Subgroups — C₂₃×#29129 #19176755

Order 5888.

674 conjugacy classes of subgroups.

The subgroup lattice has 292 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	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2100	C₂	1	2	yes	maximal
4	C₂₃×#2043	C₂	1	2	yes	maximal
5	C₂₃×#1935	C₂	1	4	yes	maximal
6	C₂₃×#1805	C₂	1	2	yes	maximal
7	C₂₃×#1805	C₂	1	2	yes	maximal
8	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×#103	C₂²	1	4	yes	—
13	C₂×C₂₃×#103	C₂²	1	4	yes	—
14	C₂×C₂₃×#97	C₂²	1	2	yes	—
15	C₂₃×#391	C₂²	1	2	yes	—
16	C₂₃×#389	—	2	2	no	—
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×#384	C₂²	1	1	yes	characteristic
22	C₂₃×#382	C₂²	1	1	yes	characteristic
23	C₂₃×#382	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
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×Q₈×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
31	C₂₃×C₄⋊Q₈	—	2	4	no	—
32	C₄×Q₈×C₂₃	—	2	4	no	—
33	C₄×Q₈×C₂₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
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	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₂₃	—	2	2	no	—
43	C₂×C₄²×C₂₃	D₄	1	2	yes	—
44	C₂×C₄²×C₂₃	D₄	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×C₄⋊C₈	—	2	4	no	—
49	C₂₃×#103	—	2	8	no	—
50	C₂₃×#103	—	4	4	no	—
51	C₂₃×#97	—	4	2	no	—
52	C₂₃×#95	—	2	4	no	—
53	C₂₃×#95	—	2	1	no	—
54	C₂₃×#95	—	2	1	no	—
55	C₂₃×#95	—	2	1	no	—
56	C₂₃×#95	—	2	1	no	—
57	C₂₃×#95	—	2	2	no	—
58	C₂₃×#95	—	2	2	no	—
59	C₂×Q₈×C₂₃	—	2	2	no	—
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	#41	1	4	yes	—
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	—	2	2	no	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	2	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	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	—	2	8	no	—
78	C₄⋊C₄×C₂₃	—	2	4	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	—	4	4	no	—
84	C₄⋊C₄×C₂₃	—	4	4	no	—
85	C₄⋊C₄×C₂₃	—	4	4	no	—
86	C₄⋊C₄×C₂₃	—	4	4	no	—
87	C₄²×C₂₃	—	2	4	no	—
88	C₄²×C₂₃	—	2	2	no	—
89	C₄²×C₂₃	—	2	1	no	—
90	C₄²×C₂₃	—	2	1	no	—
91	C₄²×C₂₃	—	4	2	no	—
92	C₄²×C₂₃	—	4	4	no	—
93	C₄²×C₂₃	#41	1	4	yes	—
94	#29129	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	4	no	—
97	Q₈×C₂₃	—	2	4	no	—
98	Q₈×C₂₃	—	4	2	no	—
99	Q₈×C₂₃	—	4	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	—	2	4	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	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	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	4	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	#124	1	2	yes	—
119	C₂×C₄×C₂₃	#124	1	2	yes	—
120	C₂×C₄×C₂₃	#137	1	4	yes	—
121	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
125	C₈×C₂₃	—	4	4	no	—
126	#2864	C₂×C₂₃	1	1	yes	characteristic
127	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
128	#2100	C₂×C₂₃	1	2	yes	—
129	#2043	C₂×C₂₃	1	2	yes	—
130	#1935	C₂×C₂₃	1	4	yes	—
131	#1805	C₂×C₂₃	1	2	yes	—
132	#1805	C₂×C₂₃	1	2	yes	—
133	#1767	C₂×C₂₃	1	1	yes	characteristic
134	C₂²×C₂₃	#525	1	1	yes	characteristic
135	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
136	C₂²×C₂₃	#537	1	1	yes	characteristic
137	C₂²×C₂₃	C₂×#137	1	2	yes	—
138	C₂²×C₂₃	#575	1	2	yes	—
139	C₄×C₂₃	—	2	4	no	—
140	C₄×C₂₃	—	2	2	no	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	4	4	no	—
143	C₄×C₂₃	—	4	2	no	—
144	C₄×C₂₃	—	8	2	no	—
145	C₄×C₂₃	—	8	2	no	—
146	C₄×C₂₃	—	8	2	no	—
147	C₄×C₂₃	#552	1	2	yes	—
148	C₄×C₂₃	#553	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₂×#103	C₂²×C₂₃	1	4	yes	—
153	C₂×#103	C₂²×C₂₃	1	4	yes	—
154	C₂×#97	C₂²×C₂₃	1	2	yes	—
155	#391	C₂²×C₂₃	1	2	yes	—
156	#389	—	2	2	no	—
157	#389	C₂²×C₂₃	1	2	yes	—
158	#389	C₂²×C₂₃	1	2	yes	—
159	#387	—	2	2	no	—
160	#387	—	2	2	no	—
161	#384	C₂²×C₂₃	1	1	yes	characteristic
162	#382	C₂²×C₂₃	1	1	yes	characteristic
163	#382	C₂²×C₂₃	1	1	yes	characteristic
164	C₄×C₄⋊C₄	—	2	2	no	—
165	C₄³	C₂²×C₂₃	1	1	yes	characteristic
166	#340	C₂²×C₂₃	1	4	yes	—
167	#340	C₂²×C₂₃	1	4	yes	—
168	#336	C₂²×C₂₃	1	2	yes	—
169	C₂×C₂₃	#2562	1	1	yes	characteristic
170	C₂×C₂₃	#3117	1	2	yes	—
171	C₂×C₂₃	#3135	1	1	yes	characteristic
172	C₂×C₂₃	#3139	1	1	yes	characteristic
173	C₂×C₂₃	#3252	1	2	yes	—
174	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
175	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
176	C₄⋊Q₈	—	2	4	no	—
177	C₄×Q₈	—	2	4	no	—
178	C₄×Q₈	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	D₄×C₂₃	1	2	yes	—
189	C₂×C₄²	D₄×C₂₃	1	2	yes	—
190	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	4	no	—
194	#103	—	2	8	no	—
195	#103	—	4	4	no	—
196	#97	—	4	2	no	—
197	#95	—	2	4	no	—
198	#95	—	2	1	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	C₂₃	#29129	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	#41×C₂₃	1	4	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	2	no	—
238	C₄²	—	4	4	no	—
239	C₄²	#41×C₂₃	1	4	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	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂₃×#124	1	2	yes	—
264	C₂×C₄	C₂₃×#124	1	2	yes	—
265	C₂×C₄	C₂₃×#137	1	4	yes	—
266	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
267	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
268	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
269	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
270	C₈	—	4	4	no	—
271	C₂²	C₂₃×#525	1	1	yes	characteristic
272	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
273	C₂²	C₂₃×#537	1	1	yes	characteristic
274	C₂²	C₂×C₂₃×#137	1	2	yes	—
275	C₂²	C₂₃×#575	1	2	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	4	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	C₂₃×#552	1	2	yes	—
285	C₄	C₂₃×#553	1	2	yes	—
286	C₂	C₂₃×#2562	1	1	yes	characteristic
287	C₂	C₂₃×#3117	1	2	yes	—
288	C₂	C₂₃×#3135	1	1	yes	characteristic
289	C₂	C₂₃×#3139	1	1	yes	characteristic
290	C₂	C₂₃×#3252	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
94. #29129	C₂₃	direct (N × H)
204. C₂₃	#29129	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
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#2100	2
C₂₃×#2043	2
C₂₃×#1935	2
C₂₃×#1805	2
C₂₃×#1767	2
#29129	23

Subgroups — C23×#29129 #19176755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29129 #19176755