Subgroups — C₂₃×#29145 #19176771

Order 5888.

894 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2023	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1961	C₂	1	2	yes	maximal
5	C₂₃×#1836	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1804	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₂₃	—	2	2	no	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×#102	C₂²	1	8	yes	—
14	C₂×C₂₃×#97	C₂²	1	2	yes	—
15	C₂₃×#391	—	2	2	no	—
16	C₂₃×#391	—	2	2	no	—
17	C₂₃×#390	—	2	2	no	—
18	C₂₃×#390	C₂²	1	2	yes	—
19	C₂₃×#384	C₂²	1	1	yes	characteristic
20	C₂₃×#382	C₂²	1	2	yes	—
21	C₂₃×#382	C₂²	1	2	yes	—
22	C₄³×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#340	C₂²	1	8	yes	—
24	C₂₃×#336	C₂²	1	2	yes	—
25	C₂²×Q₈×C₂₃	—	2	2	no	—
26	C₂²×D₄×C₂₃	—	2	2	no	—
27	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
30	C₂₃×C₄⋊Q₈	—	4	4	no	—
31	C₂₃×#127	—	2	2	no	—
32	C₂₃×#127	Q₈	1	4	yes	complemented
33	C₄×D₄×C₂₃	—	2	8	no	—
34	C₄×D₄×C₂₃	—	4	8	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
41	C₂×C₄²×C₂₃	D₄	1	2	yes	—
42	C₂×C₄²×C₂₃	D₄	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂₃×C₄⋊C₈	—	2	4	no	—
47	C₂₃×#102	—	4	8	no	—
48	C₂₃×#97	—	2	4	no	—
49	C₂₃×#95	—	2	4	no	—
50	C₂₃×#95	—	2	2	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂⁴×C₂₃	—	2	2	no	—
53	C₂×Q₈×C₂₃	—	4	4	no	—
54	C₂×Q₈×C₂₃	—	4	4	no	—
55	C₂×Q₈×C₂₃	—	4	4	no	—
56	C₂×D₄×C₂₃	—	2	8	no	—
57	C₂×D₄×C₂₃	—	2	4	no	—
58	C₂×D₄×C₂₃	—	2	2	no	—
59	C₂×D₄×C₂₃	—	2	4	no	—
60	C₂×D₄×C₂₃	—	4	4	no	—
61	C₂×D₄×C₂₃	—	4	4	no	—
62	C₂×D₄×C₂₃	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	2	4	no	—
65	C₂²×C₄×C₂₃	—	4	8	no	—
66	C₂²×C₄×C₂₃	—	4	4	no	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	2	8	no	—
77	C₄⋊C₄×C₂₃	—	4	4	no	—
78	C₄⋊C₄×C₂₃	—	4	4	no	—
79	C₂²⋊C₄×C₂₃	—	4	4	no	—
80	C₂²⋊C₄×C₂₃	—	4	8	no	—
81	C₄²×C₂₃	—	2	4	no	—
82	C₄²×C₂₃	—	2	2	no	—
83	C₄²×C₂₃	—	4	2	no	—
84	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
85	C₄²×C₂₃	#41	1	2	yes	—
86	C₄²×C₂₃	#41	1	4	yes	—
87	#29145	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	2	4	no	—
89	C₂³×C₂₃	—	4	2	no	—
90	C₂³×C₂₃	—	4	2	no	—
91	C₂³×C₂₃	—	4	4	no	—
92	C₂³×C₂₃	—	4	4	no	—
93	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₂₃	—	8	8	no	—
95	D₄×C₂₃	—	2	8	no	—
96	D₄×C₂₃	—	4	16	no	—
97	D₄×C₂₃	—	4	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	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	—	8	8	no	—
110	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
111	C₂×C₄×C₂₃	#124	1	2	yes	—
112	C₂×C₄×C₂₃	#124	1	2	yes	—
113	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
114	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
116	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
118	C₈×C₂₃	—	4	4	no	—
119	#2862	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
121	#2023	C₂×C₂₃	1	4	yes	complemented
122	#1961	C₂×C₂₃	1	2	yes	—
123	#1836	C₂×C₂₃	1	2	yes	complemented
124	#1804	C₂×C₂₃	1	4	yes	complemented
125	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂²×C₂₃	—	4	4	no	—
127	C₂²×C₂₃	—	4	8	no	—
128	C₂²×C₂₃	—	8	2	no	—
129	C₂²×C₂₃	—	8	2	no	—
130	C₂²×C₂₃	—	8	4	no	—
131	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
133	C₂²×C₂₃	#535	1	1	yes	characteristic
134	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
135	C₂²×C₂₃	#575	1	2	yes	—
136	C₄×C₂₃	—	2	4	no	—
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	2	4	no	—
140	C₄×C₂₃	—	8	2	no	—
141	C₄×C₂₃	—	8	4	no	—
142	C₄×C₂₃	#554	1	2	yes	—
143	C₄×C₂₃	#555	1	2	yes	—
144	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
145	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₄×D₄	—	2	2	no	—
147	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
148	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
149	C₂×#102	C₂²×C₂₃	1	8	yes	—
150	C₂×#97	C₂²×C₂₃	1	2	yes	—
151	#391	—	2	2	no	—
152	#391	—	2	2	no	—
153	#390	—	2	2	no	—
154	#390	C₂²×C₂₃	1	2	yes	—
155	#384	C₂²×C₂₃	1	1	yes	characteristic
156	#382	C₂²×C₂₃	1	2	yes	—
157	#382	C₂²×C₂₃	1	2	yes	—
158	C₄³	C₂²×C₂₃	1	1	yes	characteristic
159	#340	C₂²×C₂₃	1	8	yes	—
160	#336	C₂²×C₂₃	1	2	yes	—
161	C₂×C₂₃	—	8	4	no	—
162	C₂×C₂₃	#2563	1	1	yes	characteristic
163	C₂×C₂₃	#3077	1	2	yes	—
164	C₂×C₂₃	#3135	1	1	yes	characteristic
165	C₂×C₂₃	#3138	1	1	yes	characteristic
166	C₂×C₂₃	#3213	1	2	yes	—
167	C₂²×Q₈	—	2	2	no	—
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
170	C₂³×C₄	—	2	2	no	—
171	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
172	C₄⋊Q₈	—	4	4	no	—
173	#127	—	2	2	no	—
174	#127	Q₈×C₂₃	1	4	yes	complemented
175	C₄×D₄	—	2	8	no	—
176	C₄×D₄	—	4	8	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
181	C₂×C₂²⋊C₄	—	2	4	no	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₄²	D₄×C₂₃	1	2	yes	—
184	C₂×C₄²	D₄×C₂₃	1	2	yes	—
185	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	#102	—	4	8	no	—
190	#97	—	2	4	no	—
191	#95	—	2	4	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	C₂₃	#29145	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	—	2	2	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	C₂×Q₈×C₂₃	1	4	yes	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₂²⋊C₄	—	4	8	no	—
224	C₄²	—	2	4	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	4	2	no	—
227	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
228	C₄²	#41×C₂₃	1	2	yes	—
229	C₄²	#41×C₂₃	1	4	yes	—
230	C₂³	—	2	4	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	4	2	no	—
235	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
236	Q₈	—	8	8	no	—
237	D₄	—	2	8	no	—
238	D₄	—	4	16	no	—
239	D₄	—	4	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
253	C₂×C₄	C₂₃×#124	1	2	yes	—
254	C₂×C₄	C₂₃×#124	1	2	yes	—
255	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
256	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
258	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
260	C₈	—	4	4	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	8	no	—
263	C₂²	—	8	4	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
268	C₂²	C₂₃×#535	1	1	yes	characteristic
269	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
270	C₂²	C₂₃×#575	1	2	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	8	4	no	—
276	C₄	—	8	2	no	—
277	C₄	C₂₃×#554	1	2	yes	—
278	C₄	C₂₃×#555	1	2	yes	—
279	C₂	—	8	4	no	—
280	C₂	C₂₃×#2563	1	1	yes	characteristic
281	C₂	C₂₃×#3077	1	2	yes	—
282	C₂	C₂₃×#3135	1	1	yes	characteristic
283	C₂	C₂₃×#3138	1	1	yes	characteristic
284	C₂	C₂₃×#3213	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
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
1. C₂₃×#2862	C₂	semidirect
3. C₂₃×#2023	C₂	semidirect
5. C₂₃×#1836	C₂	semidirect
6. C₂₃×#1804	C₂	semidirect
7. C₂₃×#1767	C₂	semidirect
32. C₂₃×#127	Q₈	semidirect
87. #29145	C₂₃	direct (N × H)
119. #2862	C₂×C₂₃	semidirect
121. #2023	C₂×C₂₃	semidirect
123. #1836	C₂×C₂₃	semidirect
124. #1804	C₂×C₂₃	semidirect
125. #1767	C₂×C₂₃	semidirect
174. #127	Q₈×C₂₃	semidirect
194. C₂₃	#29145	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×#2862	2
C₄×C₂₃×#127	2
C₂₃×#2023	2
C₂₃×#1961	2
C₂₃×#1836	2
C₂₃×#1804	2
C₂₃×#1767	2
#29145	23

Subgroups — C23×#29145 #19176771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29145 #19176771