Subgroups — C₂₃×#9664 #19157290

Order 5888.

482 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×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1984	C₂	1	2	yes	maximal
3	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₄×Q₁₆×C₂₃	—	2	2	no	—
9	C₄×Q₁₆×C₂₃	—	2	2	no	—
10	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×#103	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#336	—	2	4	no	—
17	C₂₃×#334	—	2	2	no	—
18	C₂₃×Q₈⋊C₈	—	2	4	no	—
19	C₂₃×#322	—	2	2	no	—
20	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×Q₁₆×C₂₃	—	2	4	no	—
22	C₂×Q₁₆×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	—	2	1	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	D₄	1	2	yes	—
28	C₄×Q₈×C₂₃	—	2	2	no	—
29	C₄×Q₈×C₂₃	—	2	2	no	—
30	C₄×Q₈×C₂₃	—	2	2	no	—
31	C₄×Q₈×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₂₃	—	2	2	no	—
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×#107	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×#103	—	2	4	no	—
40	C₄×C₈×C₂₃	—	2	1	no	—
41	C₄×C₈×C₂₃	—	2	1	no	—
42	C₄×C₈×C₂₃	—	2	1	no	—
43	C₄×C₈×C₂₃	D₄	1	2	yes	—
44	C₄×C₈×C₂₃	D₄	1	2	yes	—
45	C₄×C₈×C₂₃	D₄	1	2	yes	—
46	C₂×Q₈×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	2	2	no	—
48	C₂×Q₈×C₂₃	—	2	2	no	—
49	C₂×Q₈×C₂₃	—	2	2	no	—
50	C₂×Q₈×C₂₃	—	2	2	no	—
51	C₂×Q₈×C₂₃	M₁₆	1	4	yes	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
57	Q₁₆×C₂₃	—	2	8	no	—
58	C₂×C₈×C₂₃	—	2	4	no	—
59	C₂×C₈×C₂₃	—	2	2	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	4	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	4	4	no	—
66	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
68	C₄⋊C₄×C₂₃	—	2	2	no	—
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	—	4	2	no	—
73	C₄²×C₂₃	—	4	2	no	—
74	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₂₃	#41	1	1	yes	characteristic
78	#9664	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
80	Q₈×C₂₃	—	2	4	no	—
81	Q₈×C₂₃	—	2	4	no	—
82	Q₈×C₂₃	—	4	2	no	—
83	Q₈×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	4	no	—
86	C₂×C₄×C₂₃	—	4	2	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₂₃	#131	1	2	yes	—
98	C₂×C₄×C₂₃	#137	1	4	yes	—
99	C₈×C₂₃	—	2	4	no	—
100	C₈×C₂₃	—	2	2	no	—
101	C₈×C₂₃	—	2	2	no	—
102	C₈×C₂₃	—	4	2	no	—
103	C₈×C₂₃	—	8	2	no	—
104	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
105	#1984	C₂×C₂₃	1	2	yes	—
106	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
107	C₂×Q₈⋊C₈	C₂×C₂₃	1	2	yes	—
108	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
109	C₂²×C₂₃	#390	1	1	yes	characteristic
110	C₂²×C₂₃	#435	1	2	yes	—
111	C₂²×C₂₃	#441	1	2	yes	—
112	C₂²×C₂₃	#443	1	2	yes	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	—	4	4	no	—
116	C₄×C₂₃	—	8	2	no	—
117	C₄×C₂₃	#407	1	1	yes	characteristic
118	C₄×C₂₃	#408	1	1	yes	characteristic
119	C₄×C₂₃	#435	1	2	yes	—
120	C₄×C₂₃	#470	1	2	yes	—
121	C₄×C₂₃	#497	1	2	yes	—
122	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
123	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
124	C₄×Q₁₆	—	2	2	no	—
125	C₄×Q₁₆	—	2	2	no	—
126	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
127	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
128	C₂×#103	C₄×C₂₃	1	2	yes	—
129	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
132	#336	—	2	4	no	—
133	#334	—	2	2	no	—
134	Q₈⋊C₈	—	2	4	no	—
135	#322	—	2	2	no	—
136	C₂×C₂₃	#1587	1	4	yes	—
137	C₂×C₂₃	#1955	1	1	yes	characteristic
138	C₂×C₂₃	#1971	1	1	yes	characteristic
139	C₂×C₂₃	#1974	1	1	yes	characteristic
140	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
141	C₂×Q₁₆	—	2	4	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₈	D₄×C₂₃	1	2	yes	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
154	C₂×C₄²	—	2	2	no	—
155	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
156	#107	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	#103	—	2	4	no	—
160	C₄×C₈	—	2	1	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	D₄×C₂₃	1	2	yes	—
164	C₄×C₈	D₄×C₂₃	1	2	yes	—
165	C₄×C₈	D₄×C₂₃	1	2	yes	—
166	C₂₃	#9664	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	M₁₆×C₂₃	1	4	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²×C₄×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	Q₁₆	—	2	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
188	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
196	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
197	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
198	C₄²	#41×C₂₃	1	1	yes	characteristic
199	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
200	Q₈	—	2	4	no	—
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
213	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
214	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
215	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
216	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
217	C₂×C₄	C₂₃×#131	1	2	yes	—
218	C₂×C₄	C₂₃×#137	1	4	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₈	—	8	2	no	—
224	C₂²	C₂₃×#390	1	1	yes	characteristic
225	C₂²	C₂₃×#435	1	2	yes	—
226	C₂²	C₂₃×#441	1	2	yes	—
227	C₂²	C₂₃×#443	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	4	no	—
231	C₄	—	8	2	no	—
232	C₄	C₂₃×#407	1	1	yes	characteristic
233	C₄	C₂₃×#408	1	1	yes	characteristic
234	C₄	C₂₃×#435	1	2	yes	—
235	C₄	C₂₃×#470	1	2	yes	—
236	C₄	C₂₃×#497	1	2	yes	—
237	C₂	C₂₃×#1587	1	4	yes	—
238	C₂	C₂₃×#1955	1	1	yes	characteristic
239	C₂	C₂₃×#1971	1	1	yes	characteristic
240	C₂	C₂₃×#1974	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, prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
78. #9664	C₂₃	direct (N × H)
166. C₂₃	#9664	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×C₄×Q₁₆×C₂₃	2
C₂₃×#1984	2
C₂×C₂₃×#334	2
C₂×C₂₃×Q₈⋊C₈	2
C₂×C₂₃×#322	2
#9664	23

Subgroups — C23×#9664 #19157290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9664 #19157290