Subgroups — C₂₃×#33319 #19180945

Order 5888.

806 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#501	C₂	1	2	yes	maximal
2	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2660	C₂	1	2	yes	maximal
5	C₂₃×#1847	C₂	1	4	yes	maximal
6	C₂₃×#1846	C₂	1	2	yes	maximal
7	C₂₃×#1845	C₂	1	2	yes	maximal
8	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₂₃×#125	C₂²	1	2	yes	—
11	C₂₃×#501	—	2	8	no	—
12	C₂₃×#500	—	2	4	no	—
13	C₂₃×#499	—	2	4	no	—
14	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#107	C₂²	1	2	yes	—
16	C₂×C₂₃×#107	C₂²	1	4	yes	—
17	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#107	C₂²	1	2	yes	—
19	C₂×C₂₃×#106	C₂²	1	4	yes	—
20	C₂×C₂₃×#106	C₂²	1	2	yes	—
21	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#398	—	2	4	no	—
26	C₂₃×#395	—	2	4	no	—
27	C₂₃×#384	C₂²	1	4	yes	—
28	C₂₃×#382	C₂²	1	4	yes	—
29	C₂₃×#376	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
34	C₂₃×C₄⋊Q₈	—	2	4	no	—
35	C₂₃×C₄⋊Q₈	—	2	4	no	—
36	C₂₃×#125	—	2	4	no	—
37	C₂₃×#125	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂₃×#107	—	2	2	no	—
48	C₂₃×#107	—	2	2	no	—
49	C₂₃×#107	—	2	4	no	—
50	C₂₃×#107	—	2	4	no	—
51	C₂₃×#107	—	2	8	no	—
52	C₂₃×#106	—	2	4	no	—
53	C₂₃×#106	—	2	8	no	—
54	C₂₃×#97	D₄	1	4	yes	—
55	C₂₃×#97	Q₈	1	8	yes	complemented
56	C₄×C₈×C₂₃	D₄	1	4	yes	—
57	C₄×C₈×C₂₃	Q₈	1	8	yes	complemented
58	C₂₃×#95	—	2	8	no	—
59	C₂₃×#95	D₄	1	2	yes	—
60	C₂₃×#95	D₄	1	2	yes	—
61	C₂×Q₈×C₂₃	—	4	4	no	—
62	C₂×Q₈×C₂₃	—	8	4	no	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	2	4	no	—
65	C₂²×C₄×C₂₃	—	2	2	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
74	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
76	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
77	C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	—	4	4	no	—
84	C₄⋊C₄×C₂₃	—	4	4	no	—
85	C₄⋊C₄×C₂₃	—	4	4	no	—
86	C₄⋊C₄×C₂₃	—	8	8	no	—
87	C₄²×C₂₃	C₂×D₄	1	2	yes	—
88	C₄²×C₂₃	C₂×D₄	1	2	yes	—
89	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
90	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
91	#33319	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
93	Q₈×C₂₃	—	8	8	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	8	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	#127	1	2	yes	—
105	C₂×C₄×C₂₃	#127	1	2	yes	—
106	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
107	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
110	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
111	C₈×C₂₃	C₄⋊Q₈	1	4	yes	—
112	C₈×C₂₃	C₄⋊Q₈	1	8	yes	—
113	C₈×C₂₃	C₄⋊Q₈	1	4	yes	—
114	C₂×#501	C₂×C₂₃	1	2	yes	—
115	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
116	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
117	#2660	C₂×C₂₃	1	2	yes	—
118	#1847	C₂×C₂₃	1	4	yes	—
119	#1846	C₂×C₂₃	1	2	yes	—
120	#1845	C₂×C₂₃	1	2	yes	—
121	#1726	C₂×C₂₃	1	1	yes	characteristic
122	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
123	C₂²×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
124	C₂²×C₂₃	#576	1	2	yes	—
125	C₂²×C₂₃	#578	1	2	yes	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	8	no	—
129	C₄×C₂₃	—	8	2	no	—
130	C₄×C₂₃	—	8	4	no	—
131	C₄×C₂₃	—	8	2	no	—
132	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
133	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
134	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
136	C₂×#125	C₂²×C₂₃	1	2	yes	—
137	#501	—	2	8	no	—
138	#500	—	2	4	no	—
139	#499	—	2	4	no	—
140	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×#107	C₂²×C₂₃	1	2	yes	—
142	C₂×#107	C₂²×C₂₃	1	4	yes	—
143	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×#107	C₂²×C₂₃	1	2	yes	—
145	C₂×#106	C₂²×C₂₃	1	4	yes	—
146	C₂×#106	C₂²×C₂₃	1	2	yes	—
147	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×#97	C₂²×C₂₃	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
151	#398	—	2	4	no	—
152	#395	—	2	4	no	—
153	#384	C₂²×C₂₃	1	4	yes	—
154	#382	C₂²×C₂₃	1	4	yes	—
155	#376	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×C₂₃	#2868	1	1	yes	characteristic
157	C₂×C₂₃	#3155	1	1	yes	characteristic
158	C₂×C₂₃	#3156	1	1	yes	characteristic
159	C₂×C₂₃	#3164	1	4	yes	—
160	C₂²×Q₈	—	2	2	no	—
161	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
164	C₄⋊Q₈	—	2	4	no	—
165	C₄⋊Q₈	—	2	4	no	—
166	#125	—	2	4	no	—
167	#125	—	2	4	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	8	no	—
172	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
175	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
177	#107	—	2	2	no	—
178	#107	—	2	2	no	—
179	#107	—	2	4	no	—
180	#107	—	2	4	no	—
181	#107	—	2	8	no	—
182	#106	—	2	4	no	—
183	#106	—	2	8	no	—
184	#97	D₄×C₂₃	1	4	yes	—
185	#97	Q₈×C₂₃	1	8	yes	complemented
186	C₄×C₈	D₄×C₂₃	1	4	yes	—
187	C₄×C₈	Q₈×C₂₃	1	8	yes	complemented
188	#95	—	2	8	no	—
189	#95	D₄×C₂₃	1	2	yes	—
190	#95	D₄×C₂₃	1	2	yes	—
191	C₂₃	#33319	1	1	yes	characteristic, direct product factor, Sylow
192	C₂×Q₈	—	4	4	no	—
193	C₂×Q₈	—	8	4	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
200	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
201	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
204	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
205	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
206	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
207	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	8	8	no	—
218	C₄²	C₂×D₄×C₂₃	1	2	yes	—
219	C₄²	C₂×D₄×C₂₃	1	2	yes	—
220	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
221	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
222	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
223	Q₈	—	8	8	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂₃×#127	1	2	yes	—
235	C₂×C₄	C₂₃×#127	1	2	yes	—
236	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
237	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
240	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
241	C₈	C₂₃×C₄⋊Q₈	1	4	yes	—
242	C₈	C₂₃×C₄⋊Q₈	1	4	yes	—
243	C₈	C₂₃×C₄⋊Q₈	1	8	yes	—
244	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
245	C₂²	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
246	C₂²	C₂₃×#576	1	2	yes	—
247	C₂²	C₂₃×#578	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	8	no	—
251	C₄	—	8	4	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
255	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
256	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
257	C₂	C₂₃×#2868	1	1	yes	characteristic
258	C₂	C₂₃×#3155	1	1	yes	characteristic
259	C₂	C₂₃×#3156	1	1	yes	characteristic
260	C₂	C₂₃×#3164	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
55. C₂₃×#97	Q₈	semidirect
57. C₄×C₈×C₂₃	Q₈	semidirect
91. #33319	C₂₃	direct (N × H)
185. #97	Q₈×C₂₃	semidirect
187. C₄×C₈	Q₈×C₂₃	semidirect
191. C₂₃	#33319	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×#501	2
C₂×C₂₃×#500	2
C₂×C₂₃×#499	2
C₂₃×#2660	2
C₂₃×#1847	2
C₂₃×#1846	2
C₂₃×#1845	2
C₂₃×#1726	2
#33319	23

Subgroups — C23×#33319 #19180945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33319 #19180945