Subgroups — C₂₃×#9788 #19157414

Order 5888.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1528	C₂	1	2	yes	maximal
4	C₂₃×#1470	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#125	C₂²	1	2	yes	—
7	C₂₃×#499	—	2	1	no	—
8	C₂₃×#499	—	2	1	no	—
9	C₂₃×#499	—	2	1	no	—
10	C₂₃×#499	—	2	1	no	—
11	C₂₃×#432	C₂²	1	2	yes	—
12	C₂₃×#431	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
14	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
15	C₂×C₂₃×#106	C₄	1	2	yes	—
16	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#334	—	2	1	no	—
20	C₂₃×#330	—	2	4	no	—
21	C₂₃×#324	—	2	4	no	—
22	C₂₃×#322	—	2	1	no	—
23	C₈²×C₂₃	—	2	1	no	—
24	C₂×M₁₆×C₂₃	—	2	1	no	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂₃×#125	—	2	2	no	—
28	C₂₃×#125	—	2	2	no	—
29	C₂₃×#125	—	2	2	no	—
30	C₂₃×#125	—	2	2	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×#107	—	2	1	no	—
37	C₂₃×#107	—	2	1	no	—
38	C₂₃×#107	—	2	1	no	—
39	C₂₃×#107	—	2	1	no	—
40	C₂₃×#106	—	2	2	no	—
41	C₂₃×#106	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	1	no	—
44	C₂₃×C₂²⋊C₈	—	2	2	no	—
45	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
46	C₂₃×#97	—	2	2	no	—
47	C₂₃×#97	D₄	1	2	yes	—
48	C₄×C₈×C₂₃	—	2	1	no	—
49	C₄×C₈×C₂₃	—	2	1	no	—
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
59	M₁₆×C₂₃	—	2	4	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₄⋊C₄×C₂₃	—	2	2	no	—
71	C₄⋊C₄×C₂₃	—	2	2	no	—
72	C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₄⋊C₄×C₂₃	—	4	2	no	—
74	C₄⋊C₄×C₂₃	—	4	2	no	—
75	C₄⋊C₄×C₂₃	—	4	2	no	—
76	C₄⋊C₄×C₂₃	—	4	2	no	—
77	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₂₃	#41	1	1	yes	characteristic
81	#9788	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	4	1	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	4	no	—
90	C₂×C₄×C₂₃	—	8	2	no	—
91	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
97	C₈×C₂₃	—	2	4	no	—
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	—	2	1	no	—
100	C₈×C₂₃	—	4	2	no	—
101	C₈×C₂₃	—	8	2	no	—
102	C₈×C₂₃	#101	1	2	yes	—
103	C₈×C₂₃	#101	1	2	yes	—
104	C₂×#499	C₂×C₂₃	1	1	yes	characteristic
105	#1568	C₂×C₂₃	1	1	yes	characteristic
106	#1528	C₂×C₂₃	1	2	yes	—
107	#1470	C₂×C₂₃	1	2	yes	—
108	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	—	2	1	no	—
111	C₂²×C₂₃	#390	1	1	yes	characteristic
112	C₂²×C₂₃	#443	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	—	2	1	no	—
118	C₄×C₂₃	—	8	2	no	—
119	C₄×C₂₃	—	8	2	no	—
120	C₄×C₂₃	C₂×#101	1	1	yes	characteristic
121	C₄×C₂₃	C₂×#101	1	1	yes	characteristic
122	C₂×#125	C₂²×C₂₃	1	2	yes	—
123	#499	—	2	1	no	—
124	#499	—	2	1	no	—
125	#499	—	2	1	no	—
126	#499	—	2	1	no	—
127	#432	C₂²×C₂₃	1	2	yes	—
128	#431	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
130	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
131	C₂×#106	C₄×C₂₃	1	2	yes	—
132	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
134	#335	—	2	1	no	—
135	#334	—	2	1	no	—
136	#330	—	2	4	no	—
137	#324	—	2	4	no	—
138	#322	—	2	1	no	—
139	C₈²	—	2	1	no	—
140	C₂×C₂₃	—	2	2	no	—
141	C₂×C₂₃	#1966	1	1	yes	characteristic
142	C₂×C₂₃	#1974	1	1	yes	characteristic
143	C₂×C₂₃	#1976	1	1	yes	characteristic
144	C₂×M₁₆	—	2	1	no	—
145	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
147	#125	—	2	2	no	—
148	#125	—	2	2	no	—
149	#125	—	2	2	no	—
150	#125	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
156	#107	—	2	1	no	—
157	#107	—	2	1	no	—
158	#107	—	2	1	no	—
159	#107	—	2	1	no	—
160	#106	—	2	2	no	—
161	#106	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₂²⋊C₈	—	2	2	no	—
165	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
166	#97	—	2	2	no	—
167	#97	D₄×C₂₃	1	2	yes	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
171	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
172	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
173	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
174	C₂₃	#9788	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
180	M₁₆	—	2	4	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₄²	#41×C₂₃	1	1	yes	characteristic
202	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
214	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
215	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
216	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
217	C₈	—	2	4	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	4	2	no	—
221	C₈	—	8	2	no	—
222	C₈	C₂₃×#101	1	2	yes	—
223	C₈	C₂₃×#101	1	2	yes	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	C₂₃×#390	1	1	yes	characteristic
227	C₂²	C₂₃×#443	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	8	2	no	—
234	C₄	—	8	2	no	—
235	C₄	C₂×C₂₃×#101	1	1	yes	characteristic
236	C₄	C₂×C₂₃×#101	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	C₂₃×#1966	1	1	yes	characteristic
239	C₂	C₂₃×#1974	1	1	yes	characteristic
240	C₂	C₂₃×#1976	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#390
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ 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
81. #9788	C₂₃	direct (N × H)
174. C₂₃	#9788	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#499	2
C₂₃×#1568	2
C₂₃×#1528	2
C₂₃×#1470	2
C₈×M₁₆×C₂₃	2
#9788	23

Subgroups — C23×#9788 #19157414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9788 #19157414