Subgroups — C₂₃×#21151 #19168777

Order 5888.

1038 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	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2296	C₂	1	2	yes	maximal
5	C₂₃×#1942	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1941	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1735	C₂	1	2	yes	maximal
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	2	yes	—
11	C₂×C₂₃×#117	C₂²	1	2	yes	—
12	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#427	—	2	4	no	—
14	C₂₃×#427	C₄	1	8	yes	complemented
15	C₂×C₂₃×#107	C₂²	1	2	yes	—
16	C₂×C₂₃×#107	C₂²	1	2	yes	—
17	C₂×C₂₃×#106	C₂²	1	2	yes	—
18	C₂×C₂₃×#106	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	—	2	4	no	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#389	C₂²	1	8	yes	—
23	C₂₃×#385	—	2	4	no	—
24	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
25	C₂₃×#340	C₂²	1	8	yes	—
26	C₂³×C₄×C₂₃	D₄	1	2	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₂₃	D₄	1	4	yes	—
32	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
33	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂₃×#117	—	2	4	no	—
38	C₂₃×#117	—	2	4	no	—
39	C₂₃×#117	—	2	4	no	—
40	C₂₃×#117	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	—	2	2	no	—
47	C₂×C₄²×C₂₃	—	2	2	no	—
48	C₂×C₄²×C₂₃	—	2	2	no	—
49	C₂×C₄²×C₂₃	—	2	2	no	—
50	C₂×C₄²×C₂₃	D₄	1	2	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
52	C₂₃×#107	—	2	4	no	—
53	C₂₃×#107	C₂×C₄	1	8	yes	complemented
54	C₂₃×#106	—	2	4	no	—
55	C₂₃×#106	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₂₃	—	2	4	no	—
57	C₄×C₈×C₂₃	—	2	4	no	—
58	C₂₃×#95	—	2	8	no	—
59	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	—	2	8	no	—
62	C₂²×C₄×C₂₃	—	2	4	no	—
63	C₂²×C₄×C₂₃	—	2	4	no	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₂₃	#41	1	2	yes	—
71	C₂²×C₄×C₂₃	#41	1	2	yes	—
72	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₂₃	—	2	8	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
84	C₄⋊C₄×C₂₃	—	2	4	no	—
85	C₄⋊C₄×C₂₃	—	2	4	no	—
86	C₄⋊C₄×C₂₃	—	2	4	no	—
87	C₄⋊C₄×C₂₃	—	4	16	no	—
88	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
89	C₂²⋊C₄×C₂₃	—	2	8	no	—
90	C₂²⋊C₄×C₂₃	—	4	4	no	—
91	C₄²×C₂₃	—	2	8	no	—
92	C₄²×C₂₃	—	2	4	no	—
93	C₄²×C₂₃	—	2	2	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	4	4	no	—
96	#21151	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	2	4	no	—
98	C₂³×C₂₃	C₄×D₄	1	2	yes	—
99	C₂³×C₂₃	#121	1	2	yes	—
100	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	—	2	16	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	2	8	no	—
106	C₂×C₄×C₂₃	—	4	8	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	8	no	—
109	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₂₃	#121	1	2	yes	—
113	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₂₃	#125	1	4	yes	—
115	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₂₃	#135	1	8	yes	—
117	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
123	C₈×C₂₃	—	2	4	no	—
124	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
127	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
128	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
129	#2296	C₂×C₂₃	1	2	yes	—
130	#1942	C₂×C₂₃	1	4	yes	complemented
131	#1941	C₂×C₂₃	1	4	yes	complemented
132	#1735	C₂×C₂₃	1	2	yes	—
133	C₂²×C₂₃	—	2	4	no	—
134	C₂²×C₂₃	—	2	4	no	—
135	C₂²×C₂₃	—	2	4	no	—
136	C₂²×C₂₃	#389	1	4	yes	—
137	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
139	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₂²×C₂₃	C₂×#135	1	4	yes	—
141	C₄×C₂₃	—	2	8	no	—
142	C₄×C₂₃	—	4	8	no	—
143	C₄×C₂₃	—	8	4	no	—
144	C₄×C₂₃	#389	1	4	yes	—
145	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
147	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
148	C₄×C₂₃	C₂×#125	1	1	yes	characteristic
149	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×#117	C₂²×C₂₃	1	2	yes	—
152	C₂×#117	C₂²×C₂₃	1	2	yes	—
153	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
154	#427	—	2	4	no	—
155	#427	C₄×C₂₃	1	8	yes	complemented
156	C₂×#107	C₂²×C₂₃	1	2	yes	—
157	C₂×#107	C₂²×C₂₃	1	2	yes	—
158	C₂×#106	C₂²×C₂₃	1	2	yes	—
159	C₂×#106	C₂²×C₂₃	1	2	yes	—
160	C₂×C₄×C₈	—	2	4	no	—
161	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
163	#389	C₂²×C₂₃	1	8	yes	—
164	#385	—	2	4	no	—
165	C₄×C₄⋊C₄	—	2	4	no	—
166	#340	C₂²×C₂₃	1	8	yes	—
167	C₂×C₂₃	—	2	4	no	—
168	C₂×C₂₃	C₂×#389	1	1	yes	characteristic
169	C₂×C₂₃	C₄×#135	1	2	yes	—
170	C₂×C₂₃	#3050	1	2	yes	—
171	C₂×C₂₃	#3160	1	2	yes	—
172	C₂³×C₄	D₄×C₂₃	1	2	yes	—
173	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
177	C₂²×C₈	D₄×C₂₃	1	4	yes	—
178	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
179	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
180	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
181	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	#117	—	2	4	no	—
184	#117	—	2	4	no	—
185	#117	—	2	4	no	—
186	#117	C₂×C₄×C₂₃	1	8	yes	—
187	C₂×C₄⋊C₄	—	2	8	no	—
188	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
189	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	D₄×C₂₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
198	#107	—	2	4	no	—
199	#107	C₂×C₄×C₂₃	1	8	yes	complemented
200	#106	—	2	4	no	—
201	#106	C₂×C₄×C₂₃	1	8	yes	complemented
202	C₄×C₈	—	2	4	no	—
203	C₄×C₈	—	2	4	no	—
204	#95	—	2	8	no	—
205	C₂₃	#21151	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
213	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
217	C₂²×C₄	#41×C₂₃	1	2	yes	—
218	C₂²×C₄	#41×C₂₃	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	C₄⋊C₄×C₂₃	1	16	yes	complemented
224	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
225	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
227	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
228	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
229	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
230	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	16	no	—
235	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	C₄×D₄×C₂₃	1	2	yes	—
245	C₂³	C₂₃×#121	1	2	yes	—
246	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
247	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
256	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
257	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
258	C₂×C₄	C₂₃×#121	1	2	yes	—
259	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
260	C₂×C₄	C₂₃×#125	1	4	yes	—
261	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
262	C₂×C₄	C₂₃×#135	1	8	yes	—
263	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
268	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
269	C₈	—	2	4	no	—
270	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
271	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₂₃×#389	1	4	yes	—
276	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
277	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
278	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
279	C₂²	C₂×C₂₃×#135	1	4	yes	—
280	C₄	—	2	8	no	—
281	C₄	—	4	8	no	—
282	C₄	—	8	4	no	—
283	C₄	C₂₃×#389	1	4	yes	—
284	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
285	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
286	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
287	C₄	C₂×C₂₃×#125	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₂₃×#389	1	1	yes	characteristic
290	C₂	C₄×C₂₃×#135	1	2	yes	—
291	C₂	C₂₃×#3050	1	2	yes	—
292	C₂	C₂₃×#3160	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1942	C₂	semidirect
6. C₂₃×#1941	C₂	semidirect
14. C₂₃×#427	C₄	semidirect
53. C₂₃×#107	C₂×C₄	semidirect
55. C₂₃×#106	C₂×C₄	semidirect
76. C₂×C₈×C₂₃	C₄⋊C₄	semidirect
96. #21151	C₂₃	direct (N × H)
130. #1942	C₂×C₂₃	semidirect
131. #1941	C₂×C₂₃	semidirect
155. #427	C₄×C₂₃	semidirect
199. #107	C₂×C₄×C₂₃	semidirect
201. #106	C₂×C₄×C₂₃	semidirect
205. C₂₃	#21151	direct (N × H)
223. C₂×C₈	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#427	2
C₂²×C₄×C₈×C₂₃	2
C₂₃×#2296	2
C₂₃×#1942	2
C₂₃×#1941	2
C₂₃×#1735	2
#21151	23

Subgroups — C23×#21151 #19168777

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21151 #19168777