Subgroups — C₂₃×#12833 #19160459

Order 5888.

358 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3407	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1815	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1814	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1386	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
9	C₂₃×#504	—	2	2	no	—
10	C₂₃×#504	C₄	1	4	yes	complemented
11	C₂×C₂₃×M₃₂	—	2	2	no	—
12	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
16	C₂₃×#427	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
18	C₂₃×#421	—	2	1	no	—
19	C₂₃×#417	—	2	1	no	—
20	C₂₃×#365	—	2	2	no	—
21	C₂₃×#365	—	2	2	no	—
22	C₂₃×#344	—	2	1	no	—
23	C₂₃×#343	—	2	1	no	—
24	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
25	C₂₃×#131	—	2	1	no	—
26	C₂₃×#131	—	2	1	no	—
27	C₂₃×#131	C₂×C₄	1	2	yes	—
28	C₂₃×#131	Q₈	1	2	yes	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂×M₁₆×C₂₃	—	2	1	no	—
31	C₂×M₁₆×C₂₃	—	2	1	no	—
32	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	—	2	1	no	—
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#117	—	2	1	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
37	C₂₃×M₃₂	—	2	1	no	—
38	C₂₃×M₃₂	—	2	4	no	—
39	C₂₃×M₃₂	C₂×C₄	1	2	yes	—
40	C₂₃×M₃₂	D₄	1	2	yes	—
41	C₂₃×M₃₂	D₄	1	2	yes	—
42	C₂×C₁₆×C₂₃	—	2	1	no	—
43	C₂×C₁₆×C₂₃	—	2	4	no	—
44	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
45	C₂₃×#108	—	2	1	no	—
46	C₂₃×#108	—	2	1	no	—
47	C₂₃×#108	—	2	1	no	—
48	C₂₃×#108	—	2	1	no	—
49	C₂₃×#107	—	2	1	no	—
50	C₂₃×#106	—	2	1	no	—
51	C₂₃×#106	—	2	1	no	—
52	C₂₃×#106	—	2	1	no	—
53	C₂₃×#104	—	4	2	no	—
54	C₂₃×#103	—	4	1	no	—
55	C₂₃×#102	—	4	1	no	—
56	#41×C₂₃	—	2	1	no	—
57	#41×C₂₃	—	2	1	no	—
58	#41×C₂₃	C₄⋊C₄	1	2	yes	—
59	#41×C₂₃	SD₁₆	1	2	yes	—
60	C₂×Q₈×C₂₃	SD₁₆	1	1	yes	characteristic
61	C₂×D₄×C₂₃	—	2	1	no	—
62	C₂×D₄×C₂₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	M₁₆×C₂₃	—	2	1	no	—
67	M₁₆×C₂₃	—	2	2	no	—
68	M₁₆×C₂₃	—	2	2	no	—
69	M₁₆×C₂₃	—	4	1	no	—
70	M₁₆×C₂₃	—	4	1	no	—
71	M₁₆×C₂₃	—	4	1	no	—
72	M₁₆×C₂₃	—	4	1	no	—
73	M₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₂₃	—	4	1	no	—
86	C₄⋊C₄×C₂₃	—	4	1	no	—
87	C₄⋊C₄×C₂₃	—	4	1	no	—
88	C₂²⋊C₄×C₂₃	—	4	1	no	—
89	C₄²×C₂₃	—	4	1	no	—
90	C₁₆×C₂₃	—	2	2	no	—
91	C₁₆×C₂₃	C₂²⋊C₄	1	4	yes	—
92	#12833	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	1	no	—
94	C₂³×C₂₃	#135	1	1	yes	characteristic
95	Q₈×C₂₃	—	2	1	no	—
96	Q₈×C₂₃	#106	1	2	yes	—
97	D₄×C₂₃	—	2	2	no	—
98	D₄×C₂₃	—	2	2	no	—
99	D₄×C₂₃	—	2	1	no	—
100	D₄×C₂₃	#106	1	2	yes	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	8	1	no	—
105	C₂×C₄×C₂₃	—	8	1	no	—
106	C₂×C₄×C₂₃	—	16	1	no	—
107	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
113	C₈×C₂₃	—	2	2	no	—
114	C₈×C₂₃	—	8	1	no	—
115	C₈×C₂₃	—	8	1	no	—
116	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₈×C₂₃	#117	1	1	yes	characteristic
118	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
120	#3407	C₂×C₂₃	1	1	yes	characteristic
121	#2155	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂×#365	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#1815	C₂×C₂₃	1	1	yes	characteristic
124	#1814	C₂×C₂₃	1	1	yes	characteristic
125	#1388	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#1386	C₂×C₂₃	1	1	yes	characteristic, complemented
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	4	1	no	—
129	C₂²×C₂₃	—	8	1	no	—
130	C₂²×C₂₃	#427	1	1	yes	characteristic
131	C₂²×C₂₃	#465	1	1	yes	characteristic
132	C₂²×C₂₃	#465	1	1	yes	characteristic
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	16	1	no	—
135	C₄×C₂₃	—	16	1	no	—
136	C₄×C₂₃	#380	1	1	yes	characteristic
137	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
138	C₄×C₂₃	#465	1	1	yes	characteristic
139	C₄×C₂₃	#465	1	1	yes	characteristic
140	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
141	#504	—	2	2	no	—
142	#504	C₄×C₂₃	1	4	yes	complemented
143	C₂×M₃₂	—	2	2	no	—
144	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
148	#427	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
150	#421	—	2	1	no	—
151	#417	—	2	1	no	—
152	#365	—	2	2	no	—
153	#365	—	2	2	no	—
154	#344	—	2	1	no	—
155	#343	—	2	1	no	—
156	C₂×C₂₃	—	2	1	no	—
157	C₂×C₂₃	—	2	1	no	—
158	C₂×C₂₃	—	2	1	no	—
159	C₂×C₂₃	—	4	2	no	—
160	C₂×C₂₃	#1807	1	1	yes	characteristic
161	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
162	#131	—	2	1	no	—
163	#131	—	2	1	no	—
164	#131	C₂×C₄×C₂₃	1	2	yes	—
165	#131	Q₈×C₂₃	1	2	yes	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
172	#117	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	4	no	—
176	M₃₂	C₂×C₄×C₂₃	1	2	yes	—
177	M₃₂	D₄×C₂₃	1	2	yes	—
178	M₃₂	D₄×C₂₃	1	2	yes	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
182	#108	—	2	1	no	—
183	#108	—	2	1	no	—
184	#108	—	2	1	no	—
185	#108	—	2	1	no	—
186	#107	—	2	1	no	—
187	#106	—	2	1	no	—
188	#106	—	2	1	no	—
189	#106	—	2	1	no	—
190	#104	—	4	2	no	—
191	#103	—	4	1	no	—
192	#102	—	4	1	no	—
193	C₂₃	#12833	1	1	yes	characteristic, direct product factor, Sylow
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	#41	C₄⋊C₄×C₂₃	1	2	yes	—
197	#41	SD₁₆×C₂₃	1	2	yes	—
198	C₂×Q₈	SD₁₆×C₂₃	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	SD₁₆×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
221	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
222	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₂²⋊C₄	—	4	1	no	—
227	C₄²	—	4	1	no	—
228	C₁₆	—	2	2	no	—
229	C₁₆	C₂²⋊C₄×C₂₃	1	4	yes	—
230	C₂³	—	2	1	no	—
231	C₂³	C₂₃×#135	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	C₂₃×#106	1	2	yes	—
234	D₄	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	2	1	no	—
237	D₄	C₂₃×#106	1	2	yes	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	16	1	no	—
244	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
246	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
247	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	1	no	—
252	C₈	—	8	1	no	—
253	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
254	C₈	C₂₃×#117	1	1	yes	characteristic
255	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
256	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₂₃×#427	1	1	yes	characteristic
261	C₂²	C₂₃×#465	1	1	yes	characteristic
262	C₂²	C₂₃×#465	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	16	1	no	—
265	C₄	—	16	1	no	—
266	C₄	C₂₃×#380	1	1	yes	characteristic
267	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
268	C₄	C₂₃×#465	1	1	yes	characteristic
269	C₄	C₂₃×#465	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂₃×#1807	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#1807
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2155	C₂	semidirect
3. C₂×C₂₃×#365	C₂	semidirect
6. C₂₃×#1388	C₂	semidirect
7. C₂₃×#1386	C₂	semidirect
10. C₂₃×#504	C₄	semidirect
92. #12833	C₂₃	direct (N × H)
121. #2155	C₂×C₂₃	semidirect
122. C₂×#365	C₂×C₂₃	semidirect
125. #1388	C₂×C₂₃	semidirect
126. #1386	C₂×C₂₃	semidirect
142. #504	C₄×C₂₃	semidirect
193. C₂₃	#12833	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×#3407	2
C₂₃×#2155	2
C₂×C₂₃×#365	2
C₂₃×#1815	2
C₂₃×#1814	2
C₂₃×#1388	2
C₂₃×#1386	2
#12833	23

Subgroups — C23×#12833 #19160459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12833 #19160459