Subgroups — C₂₃×#33718 #19181344

Order 5888.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 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	4	yes	maximal, complemented
2	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2251	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2248	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2247	C₂	1	2	yes	maximal, complemented
7	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#135	—	2	1	no	—
9	C₂×C₂₃×#135	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₂₃	—	2	1	no	—
11	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#532	—	2	1	no	—
13	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
14	C₂₃×#510	—	2	16	no	—
15	C₂×C₂₃×Q₃₂	C₂²	1	8	yes	—
16	C₂×C₂₃×SD₃₂	C₂²	1	8	yes	—
17	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
18	C₂₃×#495	—	2	1	no	—
19	C₂₃×#495	C₂²	1	2	yes	—
20	C₂₃×#494	—	2	1	no	—
21	C₂₃×#494	—	2	1	no	—
22	C₂₃×#494	C₂²	1	1	yes	characteristic
23	C₂₃×#494	C₂²	1	2	yes	—
24	C₂₃×#494	C₂²	1	2	yes	—
25	C₂₃×#493	C₂²	1	1	yes	characteristic
26	C₂₃×#492	—	2	1	no	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#346	C₂²	1	2	yes	—
29	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
30	C₂×#41×C₂₃	—	2	2	no	—
31	C₂²×Q₈×C₂₃	—	2	2	no	—
32	C₂₃×#135	—	2	8	no	—
33	C₂₃×#135	—	4	4	no	—
34	C₂×Q₁₆×C₂₃	—	2	8	no	—
35	C₂×Q₁₆×C₂₃	—	2	2	no	—
36	C₂×Q₁₆×C₂₃	—	2	2	no	—
37	C₂×Q₁₆×C₂₃	—	2	2	no	—
38	C₂×Q₁₆×C₂₃	—	2	1	no	—
39	C₂×Q₁₆×C₂₃	—	4	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	2	no	—
44	C₂×D₈×C₂₃	—	2	1	no	—
45	C₂×D₈×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂₃×C₄⋊Q₈	—	2	1	no	—
49	C₂₃×C₄⋊Q₈	—	2	2	no	—
50	C₂₃×C₄⋊Q₈	—	2	2	no	—
51	C₂₃×C₄⋊Q₈	—	4	2	no	—
52	C₂₃×#127	—	2	1	no	—
53	C₂₃×#124	—	2	2	no	—
54	C₂₃×#121	—	4	2	no	—
55	C₄×D₄×C₂₃	—	4	2	no	—
56	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
57	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
58	C₂₃×Q₃₂	—	2	16	no	—
59	C₂₃×SD₃₂	—	2	16	no	—
60	C₂₃×M₃₂	D₄	1	8	yes	complemented
61	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
62	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	#41×C₂₃	—	4	4	no	—
67	#41×C₂₃	—	4	4	no	—
68	C₂×Q₈×C₂₃	—	2	2	no	—
69	C₂×Q₈×C₂₃	—	2	4	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	C₂×D₄×C₂₃	—	4	2	no	—
76	C₂²×C₄×C₂₃	—	4	2	no	—
77	C₂²×C₄×C₂₃	—	4	2	no	—
78	C₂²×C₄×C₂₃	D₈	1	2	yes	—
79	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₂₃	—	2	8	no	—
81	Q₁₆×C₂₃	—	2	4	no	—
82	Q₁₆×C₂₃	—	4	4	no	—
83	Q₁₆×C₂₃	—	4	4	no	—
84	Q₁₆×C₂₃	—	4	4	no	—
85	Q₁₆×C₂₃	—	4	2	no	—
86	SD₁₆×C₂₃	—	4	4	no	—
87	SD₁₆×C₂₃	—	4	4	no	—
88	D₈×C₂₃	—	2	4	no	—
89	D₈×C₂₃	—	4	2	no	—
90	C₂×C₈×C₂₃	D₈	1	4	yes	—
91	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₂₃	—	4	4	no	—
97	C₄⋊C₄×C₂₃	—	4	4	no	—
98	C₄⋊C₄×C₂₃	—	4	2	no	—
99	C₂²⋊C₄×C₂₃	—	4	4	no	—
100	C₄²×C₂₃	D₈	1	2	yes	—
101	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₁₆×C₂₃	C₂×D₄	1	8	yes	—
104	#33718	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	—	4	2	no	—
106	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₂₃	—	4	2	no	—
108	Q₈×C₂₃	—	4	4	no	—
109	Q₈×C₂₃	—	4	2	no	—
110	Q₈×C₂₃	—	4	4	no	—
111	Q₈×C₂₃	—	8	4	no	—
112	D₄×C₂₃	—	4	2	no	—
113	D₄×C₂₃	—	4	2	no	—
114	D₄×C₂₃	—	8	4	no	—
115	D₄×C₂₃	—	8	4	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	4	4	no	—
119	C₂×C₄×C₂₃	—	8	4	no	—
120	C₂×C₄×C₂₃	—	8	4	no	—
121	C₂×C₄×C₂₃	#127	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₂₃	#127	1	2	yes	—
128	C₈×C₂₃	C₂×D₈	1	2	yes	—
129	C₈×C₂₃	C₂×D₈	1	2	yes	—
130	C₈×C₂₃	C₂²×D₄	1	2	yes	—
131	C₂×#510	C₂×C₂₃	1	4	yes	complemented
132	#3147	C₂×C₂₃	1	1	yes	characteristic
133	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#2251	C₂×C₂₃	1	4	yes	complemented
135	#2248	C₂×C₂₃	1	2	yes	complemented
136	#2247	C₂×C₂₃	1	2	yes	complemented
137	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
138	C₂²×C₂₃	—	2	2	no	—
139	C₂²×C₂₃	—	8	2	no	—
140	C₂²×C₂₃	—	8	4	no	—
141	C₂²×C₂₃	#493	1	2	yes	—
142	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	8	2	no	—
145	C₄×C₂₃	—	8	4	no	—
146	C₄×C₂₃	#493	1	2	yes	—
147	C₄×C₂₃	#510	1	4	yes	complemented
148	C₄×C₂₃	C₂×#127	1	1	yes	characteristic
149	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
150	C₂×#135	—	2	1	no	—
151	C₂×#135	C₂²×C₂₃	1	2	yes	—
152	C₂²×Q₁₆	—	2	1	no	—
153	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
154	#532	—	2	1	no	—
155	C₂×C₄⋊Q₈	—	2	1	no	—
156	#510	—	2	16	no	—
157	C₂×Q₃₂	C₂²×C₂₃	1	8	yes	—
158	C₂×SD₃₂	C₂²×C₂₃	1	8	yes	—
159	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
160	#495	—	2	1	no	—
161	#495	C₂²×C₂₃	1	2	yes	—
162	#494	—	2	1	no	—
163	#494	—	2	1	no	—
164	#494	C₂²×C₂₃	1	2	yes	—
165	#494	C₂²×C₂₃	1	2	yes	—
166	#494	C₂²×C₂₃	1	1	yes	characteristic
167	#493	C₂²×C₂₃	1	1	yes	characteristic
168	#492	—	2	1	no	—
169	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
170	#346	C₂²×C₂₃	1	2	yes	—
171	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
172	C₂×C₂₃	—	2	2	no	—
173	C₂×C₂₃	—	16	2	no	—
174	C₂×C₂₃	C₂×#493	1	1	yes	characteristic
175	C₂×C₂₃	C₂×#510	1	2	yes	—
176	C₂×#41	—	2	2	no	—
177	C₂²×Q₈	—	2	2	no	—
178	#135	—	2	8	no	—
179	#135	—	4	4	no	—
180	C₂×Q₁₆	—	2	8	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	4	4	no	—
186	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
187	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
188	C₂×SD₁₆	—	2	2	no	—
189	C₂×SD₁₆	—	2	2	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	C₂³×C₂₃	1	2	yes	—
192	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	—	4	2	no	—
198	#127	—	2	1	no	—
199	#124	—	2	2	no	—
200	#121	—	4	2	no	—
201	C₄×D₄	—	4	2	no	—
202	C₂×C₄⋊C₄	—	4	2	no	—
203	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
204	Q₃₂	—	2	16	no	—
205	SD₃₂	—	2	16	no	—
206	M₃₂	D₄×C₂₃	1	8	yes	complemented
207	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
208	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
209	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
212	C₂₃	#33718	1	1	yes	characteristic, direct product factor, Sylow
213	#41	—	4	4	no	—
214	#41	—	4	4	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	D₈×C₂₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
227	Q₁₆	—	2	8	no	—
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	4	4	no	—
230	Q₁₆	—	4	4	no	—
231	Q₁₆	—	4	4	no	—
232	Q₁₆	—	4	2	no	—
233	SD₁₆	—	4	4	no	—
234	SD₁₆	—	4	4	no	—
235	D₈	—	2	4	no	—
236	D₈	—	4	2	no	—
237	C₂×C₈	D₈×C₂₃	1	4	yes	—
238	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
240	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₄²	D₈×C₂₃	1	2	yes	—
248	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
250	C₁₆	C₂×D₄×C₂₃	1	8	yes	—
251	C₂³	—	4	2	no	—
252	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	4	2	no	—
256	Q₈	—	4	4	no	—
257	Q₈	—	8	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	4	no	—
261	D₄	—	8	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	C₂₃×#127	1	2	yes	—
268	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
270	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
271	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
273	C₈	C₂₃×#127	1	2	yes	—
274	C₈	C₂×D₈×C₂₃	1	2	yes	—
275	C₈	C₂×D₈×C₂₃	1	2	yes	—
276	C₈	C₂²×D₄×C₂₃	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	4	no	—
280	C₂²	C₂₃×#493	1	2	yes	—
281	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₂₃×#493	1	2	yes	—
286	C₄	C₂₃×#510	1	4	yes	complemented
287	C₄	C₂×C₂₃×#127	1	1	yes	characteristic
288	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₂₃×#493	1	1	yes	characteristic
292	C₂	C₂×C₂₃×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#510	C₂	semidirect
3. C₂×C₂₃×#494	C₂	semidirect
4. C₂₃×#2251	C₂	semidirect
5. C₂₃×#2248	C₂	semidirect
6. C₂₃×#2247	C₂	semidirect
7. C₄×C₂₃×M₃₂	C₂	semidirect
60. C₂₃×M₃₂	D₄	semidirect
104. #33718	C₂₃	direct (N × H)
131. C₂×#510	C₂×C₂₃	semidirect
133. C₂×#494	C₂×C₂₃	semidirect
134. #2251	C₂×C₂₃	semidirect
135. #2248	C₂×C₂₃	semidirect
136. #2247	C₂×C₂₃	semidirect
137. C₄×M₃₂	C₂×C₂₃	semidirect
147. C₄×C₂₃	#510	semidirect
206. M₃₂	D₄×C₂₃	semidirect
212. C₂₃	#33718	direct (N × H)
286. C₄	C₂₃×#510	semidirect

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
C₂₃×#3147	2
C₂×C₂₃×#494	2
C₂₃×#2251	2
C₂₃×#2248	2
C₂₃×#2247	2
C₄×C₂₃×M₃₂	2
#33718	23

Subgroups — C23×#33718 #19181344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33718 #19181344