Subgroups — C₂₃×#33969 #19181595

Order 5888.

504 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3393	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2251	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2249	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2211	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2210	C₂	1	2	yes	maximal
6	C₂₃×#2209	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#556	—	2	2	no	—
11	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
14	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
17	C₂₃×#495	C₂²	1	2	yes	—
18	C₂₃×#495	C₂²	1	2	yes	—
19	C₂₃×#494	C₂²	1	2	yes	—
20	C₂₃×#464	—	2	2	no	—
21	C₂₃×#464	—	2	2	no	—
22	C₂₃×#461	—	2	2	no	—
23	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
26	C₄×D₈×C₂₃	C₂²	1	2	yes	—
27	C₂₃×#366	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₂₃×#358	C₂²	1	2	yes	—
31	C₂₃×#358	C₂²	1	2	yes	—
32	C₂₃×#357	C₂²	1	2	yes	—
33	C₂₃×#346	C₂²	1	2	yes	—
34	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₂₃	—	2	2	no	—
36	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
38	C₂×SD₁₆×C₂₃	—	2	2	no	—
39	C₂×SD₁₆×C₂₃	—	2	2	no	—
40	C₂×D₈×C₂₃	C₂³	1	2	yes	—
41	C₂₃×C₄⋊Q₈	—	2	2	no	—
42	C₂₃×#126	—	4	2	no	—
43	C₂₃×#126	—	4	2	no	—
44	C₂₃×#124	—	2	2	no	—
45	C₂₃×#124	—	2	2	no	—
46	C₂₃×C₂²⋊Q₈	—	4	2	no	—
47	C₄×Q₈×C₂₃	—	2	2	no	—
48	C₄×Q₈×C₂₃	—	2	2	no	—
49	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
50	C₄×D₄×C₂₃	—	2	2	no	—
51	C₂₃×#117	—	4	2	no	—
52	C₂₃×Q₃₂	—	2	2	no	—
53	C₂₃×Q₃₂	—	4	1	no	—
54	C₂₃×Q₃₂	—	4	2	no	—
55	C₂₃×SD₃₂	—	2	4	no	—
56	C₂₃×SD₃₂	—	4	2	no	—
57	C₂₃×D₁₆	—	4	1	no	—
58	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
60	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
61	C₂₃×#107	C₂³	1	2	yes	—
62	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
63	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
64	C₂₃×#103	—	2	2	no	—
65	C₂₃×#103	—	2	2	no	—
66	C₂₃×#102	—	2	2	no	—
67	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₂₃	C₂³	1	2	yes	—
69	C₂×Q₈×C₂₃	—	2	2	no	—
70	C₂×Q₈×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
72	C₂×D₄×C₂₃	—	2	2	no	—
73	C₂²×C₄×C₂₃	—	4	2	no	—
74	Q₁₆×C₂₃	—	2	2	no	—
75	Q₁₆×C₂₃	—	2	2	no	—
76	Q₁₆×C₂₃	—	2	2	no	—
77	Q₁₆×C₂₃	—	2	2	no	—
78	Q₁₆×C₂₃	—	8	2	no	—
79	SD₁₆×C₂₃	—	8	2	no	—
80	SD₁₆×C₂₃	—	8	2	no	—
81	D₈×C₂₃	—	2	2	no	—
82	D₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	4	2	no	—
88	C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₄⋊C₄×C₂₃	—	4	2	no	—
90	C₄⋊C₄×C₂₃	—	4	2	no	—
91	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
92	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
93	C₂²⋊C₄×C₂₃	—	4	2	no	—
94	C₂²⋊C₄×C₂₃	—	4	2	no	—
95	C₂²⋊C₄×C₂₃	—	4	2	no	—
96	C₂²⋊C₄×C₂₃	—	4	2	no	—
97	C₂²⋊C₄×C₂₃	—	4	2	no	—
98	C₄²×C₂₃	—	4	2	no	—
99	C₄²×C₂₃	—	4	2	no	—
100	C₄²×C₂₃	C₂×D₄	1	2	yes	—
101	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₁₆×C₂₃	—	2	1	no	—
103	C₁₆×C₂₃	—	2	2	no	—
104	C₁₆×C₂₃	#41	1	2	yes	—
105	#33969	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₂₃	—	4	2	no	—
107	Q₈×C₂₃	—	4	1	no	—
108	Q₈×C₂₃	—	4	2	no	—
109	Q₈×C₂₃	—	4	2	no	—
110	Q₈×C₂₃	—	4	2	no	—
111	Q₈×C₂₃	—	4	2	no	—
112	D₄×C₂₃	—	4	2	no	—
113	D₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	8	4	no	—
118	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₂₃	—	2	1	no	—
124	C₈×C₂₃	—	2	2	no	—
125	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
126	C₈×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
127	#3393	C₂×C₂₃	1	2	yes	complemented
128	#2251	C₂×C₂₃	1	2	yes	complemented
129	#2249	C₂×C₂₃	1	1	yes	characteristic
130	#2211	C₂×C₂₃	1	2	yes	complemented
131	#2210	C₂×C₂₃	1	2	yes	—
132	#2209	C₂×C₂₃	1	2	yes	complemented
133	#2183	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#2175	C₂×C₂₃	1	2	yes	complemented
135	C₄×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₂²×C₂₃	—	8	2	no	—
137	C₂²×C₂₃	—	8	2	no	—
138	C₂²×C₂₃	—	8	2	no	—
139	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₂₃	—	2	2	no	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	8	2	no	—
145	C₄×C₂₃	—	8	2	no	—
146	C₄×C₂₃	—	8	2	no	—
147	C₄×C₂₃	#550	1	1	yes	characteristic
148	C₄×C₂₃	#576	1	1	yes	characteristic
149	#556	—	2	2	no	—
150	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
153	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
154	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
155	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
156	#495	C₂²×C₂₃	1	2	yes	—
157	#495	C₂²×C₂₃	1	2	yes	—
158	#494	C₂²×C₂₃	1	2	yes	—
159	#464	—	2	2	no	—
160	#464	—	2	2	no	—
161	#461	—	2	2	no	—
162	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
163	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
164	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
165	C₄×D₈	C₂²×C₂₃	1	2	yes	—
166	#366	C₂²×C₂₃	1	1	yes	characteristic
167	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
168	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
169	#358	C₂²×C₂₃	1	2	yes	—
170	#358	C₂²×C₂₃	1	2	yes	—
171	#357	C₂²×C₂₃	1	2	yes	—
172	#346	C₂²×C₂₃	1	2	yes	—
173	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
174	C₂×C₂₃	—	16	2	no	—
175	C₂×C₂₃	#3391	1	1	yes	characteristic
176	C₂×C₂₃	#3414	1	1	yes	characteristic
177	C₂×C₂₃	#3417	1	1	yes	characteristic
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
180	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
181	C₂×SD₁₆	—	2	2	no	—
182	C₂×SD₁₆	—	2	2	no	—
183	C₂×D₈	C₂³×C₂₃	1	2	yes	—
184	C₄⋊Q₈	—	2	2	no	—
185	#126	—	4	2	no	—
186	#126	—	4	2	no	—
187	#124	—	2	2	no	—
188	#124	—	2	2	no	—
189	C₂²⋊Q₈	—	4	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
193	C₄×D₄	—	2	2	no	—
194	#117	—	4	2	no	—
195	Q₃₂	—	2	2	no	—
196	Q₃₂	—	4	1	no	—
197	Q₃₂	—	4	2	no	—
198	SD₃₂	—	2	4	no	—
199	SD₃₂	—	4	2	no	—
200	D₁₆	—	4	1	no	—
201	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
202	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
203	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
204	#107	C₂³×C₂₃	1	2	yes	—
205	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
206	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
207	#103	—	2	2	no	—
208	#103	—	2	2	no	—
209	#102	—	2	2	no	—
210	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₂₃	1	2	yes	—
212	C₂₃	#33969	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
216	C₂×D₄	—	2	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	Q₁₆	—	2	2	no	—
219	Q₁₆	—	2	2	no	—
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	2	no	—
222	Q₁₆	—	8	2	no	—
223	SD₁₆	—	8	2	no	—
224	SD₁₆	—	8	2	no	—
225	D₈	—	2	2	no	—
226	D₈	—	2	2	no	—
227	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
228	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
236	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	C₂×D₄×C₂₃	1	2	yes	—
245	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	#41×C₂₃	1	2	yes	—
249	C₂³	—	4	2	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
263	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	2	no	—
268	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
269	C₈	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	C₂₃×#550	1	1	yes	characteristic
282	C₄	C₂₃×#576	1	1	yes	characteristic
283	C₂	—	16	2	no	—
284	C₂	C₂₃×#3391	1	1	yes	characteristic
285	C₂	C₂₃×#3414	1	1	yes	characteristic
286	C₂	C₂₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×C₂₃×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ C₈×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3393	C₂	semidirect
2. C₂₃×#2251	C₂	semidirect
4. C₂₃×#2211	C₂	semidirect
6. C₂₃×#2209	C₂	semidirect
7. C₂₃×#2183	C₂	semidirect
8. C₂₃×#2175	C₂	semidirect
9. C₄×C₂₃×Q₃₂	C₂	semidirect
105. #33969	C₂₃	direct (N × H)
127. #3393	C₂×C₂₃	semidirect
128. #2251	C₂×C₂₃	semidirect
130. #2211	C₂×C₂₃	semidirect
132. #2209	C₂×C₂₃	semidirect
133. #2183	C₂×C₂₃	semidirect
134. #2175	C₂×C₂₃	semidirect
135. C₄×Q₃₂	C₂×C₂₃	semidirect
212. C₂₃	#33969	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×#3393	2
C₂₃×#2251	2
C₂₃×#2249	2
C₂₃×#2211	2
C₂₃×#2210	2
C₂₃×#2209	2
C₂₃×#2183	2
C₂₃×#2175	2
C₄×C₂₃×Q₃₂	2
#33969	23

Subgroups — C23×#33969 #19181595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33969 #19181595