Subgroups — C₂₃×#8356 #19155982

Order 5888.

686 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3502	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1984	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1801	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1550	C₂	1	2	yes	maximal
5	C₂₃×#1493	C₂	1	2	yes	maximal
6	Q₈²×C₂₃	—	2	2	no	—
7	C₂₃×#557	—	2	2	no	—
8	C₂₃×#557	—	2	4	no	—
9	C₂₃×#544	—	2	1	no	—
10	C₂₃×#544	C₄	1	4	yes	—
11	C₂₃×#533	—	2	2	no	—
12	C₂₃×#533	C₂²	1	2	yes	—
13	C₂₃×#519	—	2	1	no	—
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#423	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#101	—	2	1	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#343	—	2	1	no	—
20	C₂₃×#336	—	2	2	no	—
21	C₂₃×#333	—	2	2	no	—
22	C₂₃×#332	—	2	2	no	—
23	C₂₃×Q₈⋊C₈	—	2	4	no	—
24	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	1	no	—
27	C₂²×C₈×C₂₃	—	2	1	no	—
28	C₂₃×C₄⋊Q₈	—	2	2	no	—
29	C₂₃×C₄⋊Q₈	—	2	4	no	—
30	C₂₃×C₄⋊Q₈	—	2	2	no	—
31	C₂₃×C₄⋊Q₈	—	2	8	no	—
32	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
33	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
34	C₂₃×#125	—	2	4	no	—
35	C₂₃×#125	—	2	4	no	—
36	C₂₃×#125	—	2	8	no	—
37	C₂₃×#125	—	2	2	no	—
38	C₂₃×C₂²⋊Q₈	—	2	4	no	—
39	C₂₃×C₂²⋊Q₈	—	2	2	no	—
40	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₂₃	—	2	4	no	—
42	C₄×Q₈×C₂₃	—	2	4	no	—
43	C₄×Q₈×C₂₃	—	2	4	no	—
44	C₄×Q₈×C₂₃	—	2	4	no	—
45	C₄×Q₈×C₂₃	—	2	2	no	—
46	C₄×Q₈×C₂₃	—	2	2	no	—
47	C₄×Q₈×C₂₃	—	2	2	no	—
48	C₄×Q₈×C₂₃	D₄	1	4	yes	—
49	C₂₃×#117	—	2	4	no	—
50	C₂₃×#117	—	2	2	no	—
51	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₂₃	D₄	1	2	yes	—
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×C₄⋊C₈	—	2	2	no	—
56	C₂₃×C₄⋊C₈	—	2	2	no	—
57	C₂₃×C₄⋊C₈	—	2	2	no	—
58	C₂₃×#101	—	4	2	no	—
59	C₄×C₈×C₂₃	—	2	2	no	—
60	C₂×Q₈×C₂₃	—	2	4	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	C₂²⋊C₄	1	4	yes	—
65	C₂×Q₈×C₂₃	SD₁₆	1	2	yes	—
66	C₂×Q₈×C₂₃	Q₁₆	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₂₃	—	4	2	no	—
73	M₁₆×C₂₃	—	4	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	—	4	2	no	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	—	2	8	no	—
84	C₄⋊C₄×C₂₃	—	2	4	no	—
85	C₄⋊C₄×C₂₃	—	2	4	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₂₃	—	2	4	no	—
93	C₄²×C₂₃	—	2	8	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	2	no	—
96	C₄²×C₂₃	—	2	4	no	—
97	C₄²×C₂₃	C₂×D₄	1	2	yes	—
98	C₄²×C₂₃	C₂×D₄	1	2	yes	—
99	#8356	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	Q₈×C₂₃	—	2	4	no	—
102	Q₈×C₂₃	—	2	4	no	—
103	Q₈×C₂₃	—	4	4	no	—
104	Q₈×C₂₃	—	4	2	no	—
105	Q₈×C₂₃	—	4	2	no	—
106	Q₈×C₂₃	—	8	4	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	4	1	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	#103	1	4	yes	—
117	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#120	1	2	yes	—
120	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₂₃	#137	1	2	yes	—
124	C₈×C₂₃	—	4	2	no	—
125	C₈×C₂₃	—	8	2	no	—
126	C₈×C₂₃	—	8	1	no	—
127	#3502	C₂×C₂₃	1	1	yes	characteristic
128	#1984	C₂×C₂₃	1	1	yes	characteristic
129	#1801	C₂×C₂₃	1	1	yes	characteristic
130	#1550	C₂×C₂₃	1	2	yes	—
131	#1493	C₂×C₂₃	1	2	yes	—
132	C₂²×C₂₃	—	2	1	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	#379	1	1	yes	characteristic
135	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
136	C₂²×C₂₃	#419	1	1	yes	characteristic
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	4	no	—
139	C₄×C₂₃	—	2	2	no	—
140	C₄×C₂₃	—	4	4	no	—
141	C₄×C₂₃	—	4	2	no	—
142	C₄×C₂₃	—	4	4	no	—
143	C₄×C₂₃	#448	1	2	yes	—
144	C₄×C₂₃	#451	1	2	yes	—
145	C₄×C₂₃	#454	1	1	yes	characteristic
146	C₄×C₂₃	#456	1	1	yes	characteristic
147	Q₈²	—	2	2	no	—
148	#557	—	2	2	no	—
149	#557	—	2	4	no	—
150	#544	—	2	1	no	—
151	#544	C₄×C₂₃	1	4	yes	—
152	#533	—	2	2	no	—
153	#533	C₂²×C₂₃	1	2	yes	—
154	#519	—	2	1	no	—
155	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
156	#423	C₂²×C₂₃	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×#101	—	2	1	no	—
159	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
160	#343	—	2	1	no	—
161	#336	—	2	2	no	—
162	#333	—	2	2	no	—
163	#332	—	2	2	no	—
164	Q₈⋊C₈	—	2	4	no	—
165	C₂×C₂₃	—	2	1	no	—
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	#1787	1	1	yes	characteristic
168	C₂×C₂₃	#1791	1	1	yes	characteristic
169	C₂×C₂₃	#1793	1	1	yes	characteristic
170	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	4	no	—
177	C₄⋊Q₈	—	2	8	no	—
178	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
179	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
180	#125	—	2	4	no	—
181	#125	—	2	4	no	—
182	#125	—	2	8	no	—
183	#125	—	2	2	no	—
184	C₂²⋊Q₈	—	2	4	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
187	C₄×Q₈	—	2	4	no	—
188	C₄×Q₈	—	2	4	no	—
189	C₄×Q₈	—	2	4	no	—
190	C₄×Q₈	—	2	4	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×Q₈	D₄×C₂₃	1	4	yes	—
195	#117	—	2	4	no	—
196	#117	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
198	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₂₃	1	2	yes	—
200	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	2	2	no	—
204	#101	—	4	2	no	—
205	C₄×C₈	—	2	2	no	—
206	C₂₃	#8356	1	1	yes	characteristic, direct product factor, Sylow
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
212	C₂×Q₈	SD₁₆×C₂₃	1	2	yes	—
213	C₂×Q₈	Q₁₆×C₂₃	1	2	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
215	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₄²	—	2	8	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	4	no	—
244	C₄²	C₂×D₄×C₂₃	1	2	yes	—
245	C₄²	C₂×D₄×C₂₃	1	2	yes	—
246	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	2	4	no	—
249	Q₈	—	4	4	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	8	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₂₃×#103	1	4	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
264	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
265	C₂×C₄	C₂₃×#120	1	2	yes	—
266	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
267	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
269	C₂×C₄	C₂₃×#137	1	2	yes	—
270	C₈	—	4	2	no	—
271	C₈	—	8	2	no	—
272	C₈	—	8	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	C₂₃×#379	1	1	yes	characteristic
276	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
277	C₂²	C₂₃×#419	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	C₂₃×#448	1	2	yes	—
285	C₄	C₂₃×#451	1	2	yes	—
286	C₄	C₂₃×#454	1	1	yes	characteristic
287	C₄	C₂₃×#456	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	C₂₃×#1787	1	1	yes	characteristic
291	C₂	C₂₃×#1791	1	1	yes	characteristic
292	C₂	C₂₃×#1793	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#379
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
99. #8356	C₂₃	direct (N × H)
206. C₂₃	#8356	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×#3502	2
C₂₃×#1984	2
C₂₃×#1801	2
C₂₃×#1550	2
C₂₃×#1493	2
#8356	23

Subgroups — C23×#8356 #19155982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8356 #19155982