Subgroups — C₂₃×#9648 #19157274

Order 5888.

726 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#127	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₂₃	—	2	2	no	—
7	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₃×#97	—	2	2	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#390	—	2	2	no	—
12	C₂₃×#390	C₄	1	2	yes	—
13	C₂₃×#384	C₄	1	1	yes	characteristic
14	C₄³×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#336	—	2	2	no	—
16	C₂₃×D₄⋊C₈	—	2	8	no	—
17	C₂²×D₄×C₂₃	—	2	2	no	—
18	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	—	2	2	no	—
22	C₂₃×#127	—	2	4	no	—
23	C₄×D₄×C₂₃	—	2	4	no	—
24	C₄×D₄×C₂₃	—	2	4	no	—
25	C₄×D₄×C₂₃	—	4	8	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
29	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	D₄	1	2	yes	—
33	C₂×C₄²×C₂₃	D₄	1	2	yes	—
34	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×C₄⋊C₈	—	2	4	no	—
36	C₂₃×#97	—	2	8	no	—
37	C₄×C₈×C₂₃	—	2	2	no	—
38	C₄×C₈×C₂₃	D₄	1	4	yes	complemented
39	C₂⁴×C₂₃	—	2	2	no	—
40	C₂×D₄×C₂₃	—	2	8	no	—
41	C₂×D₄×C₂₃	—	2	4	no	—
42	C₂×D₄×C₂₃	—	2	2	no	—
43	C₂×D₄×C₂₃	—	2	4	no	—
44	C₂×D₄×C₂₃	—	4	4	no	—
45	C₂×D₄×C₂₃	—	4	4	no	—
46	C₂×D₄×C₂₃	M₁₆	1	4	yes	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	4	2	no	—
49	C₂²×C₄×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	—	4	8	no	—
51	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	—	2	8	no	—
57	C₂×C₈×C₂₃	—	2	4	no	—
58	C₂×C₈×C₂₃	—	4	2	no	—
59	C₂×C₈×C₂₃	—	4	4	no	—
60	C₄⋊C₄×C₂₃	—	2	4	no	—
61	C₄⋊C₄×C₂₃	—	4	4	no	—
62	C₂²⋊C₄×C₂₃	—	4	8	no	—
63	C₂²⋊C₄×C₂₃	—	4	4	no	—
64	C₄²×C₂₃	—	2	4	no	—
65	C₄²×C₂₃	—	2	2	no	—
66	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	—	2	4	no	—
68	C₄²×C₂₃	C₂×D₄	1	2	yes	—
69	C₄²×C₂₃	#41	1	2	yes	—
70	#9648	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₂₃	—	2	4	no	—
72	C₂³×C₂₃	—	4	2	no	—
73	C₂³×C₂₃	—	4	2	no	—
74	C₂³×C₂₃	—	4	4	no	—
75	C₂³×C₂₃	—	4	4	no	—
76	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	D₄×C₂₃	—	2	8	no	—
78	D₄×C₂₃	—	4	16	no	—
79	D₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	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	—	4	2	no	—
87	C₂×C₄×C₂₃	—	4	4	no	—
88	C₂×C₄×C₂₃	—	8	4	no	—
89	C₂×C₄×C₂₃	—	8	4	no	—
90	C₂×C₄×C₂₃	#104	1	4	yes	—
91	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	#121	1	2	yes	—
93	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₂₃	#131	1	2	yes	—
97	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
98	C₈×C₂₃	—	2	8	no	—
99	C₈×C₂₃	—	8	2	no	—
100	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
101	#1831	C₂×C₂₃	1	1	yes	characteristic, complemented
102	C₄×#97	C₂×C₂₃	1	1	yes	characteristic, complemented
103	C₂×D₄⋊C₈	C₂×C₂₃	1	4	yes	complemented
104	C₂²×C₂₃	—	4	4	no	—
105	C₂²×C₂₃	—	4	8	no	—
106	C₂²×C₂₃	—	8	2	no	—
107	C₂²×C₂₃	—	8	2	no	—
108	C₂²×C₂₃	—	8	4	no	—
109	C₂²×C₂₃	#390	1	1	yes	characteristic
110	C₂²×C₂₃	#407	1	1	yes	characteristic
111	C₂²×C₂₃	#408	1	1	yes	characteristic
112	C₂²×C₂₃	#418	1	2	yes	—
113	C₂²×C₂₃	C₂×#104	1	2	yes	—
114	C₄×C₂₃	—	2	2	no	—
115	C₄×C₂₃	—	2	4	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	—	8	2	no	—
118	C₄×C₂₃	#435	1	4	yes	—
119	C₄×C₂₃	#490	1	2	yes	—
120	C₄×C₂₃	#496	1	2	yes	—
121	C₂×#127	C₄×C₂₃	1	1	yes	characteristic
122	C₂×C₄×D₄	—	2	2	no	—
123	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
124	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
125	C₂×#97	—	2	2	no	—
126	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
127	#390	—	2	2	no	—
128	#390	C₄×C₂₃	1	2	yes	—
129	#384	C₄×C₂₃	1	1	yes	characteristic
130	C₄³	C₂²×C₂₃	1	1	yes	characteristic
131	#336	—	2	2	no	—
132	D₄⋊C₈	—	2	8	no	—
133	C₂×C₂₃	—	8	4	no	—
134	C₂×C₂₃	#1484	1	2	yes	—
135	C₂×C₂₃	#1490	1	2	yes	—
136	C₂×C₂₃	#1955	1	1	yes	characteristic
137	C₂×C₂₃	#1961	1	1	yes	characteristic
138	C₂×C₂₃	#1962	1	1	yes	characteristic
139	C₂²×D₄	—	2	2	no	—
140	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
141	C₂³×C₄	—	2	2	no	—
142	C₂²×C₈	—	2	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	#127	—	2	4	no	—
145	C₄×D₄	—	2	4	no	—
146	C₄×D₄	—	2	4	no	—
147	C₄×D₄	—	4	8	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
150	C₂×C₂²⋊C₄	—	2	4	no	—
151	C₂×C₂²⋊C₄	—	2	2	no	—
152	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄²	D₄×C₂₃	1	2	yes	—
155	C₂×C₄²	D₄×C₂₃	1	2	yes	—
156	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
157	C₄⋊C₈	—	2	4	no	—
158	#97	—	2	8	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	D₄×C₂₃	1	4	yes	complemented
161	C₂₃	#9648	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	—	2	2	no	—
163	C₂×D₄	—	2	8	no	—
164	C₂×D₄	—	2	4	no	—
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	M₁₆×C₂₃	1	4	yes	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	8	no	—
174	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
175	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	4	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₂²⋊C₄	—	4	8	no	—
186	C₂²⋊C₄	—	4	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	4	no	—
191	C₄²	C₂×D₄×C₂₃	1	2	yes	—
192	C₄²	#41×C₂₃	1	2	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	4	2	no	—
195	C₂³	—	4	2	no	—
196	C₂³	—	4	4	no	—
197	C₂³	—	4	4	no	—
198	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
199	D₄	—	2	8	no	—
200	D₄	—	4	16	no	—
201	D₄	—	4	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	C₂₃×#104	1	4	yes	—
213	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
214	C₂×C₄	C₂₃×#121	1	2	yes	—
215	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
216	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
217	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
218	C₂×C₄	C₂₃×#131	1	2	yes	—
219	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
220	C₈	—	2	8	no	—
221	C₈	—	8	2	no	—
222	C₂²	—	4	4	no	—
223	C₂²	—	4	8	no	—
224	C₂²	—	8	2	no	—
225	C₂²	—	8	2	no	—
226	C₂²	—	8	4	no	—
227	C₂²	C₂₃×#390	1	1	yes	characteristic
228	C₂²	C₂₃×#407	1	1	yes	characteristic
229	C₂²	C₂₃×#408	1	1	yes	characteristic
230	C₂²	C₂₃×#418	1	2	yes	—
231	C₂²	C₂×C₂₃×#104	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	C₂₃×#435	1	4	yes	—
237	C₄	C₂₃×#490	1	2	yes	—
238	C₄	C₂₃×#496	1	2	yes	—
239	C₂	—	8	4	no	—
240	C₂	C₂₃×#1484	1	2	yes	—
241	C₂	C₂₃×#1490	1	2	yes	—
242	C₂	C₂₃×#1955	1	1	yes	characteristic
243	C₂	C₂₃×#1961	1	1	yes	characteristic
244	C₂	C₂₃×#1962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂×C₂₃×#127	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1831	C₂	semidirect
3. C₄×C₂₃×#97	C₂	semidirect
4. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
38. C₄×C₈×C₂₃	D₄	semidirect
70. #9648	C₂₃	direct (N × H)
101. #1831	C₂×C₂₃	semidirect
102. C₄×#97	C₂×C₂₃	semidirect
103. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
160. C₄×C₈	D₄×C₂₃	semidirect
161. C₂₃	#9648	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#127	2
C₂₃×#1831	2
C₄×C₂₃×#97	2
C₂×C₂₃×D₄⋊C₈	2
#9648	23

Subgroups — C23×#9648 #19157274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9648 #19157274