Subgroups — C₂₃×#10841 #19158467

Order 5888.

1126 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#384	C₂	1	2	yes	maximal
3	C₂×C₂₃×#340	C₂	1	4	yes	maximal
4	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
5	C₂²×C₄⋊C₄×C₂₃	—	2	4	no	—
6	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	—	2	4	no	—
9	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	complemented
10	C₂₃×#384	—	2	4	no	—
11	C₂₃×#384	C₄	1	8	yes	complemented
12	C₂₃×#340	—	2	8	no	—
13	C₂³×C₄×C₂₃	—	2	4	no	—
14	C₂³×C₄×C₂₃	D₄	1	2	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	—	2	4	no	—
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₂₃	D₄	1	4	yes	—
20	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
21	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
25	C₂×C₄⋊C₄×C₂₃	—	4	8	no	—
26	C₂×C₄⋊C₄×C₂₃	—	4	8	no	—
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₂₃	D₄	1	4	yes	—
31	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
32	C₄×C₈×C₂₃	—	2	8	no	—
33	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	—	2	8	no	—
35	C₂²×C₄×C₂₃	—	4	4	no	—
36	C₂²×C₄×C₂₃	—	4	4	no	—
37	C₂²×C₄×C₂₃	—	4	8	no	—
38	C₂²×C₄×C₂₃	—	4	8	no	—
39	C₂²×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
40	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₂₃	D₈	1	4	yes	—
42	C₂²×C₄×C₂₃	SD₁₆	1	8	yes	—
43	C₂²×C₄×C₂₃	Q₁₆	1	4	yes	—
44	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₂₃	#41	1	4	yes	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	—	2	4	no	—
53	C₂×C₈×C₂₃	—	2	8	no	—
54	C₂×C₈×C₂₃	—	2	8	no	—
55	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	complemented
56	C₄⋊C₄×C₂₃	—	2	16	no	—
57	C₄⋊C₄×C₂₃	—	4	32	no	—
58	C₄⋊C₄×C₂₃	—	4	8	no	—
59	C₄²×C₂₃	C₄²	1	4	yes	—
60	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
62	#10841	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂³×C₂₃	#124	1	2	yes	—
66	C₂³×C₂₃	#127	1	1	yes	characteristic
67	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂³×C₂₃	C₂×D₈	1	2	yes	—
69	C₂³×C₂₃	C₂×SD₁₆	1	4	yes	—
70	C₂³×C₂₃	C₂×Q₁₆	1	2	yes	—
71	C₂×C₄×C₂₃	—	4	8	no	—
72	C₂×C₄×C₂₃	—	4	16	no	—
73	C₂×C₄×C₂₃	—	8	8	no	—
74	C₂×C₄×C₂₃	—	8	4	no	—
75	C₂×C₄×C₂₃	—	8	4	no	—
76	C₂×C₄×C₂₃	#95	1	8	yes	—
77	C₂×C₄×C₂₃	#102	1	16	yes	—
78	C₂×C₄×C₂₃	#103	1	16	yes	—
79	C₂×C₄×C₂₃	#106	1	8	yes	—
80	C₂×C₄×C₂₃	#107	1	8	yes	—
81	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₂₃	#117	1	2	yes	—
86	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
87	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
88	C₂×C₄×C₂₃	#121	1	4	yes	—
89	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
90	C₂×C₄×C₂₃	#125	1	2	yes	—
91	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
92	C₈×C₂₃	—	2	8	no	—
93	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
94	C₂×#384	C₂×C₂₃	1	2	yes	—
95	C₂×#340	C₂×C₂₃	1	4	yes	—
96	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
97	C₂²×C₂₃	#382	1	1	yes	characteristic
98	C₂²×C₂₃	#384	1	1	yes	characteristic
99	C₂²×C₂₃	#390	1	2	yes	—
100	C₂²×C₂₃	#391	1	2	yes	—
101	C₂²×C₂₃	C₂×#102	1	4	yes	—
102	C₂²×C₂₃	C₂×#103	1	4	yes	—
103	C₂²×C₂₃	C₂×#106	1	2	yes	—
104	C₂²×C₂₃	C₂×#107	1	2	yes	—
105	C₂²×C₂₃	#486	1	4	yes	—
106	C₂²×C₂₃	#487	1	4	yes	—
107	C₂²×C₂₃	#492	1	2	yes	—
108	C₂²×C₂₃	#493	1	1	yes	characteristic
109	C₂²×C₂₃	#494	1	1	yes	characteristic
110	C₂²×C₂₃	#498	1	2	yes	—
111	C₂²×C₂₃	#500	1	2	yes	—
112	C₄×C₂₃	—	8	8	no	—
113	C₄×C₂₃	#340	1	16	yes	complemented
114	C₄×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
115	C₄×C₂₃	#380	1	2	yes	—
116	C₄×C₂₃	#389	1	4	yes	—
117	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
118	C₂²×C₄⋊C₄	—	2	4	no	—
119	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
120	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
121	C₂×C₄×C₈	—	2	4	no	—
122	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	complemented
123	#384	—	2	4	no	—
124	#384	C₄×C₂₃	1	8	yes	complemented
125	#340	—	2	8	no	—
126	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
127	C₂×C₂₃	C₂×#340	1	2	yes	—
128	C₂×C₂₃	#1834	1	2	yes	—
129	C₂×C₂₃	#1844	1	1	yes	characteristic
130	C₂×C₂₃	#1845	1	1	yes	characteristic
131	C₂×C₂₃	#1956	1	2	yes	—
132	C₂×C₂₃	#1957	1	2	yes	—
133	C₂×C₂₃	#1990	1	4	yes	—
134	C₂³×C₄	—	2	4	no	—
135	C₂³×C₄	D₄×C₂₃	1	2	yes	—
136	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
137	C₂²×C₈	—	2	4	no	—
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
140	C₂²×C₈	D₄×C₂₃	1	4	yes	—
141	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
142	C₂×C₄⋊C₄	—	2	8	no	—
143	C₂×C₄⋊C₄	—	2	8	no	—
144	C₂×C₄⋊C₄	—	2	16	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	4	8	no	—
147	C₂×C₄⋊C₄	—	4	8	no	—
148	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
149	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
150	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
151	C₂×C₄²	D₄×C₂₃	1	4	yes	—
152	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
153	C₄×C₈	—	2	8	no	—
154	C₂₃	#10841	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	—	4	8	no	—
161	C₂²×C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
162	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
163	C₂²×C₄	D₈×C₂₃	1	4	yes	—
164	C₂²×C₄	SD₁₆×C₂₃	1	8	yes	—
165	C₂²×C₄	Q₁₆×C₂₃	1	4	yes	—
166	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
168	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
172	C₂²×C₄	#41×C₂₃	1	4	yes	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	complemented
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	4	32	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄²	C₄²×C₂₃	1	4	yes	—
182	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
183	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
184	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
185	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
186	C₂³	C₂₃×#124	1	2	yes	—
187	C₂³	C₂₃×#127	1	1	yes	characteristic
188	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
189	C₂³	C₂×D₈×C₂₃	1	2	yes	—
190	C₂³	C₂×SD₁₆×C₂₃	1	4	yes	—
191	C₂³	C₂×Q₁₆×C₂₃	1	2	yes	—
192	C₂×C₄	—	4	8	no	—
193	C₂×C₄	—	4	16	no	—
194	C₂×C₄	—	8	8	no	—
195	C₂×C₄	—	8	4	no	—
196	C₂×C₄	—	8	4	no	—
197	C₂×C₄	C₂₃×#95	1	8	yes	—
198	C₂×C₄	C₂₃×#102	1	16	yes	—
199	C₂×C₄	C₂₃×#103	1	16	yes	—
200	C₂×C₄	C₂₃×#106	1	8	yes	—
201	C₂×C₄	C₂₃×#107	1	8	yes	—
202	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
206	C₂×C₄	C₂₃×#117	1	2	yes	—
207	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
208	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
209	C₂×C₄	C₂₃×#121	1	4	yes	—
210	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
211	C₂×C₄	C₂₃×#125	1	2	yes	—
212	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
213	C₈	—	2	8	no	—
214	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
215	C₂²	C₂₃×#382	1	1	yes	characteristic
216	C₂²	C₂₃×#384	1	1	yes	characteristic
217	C₂²	C₂₃×#390	1	2	yes	—
218	C₂²	C₂₃×#391	1	2	yes	—
219	C₂²	C₂×C₂₃×#102	1	4	yes	—
220	C₂²	C₂×C₂₃×#103	1	4	yes	—
221	C₂²	C₂×C₂₃×#106	1	2	yes	—
222	C₂²	C₂×C₂₃×#107	1	2	yes	—
223	C₂²	C₂₃×#486	1	4	yes	—
224	C₂²	C₂₃×#487	1	4	yes	—
225	C₂²	C₂₃×#492	1	2	yes	—
226	C₂²	C₂₃×#493	1	1	yes	characteristic
227	C₂²	C₂₃×#494	1	1	yes	characteristic
228	C₂²	C₂₃×#498	1	2	yes	—
229	C₂²	C₂₃×#500	1	2	yes	—
230	C₄	—	8	8	no	—
231	C₄	C₂₃×#340	1	16	yes	complemented
232	C₄	C₄×C₄⋊C₄×C₂₃	1	2	yes	—
233	C₄	C₂₃×#380	1	2	yes	—
234	C₄	C₂₃×#389	1	4	yes	—
235	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
236	C₂	C₂×C₂₃×#340	1	2	yes	—
237	C₂	C₂₃×#1834	1	2	yes	—
238	C₂	C₂₃×#1844	1	1	yes	characteristic
239	C₂	C₂₃×#1845	1	1	yes	characteristic
240	C₂	C₂₃×#1956	1	2	yes	—
241	C₂	C₂₃×#1957	1	2	yes	—
242	C₂	C₂₃×#1990	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₄×C₈×C₂₃	C₄	semidirect
11. C₂₃×#384	C₄	semidirect
55. C₂×C₈×C₂₃	C₄⋊C₄	semidirect
62. #10841	C₂₃	direct (N × H)
113. C₄×C₂₃	#340	semidirect
122. C₂×C₄×C₈	C₄×C₂₃	semidirect
124. #384	C₄×C₂₃	semidirect
154. C₂₃	#10841	direct (N × H)
177. C₂×C₈	C₄⋊C₄×C₂₃	semidirect
231. C₄	C₂₃×#340	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₂₃	2
C₂×C₂₃×#384	2
C₂×C₂₃×#340	2
#10841	23

Subgroups — C23×#10841 #19158467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10841 #19158467