Subgroups — C₂₃×#9706 #19157332

Order 5888.

850 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2867	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#331	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#127	—	2	1	no	—
6	C₂×C₂₃×#127	—	2	2	no	—
7	C₂×C₂₃×#127	—	2	1	no	—
8	C₂×C₂₃×#127	C₄	1	4	yes	—
9	C₂×C₂₃×#127	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×#97	—	2	1	no	—
13	C₄³×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#336	—	2	1	no	—
15	C₂₃×#336	—	2	2	no	—
16	C₂₃×#331	—	2	8	no	—
17	C₂²×D₄×C₂₃	—	2	4	no	—
18	C₂²×D₄×C₂₃	—	2	4	no	—
19	C₂²×D₄×C₂₃	—	2	2	no	—
20	C₂²×D₄×C₂₃	—	2	2	no	—
21	C₂²×D₄×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂₃×#127	—	2	4	no	—
25	C₂₃×#127	—	2	4	no	—
26	C₂₃×#127	—	4	8	no	—
27	C₂₃×#127	—	4	4	no	—
28	C₂₃×#127	—	4	8	no	—
29	C₂₃×#127	—	4	2	no	—
30	C₂₃×#127	—	4	2	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	D₄	1	2	yes	—
34	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×C₄⋊C₈	—	2	4	no	—
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×C₄⋊C₈	D₄	1	4	yes	complemented
39	C₂₃×#97	—	4	2	no	—
40	C₂⁴×C₂₃	—	2	4	no	—
41	C₂×D₄×C₂₃	—	2	8	no	—
42	C₂×D₄×C₂₃	—	4	8	no	—
43	C₂×D₄×C₂₃	—	4	8	no	—
44	C₂×D₄×C₂₃	—	4	8	no	—
45	C₂×D₄×C₂₃	—	4	8	no	—
46	C₂×D₄×C₂₃	—	4	8	no	—
47	C₂×D₄×C₂₃	—	4	8	no	—
48	C₂×D₄×C₂₃	—	4	8	no	—
49	C₂×D₄×C₂₃	—	4	4	no	—
50	C₂×D₄×C₂₃	—	4	2	no	—
51	C₂×D₄×C₂₃	—	4	2	no	—
52	C₂×D₄×C₂₃	—	4	2	no	—
53	C₂×D₄×C₂₃	—	4	2	no	—
54	C₂×D₄×C₂₃	—	4	4	no	—
55	C₂×D₄×C₂₃	—	4	8	no	—
56	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	4	4	no	—
66	C₄²×C₂₃	—	2	4	no	—
67	C₄²×C₂₃	—	2	2	no	—
68	C₄²×C₂₃	—	2	4	no	—
69	C₄²×C₂₃	D₈	1	2	yes	—
70	C₄²×C₂₃	SD₁₆	1	2	yes	—
71	C₄²×C₂₃	C₂×D₄	1	2	yes	—
72	C₄²×C₂₃	#41	1	2	yes	—
73	#9706	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	—	4	4	no	—
75	C₂³×C₂₃	—	4	8	no	—
76	C₂³×C₂₃	—	4	4	no	—
77	C₂³×C₂₃	—	4	8	no	—
78	C₂³×C₂₃	—	4	4	no	—
79	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	D₄×C₂₃	—	8	8	no	—
81	D₄×C₂₃	—	8	16	no	—
82	D₄×C₂₃	—	8	16	no	—
83	D₄×C₂₃	—	8	4	no	—
84	D₄×C₂₃	—	8	4	no	—
85	D₄×C₂₃	—	8	4	no	—
86	D₄×C₂₃	—	8	4	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	#100	1	4	yes	—
93	C₂×C₄×C₂₃	#102	1	4	yes	—
94	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₂₃	#121	1	2	yes	—
96	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
102	C₈×C₂₃	—	4	4	no	—
103	C₈×C₂₃	—	8	2	no	—
104	#2867	C₂×C₂₃	1	1	yes	characteristic
105	#1831	C₂×C₂₃	1	1	yes	characteristic, complemented
106	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
107	C₂×#331	C₂×C₂₃	1	4	yes	complemented
108	C₂²×C₂₃	—	8	8	no	—
109	C₂²×C₂₃	—	8	8	no	—
110	C₂²×C₂₃	—	8	4	no	—
111	C₂²×C₂₃	—	8	4	no	—
112	C₂²×C₂₃	—	8	4	no	—
113	C₂²×C₂₃	#390	1	1	yes	characteristic
114	C₂²×C₂₃	C₂×#100	1	2	yes	—
115	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
116	C₂²×C₂₃	#418	1	2	yes	—
117	C₂²×C₂₃	#418	1	1	yes	characteristic
118	C₄×C₂₃	—	2	4	no	—
119	C₄×C₂₃	—	2	4	no	—
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	#459	1	2	yes	—
122	C₄×C₂₃	#460	1	2	yes	—
123	C₄×C₂₃	#490	1	2	yes	—
124	C₄×C₂₃	#496	1	2	yes	—
125	C₂×#127	—	2	1	no	—
126	C₂×#127	—	2	2	no	—
127	C₂×#127	—	2	1	no	—
128	C₂×#127	C₄×C₂₃	1	4	yes	—
129	C₂×#127	C₂²×C₂₃	1	2	yes	—
130	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
132	C₂×#97	—	2	1	no	—
133	C₄³	C₂²×C₂₃	1	1	yes	characteristic
134	#336	—	2	2	no	—
135	#336	—	2	1	no	—
136	#331	—	2	8	no	—
137	C₂×C₂₃	—	16	4	no	—
138	C₂×C₂₃	#1545	1	2	yes	—
139	C₂×C₂₃	#1555	1	2	yes	—
140	C₂×C₂₃	#1961	1	1	yes	characteristic
141	C₂×C₂₃	#1965	1	1	yes	characteristic
142	C₂×C₂₃	#1982	1	1	yes	characteristic
143	C₂²×D₄	—	2	4	no	—
144	C₂²×D₄	—	2	4	no	—
145	C₂²×D₄	—	2	2	no	—
146	C₂²×D₄	—	2	2	no	—
147	C₂²×D₄	C₂×C₄×C₂₃	1	4	yes	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	#127	—	2	4	no	—
151	#127	—	2	4	no	—
152	#127	—	4	8	no	—
153	#127	—	4	4	no	—
154	#127	—	4	8	no	—
155	#127	—	4	2	no	—
156	#127	—	4	2	no	—
157	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
158	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
159	C₂×C₄²	D₄×C₂₃	1	2	yes	—
160	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	D₄×C₂₃	1	4	yes	complemented
165	#97	—	4	2	no	—
166	C₂₃	#9706	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	—	2	4	no	—
168	C₂×D₄	—	2	8	no	—
169	C₂×D₄	—	4	8	no	—
170	C₂×D₄	—	4	8	no	—
171	C₂×D₄	—	4	8	no	—
172	C₂×D₄	—	4	8	no	—
173	C₂×D₄	—	4	8	no	—
174	C₂×D₄	—	4	8	no	—
175	C₂×D₄	—	4	8	no	—
176	C₂×D₄	—	4	4	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	8	no	—
183	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	4	no	—
196	C₄²	D₈×C₂₃	1	2	yes	—
197	C₄²	SD₁₆×C₂₃	1	2	yes	—
198	C₄²	C₂×D₄×C₂₃	1	2	yes	—
199	C₄²	#41×C₂₃	1	2	yes	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	8	no	—
204	C₂³	—	4	4	no	—
205	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	D₄	—	8	8	no	—
207	D₄	—	8	16	no	—
208	D₄	—	8	16	no	—
209	D₄	—	8	4	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	C₂₃×#100	1	4	yes	—
219	C₂×C₄	C₂₃×#102	1	4	yes	—
220	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
221	C₂×C₄	C₂₃×#121	1	2	yes	—
222	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
223	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
224	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
226	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
227	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
228	C₈	—	4	4	no	—
229	C₈	—	8	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	—	8	4	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	8	no	—
234	C₂²	—	8	8	no	—
235	C₂²	C₂₃×#390	1	1	yes	characteristic
236	C₂²	C₂×C₂₃×#100	1	2	yes	—
237	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
238	C₂²	C₂₃×#418	1	2	yes	—
239	C₂²	C₂₃×#418	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	2	4	no	—
243	C₄	C₂₃×#459	1	2	yes	—
244	C₄	C₂₃×#460	1	2	yes	—
245	C₄	C₂₃×#490	1	2	yes	—
246	C₄	C₂₃×#496	1	2	yes	—
247	C₂	—	16	4	no	—
248	C₂	C₂₃×#1545	1	2	yes	—
249	C₂	C₂₃×#1555	1	2	yes	—
250	C₂	C₂₃×#1961	1	1	yes	characteristic
251	C₂	C₂₃×#1965	1	1	yes	characteristic
252	C₂	C₂₃×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂₃×#2867	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×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. C₂₃×#1767	C₂	semidirect
4. C₂×C₂₃×#331	C₂	semidirect
38. C₂₃×C₄⋊C₈	D₄	semidirect
73. #9706	C₂₃	direct (N × H)
105. #1831	C₂×C₂₃	semidirect
106. #1767	C₂×C₂₃	semidirect
107. C₂×#331	C₂×C₂₃	semidirect
164. C₄⋊C₈	D₄×C₂₃	semidirect
166. C₂₃	#9706	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×#2867	2
C₂₃×#1831	2
C₂₃×#1767	2
C₂×C₂₃×#331	2
#9706	23

Subgroups — C23×#9706 #19157332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9706 #19157332