Subgroups — C₂₃×#29227 #19176853

Order 5888.

626 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2101	C₂	1	4	yes	maximal
4	C₂₃×#1992	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1992	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1989	C₂	1	4	yes	maximal
7	C₂₃×#1836	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1836	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×#125	C₂²	1	2	yes	—
11	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
15	C₂₃×#389	—	2	2	no	—
16	C₂₃×#389	—	2	2	no	—
17	C₂₃×#389	C₂²	1	2	yes	—
18	C₂₃×#387	—	2	2	no	—
19	C₂₃×#384	C₂²	1	1	yes	characteristic
20	C₂₃×#384	C₂²	1	1	yes	characteristic
21	C₂₃×#382	C₂²	1	1	yes	characteristic
22	C₂₃×#382	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₄³×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#340	C₂²	1	4	yes	—
27	C₂₃×#340	C₂²	1	4	yes	—
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂₃×#340	C₂²	1	4	yes	—
30	C₂₃×#336	C₂²	1	2	yes	—
31	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂₃×#125	—	2	2	no	—
34	C₂₃×#125	—	2	8	no	—
35	C₂₃×#125	Q₈	1	4	yes	—
36	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
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	—	2	2	no	—
47	C₂×C₄²×C₂₃	D₄	1	2	yes	—
48	C₂×C₄²×C₂₃	D₄	1	2	yes	—
49	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×C₄⋊C₈	—	2	4	no	—
53	C₂₃×#97	—	2	2	no	—
54	C₂₃×#97	—	2	2	no	—
55	C₂₃×#95	—	2	2	no	—
56	C₂₃×#95	—	2	2	no	—
57	C₂₃×#95	—	2	2	no	—
58	C₂₃×#95	—	2	4	no	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	2	2	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₄⋊C₄×C₂₃	—	2	8	no	—
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	4	4	no	—
80	C₄⋊C₄×C₂₃	—	4	4	no	—
81	C₄⋊C₄×C₂₃	—	4	4	no	—
82	C₄⋊C₄×C₂₃	—	4	4	no	—
83	C₄⋊C₄×C₂₃	—	4	4	no	—
84	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
85	C₄²×C₂₃	—	2	2	no	—
86	C₄²×C₂₃	—	4	2	no	—
87	C₄²×C₂₃	—	4	2	no	—
88	C₄²×C₂₃	—	4	4	no	—
89	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₂₃	#41	1	2	yes	—
91	C₄²×C₂₃	#41	1	4	yes	—
92	#29227	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	—	4	4	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
112	C₂×C₄×C₂₃	#124	1	2	yes	—
113	C₂×C₄×C₂₃	#124	1	2	yes	—
114	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
117	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
118	C₈×C₂₃	—	4	2	no	—
119	C₈×C₂₃	—	4	2	no	—
120	#2859	C₂×C₂₃	1	1	yes	characteristic
121	C₄×#125	C₂×C₂₃	1	1	yes	characteristic
122	#2101	C₂×C₂₃	1	4	yes	—
123	#1992	C₂×C₂₃	1	1	yes	characteristic
124	#1992	C₂×C₂₃	1	1	yes	characteristic
125	#1989	C₂×C₂₃	1	4	yes	—
126	#1836	C₂×C₂₃	1	1	yes	characteristic
127	#1836	C₂×C₂₃	1	1	yes	characteristic
128	#1767	C₂×C₂₃	1	1	yes	characteristic
129	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
131	C₂²×C₂₃	#536	1	1	yes	characteristic
132	C₂²×C₂₃	#575	1	2	yes	—
133	C₂²×C₂₃	#575	1	2	yes	—
134	C₄×C₂₃	—	2	4	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	4	4	no	—
138	C₄×C₂₃	—	4	2	no	—
139	C₄×C₂₃	—	8	2	no	—
140	C₄×C₂₃	—	8	2	no	—
141	C₄×C₂₃	—	8	2	no	—
142	C₄×C₂₃	#556	1	2	yes	—
143	C₄×C₂₃	#557	1	2	yes	—
144	C₂×#125	C₂²×C₂₃	1	2	yes	—
145	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
147	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
149	#389	—	2	2	no	—
150	#389	—	2	2	no	—
151	#389	C₂²×C₂₃	1	2	yes	—
152	#387	—	2	2	no	—
153	#384	C₂²×C₂₃	1	1	yes	characteristic
154	#384	C₂²×C₂₃	1	1	yes	characteristic
155	#382	C₂²×C₂₃	1	1	yes	characteristic
156	#382	C₂²×C₂₃	1	2	yes	—
157	C₄×C₄⋊C₄	—	2	2	no	—
158	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
159	C₄³	C₂²×C₂₃	1	1	yes	characteristic
160	#340	C₂²×C₂₃	1	4	yes	—
161	#340	C₂²×C₂₃	1	4	yes	—
162	#340	C₂²×C₂₃	1	4	yes	—
163	#340	C₂²×C₂₃	1	4	yes	—
164	#336	C₂²×C₂₃	1	2	yes	—
165	C₂×C₂₃	#2568	1	1	yes	characteristic
166	C₂×C₂₃	#3080	1	2	yes	—
167	C₂×C₂₃	#3135	1	1	yes	characteristic
168	C₂×C₂₃	#3142	1	1	yes	characteristic
169	C₂×C₂₃	#3240	1	2	yes	—
170	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
171	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
172	#125	—	2	2	no	—
173	#125	—	2	8	no	—
174	#125	Q₈×C₂₃	1	4	yes	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
182	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	D₄×C₂₃	1	2	yes	—
187	C₂×C₄²	D₄×C₂₃	1	2	yes	—
188	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
191	C₄⋊C₈	—	2	4	no	—
192	#97	—	2	2	no	—
193	#97	—	2	2	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	4	no	—
198	C₂₃	#29227	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	2	no	—
227	C₄²	—	4	2	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
230	C₄²	#41×C₂₃	1	2	yes	—
231	C₄²	#41×C₂₃	1	4	yes	—
232	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
251	C₂×C₄	C₂₃×#124	1	2	yes	—
252	C₂×C₄	C₂₃×#124	1	2	yes	—
253	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
256	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
260	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
261	C₂²	C₂₃×#536	1	1	yes	characteristic
262	C₂²	C₂₃×#575	1	2	yes	—
263	C₂²	C₂₃×#575	1	2	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	4	no	—
269	C₄	—	8	2	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	2	no	—
272	C₄	C₂₃×#556	1	2	yes	—
273	C₄	C₂₃×#557	1	2	yes	—
274	C₂	C₂₃×#2568	1	1	yes	characteristic
275	C₂	C₂₃×#3080	1	2	yes	—
276	C₂	C₂₃×#3135	1	1	yes	characteristic
277	C₂	C₂₃×#3142	1	1	yes	characteristic
278	C₂	C₂₃×#3240	1	2	yes	—

	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
92. #29227	C₂₃	direct (N × H)
198. C₂₃	#29227	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×#2859	2
C₄×C₂₃×#125	2
C₂₃×#2101	2
C₂₃×#1992	2
C₂₃×#1989	2
C₂₃×#1836	2
C₂₃×#1767	2
#29227	23

Subgroups — C23×#29227 #19176853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29227 #19176853