Subgroups — C₂₃×#33721 #19181347

Order 5888.

794 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#510	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#508	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3156	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2251	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2249	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2247	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#137	—	2	2	no	—
10	C₂×C₂₃×#135	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#536	—	2	2	no	—
13	C₂₃×#510	—	2	8	no	—
14	C₂₃×#508	—	2	4	no	—
15	C₂×C₂₃×Q₃₂	—	2	4	no	—
16	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
18	C₂×C₂₃×Q₃₂	C₂²	1	4	yes	—
19	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
20	C₂×C₂₃×SD₃₂	C₂²	1	4	yes	—
21	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
22	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#497	—	2	4	no	—
26	C₂₃×#495	C₂²	1	4	yes	—
27	C₂₃×#494	C₂²	1	4	yes	—
28	C₂₃×#405	C₂²	1	1	yes	characteristic
29	C₂₃×#346	C₂²	1	2	yes	—
30	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
31	C₂×#41×C₂₃	—	2	2	no	—
32	C₂²×Q₈×C₂₃	—	2	2	no	—
33	C₂₃×#137	—	4	8	no	—
34	C₂₃×#135	—	2	4	no	—
35	C₂₃×#135	—	2	4	no	—
36	C₂×Q₁₆×C₂₃	—	2	4	no	—
37	C₂×Q₁₆×C₂₃	—	2	4	no	—
38	C₂×Q₁₆×C₂₃	—	2	2	no	—
39	C₂×Q₁₆×C₂₃	—	2	4	no	—
40	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
41	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
42	C₂×SD₁₆×C₂₃	—	2	2	no	—
43	C₂×SD₁₆×C₂₃	—	2	4	no	—
44	C₂×D₈×C₂₃	C₂³	1	2	yes	—
45	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×C₄⋊Q₈	—	2	4	no	—
48	C₂₃×#124	—	2	4	no	—
49	C₂₃×#123	—	4	4	no	—
50	C₂₃×C₂²⋊Q₈	—	4	4	no	—
51	C₂₃×#117	D₄	1	1	yes	characteristic
52	C₂₃×Q₃₂	—	2	4	no	—
53	C₂₃×Q₃₂	—	2	4	no	—
54	C₂₃×Q₃₂	—	2	2	no	—
55	C₂₃×Q₃₂	—	2	8	no	—
56	C₂₃×SD₃₂	—	2	4	no	—
57	C₂₃×SD₃₂	—	2	8	no	—
58	C₂₃×D₁₆	—	2	2	no	—
59	C₂₃×M₃₂	D₄	1	4	yes	—
60	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
61	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×#97	D₄	1	2	yes	—
65	C₄×C₈×C₂₃	C₂³	1	2	yes	—
66	#41×C₂₃	—	4	4	no	—
67	#41×C₂₃	—	4	4	no	—
68	C₂×Q₈×C₂₃	—	2	4	no	—
69	C₂×Q₈×C₂₃	—	2	2	no	—
70	C₂×Q₈×C₂₃	—	4	2	no	—
71	C₂×Q₈×C₂₃	—	4	4	no	—
72	C₂×Q₈×C₂₃	—	4	4	no	—
73	C₂×D₄×C₂₃	—	2	2	no	—
74	C₂×D₄×C₂₃	—	4	2	no	—
75	C₂²×C₄×C₂₃	—	4	2	no	—
76	C₂²×C₄×C₂₃	—	4	2	no	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₂₃	—	2	4	no	—
79	Q₁₆×C₂₃	—	2	2	no	—
80	Q₁₆×C₂₃	—	2	4	no	—
81	Q₁₆×C₂₃	—	2	2	no	—
82	Q₁₆×C₂₃	—	4	4	no	—
83	Q₁₆×C₂₃	—	4	8	no	—
84	SD₁₆×C₂₃	—	4	4	no	—
85	SD₁₆×C₂₃	—	4	8	no	—
86	D₈×C₂₃	—	2	2	no	—
87	D₈×C₂₃	—	2	2	no	—
88	M₁₆×C₂₃	D₈	1	4	yes	—
89	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₂₃	—	4	8	no	—
95	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
96	C₂²⋊C₄×C₂₃	—	4	8	no	—
97	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
98	C₄²×C₂₃	C₂×D₄	1	2	yes	—
99	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
100	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
101	C₁₆×C₂₃	C₂×D₄	1	4	yes	—
102	#33721	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₃	—	4	2	no	—
104	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₂₃	—	4	2	no	—
106	Q₈×C₂₃	—	4	4	no	—
107	Q₈×C₂₃	—	4	4	no	—
108	Q₈×C₂₃	—	4	2	no	—
109	Q₈×C₂₃	—	8	4	no	—
110	D₄×C₂₃	—	4	2	no	—
111	D₄×C₂₃	—	4	2	no	—
112	D₄×C₂₃	—	8	4	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	8	4	no	—
116	C₂×C₄×C₂₃	—	8	4	no	—
117	C₂×C₄×C₂₃	#127	1	2	yes	—
118	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
119	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₂₃	#127	1	2	yes	—
123	C₈×C₂₃	C₂×D₈	1	4	yes	—
124	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×#510	C₂×C₂₃	1	2	yes	complemented
127	C₂×#508	C₂×C₂₃	1	1	yes	characteristic
128	C₂²×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#3156	C₂×C₂₃	1	2	yes	complemented
130	#2251	C₂×C₂₃	1	4	yes	complemented
131	#2249	C₂×C₂₃	1	2	yes	complemented
132	#2247	C₂×C₂₃	1	2	yes	complemented
133	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	—	8	2	no	—
136	C₂²×C₂₃	—	8	4	no	—
137	C₂²×C₂₃	#493	1	2	yes	—
138	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₂₃	—	2	4	no	—
140	C₄×C₂₃	—	8	2	no	—
141	C₄×C₂₃	—	8	4	no	—
142	C₄×C₂₃	#493	1	2	yes	—
143	C₄×C₂₃	C₂×#127	1	1	yes	characteristic
144	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
145	C₂×#137	—	2	2	no	—
146	C₂×#135	C₂²×C₂₃	1	2	yes	—
147	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
148	#536	—	2	2	no	—
149	#510	—	2	8	no	—
150	#508	—	2	4	no	—
151	C₂×Q₃₂	—	2	4	no	—
152	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
153	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
154	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×Q₃₂	C₂²×C₂₃	1	4	yes	—
156	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
157	C₂×SD₃₂	C₂²×C₂₃	1	4	yes	—
158	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
159	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
160	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
161	#497	—	2	4	no	—
162	#495	C₂²×C₂₃	1	4	yes	—
163	#494	C₂²×C₂₃	1	4	yes	—
164	#405	C₂²×C₂₃	1	1	yes	characteristic
165	#346	C₂²×C₂₃	1	2	yes	—
166	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	16	2	no	—
169	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
170	C₂×C₂₃	#3417	1	2	yes	—
171	C₂×#41	—	2	2	no	—
172	C₂²×Q₈	—	2	2	no	—
173	#137	—	4	8	no	—
174	#135	—	2	4	no	—
175	#135	—	2	4	no	—
176	C₂×Q₁₆	—	2	4	no	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×Q₁₆	—	2	2	no	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
181	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
182	C₂×SD₁₆	—	2	2	no	—
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×D₈	C₂³×C₂₃	1	2	yes	—
185	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	C₄⋊Q₈	—	2	4	no	—
188	#124	—	2	4	no	—
189	#123	—	4	4	no	—
190	C₂²⋊Q₈	—	4	4	no	—
191	#117	D₄×C₂₃	1	1	yes	characteristic
192	Q₃₂	—	2	4	no	—
193	Q₃₂	—	2	4	no	—
194	Q₃₂	—	2	2	no	—
195	Q₃₂	—	2	8	no	—
196	SD₃₂	—	2	4	no	—
197	SD₃₂	—	2	8	no	—
198	D₁₆	—	2	2	no	—
199	M₃₂	D₄×C₂₃	1	4	yes	—
200	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
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	#97	D₄×C₂₃	1	2	yes	—
205	C₄×C₈	C₂³×C₂₃	1	2	yes	—
206	C₂₃	#33721	1	1	yes	characteristic, direct product factor, Sylow
207	#41	—	4	4	no	—
208	#41	—	4	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
219	Q₁₆	—	2	4	no	—
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	2	2	no	—
223	Q₁₆	—	4	4	no	—
224	Q₁₆	—	4	8	no	—
225	SD₁₆	—	4	4	no	—
226	SD₁₆	—	4	8	no	—
227	D₈	—	2	2	no	—
228	D₈	—	2	2	no	—
229	M₁₆	D₈×C₂₃	1	4	yes	—
230	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
234	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
235	C₄⋊C₄	—	4	8	no	—
236	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
237	C₂²⋊C₄	—	4	8	no	—
238	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
239	C₄²	C₂×D₄×C₂₃	1	2	yes	—
240	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
241	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
242	C₁₆	C₂×D₄×C₂₃	1	4	yes	—
243	C₂³	—	4	2	no	—
244	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	4	2	no	—
247	Q₈	—	4	2	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	8	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	C₂₃×#127	1	2	yes	—
258	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
259	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
260	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
262	C₈	C₂₃×#127	1	2	yes	—
263	C₈	C₂×D₈×C₂₃	1	4	yes	—
264	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
265	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	8	2	no	—
268	C₂²	—	8	4	no	—
269	C₂²	C₂₃×#493	1	2	yes	—
270	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₂₃×#493	1	2	yes	—
275	C₄	C₂×C₂₃×#127	1	1	yes	characteristic
276	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
280	C₂	C₂₃×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#510	C₂	semidirect
3. C₂²×C₂₃×Q₃₂	C₂	semidirect
4. C₂₃×#3156	C₂	semidirect
5. C₂₃×#2251	C₂	semidirect
6. C₂₃×#2249	C₂	semidirect
7. C₂₃×#2247	C₂	semidirect
8. C₂₃×#2108	C₂	semidirect
102. #33721	C₂₃	direct (N × H)
126. C₂×#510	C₂×C₂₃	semidirect
128. C₂²×Q₃₂	C₂×C₂₃	semidirect
129. #3156	C₂×C₂₃	semidirect
130. #2251	C₂×C₂₃	semidirect
131. #2249	C₂×C₂₃	semidirect
132. #2247	C₂×C₂₃	semidirect
133. #2108	C₂×C₂₃	semidirect
206. C₂₃	#33721	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×#510	2
C₂×C₂₃×#508	2
C₂²×C₂₃×Q₃₂	2
C₂₃×#3156	2
C₂₃×#2251	2
C₂₃×#2249	2
C₂₃×#2247	2
C₂₃×#2108	2
#33721	23

Subgroups — C23×#33721 #19181347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33721 #19181347