Subgroups — C₂₃×#11364 #19158990

Order 5888.

606 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₂₃×#1740	C₂	1	2	yes	maximal
3	C₂₃×#1735	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1733	C₂	1	2	yes	maximal
5	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#117	C₂²	1	2	yes	—
9	C₂×C₂₃×#117	C₂²	1	2	yes	—
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂₃×#427	—	2	2	no	—
12	C₂₃×#427	—	2	2	no	—
13	C₂×C₂₃×#107	C₄	1	2	yes	complemented
14	C₂×C₂₃×#106	C₄	1	2	yes	complemented
15	C₂₃×#342	—	2	4	no	—
16	C₂₃×#340	—	2	2	no	—
17	C₂₃×#339	—	2	4	no	—
18	C₂₃×#336	—	2	2	no	—
19	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	—	2	1	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	1	no	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	D₄	1	2	yes	—
30	C₂²×C₈×C₂₃	D₄	1	2	yes	—
31	C₂₃×#117	—	2	2	no	—
32	C₂₃×#117	—	2	2	no	—
33	C₂₃×#117	—	2	2	no	—
34	C₂₃×#117	—	2	2	no	—
35	C₂₃×#117	—	2	2	no	—
36	C₂₃×#117	—	2	2	no	—
37	C₂₃×#117	—	2	2	no	—
38	C₂₃×#117	—	2	2	no	—
39	C₂₃×#117	—	2	2	no	—
40	C₂₃×#117	Q₈	1	4	yes	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
47	C₂×C₄²×C₂₃	—	2	2	no	—
48	C₂×C₄²×C₂₃	—	2	1	no	—
49	C₂×C₄²×C₂₃	—	2	2	no	—
50	C₂₃×#107	—	2	4	no	—
51	C₂₃×#106	—	2	4	no	—
52	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	4	2	no	—
56	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₂₃	#41	1	2	yes	—
62	C₂²×C₄×C₂₃	#41	1	2	yes	—
63	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	#41	1	2	yes	—
65	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
66	M₁₆×C₂₃	—	2	4	no	—
67	M₁₆×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	4	2	no	—
82	C₄⋊C₄×C₂₃	—	4	2	no	—
83	C₂²⋊C₄×C₂₃	—	2	4	no	—
84	C₂²⋊C₄×C₂₃	—	2	4	no	—
85	C₂²⋊C₄×C₂₃	—	4	2	no	—
86	C₂²⋊C₄×C₂₃	—	4	2	no	—
87	C₂²⋊C₄×C₂₃	—	4	2	no	—
88	C₄²×C₂₃	—	2	4	no	—
89	C₄²×C₂₃	—	4	2	no	—
90	C₄²×C₂₃	—	4	1	no	—
91	C₄²×C₂₃	—	4	2	no	—
92	C₄²×C₂₃	—	4	2	no	—
93	C₄²×C₂₃	—	4	1	no	—
94	#11364	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	—	4	2	no	—
96	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
97	C₂³×C₂₃	#121	1	1	yes	characteristic
98	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₂₃	#123	1	2	yes	—
100	C₂³×C₂₃	#127	1	1	yes	characteristic
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	8	4	no	—
106	C₂×C₄×C₂₃	—	8	4	no	—
107	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#117	1	2	yes	—
109	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#123	1	2	yes	—
117	C₂×C₄×C₂₃	#123	1	2	yes	—
118	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#125	1	2	yes	—
121	C₂×C₄×C₂₃	#125	1	2	yes	—
122	C₂×C₄×C₂₃	#126	1	2	yes	—
123	C₂×C₄×C₂₃	#126	1	2	yes	—
124	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
125	C₈×C₂₃	—	2	2	no	—
126	C₈×C₂₃	—	2	2	no	—
127	C₈×C₂₃	—	4	2	no	—
128	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
129	#1740	C₂×C₂₃	1	2	yes	—
130	#1735	C₂×C₂₃	1	1	yes	characteristic
131	#1733	C₂×C₂₃	1	2	yes	—
132	#1729	C₂×C₂₃	1	1	yes	characteristic
133	C₂²×C₂₃	—	2	2	no	—
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	4	2	no	—
137	C₂²×C₂₃	—	4	2	no	—
138	C₂²×C₂₃	#387	1	2	yes	—
139	C₂²×C₂₃	#390	1	1	yes	characteristic
140	C₂²×C₂₃	#397	1	1	yes	characteristic
141	C₂²×C₂₃	#398	1	1	yes	characteristic
142	C₂²×C₂₃	#399	1	2	yes	—
143	C₄×C₂₃	—	8	2	no	—
144	C₄×C₂₃	—	8	4	no	—
145	C₄×C₂₃	—	8	4	no	—
146	C₄×C₂₃	#381	1	1	yes	characteristic
147	C₄×C₂₃	#387	1	2	yes	—
148	C₄×C₂₃	#391	1	1	yes	characteristic
149	C₄×C₂₃	#397	1	1	yes	characteristic
150	C₄×C₂₃	#398	1	1	yes	characteristic
151	C₄×C₂₃	#400	1	2	yes	—
152	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
153	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×#117	C₂²×C₂₃	1	2	yes	—
155	C₂×#117	C₂²×C₂₃	1	2	yes	—
156	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
157	#427	—	2	2	no	—
158	#427	—	2	2	no	—
159	C₂×#107	C₄×C₂₃	1	2	yes	complemented
160	C₂×#106	C₄×C₂₃	1	2	yes	complemented
161	#342	—	2	4	no	—
162	#340	—	2	2	no	—
163	#339	—	2	4	no	—
164	#336	—	2	2	no	—
165	C₂×C₂₃	—	2	1	no	—
166	C₂×C₂₃	—	2	2	no	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	2	1	no	—
169	C₂×C₂₃	#1445	1	1	yes	characteristic
170	C₂×C₂₃	#1976	1	2	yes	—
171	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₂₃	1	2	yes	—
182	C₂²×C₈	D₄×C₂₃	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	Q₈×C₂₃	1	4	yes	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	2	no	—
202	#107	—	2	4	no	—
203	#106	—	2	4	no	—
204	C₂₃	#11364	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
214	C₂²×C₄	#41×C₂₃	1	2	yes	—
215	C₂²×C₄	#41×C₂₃	1	2	yes	—
216	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	#41×C₂₃	1	2	yes	—
218	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
230	C₂×C₈	C₂²⋊C₄×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	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
249	C₂³	C₂₃×#121	1	1	yes	characteristic
250	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
251	C₂³	C₂₃×#123	1	2	yes	—
252	C₂³	C₂₃×#127	1	1	yes	characteristic
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂₃×#117	1	2	yes	—
261	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
263	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
264	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
265	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
266	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
267	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
268	C₂×C₄	C₂₃×#123	1	2	yes	—
269	C₂×C₄	C₂₃×#123	1	2	yes	—
270	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
271	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
272	C₂×C₄	C₂₃×#125	1	2	yes	—
273	C₂×C₄	C₂₃×#125	1	2	yes	—
274	C₂×C₄	C₂₃×#126	1	2	yes	—
275	C₂×C₄	C₂₃×#126	1	2	yes	—
276	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₂₃×#387	1	2	yes	—
286	C₂²	C₂₃×#390	1	1	yes	characteristic
287	C₂²	C₂₃×#397	1	1	yes	characteristic
288	C₂²	C₂₃×#398	1	1	yes	characteristic
289	C₂²	C₂₃×#399	1	2	yes	—
290	C₄	—	8	2	no	—
291	C₄	—	8	4	no	—
292	C₄	—	8	4	no	—
293	C₄	C₂₃×#381	1	1	yes	characteristic
294	C₄	C₂₃×#387	1	2	yes	—
295	C₄	C₂₃×#391	1	1	yes	characteristic
296	C₄	C₂₃×#397	1	1	yes	characteristic
297	C₄	C₂₃×#398	1	1	yes	characteristic
298	C₄	C₂₃×#400	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	C₂₃×#1445	1	1	yes	characteristic
304	C₂	C₂₃×#1976	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#390
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₂₃×#107	C₄	semidirect
14. C₂×C₂₃×#106	C₄	semidirect
94. #11364	C₂₃	direct (N × H)
159. C₂×#107	C₄×C₂₃	semidirect
160. C₂×#106	C₄×C₂₃	semidirect
204. C₂₃	#11364	direct (N × H)

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
C₂₃×#1740	2
C₂₃×#1735	2
C₂₃×#1733	2
C₂₃×#1729	2
#11364	23

Subgroups — C23×#11364 #19158990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11364 #19158990