Subgroups — C₂₃×#3116 #15670735

Order 2944.

386 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#485	C₂	1	2	yes	maximal, complemented
4	C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#452	C₂	1	2	yes	maximal
9	C₂₃×#441	C₂	1	2	yes	maximal, complemented
10	C₂₃×#440	C₂	1	2	yes	maximal, complemented
11	C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#125	—	2	1	no	—
18	C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₄×Q₈×C₂₃	—	2	1	no	—
21	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
22	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₂₃	—	2	1	no	—
25	C₄×D₄×C₂₃	—	2	1	no	—
26	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#117	—	2	1	no	—
29	C₂₃×#117	—	2	1	no	—
30	C₂₃×#117	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	—	2	1	no	—
32	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#107	C₂²	1	2	yes	—
34	C₂₃×#106	C₂²	1	2	yes	—
35	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂₃×#103	C₂²	1	2	yes	—
38	C₂₃×#103	C₂²	1	2	yes	—
39	C₂₃×#103	C₂²	1	2	yes	—
40	C₂₃×#102	C₂²	1	2	yes	—
41	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₂₃×#97	C₂²	1	2	yes	—
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	—	2	1	no	—
46	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	Q₁₆×C₂₃	—	2	4	no	—
57	SD₁₆×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	C₂³	1	2	yes	—
59	C₂×C₈×C₂₃	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	—	2	2	no	—
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	—	2	1	no	—
75	C₄²×C₂₃	—	2	1	no	—
76	C₄²×C₂₃	—	2	1	no	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	D₄	1	1	yes	characteristic
79	C₄²×C₂₃	D₄	1	1	yes	characteristic
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₂³×C₂₃	—	2	1	no	—
83	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₂₃	—	2	1	no	—
85	Q₈×C₂₃	—	2	2	no	—
86	Q₈×C₂₃	—	2	2	no	—
87	Q₈×C₂₃	—	4	2	no	—
88	Q₈×C₂₃	#41	1	2	yes	—
89	D₄×C₂₃	—	2	4	no	—
90	D₄×C₂₃	—	2	1	no	—
91	D₄×C₂₃	#41	1	2	yes	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	1	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	2	no	—
101	C₂×C₄×C₂₃	—	4	1	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
110	C₈×C₂₃	—	2	2	no	—
111	C₈×C₂₃	—	2	2	no	—
112	#3116	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	4	1	no	—
118	C₂²×C₂₃	—	4	1	no	—
119	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	2	1	no	—
125	C₄×C₂₃	—	4	2	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	#137	1	2	yes	—
130	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
131	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	#533	C₂×C₂₃	1	1	yes	characteristic, complemented
133	C₄×#41	C₂×C₂₃	1	1	yes	characteristic
134	#485	C₂×C₂₃	1	2	yes	complemented
135	#477	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#476	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#462	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#461	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#452	C₂×C₂₃	1	2	yes	—
140	#441	C₂×C₂₃	1	2	yes	complemented
141	#440	C₂×C₂₃	1	2	yes	complemented
142	#432	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂×C₂₃	—	4	1	no	—
144	C₂×C₂₃	—	4	2	no	—
145	C₂×C₂₃	#525	1	1	yes	characteristic
146	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
147	C₂×C₂₃	#575	1	1	yes	characteristic
148	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
150	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
151	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
153	#125	—	2	1	no	—
154	C₂²⋊Q₈	—	2	1	no	—
155	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
158	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
159	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
163	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
164	#117	—	2	1	no	—
165	#117	—	2	1	no	—
166	#117	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
169	#107	C₂²×C₂₃	1	2	yes	—
170	#106	C₂²×C₂₃	1	2	yes	—
171	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
172	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
173	#103	C₂²×C₂₃	1	2	yes	—
174	#103	C₂²×C₂₃	1	2	yes	—
175	#103	C₂²×C₂₃	1	2	yes	—
176	#102	C₂²×C₂₃	1	2	yes	—
177	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
178	#97	C₂²×C₂₃	1	2	yes	—
179	C₂₃	#3116	1	1	yes	characteristic, direct product factor, Sylow
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
184	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
193	Q₁₆	—	2	4	no	—
194	SD₁₆	—	2	4	no	—
195	C₂×C₈	C₂³×C₂₃	1	2	yes	—
196	C₂×C₈	C₂³×C₂₃	1	2	yes	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
205	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
206	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	2	no	—
215	C₄²	D₄×C₂₃	1	1	yes	characteristic
216	C₄²	D₄×C₂₃	1	1	yes	characteristic
217	C₄²	C₂³×C₂₃	1	1	yes	characteristic
218	C₄²	C₂³×C₂₃	1	1	yes	characteristic
219	C₂³	—	2	1	no	—
220	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
221	Q₈	—	2	1	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	4	2	no	—
225	Q₈	#41×C₂₃	1	2	yes	—
226	D₄	—	2	4	no	—
227	D₄	—	2	1	no	—
228	D₄	#41×C₂₃	1	2	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	C₂₃×#137	1	2	yes	—
266	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
267	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	—	4	2	no	—
270	C₂	C₂₃×#525	1	1	yes	characteristic
271	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
272	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₂₃×#533	C₂	semidirect
3. C₂₃×#485	C₂	semidirect
4. C₂₃×#477	C₂	semidirect
5. C₂₃×#476	C₂	semidirect
6. C₂₃×#462	C₂	semidirect
7. C₂₃×#461	C₂	semidirect
9. C₂₃×#441	C₂	semidirect
10. C₂₃×#440	C₂	semidirect
11. C₂₃×#432	C₂	semidirect
112. #3116	C₂₃	direct (N × H)
132. #533	C₂×C₂₃	semidirect
134. #485	C₂×C₂₃	semidirect
135. #477	C₂×C₂₃	semidirect
136. #476	C₂×C₂₃	semidirect
137. #462	C₂×C₂₃	semidirect
138. #461	C₂×C₂₃	semidirect
140. #441	C₂×C₂₃	semidirect
141. #440	C₂×C₂₃	semidirect
142. #432	C₂×C₂₃	semidirect
179. C₂₃	#3116	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×#533	2
C₄×#41×C₂₃	2
C₂₃×#485	2
C₂₃×#477	2
C₂₃×#476	2
C₂₃×#462	2
C₂₃×#461	2
C₂₃×#452	2
C₂₃×#441	2
C₂₃×#440	2
C₂₃×#432	2
#3116	23

Subgroups — C23×#3116 #15670735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3116 #15670735