Subgroups — C₂₃×#29225 #19176851

Order 5888.

714 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2100	C₂	1	4	yes	maximal
4	C₂₃×#1992	C₂	1	2	yes	maximal
5	C₂₃×#1988	C₂	1	4	yes	maximal
6	C₂₃×#1836	C₂	1	2	yes	maximal
7	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×#97	C₂²	1	2	yes	—
13	C₂₃×#389	C₂²	1	2	yes	—
14	C₂₃×#387	—	2	2	no	—
15	C₂₃×#384	—	2	2	no	—
16	C₂₃×#384	C₂²	1	2	yes	—
17	C₂₃×#384	C₂²	1	2	yes	—
18	C₂₃×#382	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
20	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₄³×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#340	C₂²	1	8	yes	—
23	C₂₃×#340	C₂²	1	8	yes	—
24	C₂₃×#336	C₂²	1	2	yes	—
25	C₂²×Q₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
27	C₂₃×C₄⋊Q₈	—	2	8	no	—
28	C₂₃×C₄⋊Q₈	—	4	8	no	—
29	C₂₃×#125	—	2	2	no	—
30	C₂₃×#125	Q₈	1	4	yes	complemented
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₂₃	—	2	2	no	—
40	C₂×C₄²×C₂₃	D₄	1	2	yes	—
41	C₂×C₄²×C₂₃	D₄	1	2	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×C₄⋊C₈	—	2	4	no	—
46	C₂₃×#97	—	2	4	no	—
47	C₂₃×#95	—	2	2	no	—
48	C₂₃×#95	—	2	2	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂×Q₈×C₂₃	—	4	4	no	—
51	C₂×Q₈×C₂₃	—	4	8	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₂₃	—	2	4	no	—
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₄⋊C₄×C₂₃	—	2	8	no	—
64	C₄⋊C₄×C₂₃	—	2	2	no	—
65	C₄⋊C₄×C₂₃	—	2	8	no	—
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₄⋊C₄×C₂₃	—	4	8	no	—
68	C₄⋊C₄×C₂₃	—	4	8	no	—
69	C₄⋊C₄×C₂₃	—	4	4	no	—
70	C₄⋊C₄×C₂₃	—	4	4	no	—
71	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
72	C₄²×C₂₃	—	2	4	no	—
73	C₄²×C₂₃	—	2	4	no	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₄²×C₂₃	—	4	4	no	—
76	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
77	C₄²×C₂₃	#41	1	2	yes	—
78	C₄²×C₂₃	#41	1	4	yes	—
79	#29225	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
81	Q₈×C₂₃	—	8	8	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	4	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	4	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	4	8	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
96	C₂×C₄×C₂₃	#124	1	2	yes	—
97	C₂×C₄×C₂₃	#124	1	2	yes	—
98	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
99	C₂×C₄×C₂₃	#137	1	4	yes	—
100	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
103	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
104	C₈×C₂₃	—	4	4	no	—
105	#2868	C₂×C₂₃	1	1	yes	characteristic
106	C₄×#125	C₂×C₂₃	1	1	yes	characteristic
107	#2100	C₂×C₂₃	1	4	yes	—
108	#1992	C₂×C₂₃	1	2	yes	—
109	#1988	C₂×C₂₃	1	4	yes	—
110	#1836	C₂×C₂₃	1	2	yes	—
111	#1767	C₂×C₂₃	1	1	yes	characteristic
112	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
114	C₂²×C₂₃	#536	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
116	C₂²×C₂₃	C₂×#137	1	2	yes	—
117	C₄×C₂₃	—	2	4	no	—
118	C₄×C₂₃	—	2	2	no	—
119	C₄×C₂₃	—	2	4	no	—
120	C₄×C₂₃	—	4	4	no	—
121	C₄×C₂₃	—	8	4	no	—
122	C₄×C₂₃	—	8	2	no	—
123	C₄×C₂₃	#490	1	4	yes	—
124	C₄×C₂₃	#556	1	2	yes	—
125	C₄×C₂₃	#557	1	2	yes	—
126	C₂×C₄⋊Q₈	—	2	2	no	—
127	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
128	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
130	C₂×#97	C₂²×C₂₃	1	2	yes	—
131	#389	C₂²×C₂₃	1	2	yes	—
132	#387	—	2	2	no	—
133	#384	—	2	2	no	—
134	#384	C₂²×C₂₃	1	2	yes	—
135	#384	C₂²×C₂₃	1	2	yes	—
136	#382	C₂²×C₂₃	1	1	yes	characteristic
137	C₄×C₄⋊C₄	—	2	2	no	—
138	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
139	C₄³	C₂²×C₂₃	1	1	yes	characteristic
140	#340	C₂²×C₂₃	1	8	yes	—
141	#340	C₂²×C₂₃	1	8	yes	—
142	#336	C₂²×C₂₃	1	2	yes	—
143	C₂×C₂₃	#2568	1	1	yes	characteristic
144	C₂×C₂₃	#3079	1	2	yes	—
145	C₂×C₂₃	C₂×#490	1	1	yes	characteristic
146	C₂×C₂₃	#3139	1	1	yes	characteristic
147	C₂×C₂₃	#3239	1	2	yes	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
150	C₄⋊Q₈	—	2	8	no	—
151	C₄⋊Q₈	—	4	8	no	—
152	#125	—	2	2	no	—
153	#125	Q₈×C₂₃	1	4	yes	complemented
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	D₄×C₂₃	1	2	yes	—
164	C₂×C₄²	D₄×C₂₃	1	2	yes	—
165	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	C₄⋊C₈	—	2	4	no	—
169	#97	—	2	4	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	C₂₃	#29225	1	1	yes	characteristic, direct product factor, Sylow
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	8	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	8	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	8	no	—
192	C₄⋊C₄	—	4	8	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	4	4	no	—
200	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
201	C₄²	#41×C₂₃	1	2	yes	—
202	C₄²	#41×C₂₃	1	4	yes	—
203	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
204	Q₈	—	8	8	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	8	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
219	C₂×C₄	C₂₃×#124	1	2	yes	—
220	C₂×C₄	C₂₃×#124	1	2	yes	—
221	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
222	C₂×C₄	C₂₃×#137	1	4	yes	—
223	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
224	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
226	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
229	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
230	C₂²	C₂₃×#536	1	1	yes	characteristic
231	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
232	C₂²	C₂×C₂₃×#137	1	2	yes	—
233	C₄	—	2	4	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	4	no	—
239	C₄	C₂₃×#490	1	4	yes	—
240	C₄	C₂₃×#556	1	2	yes	—
241	C₄	C₂₃×#557	1	2	yes	—
242	C₂	C₂₃×#2568	1	1	yes	characteristic
243	C₂	C₂₃×#3079	1	2	yes	—
244	C₂	C₂×C₂₃×#490	1	1	yes	characteristic
245	C₂	C₂₃×#3139	1	1	yes	characteristic
246	C₂	C₂₃×#3239	1	2	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 ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
30. C₂₃×#125	Q₈	semidirect
79. #29225	C₂₃	direct (N × H)
153. #125	Q₈×C₂₃	semidirect
173. C₂₃	#29225	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×#2868	2
C₄×C₂₃×#125	2
C₂₃×#2100	2
C₂₃×#1992	2
C₂₃×#1988	2
C₂₃×#1836	2
C₂₃×#1767	2
#29225	23

Subgroups — C23×#29225 #19176851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29225 #19176851