Subgroups — C₂₃×#29210 #19176836

Order 5888.

670 conjugacy classes of subgroups.

The subgroup lattice has 284 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×#2059	C₂	1	4	yes	maximal
4	C₂₃×#1991	C₂	1	2	yes	maximal
5	C₂₃×#1948	C₂	1	2	yes	maximal
6	C₂₃×#1947	C₂	1	2	yes	maximal
7	C₂₃×#1846	C₂	1	2	yes	maximal
8	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#125	C₂²	1	2	yes	—
11	C₂×C₂₃×#107	C₂²	1	4	yes	—
12	C₂×C₂₃×#106	C₂²	1	4	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#389	—	2	2	no	—
17	C₂₃×#389	—	2	2	no	—
18	C₂₃×#389	C₂²	1	2	yes	—
19	C₂₃×#387	—	2	2	no	—
20	C₂₃×#384	C₂²	1	2	yes	—
21	C₂₃×#382	C₂²	1	2	yes	—
22	C₂₃×#382	C₂²	1	1	yes	characteristic
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×#336	C₂²	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
31	C₂₃×#125	—	2	2	no	—
32	C₂₃×#125	—	4	2	no	—
33	C₂₃×#125	—	4	2	no	—
34	C₂₃×#125	Q₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
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	4	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₂₃	—	2	2	no	—
44	C₂×C₄²×C₂₃	D₄	1	2	yes	—
45	C₂×C₄²×C₂₃	D₄	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×#107	—	4	4	no	—
50	C₂₃×#106	—	4	4	no	—
51	C₂₃×C₄⋊C₈	Q₈	1	4	yes	—
52	C₂₃×C₄⋊C₈	Q₈	1	4	yes	—
53	C₄×C₈×C₂₃	—	2	4	no	—
54	C₂₃×#95	—	2	2	no	—
55	C₂₃×#95	—	2	2	no	—
56	C₂₃×#95	—	2	2	no	—
57	C₂₃×#95	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	2	2	no	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
72	C₄⋊C₄×C₂₃	—	2	8	no	—
73	C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	—	4	4	no	—
79	C₄⋊C₄×C₂₃	—	4	4	no	—
80	C₄⋊C₄×C₂₃	—	4	8	no	—
81	C₄⋊C₄×C₂₃	—	4	4	no	—
82	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
83	C₄²×C₂₃	—	2	4	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₄²×C₂₃	—	2	1	no	—
86	C₄²×C₂₃	—	4	2	no	—
87	C₄²×C₂₃	—	4	4	no	—
88	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
89	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
91	C₄²×C₂₃	#41	1	2	yes	—
92	#29210	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	4	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₂₃	#135	1	4	yes	—
113	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₈×C₂₃	—	2	4	no	—
120	C₈×C₂₃	—	2	4	no	—
121	#2859	C₂×C₂₃	1	1	yes	characteristic
122	C₄×#125	C₂×C₂₃	1	1	yes	characteristic
123	#2059	C₂×C₂₃	1	4	yes	—
124	#1991	C₂×C₂₃	1	2	yes	—
125	#1948	C₂×C₂₃	1	2	yes	—
126	#1947	C₂×C₂₃	1	2	yes	—
127	#1846	C₂×C₂₃	1	2	yes	—
128	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
129	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₂²×C₂₃	#535	1	1	yes	characteristic
132	C₂²×C₂₃	C₂×#135	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	—	4	4	no	—
137	C₄×C₂₃	—	4	2	no	—
138	C₄×C₂₃	—	8	4	no	—
139	C₄×C₂₃	—	8	2	no	—
140	C₄×C₂₃	#499	1	4	yes	—
141	C₄×C₂₃	#554	1	1	yes	characteristic
142	C₄×C₂₃	#554	1	1	yes	characteristic
143	C₄×C₂₃	Q₈²	1	1	yes	characteristic
144	C₄×C₂₃	Q₈²	1	1	yes	characteristic
145	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×#125	C₂²×C₂₃	1	2	yes	—
147	C₂×#107	C₂²×C₂₃	1	4	yes	—
148	C₂×#106	C₂²×C₂₃	1	4	yes	—
149	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
152	#389	—	2	2	no	—
153	#389	—	2	2	no	—
154	#389	C₂²×C₂₃	1	2	yes	—
155	#387	—	2	2	no	—
156	#384	C₂²×C₂₃	1	2	yes	—
157	#382	C₂²×C₂₃	1	1	yes	characteristic
158	#382	C₂²×C₂₃	1	2	yes	—
159	C₄×C₄⋊C₄	—	2	2	no	—
160	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
161	C₄³	C₂²×C₂₃	1	1	yes	characteristic
162	#340	C₂²×C₂₃	1	4	yes	—
163	#340	C₂²×C₂₃	1	4	yes	—
164	#336	C₂²×C₂₃	1	2	yes	—
165	C₂×C₂₃	#2567	1	1	yes	characteristic
166	C₂×C₂₃	#3076	1	2	yes	—
167	C₂×C₂₃	C₂×#499	1	1	yes	characteristic
168	C₂×C₂₃	#3166	1	1	yes	characteristic
169	C₂×C₂₃	#3212	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	—	4	2	no	—
174	#125	—	4	2	no	—
175	#125	Q₈×C₂₃	1	4	yes	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	4	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	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	D₄×C₂₃	1	2	yes	—
186	C₂×C₄²	D₄×C₂₃	1	2	yes	—
187	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	#107	—	4	4	no	—
191	#106	—	4	4	no	—
192	C₄⋊C₈	Q₈×C₂₃	1	4	yes	—
193	C₄⋊C₈	Q₈×C₂₃	1	4	yes	—
194	C₄×C₈	—	2	4	no	—
195	#95	—	2	2	no	—
196	#95	—	2	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	4	no	—
199	C₂₃	#29210	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	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	C₂×Q₈×C₂₃	1	4	yes	—
213	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	8	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	4	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	2	no	—
229	C₄²	—	4	4	no	—
230	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
231	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
232	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
233	C₄²	#41×C₂₃	1	2	yes	—
234	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
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	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	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	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
251	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
252	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
253	C₂×C₄	C₂₃×#135	1	4	yes	—
254	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	2	4	no	—
262	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
263	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
264	C₂²	C₂₃×#535	1	1	yes	characteristic
265	C₂²	C₂×C₂₃×#135	1	2	yes	—
266	C₂²	C₂₃×#575	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	4	no	—
271	C₄	—	8	2	no	—
272	C₄	—	8	4	no	—
273	C₄	C₂₃×#499	1	4	yes	—
274	C₄	C₂₃×#554	1	1	yes	characteristic
275	C₄	C₂₃×#554	1	1	yes	characteristic
276	C₄	Q₈²×C₂₃	1	1	yes	characteristic
277	C₄	Q₈²×C₂₃	1	1	yes	characteristic
278	C₂	C₂₃×#2567	1	1	yes	characteristic
279	C₂	C₂₃×#3076	1	2	yes	—
280	C₂	C₂×C₂₃×#499	1	1	yes	characteristic
281	C₂	C₂₃×#3166	1	1	yes	characteristic
282	C₂	C₂₃×#3212	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. #29210	C₂₃	direct (N × H)
199. C₂₃	#29210	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×#2059	2
C₂₃×#1991	2
C₂₃×#1948	2
C₂₃×#1947	2
C₂₃×#1846	2
C₄×C₂₃×C₄⋊C₈	2
#29210	23

Subgroups — C23×#29210 #19176836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29210 #19176836