Subgroups — C₂₃×#8679 #19156305

Order 5888.

350 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	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1835	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1518	C₂	1	2	yes	maximal
5	C₂₃×#1286	C₂	1	2	yes	maximal
6	C₂₃×#533	C₂²	1	1	yes	characteristic
7	C₂₃×#446	—	2	1	no	—
8	C₂₃×#446	C₄	1	2	yes	complemented
9	C₈×Q₈×C₂₃	—	2	1	no	—
10	C₈×Q₈×C₂₃	C₄	1	2	yes	complemented
11	C₂₃×#431	C₂²	1	2	yes	—
12	C₂₃×#424	—	2	1	no	—
13	C₂₃×#423	C₂²	1	1	yes	characteristic
14	C₂₃×#405	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#382	C₂²	1	2	yes	—
17	C₂₃×#340	—	2	1	no	—
18	C₂₃×#340	—	2	1	no	—
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#334	—	2	1	no	—
21	C₂₃×#328	—	2	4	no	—
22	C₂×M₁₆×C₂₃	—	2	1	no	—
23	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	C₂₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	C₂₃×#125	—	2	1	no	—
28	C₂₃×C₂²⋊Q₈	—	2	1	no	—
29	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
30	C₄×Q₈×C₂₃	—	2	1	no	—
31	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂₃×#117	—	2	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×C₄⋊C₈	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	1	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₂₃×C₂²⋊C₈	—	2	2	no	—
41	C₂₃×#97	—	2	2	no	—
42	C₄×C₈×C₂₃	—	2	2	no	—
43	C₄×C₈×C₂₃	—	2	2	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₂₃×#95	—	2	2	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂×Q₈×C₂₃	—	2	1	no	—
51	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
56	M₁₆×C₂₃	—	4	2	no	—
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	4	2	no	—
61	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	—	2	2	no	—
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₄⋊C₄×C₂₃	—	2	1	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	4	2	no	—
72	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
73	C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₂₃	#41	1	1	yes	characteristic
79	#8679	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	#121	1	1	yes	characteristic
81	Q₈×C₂₃	—	4	2	no	—
82	Q₈×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	4	1	no	—
86	C₂×C₄×C₂₃	—	4	1	no	—
87	C₂×C₄×C₂₃	—	8	2	no	—
88	C₂×C₄×C₂₃	—	8	2	no	—
89	C₂×C₄×C₂₃	—	8	2	no	—
90	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#135	1	2	yes	—
97	C₂×C₄×C₂₃	#137	1	2	yes	—
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	—	2	1	no	—
100	C₈×C₂₃	—	2	1	no	—
101	C₈×C₂₃	—	2	1	no	—
102	C₈×C₂₃	—	4	2	no	—
103	C₈×C₂₃	—	8	2	no	—
104	#2960	C₂×C₂₃	1	1	yes	characteristic
105	#1835	C₂×C₂₃	1	1	yes	characteristic
106	#1564	C₂×C₂₃	1	1	yes	characteristic
107	#1518	C₂×C₂₃	1	2	yes	—
108	#1286	C₂×C₂₃	1	2	yes	—
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	—	2	1	no	—
111	C₂²×C₂₃	#380	1	1	yes	characteristic
112	C₂²×C₂₃	#464	1	2	yes	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	4	2	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	—	16	2	no	—
120	C₄×C₂₃	#416	1	1	yes	characteristic
121	C₄×C₂₃	#419	1	1	yes	characteristic
122	C₄×C₂₃	#479	1	2	yes	—
123	#533	C₂²×C₂₃	1	1	yes	characteristic
124	#446	—	2	1	no	—
125	#446	C₄×C₂₃	1	2	yes	complemented
126	C₈×Q₈	—	2	1	no	—
127	C₈×Q₈	C₄×C₂₃	1	2	yes	complemented
128	#431	C₂²×C₂₃	1	2	yes	—
129	#424	—	2	1	no	—
130	#423	C₂²×C₂₃	1	1	yes	characteristic
131	#405	—	2	1	no	—
132	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
133	#382	C₂²×C₂₃	1	2	yes	—
134	#340	—	2	1	no	—
135	#340	—	2	1	no	—
136	#335	—	2	1	no	—
137	#334	—	2	1	no	—
138	#328	—	2	4	no	—
139	C₂×C₂₃	—	2	2	no	—
140	C₂×C₂₃	#1805	1	1	yes	characteristic
141	C₂×C₂₃	#1812	1	1	yes	characteristic
142	C₂×C₂₃	#1814	1	1	yes	characteristic
143	C₂×M₁₆	—	2	1	no	—
144	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
145	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
146	C₄⋊Q₈	Q₈×C₂₃	1	1	yes	characteristic
147	C₄⋊Q₈	Q₈×C₂₃	1	1	yes	characteristic
148	#125	—	2	1	no	—
149	C₂²⋊Q₈	—	2	1	no	—
150	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
153	#117	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
161	C₂²⋊C₈	—	2	2	no	—
162	#97	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
168	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
169	#95	—	2	2	no	—
170	#95	—	2	2	no	—
171	C₂₃	#8679	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
178	M₁₆	—	4	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
198	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
200	C₄²	#41×C₂₃	1	1	yes	characteristic
201	C₂³	C₂₃×#121	1	1	yes	characteristic
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
214	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
215	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
216	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
217	C₂×C₄	C₂₃×#135	1	2	yes	—
218	C₂×C₄	C₂₃×#137	1	2	yes	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	4	2	no	—
224	C₈	—	8	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	C₂₃×#380	1	1	yes	characteristic
228	C₂²	C₂₃×#464	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	16	2	no	—
236	C₄	C₂₃×#416	1	1	yes	characteristic
237	C₄	C₂₃×#419	1	1	yes	characteristic
238	C₄	C₂₃×#479	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	C₂₃×#1805	1	1	yes	characteristic
241	C₂	C₂₃×#1812	1	1	yes	characteristic
242	C₂	C₂₃×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂₃×#446	C₄	semidirect
10. C₈×Q₈×C₂₃	C₄	semidirect
79. #8679	C₂₃	direct (N × H)
125. #446	C₄×C₂₃	semidirect
127. C₈×Q₈	C₄×C₂₃	semidirect
171. C₂₃	#8679	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2960	2
C₂₃×#1835	2
C₂₃×#1564	2
C₂₃×#1518	2
C₂₃×#1286	2
#8679	23

Subgroups — C23×#8679 #19156305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8679 #19156305