Subgroups — C₂₃×#61820 #19209446

Order 5888.

1966 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3497	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×SD₁₆×C₂₃	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3362	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3227	C₂	1	6	yes	maximal, complemented
6	C₂₃×#3103	C₂	1	6	yes	maximal, complemented
7	C₂₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
9	Q₈²×C₂₃	—	2	4	no	—
10	Q₈²×C₂₃	C₂²	1	8	yes	—
11	C₂₃×#550	C₂²	1	4	yes	—
12	D₄×Q₈×C₂₃	C₂²	1	4	yes	—
13	C₂₃×#548	—	2	4	no	—
14	C₂₃×#537	—	2	3	no	—
15	C₂₃×#532	C₂²	1	6	yes	—
16	C₂×C₂₃×C₄⋊Q₈	—	2	3	no	—
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#498	C₂²	1	12	yes	—
22	C₂₃×#486	C₂²	1	12	yes	—
23	C₂₃×#481	C₂²	1	12	yes	—
24	C₂₃×#476	C₂²	1	24	yes	—
25	C₂₃×#475	C₂²	1	12	yes	—
26	C₂₃×#460	C₂²	1	12	yes	—
27	C₂₃×#448	C₂²	1	4	yes	—
28	C₈×Q₈×C₂₃	C₂²	1	8	yes	—
29	C₄×SD₁₆×C₂₃	C₂²	1	24	yes	—
30	C₂₃×#431	C₂²	1	6	yes	—
31	C₂₃×2₋¹⁺⁴	—	4	8	no	—
32	C₂×#41×C₂₃	—	2	4	no	—
33	C₂×#41×C₂₃	—	2	6	no	—
34	C₂²×Q₈×C₂₃	—	2	4	no	—
35	C₂²×Q₈×C₂₃	—	2	3	no	—
36	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
37	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
39	C₂₃×C₄⋊Q₈	—	2	24	no	—
40	C₂₃×C₄⋊Q₈	—	2	6	no	—
41	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
42	C₂₃×C₄⋊Q₈	C₂³	1	12	yes	—
43	C₂₃×#127	C₂³	1	6	yes	—
44	C₂₃×#124	—	2	6	no	—
45	C₂₃×C₂²⋊Q₈	—	2	12	no	—
46	C₂₃×#121	—	2	6	no	—
47	C₂₃×#121	—	2	12	no	—
48	C₂₃×#121	C₂³	1	4	yes	—
49	C₄×Q₈×C₂₃	—	2	8	no	—
50	C₄×Q₈×C₂₃	—	2	6	no	—
51	C₄×Q₈×C₂₃	D₄	1	4	yes	—
52	C₄×Q₈×C₂₃	C₂³	1	12	yes	—
53	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
54	C₄×D₄×C₂₃	—	2	6	no	—
55	C₄×D₄×C₂₃	C₂³	1	12	yes	—
56	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
58	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
59	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
60	C₂×C₄²×C₂₃	—	2	3	no	—
61	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
62	C₂₃×#106	C₂³	1	24	yes	—
63	C₂₃×C₄⋊C₈	C₂³	1	12	yes	—
64	C₂₃×#103	C₂³	1	8	yes	—
65	C₂₃×#102	C₂³	1	24	yes	—
66	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
67	C₄×C₈×C₂₃	C₂³	1	12	yes	—
68	#41×C₂₃	—	4	16	no	—
69	#41×C₂₃	—	4	24	no	—
70	C₂×Q₈×C₂₃	—	2	4	no	—
71	C₂×Q₈×C₂₃	—	2	1	no	—
72	C₂×Q₈×C₂₃	—	2	6	no	—
73	C₂×Q₈×C₂₃	—	2	12	no	—
74	C₂×Q₈×C₂₃	—	4	16	no	—
75	C₂×Q₈×C₂₃	—	8	6	no	—
76	C₂×Q₈×C₂₃	SD₁₆	1	8	yes	complemented
77	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
78	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
79	C₂×D₄×C₂₃	—	2	4	no	—
80	C₂×D₄×C₂₃	—	2	6	no	—
81	C₂×D₄×C₂₃	—	2	6	no	—
82	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₂₃	—	2	1	no	—
84	C₂²×C₄×C₂₃	—	2	3	no	—
85	C₂²×C₄×C₂₃	—	2	12	no	—
86	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
87	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
88	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
89	SD₁₆×C₂₃	—	2	16	no	—
90	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
91	C₄⋊C₄×C₂₃	—	2	12	no	—
92	C₄⋊C₄×C₂₃	—	2	12	no	—
93	C₄⋊C₄×C₂₃	—	2	3	no	—
94	C₄⋊C₄×C₂₃	—	2	6	no	—
95	C₄⋊C₄×C₂₃	C₂×D₄	1	12	yes	—
96	C₄⋊C₄×C₂₃	C₂⁴	1	6	yes	—
97	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
98	C₂²⋊C₄×C₂₃	—	2	4	no	—
99	C₂²⋊C₄×C₂₃	—	2	12	no	—
100	C₄²×C₂₃	—	2	12	no	—
101	C₄²×C₂₃	C₂×D₄	1	6	yes	—
102	C₄²×C₂₃	C₂⁴	1	6	yes	—
103	#61820	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	4	no	—
105	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	Q₈×C₂₃	—	2	4	no	—
107	Q₈×C₂₃	—	2	2	no	—
108	Q₈×C₂₃	—	4	12	no	—
109	Q₈×C₂₃	—	8	12	no	—
110	Q₈×C₂₃	C₂×SD₁₆	1	16	yes	—
111	Q₈×C₂₃	2₋¹⁺⁴	1	4	yes	complemented
112	D₄×C₂₃	—	2	8	no	—
113	D₄×C₂₃	—	4	8	no	—
114	D₄×C₂₃	—	4	12	no	—
115	D₄×C₂₃	—	8	6	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	6	no	—
120	C₂×C₄×C₂₃	—	2	4	no	—
121	C₂×C₄×C₂₃	—	4	3	no	—
122	C₂×C₄×C₂₃	—	4	6	no	—
123	C₂×C₄×C₂₃	—	4	24	no	—
124	C₂×C₄×C₂₃	C₂×SD₁₆	1	12	yes	—
125	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
127	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
128	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
129	C₈×C₂₃	—	2	8	no	—
130	#3497	C₂×C₂₃	1	1	yes	characteristic
131	C₂×Q₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
132	Q₈×SD₁₆	C₂×C₂₃	1	8	yes	complemented
133	#3362	C₂×C₂₃	1	8	yes	complemented
134	#3227	C₂×C₂₃	1	6	yes	complemented
135	#3103	C₂×C₂₃	1	6	yes	complemented
136	#2982	C₂×C₂₃	1	1	yes	characteristic, complemented
137	C₂²×C₂₃	—	2	1	no	—
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	4	4	no	—
140	C₂²×C₂₃	—	4	4	no	—
141	C₂²×C₂₃	—	4	4	no	—
142	C₂²×C₂₃	C₂²×SD₁₆	1	2	yes	—
143	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₂₃	—	2	6	no	—
145	C₄×C₂₃	—	2	4	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	4	6	no	—
148	C₄×C₂₃	—	4	4	no	—
149	C₄×C₂₃	C₂²×SD₁₆	1	12	yes	—
150	C₄×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
151	C₂×2₋¹⁺⁴	—	2	1	no	—
152	Q₈²	—	2	4	no	—
153	Q₈²	C₂²×C₂₃	1	8	yes	—
154	#550	C₂²×C₂₃	1	4	yes	—
155	D₄×Q₈	C₂²×C₂₃	1	4	yes	—
156	#548	—	2	4	no	—
157	#537	—	2	3	no	—
158	#532	C₂²×C₂₃	1	6	yes	—
159	C₂×C₄⋊Q₈	—	2	3	no	—
160	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
161	C₂×C₄×Q₈	—	2	1	no	—
162	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
163	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
164	#498	C₂²×C₂₃	1	12	yes	—
165	#486	C₂²×C₂₃	1	12	yes	—
166	#481	C₂²×C₂₃	1	12	yes	—
167	#476	C₂²×C₂₃	1	24	yes	—
168	#475	C₂²×C₂₃	1	12	yes	—
169	#460	C₂²×C₂₃	1	12	yes	—
170	#448	C₂²×C₂₃	1	4	yes	—
171	C₈×Q₈	C₂²×C₂₃	1	8	yes	—
172	C₄×SD₁₆	C₂²×C₂₃	1	24	yes	—
173	#431	C₂²×C₂₃	1	6	yes	—
174	C₂×C₂₃	—	2	2	no	—
175	C₂×C₂₃	—	8	4	no	—
176	C₂×C₂₃	#3486	1	1	yes	characteristic
177	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
178	C₂×C₂₃	#3585	1	1	yes	characteristic
179	2₋¹⁺⁴	—	4	8	no	—
180	C₂×#41	—	2	4	no	—
181	C₂×#41	—	2	6	no	—
182	C₂²×Q₈	—	2	4	no	—
183	C₂²×Q₈	—	2	3	no	—
184	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
185	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
186	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
187	C₄⋊Q₈	—	2	24	no	—
188	C₄⋊Q₈	—	2	6	no	—
189	C₄⋊Q₈	C₂³×C₂₃	1	12	yes	—
190	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
191	#127	C₂³×C₂₃	1	6	yes	—
192	#124	—	2	6	no	—
193	C₂²⋊Q₈	—	2	12	no	—
194	#121	—	2	6	no	—
195	#121	—	2	12	no	—
196	#121	C₂³×C₂₃	1	4	yes	—
197	C₄×Q₈	—	2	8	no	—
198	C₄×Q₈	—	2	6	no	—
199	C₄×Q₈	D₄×C₂₃	1	4	yes	—
200	C₄×Q₈	C₂³×C₂₃	1	12	yes	—
201	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
202	C₄×D₄	—	2	6	no	—
203	C₄×D₄	C₂³×C₂₃	1	12	yes	—
204	C₂×C₄⋊C₄	—	2	3	no	—
205	C₂×C₄⋊C₄	—	2	3	no	—
206	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
207	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
208	C₂×C₄²	—	2	3	no	—
209	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
210	#106	C₂³×C₂₃	1	24	yes	—
211	C₄⋊C₈	C₂³×C₂₃	1	12	yes	—
212	#103	C₂³×C₂₃	1	8	yes	—
213	#102	C₂³×C₂₃	1	24	yes	—
214	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
215	C₄×C₈	C₂³×C₂₃	1	12	yes	—
216	C₂₃	#61820	1	1	yes	characteristic, direct product factor, Sylow
217	#41	—	4	16	no	—
218	#41	—	4	24	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	2	6	no	—
222	C₂×Q₈	—	2	12	no	—
223	C₂×Q₈	—	4	16	no	—
224	C₂×Q₈	—	8	6	no	—
225	C₂×Q₈	SD₁₆×C₂₃	1	8	yes	complemented
226	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
227	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	6	no	—
230	C₂×D₄	—	2	6	no	—
231	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	3	no	—
234	C₂²×C₄	—	2	12	no	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
236	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
237	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
238	SD₁₆	—	2	16	no	—
239	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
240	C₄⋊C₄	—	2	12	no	—
241	C₄⋊C₄	—	2	12	no	—
242	C₄⋊C₄	—	2	3	no	—
243	C₄⋊C₄	—	2	6	no	—
244	C₄⋊C₄	C₂×D₄×C₂₃	1	12	yes	—
245	C₄⋊C₄	C₂⁴×C₂₃	1	6	yes	—
246	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	12	no	—
249	C₄²	—	2	12	no	—
250	C₄²	C₂×D₄×C₂₃	1	6	yes	—
251	C₄²	C₂⁴×C₂₃	1	6	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
254	Q₈	—	2	4	no	—
255	Q₈	—	2	2	no	—
256	Q₈	—	4	12	no	—
257	Q₈	—	8	12	no	—
258	Q₈	C₂×SD₁₆×C₂₃	1	16	yes	—
259	Q₈	C₂₃×2₋¹⁺⁴	1	4	yes	complemented
260	D₄	—	2	8	no	—
261	D₄	—	4	8	no	—
262	D₄	—	4	12	no	—
263	D₄	—	8	6	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	3	no	—
270	C₂×C₄	—	4	6	no	—
271	C₂×C₄	—	4	24	no	—
272	C₂×C₄	C₂×SD₁₆×C₂₃	1	12	yes	—
273	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
275	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
276	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
277	C₈	—	2	8	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₂²×SD₁₆×C₂₃	1	2	yes	—
284	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	6	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	6	no	—
290	C₄	C₂²×SD₁₆×C₂₃	1	12	yes	—
291	C₄	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	8	4	no	—
294	C₂	C₂₃×#3486	1	1	yes	characteristic
295	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
296	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3497	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈²×C₂₃	C₂	semidirect
3. Q₈×SD₁₆×C₂₃	C₂	semidirect
4. C₂₃×#3362	C₂	semidirect
5. C₂₃×#3227	C₂	semidirect
6. C₂₃×#3103	C₂	semidirect
7. C₂₃×#2982	C₂	semidirect
76. C₂×Q₈×C₂₃	SD₁₆	semidirect
103. #61820	C₂₃	direct (N × H)
111. Q₈×C₂₃	2₋¹⁺⁴	semidirect
131. C₂×Q₈²	C₂×C₂₃	semidirect
132. Q₈×SD₁₆	C₂×C₂₃	semidirect
133. #3362	C₂×C₂₃	semidirect
134. #3227	C₂×C₂₃	semidirect
135. #3103	C₂×C₂₃	semidirect
136. #2982	C₂×C₂₃	semidirect
216. C₂₃	#61820	direct (N × H)
225. C₂×Q₈	SD₁₆×C₂₃	semidirect
259. Q₈	C₂₃×2₋¹⁺⁴	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3497	2
C₂×Q₈²×C₂₃	2
Q₈×SD₁₆×C₂₃	2
C₂₃×#3362	2
C₂₃×#3227	2
C₂₃×#3103	2
C₂₃×#2982	2
#61820	23

Subgroups — C23×#61820 #19209446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61820 #19209446