
139	C₂²×C₂₃	#536	1	2	yes	—
140	C₂²×C₂₃	#537	1	1	yes	characteristic
141	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
142	C₂²×C₂₃	#575	1	2	yes	—
143	C₂²×C₂₃	#575	1	1	yes	characteristic
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	—	4	1	no	—
148	C₄×C₂₃	—	8	2	no	—
149	C₄×C₂₃	—	8	2	no	—
150	C₄×C₂₃	—	8	2	no	—
151	C₄×C₂₃	—	8	2	no	—
152	C₄×C₂₃	#563	1	1	yes	characteristic
153	C₄×C₂₃	#563	1	1	yes	characteristic
154	C₄×C₂₃	#564	1	2	yes	—
155	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#106	C₂²×C₂₃	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
160	#389	C₂²×C₂₃	1	2	yes	—
161	#389	C₂²×C₂₃	1	2	yes	—
162	#389	C₂²×C₂₃	1	2	yes	—
163	#389	C₂²×C₂₃	1	2	yes	—
164	#387	—	2	2	no	—
165	#387	—	2	2	no	—
166	#387	—	2	2	no	—
167	#387	—	2	2	no	—
168	#382	C₂²×C₂₃	1	2	yes	—
169	#382	C₂²×C₂₃	1	2	yes	—
170	C₄³	C₂²×C₂₃	1	1	yes	characteristic
171	#340	C₂²×C₂₃	1	2	yes	—
172	#340	C₂²×C₂₃	1	2	yes	—
173	#340	C₂²×C₂₃	1	2	yes	—
174	#340	C₂²×C₂₃	1	2	yes	—
175	#340	C₂²×C₂₃	1	1	yes	characteristic
176	#340	C₂²×C₂₃	1	1	yes	characteristic
177	#340	C₂²×C₂₃	1	2	yes	—
178	#336	C₂²×C₂₃	1	2	yes	—
179	#336	C₂²×C₂₃	1	2	yes	—
180	C₂×C₂₃	#2822	1	1	yes	characteristic
181	C₂×C₂₃	#3233	1	2	yes	—
182	C₂×C₂₃	#3240	1	2	yes	—
183	C₂×C₂₃	#3250	1	1	yes	characteristic
184	C₂×C₂₃	#3256	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
186	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
194	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
196	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₂₃	1	2	yes	—
198	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
201	#107	—	4	1	no	—
202	#107	—	4	1	no	—
203	#106	—	4	2	no	—
204	C₄⋊C₈	—	4	1	no	—
205	C₄⋊C₈	—	4	1	no	—
206	#95	—	2	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	C₂₃	#32851	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₂₃×#135	1	2	yes	—
273	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
275	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
276	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
277	C₈	—	4	2	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₂²	C₂₃×#536	1	2	yes	—
281	C₂²	C₂₃×#537	1	1	yes	characteristic
282	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
283	C₂²	C₂₃×#575	1	2	yes	—
284	C₂²	C₂₃×#575	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	8	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	—	8	2	no	—
293	C₄	C₂₃×#563	1	1	yes	characteristic
294	C₄	C₂₃×#563	1	1	yes	characteristic
295	C₄	C₂₃×#564	1	2	yes	—
296	C₂	C₂₃×#2822	1	1	yes	characteristic
297	C₂	C₂₃×#3233	1	2	yes	—
298	C₂	C₂₃×#3240	1	2	yes	—
299	C₂	C₂₃×#3250	1	1	yes	characteristic
300	C₂	C₂₃×#3256	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
98. #32851	C₂₃	direct (N × H)
214. C₂₃	#32851	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×#2864	2
C₂₃×#2101	2
C₂₃×#2088	2
C₂₃×#2087	2
C₂₃×#2086	2
C₂₃×#2085	2
C₂₃×#1831	2
#32851	23

Subgroups — C23×#32851 #19180477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32851 #19180477