Subgroups — C₂₃×#2360 #15669979

Order 2944.

620 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	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#389	C₂	1	2	yes	maximal, complemented
7	C₂₃×#387	C₂	1	2	yes	maximal, complemented
8	C₂₃×#385	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₂²⋊Q₈	C₄	1	8	yes	complemented
16	C₄×Q₈×C₂₃	—	2	4	no	—
17	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	1	yes	characteristic
33	C₂₃×#95	C₂²	1	2	yes	—
34	C₂₃×#95	C₂²	1	1	yes	characteristic
35	C₂₃×#95	C₂²	1	2	yes	—
36	C₂₃×#95	C₂²	1	2	yes	—
37	C₂₃×#95	C₂²	1	2	yes	—
38	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₂₃	—	2	2	no	—
40	C₂×Q₈×C₂₃	—	2	2	no	—
41	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	—	2	4	no	—
45	C₂²×C₄×C₂₃	—	2	4	no	—
46	C₂²×C₄×C₂₃	—	4	2	no	—
47	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	4	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
63	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
64	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
65	C₂²⋊C₄×C₂₃	—	2	2	no	—
66	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
67	C₂²⋊C₄×C₂₃	Q₈	1	4	yes	complemented
68	C₄²×C₂₃	—	2	4	no	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	4	2	no	—
72	C₂³×C₂₃	—	2	1	no	—
73	C₂³×C₂₃	—	2	1	no	—
74	C₂³×C₂₃	—	2	1	no	—
75	C₂³×C₂₃	—	2	1	no	—
76	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
78	C₂³×C₂₃	#41	1	2	yes	—
79	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
80	Q₈×C₂₃	—	2	8	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	8	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	4	8	no	—
104	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
109	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
112	C₂×C₄×C₂₃	#41	1	2	yes	—
113	#2360	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	—	2	2	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	C₄×Q₈	1	4	yes	complemented
121	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
122	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
125	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
126	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	—	2	4	no	—
130	C₄×C₂₃	—	2	4	no	—
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	4	1	no	—
135	C₄×C₂₃	—	4	2	no	—
136	C₄×C₂₃	—	4	1	no	—
137	C₄×C₂₃	C₄×D₄	1	4	yes	—
138	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
140	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
141	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#389	C₂×C₂₃	1	2	yes	complemented
144	#387	C₂×C₂₃	1	2	yes	complemented
145	#385	C₂×C₂₃	1	2	yes	—
146	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
147	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
148	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
149	C₂×C₂₃	—	2	4	no	—
150	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₂×C₂₃	#520	1	1	yes	characteristic
153	C₂×C₂₃	#546	1	1	yes	characteristic
154	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
155	C₂×C₂₃	#552	1	1	yes	characteristic
156	C₂×C₂₃	#554	1	1	yes	characteristic
157	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
159	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
160	C₂²⋊Q₈	C₄×C₂₃	1	8	yes	complemented
161	C₄×Q₈	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
164	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
174	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
177	#95	C₂²×C₂₃	1	1	yes	characteristic
178	#95	C₂²×C₂₃	1	2	yes	—
179	#95	C₂²×C₂₃	1	1	yes	characteristic
180	#95	C₂²×C₂₃	1	2	yes	—
181	#95	C₂²×C₂₃	1	2	yes	—
182	#95	C₂²×C₂₃	1	2	yes	—
183	C₂₃	#2360	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
194	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
196	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
198	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
199	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
201	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	complemented
209	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	complemented
210	C₄⋊C₄	D₄×C₂₃	1	4	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
213	C₂²⋊C₄	Q₈×C₂₃	1	4	yes	complemented
214	C₄²	—	2	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	4	2	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
223	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
224	C₂³	#41×C₂₃	1	2	yes	—
225	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
251	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
252	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
253	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
255	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
256	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
257	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
258	C₂×C₄	#41×C₂₃	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	C₄×Q₈×C₂₃	1	4	yes	complemented
266	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
267	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
268	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
269	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
270	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
271	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
272	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	4	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	4	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	1	no	—
282	C₄	C₄×D₄×C₂₃	1	4	yes	—
283	C₂	—	2	4	no	—
284	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
285	C₂	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
286	C₂	C₂₃×#520	1	1	yes	characteristic
287	C₂	C₂₃×#546	1	1	yes	characteristic
288	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
289	C₂	C₂₃×#552	1	1	yes	characteristic
290	C₂	C₂₃×#554	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#391	C₂	semidirect
5. C₂₃×#389	C₂	semidirect
6. C₂₃×#389	C₂	semidirect
7. C₂₃×#387	C₂	semidirect
9. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
15. C₂₃×C₂²⋊Q₈	C₄	semidirect
41. C₂×Q₈×C₂₃	C₂×C₄	semidirect
62. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
63. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
67. C₂²⋊C₄×C₂₃	Q₈	semidirect
113. #2360	C₂₃	direct (N × H)
120. C₂²×C₂₃	C₄×Q₈	semidirect
139. C₂×C₄×Q₈	C₂×C₂₃	semidirect
141. #391	C₂×C₂₃	semidirect
142. #389	C₂×C₂₃	semidirect
143. #389	C₂×C₂₃	semidirect
144. #387	C₂×C₂₃	semidirect
146. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
160. C₂²⋊Q₈	C₄×C₂₃	semidirect
183. C₂₃	#2360	direct (N × H)
187. C₂×Q₈	C₂×C₄×C₂₃	semidirect
208. C₄⋊C₄	C₂×C₄×C₂₃	semidirect
209. C₄⋊C₄	C₂×C₄×C₂₃	semidirect
213. C₂²⋊C₄	Q₈×C₂₃	semidirect
265. C₂²	C₄×Q₈×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×C₂²⋊Q₈	2
C₂×C₄×Q₈×C₂₃	2
C₂²×C₄⋊C₄×C₂₃	2
C₂₃×#391	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#385	2
C₄×C₄⋊C₄×C₂₃	2
C₄×C₂²⋊C₄×C₂₃	2
#2360	23

Subgroups — C23×#2360 #15669979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2360 #15669979