Subgroups — C₂₃×#22815 #19170441

Order 5888.

754 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2356	C₂	1	2	yes	maximal
3	C₂₃×#1971	C₂	1	2	yes	maximal
4	C₂₃×#1928	C₂	1	4	yes	maximal
5	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1805	C₂	1	2	yes	maximal
7	C₂₃×#1776	C₂	1	2	yes	maximal
8	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₄×Q₁₆×C₂₃	C₄	1	8	yes	complemented
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#103	C₂²	1	4	yes	—
17	C₂×C₂₃×#103	C₂²	1	2	yes	—
18	C₂×C₂₃×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#391	C₂²	1	4	yes	—
21	C₂₃×#387	—	2	4	no	—
22	C₂₃×#383	—	2	2	no	—
23	C₂₃×#382	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#376	—	2	2	no	—
27	C₂₃×#340	C₂²	1	4	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
30	C₂×Q₁₆×C₂₃	—	2	2	no	—
31	C₂×Q₁₆×C₂₃	—	2	2	no	—
32	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₂₃	—	2	4	no	—
37	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	—
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₂₃	—	2	2	no	—
43	C₂×C₄²×C₂₃	—	2	4	no	—
44	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×#107	—	4	2	no	—
47	C₂₃×#107	C₂×C₄	1	4	yes	—
48	C₂₃×#106	—	2	4	no	—
49	C₂₃×C₄⋊C₈	—	2	2	no	—
50	C₂₃×#103	—	4	4	no	—
51	C₂₃×#103	C₂×C₄	1	8	yes	—
52	C₂₃×#97	—	2	4	no	—
53	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
54	C₂₃×#95	—	2	2	no	—
55	C₂₃×#95	—	2	4	no	—
56	C₂₃×#95	—	2	4	no	—
57	C₂₃×#95	—	2	2	no	—
58	C₂₃×#95	—	2	4	no	—
59	C₂₃×#95	D₄	1	2	yes	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×Q₈×C₂₃	—	2	2	no	—
62	C₂×Q₈×C₂₃	—	2	2	no	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	—	2	2	no	—
65	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₂₃	—	2	2	no	—
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	—	2	4	no	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
73	Q₁₆×C₂₃	—	2	8	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	2	1	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₂₃	#41	1	2	yes	—
82	C₂×C₈×C₂₃	#41	1	2	yes	—
83	C₄⋊C₄×C₂₃	—	2	8	no	—
84	C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₄⋊C₄×C₂₃	—	2	4	no	—
86	C₄⋊C₄×C₂₃	—	2	4	no	—
87	C₄⋊C₄×C₂₃	—	4	2	no	—
88	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₂₃	—	2	8	no	—
90	C₄²×C₂₃	—	2	2	no	—
91	C₄²×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	—	4	8	no	—
93	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₂₃	#41	1	2	yes	—
95	#22815	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₂₃	—	2	4	no	—
98	Q₈×C₂₃	—	2	4	no	—
99	Q₈×C₂₃	—	4	2	no	—
100	Q₈×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	2	4	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	—	4	4	no	—
112	C₂×C₄×C₂₃	—	4	4	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
122	C₈×C₂₃	—	2	2	no	—
123	C₈×C₂₃	—	4	2	no	—
124	C₈×C₂₃	#117	1	4	yes	—
125	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
126	#2356	C₂×C₂₃	1	2	yes	—
127	#1971	C₂×C₂₃	1	2	yes	—
128	#1928	C₂×C₂₃	1	4	yes	—
129	#1846	C₂×C₂₃	1	1	yes	characteristic
130	#1805	C₂×C₂₃	1	2	yes	—
131	#1776	C₂×C₂₃	1	2	yes	—
132	#1770	C₂×C₂₃	1	1	yes	characteristic
133	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₂₃	#547	1	1	yes	characteristic
135	C₂²×C₂₃	#550	1	1	yes	characteristic
136	C₂²×C₂₃	#575	1	2	yes	—
137	C₂²×C₂₃	#578	1	2	yes	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	2	2	no	—
140	C₄×C₂₃	—	4	4	no	—
141	C₄×C₂₃	—	4	4	no	—
142	C₄×C₂₃	—	4	4	no	—
143	C₄×C₂₃	—	8	4	no	—
144	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
145	C₄×C₂₃	#520	1	1	yes	characteristic
146	C₄×C₂₃	#555	1	1	yes	characteristic
147	C₄×C₂₃	#556	1	1	yes	characteristic
148	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
150	C₄×Q₁₆	C₄×C₂₃	1	8	yes	complemented
151	C₂×#107	C₂²×C₂₃	1	2	yes	—
152	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×#106	C₂²×C₂₃	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×#103	C₂²×C₂₃	1	4	yes	—
156	C₂×#103	C₂²×C₂₃	1	2	yes	—
157	C₂×#97	C₂²×C₂₃	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
159	#391	C₂²×C₂₃	1	4	yes	—
160	#387	—	2	4	no	—
161	#383	—	2	2	no	—
162	#382	C₂²×C₂₃	1	2	yes	—
163	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
164	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
165	#376	—	2	2	no	—
166	#340	C₂²×C₂₃	1	4	yes	—
167	#336	C₂²×C₂₃	1	2	yes	—
168	C₂×C₂₃	#2354	1	1	yes	characteristic
169	C₂×C₂₃	#2943	1	1	yes	characteristic
170	C₂×C₂₃	#2950	1	1	yes	characteristic
171	C₂×C₂₃	#3352	1	2	yes	—
172	C₂×C₂₃	#3402	1	2	yes	—
173	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
174	C₂×Q₁₆	—	2	2	no	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
177	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
178	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
185	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	4	no	—
188	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
189	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	#107	—	4	2	no	—
191	#107	C₂×C₄×C₂₃	1	4	yes	—
192	#106	—	2	4	no	—
193	C₄⋊C₈	—	2	2	no	—
194	#103	—	4	4	no	—
195	#103	C₂×C₄×C₂₃	1	8	yes	—
196	#97	—	2	4	no	—
197	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
198	#95	—	2	2	no	—
199	#95	—	2	4	no	—
200	#95	—	2	4	no	—
201	#95	—	2	2	no	—
202	#95	—	2	4	no	—
203	#95	D₄×C₂₃	1	2	yes	—
204	C₂₃	#22815	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
218	Q₁₆	—	2	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
225	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
226	C₂×C₈	#41×C₂₃	1	2	yes	—
227	C₂×C₈	#41×C₂₃	1	2	yes	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
234	C₄²	—	2	8	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	8	no	—
238	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
239	C₄²	#41×C₂₃	1	2	yes	—
240	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
259	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
260	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	C₂₃×#117	1	4	yes	—
269	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
270	C₂²	C₂₃×#547	1	1	yes	characteristic
271	C₂²	C₂₃×#550	1	1	yes	characteristic
272	C₂²	C₂₃×#575	1	2	yes	—
273	C₂²	C₂₃×#578	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	8	4	no	—
280	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
281	C₄	C₂₃×#520	1	1	yes	characteristic
282	C₄	C₂₃×#555	1	1	yes	characteristic
283	C₄	C₂₃×#556	1	1	yes	characteristic
284	C₂	C₂₃×#2354	1	1	yes	characteristic
285	C₂	C₂₃×#2943	1	1	yes	characteristic
286	C₂	C₂₃×#2950	1	1	yes	characteristic
287	C₂	C₂₃×#3352	1	2	yes	—
288	C₂	C₂₃×#3402	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₄×Q₁₆×C₂₃	C₄	semidirect
95. #22815	C₂₃	direct (N × H)
150. C₄×Q₁₆	C₄×C₂₃	semidirect
204. C₂₃	#22815	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×C₄×Q₁₆×C₂₃	2
C₂₃×#2356	2
C₂₃×#1971	2
C₂₃×#1928	2
C₂₃×#1846	2
C₂₃×#1805	2
C₂₃×#1776	2
C₂₃×#1770	2
#22815	23

Subgroups — C23×#22815 #19170441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22815 #19170441