Subgroups — C₂₃×#11664 #19159290

Order 5888.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 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×#2163	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1334	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
7	C₄×Q₁₆×C₂₃	—	2	1	no	—
8	C₄×Q₁₆×C₂₃	—	2	1	no	—
9	C₄×Q₁₆×C₂₃	C₄	1	2	yes	—
10	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
12	C₂₃×#423	—	2	1	no	—
13	C₂×C₂₃×#103	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊C₁₆	—	2	1	no	—
16	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₂₃×#346	C₂²	1	2	yes	—
18	C₂₃×#334	—	2	1	no	—
19	C₂₃×#334	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂×Q₁₆×C₂₃	—	2	4	no	—
22	C₂×Q₁₆×C₂₃	—	2	1	no	—
23	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₂₃	—	4	1	no	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₂₃	—	2	2	no	—
28	C₄×Q₈×C₂₃	—	2	2	no	—
29	C₄×Q₈×C₂₃	—	4	1	no	—
30	C₄×Q₈×C₂₃	—	4	1	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
33	C₂×C₄²×C₂₃	—	4	1	no	—
34	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₂₃	—	2	2	no	—
36	C₂₃×#107	—	2	1	no	—
37	C₂₃×#107	C₂×C₄	1	2	yes	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₄⋊C₈	—	4	1	no	—
40	C₂₃×#103	—	2	2	no	—
41	C₂₃×#103	—	2	2	no	—
42	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₂₃	—	2	4	no	—
47	C₂×Q₈×C₂₃	—	2	2	no	—
48	C₂×Q₈×C₂₃	—	4	2	no	—
49	C₂×Q₈×C₂₃	—	4	1	no	—
50	C₂²×C₄×C₂₃	—	4	1	no	—
51	C₂²×C₄×C₂₃	—	4	1	no	—
52	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
55	Q₁₆×C₂₃	—	2	4	no	—
56	Q₁₆×C₂₃	—	2	2	no	—
57	Q₁₆×C₂₃	M₁₆	1	4	yes	complemented
58	M₁₆×C₂₃	—	8	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	4	1	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	—	4	1	no	—
68	C₄⋊C₄×C₂₃	—	4	2	no	—
69	C₄⋊C₄×C₂₃	—	4	2	no	—
70	C₄²×C₂₃	—	4	2	no	—
71	C₄²×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₂₃	D₈	1	1	yes	characteristic
74	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
75	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₂₃	—	4	2	no	—
77	#11664	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	#102	1	1	yes	characteristic
79	Q₈×C₂₃	—	2	4	no	—
80	Q₈×C₂₃	—	4	4	no	—
81	Q₈×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
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	4	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	8	2	no	—
90	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₂₃	—	2	1	no	—
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	—	8	2	no	—
100	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
102	#2163	C₂×C₂₃	1	1	yes	characteristic
103	#1566	C₂×C₂₃	1	1	yes	characteristic
104	#1334	C₂×C₂₃	1	4	yes	complemented
105	C₂²×C₂₃	—	2	1	no	—
106	C₂²×C₂₃	—	2	1	no	—
107	C₂²×C₂₃	#360	1	2	yes	—
108	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	4	4	no	—
114	C₄×C₂₃	—	8	2	no	—
115	C₄×C₂₃	#362	1	2	yes	—
116	C₄×C₂₃	#407	1	1	yes	characteristic
117	C₄×C₂₃	#421	1	1	yes	characteristic
118	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
119	C₂×C₄×Q₈	—	2	1	no	—
120	C₄×Q₁₆	—	2	1	no	—
121	C₄×Q₁₆	—	2	1	no	—
122	C₄×Q₁₆	C₄×C₂₃	1	2	yes	—
123	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
124	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
125	#423	—	2	1	no	—
126	C₂×#103	—	2	1	no	—
127	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
128	C₄⋊C₁₆	—	2	1	no	—
129	C₂²⋊C₁₆	—	2	1	no	—
130	#346	C₂²×C₂₃	1	2	yes	—
131	#334	—	2	1	no	—
132	#334	C₂²×C₂₃	1	2	yes	—
133	C₂×C₂₃	—	2	2	no	—
134	C₂×C₂₃	#1486	1	1	yes	characteristic
135	C₂×C₂₃	C₂×#360	1	1	yes	characteristic
136	C₂×C₂₃	C₂×#362	1	1	yes	characteristic
137	C₂²×Q₈	—	2	1	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
141	C₂×M₁₆	—	4	1	no	—
142	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
143	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	4	1	no	—
147	C₄×Q₈	—	4	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	4	1	no	—
150	C₂×C₄²	—	4	1	no	—
151	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
152	C₂×C₁₆	—	2	2	no	—
153	#107	—	2	1	no	—
154	#107	C₂×C₄×C₂₃	1	2	yes	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	4	1	no	—
157	#103	—	2	2	no	—
158	#103	—	2	2	no	—
159	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
161	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
162	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
163	C₂₃	#11664	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
173	Q₁₆	—	2	4	no	—
174	Q₁₆	—	2	2	no	—
175	Q₁₆	M₁₆×C₂₃	1	4	yes	complemented
176	M₁₆	—	8	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄²	—	4	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
191	C₄²	D₈×C₂₃	1	1	yes	characteristic
192	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
193	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₂³	C₂₃×#102	1	1	yes	characteristic
196	Q₈	—	2	4	no	—
197	Q₈	—	4	4	no	—
198	Q₈	—	4	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
208	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
209	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
210	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	8	2	no	—
217	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	C₂₃×#360	1	2	yes	—
221	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	4	no	—
227	C₄	—	8	2	no	—
228	C₄	C₂₃×#362	1	2	yes	—
229	C₄	C₂₃×#407	1	1	yes	characteristic
230	C₄	C₂₃×#421	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	C₂₃×#1486	1	1	yes	characteristic
233	C₂	C₂×C₂₃×#360	1	1	yes	characteristic
234	C₂	C₂×C₂₃×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1334	C₂	semidirect
57. Q₁₆×C₂₃	M₁₆	semidirect
77. #11664	C₂₃	direct (N × H)
104. #1334	C₂×C₂₃	semidirect
163. C₂₃	#11664	direct (N × H)
175. Q₁₆	M₁₆×C₂₃	semidirect

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×#2163	2
C₂₃×#1566	2
C₂₃×#1334	2
#11664	23

Subgroups — C23×#11664 #19159290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11664 #19159290