Subgroups — C₂₃×#11712 #19159338

Order 5888.

440 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#467	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1350	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₂₃	—	2	1	no	—
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
9	C₂×C₂₃×M₃₂	—	2	1	no	—
10	C₂₃×#467	—	2	1	no	—
11	C₂₃×#467	—	2	1	no	—
12	C₂₃×#467	C₄	1	2	yes	—
13	C₂₃×#467	C₂²	1	2	yes	—
14	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
15	C₂×C₂₃×#103	—	2	1	no	—
16	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
18	C₂₃×#341	—	2	1	no	—
19	C₂₃×#341	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₂₃	—	2	1	no	—
23	C₂×Q₁₆×C₂₃	—	4	2	no	—
24	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₂₃	—	2	4	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	4	2	no	—
28	C₂×M₁₆×C₂₃	—	4	1	no	—
29	C₂²×C₈×C₂₃	—	4	2	no	—
30	C₂²×C₈×C₂₃	—	4	1	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂₃×C₂²⋊Q₈	—	2	2	no	—
38	C₂₃×C₂²⋊Q₈	—	2	2	no	—
39	C₂₃×C₂²⋊Q₈	—	4	1	no	—
40	C₂₃×C₂²⋊Q₈	—	4	1	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
43	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
44	C₂₃×M₃₂	—	4	2	no	—
45	C₂×C₁₆×C₂₃	—	2	2	no	—
46	C₂₃×#107	—	2	1	no	—
47	C₂₃×#107	C₂×C₄	1	2	yes	—
48	C₂₃×#103	—	2	2	no	—
49	C₂₃×#103	—	2	2	no	—
50	C₂⁴×C₂₃	SD₁₆	1	1	yes	characteristic
51	C₂×Q₈×C₂₃	—	2	2	no	—
52	C₂×Q₈×C₂₃	—	4	1	no	—
53	C₂×Q₈×C₂₃	—	4	2	no	—
54	C₂×Q₈×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	4	1	no	—
58	C₂²×C₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₂₃	—	4	2	no	—
66	Q₁₆×C₂₃	—	4	2	no	—
67	M₁₆×C₂₃	—	2	4	no	—
68	M₁₆×C₂₃	—	4	4	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	—	4	4	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	4	1	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	—	4	2	no	—
86	C₂²⋊C₄×C₂₃	—	4	2	no	—
87	C₂²⋊C₄×C₂₃	—	4	2	no	—
88	C₁₆×C₂₃	—	4	2	no	—
89	#11712	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	4	1	no	—
93	C₂³×C₂₃	#102	1	1	yes	characteristic
94	C₂³×C₂₃	#102	1	1	yes	characteristic
95	C₂³×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
96	Q₈×C₂₃	—	4	2	no	—
97	Q₈×C₂₃	—	4	2	no	—
98	Q₈×C₂₃	—	8	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	8	2	no	—
109	C₂×C₄×C₂₃	—	8	2	no	—
110	C₂×C₄×C₂₃	#99	1	2	yes	—
111	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#104	1	2	yes	—
114	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
116	C₈×C₂₃	—	2	1	no	—
117	C₈×C₂₃	—	2	1	no	—
118	C₈×C₂₃	—	2	1	no	—
119	C₈×C₂₃	—	2	1	no	—
120	C₈×C₂₃	—	4	4	no	—
121	C₂×#467	C₂×C₂₃	1	1	yes	characteristic
122	#2112	C₂×C₂₃	1	1	yes	characteristic
123	C₂×#341	C₂×C₂₃	1	1	yes	characteristic
124	#1350	C₂×C₂₃	1	4	yes	complemented
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	2	1	no	—
128	C₂²×C₂₃	—	2	1	no	—
129	C₂²×C₂₃	—	2	1	no	—
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	—	4	4	no	—
133	C₂²×C₂₃	#327	1	2	yes	—
134	C₂²×C₂₃	#362	1	2	yes	—
135	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	8	2	no	—
139	C₄×C₂₃	—	8	2	no	—
140	C₄×C₂₃	#327	1	2	yes	—
141	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
142	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
143	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
144	C₂²×M₁₆	—	2	1	no	—
145	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₂²⋊Q₈	—	2	1	no	—
147	C₂×M₃₂	—	2	1	no	—
148	#467	—	2	1	no	—
149	#467	—	2	1	no	—
150	#467	C₄×C₂₃	1	2	yes	—
151	#467	C₂²×C₂₃	1	2	yes	—
152	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
153	C₂×#103	—	2	1	no	—
154	C₂²⋊C₁₆	—	2	1	no	—
155	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
156	#341	—	2	1	no	—
157	#341	C₂²×C₂₃	1	2	yes	—
158	C₂×C₂₃	—	2	2	no	—
159	C₂×C₂₃	—	2	2	no	—
160	C₂×C₂₃	—	4	1	no	—
161	C₂×C₂₃	C₂×#327	1	1	yes	characteristic
162	C₂×C₂₃	#2143	1	1	yes	characteristic
163	C₂×C₂₃	C₂×#362	1	1	yes	characteristic
164	C₂²×Q₈	—	2	1	no	—
165	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	—	4	2	no	—
168	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	4	2	no	—
172	C₂×M₁₆	—	4	1	no	—
173	C₂²×C₈	—	4	2	no	—
174	C₂²×C₈	—	4	1	no	—
175	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂²⋊Q₈	—	2	2	no	—
183	C₂²⋊Q₈	—	4	1	no	—
184	C₂²⋊Q₈	—	4	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	4	1	no	—
187	C₂×C₂²⋊C₄	—	4	1	no	—
188	M₃₂	—	4	2	no	—
189	C₂×C₁₆	—	2	2	no	—
190	#107	—	2	1	no	—
191	#107	C₂×C₄×C₂₃	1	2	yes	—
192	#103	—	2	2	no	—
193	#103	—	2	2	no	—
194	C₂₃	#11712	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	SD₁₆×C₂₃	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	Q₁₆	—	4	2	no	—
211	Q₁₆	—	4	2	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	4	no	—
214	M₁₆	—	4	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	C₂₃×#102	1	1	yes	characteristic
238	C₂³	C₂₃×#102	1	1	yes	characteristic
239	C₂³	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
240	Q₈	—	4	2	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	8	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₂₃×#99	1	2	yes	—
255	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
256	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
257	C₂×C₄	C₂₃×#104	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂₃×#327	1	2	yes	—
274	C₂²	C₂₃×#362	1	2	yes	—
275	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	C₂₃×#327	1	2	yes	—
281	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
282	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	—	4	1	no	—
286	C₂	C₂×C₂₃×#327	1	1	yes	characteristic
287	C₂	C₂₃×#2143	1	1	yes	characteristic
288	C₂	C₂×C₂₃×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1350	C₂	semidirect
89. #11712	C₂₃	direct (N × H)
124. #1350	C₂×C₂₃	semidirect
194. C₂₃	#11712	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×#467	2
C₂₃×#2112	2
C₂×C₂₃×#341	2
C₂₃×#1350	2
#11712	23

Subgroups — C23×#11712 #19159338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11712 #19159338