Subgroups — C₂₃×#9066 #19156692

Order 5888.

570 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2369	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1987	C₂	1	2	yes	maximal
4	C₂×C₂₃×#333	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1279	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₂₃	—	2	2	no	—
7	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂₃×#494	—	2	2	no	—
10	C₂₃×#494	C₄	1	4	yes	complemented
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#391	—	2	2	no	—
14	C₂₃×#389	—	2	2	no	—
15	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
16	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#340	—	2	2	no	—
18	C₂₃×#340	—	2	2	no	—
19	C₂₃×#333	—	2	2	no	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
22	C₂×Q₁₆×C₂₃	—	2	8	no	—
23	C₂×Q₁₆×C₂₃	—	4	4	no	—
24	C₂×Q₁₆×C₂₃	—	4	4	no	—
25	C₂²×C₈×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₂₃×C₄⋊Q₈	—	2	1	no	—
28	C₂₃×C₄⋊Q₈	—	2	2	no	—
29	C₂₃×C₄⋊Q₈	—	2	2	no	—
30	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
38	C₂×C₄²×C₂₃	—	2	2	no	—
39	C₂×C₄²×C₂₃	—	2	2	no	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₄×C₈×C₂₃	—	2	1	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
45	C₂₃×#95	—	2	4	no	—
46	C₂₃×#95	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	2	2	no	—
48	C₂×Q₈×C₂₃	—	2	2	no	—
49	C₂×Q₈×C₂₃	—	4	2	no	—
50	C₂×Q₈×C₂₃	—	4	2	no	—
51	C₂×Q₈×C₂₃	—	4	2	no	—
52	C₂×Q₈×C₂₃	—	4	2	no	—
53	C₂×Q₈×C₂₃	—	4	1	no	—
54	C₂×Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	Q₁₆×C₂₃	—	4	16	no	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	—	4	2	no	—
68	C₄⋊C₄×C₂₃	—	2	4	no	—
69	C₄⋊C₄×C₂₃	—	2	4	no	—
70	C₄⋊C₄×C₂₃	—	2	2	no	—
71	C₄⋊C₄×C₂₃	—	2	4	no	—
72	C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₄⋊C₄×C₂₃	—	4	2	no	—
75	C₄⋊C₄×C₂₃	—	8	2	no	—
76	C₄⋊C₄×C₂₃	Q₁₆	1	4	yes	complemented
77	C₄²×C₂₃	—	4	2	no	—
78	C₄²×C₂₃	—	4	2	no	—
79	C₄²×C₂₃	—	8	2	no	—
80	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₂₃	#41	1	2	yes	—
83	#9066	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	#120	1	1	yes	characteristic
85	Q₈×C₂₃	—	4	4	no	—
86	Q₈×C₂₃	—	4	4	no	—
87	Q₈×C₂₃	—	8	2	no	—
88	Q₈×C₂₃	—	8	2	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	8	1	no	—
102	C₂×C₄×C₂₃	—	8	1	no	—
103	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	#121	1	2	yes	—
106	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₂₃	#135	1	2	yes	—
109	C₂×C₄×C₂₃	#137	1	2	yes	—
110	C₂×C₄×C₂₃	#137	1	2	yes	—
111	C₈×C₂₃	—	2	4	no	—
112	C₈×C₂₃	—	8	2	no	—
113	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
114	#2369	C₂×C₂₃	1	1	yes	characteristic
115	#1987	C₂×C₂₃	1	2	yes	—
116	C₂×#333	C₂×C₂₃	1	1	yes	characteristic
117	#1279	C₂×C₂₃	1	2	yes	—
118	C₂²×C₂₃	#386	1	1	yes	characteristic
119	C₂²×C₂₃	#451	1	2	yes	—
120	C₂²×C₂₃	#452	1	2	yes	—
121	C₂²×C₂₃	#453	1	1	yes	characteristic
122	C₂²×C₂₃	#453	1	1	yes	characteristic
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	2	1	no	—
125	C₄×C₂₃	—	4	4	no	—
126	C₄×C₂₃	—	4	4	no	—
127	C₄×C₂₃	—	8	2	no	—
128	C₄×C₂₃	—	8	2	no	—
129	C₄×C₂₃	C₄×Q₁₆	1	2	yes	—
130	C₄×C₂₃	#441	1	2	yes	—
131	C₄×C₂₃	#462	1	2	yes	—
132	C₄×C₂₃	#464	1	2	yes	—
133	C₂²×Q₁₆	—	2	2	no	—
134	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
136	#494	—	2	2	no	—
137	#494	C₄×C₂₃	1	4	yes	complemented
138	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
140	#391	—	2	2	no	—
141	#389	—	2	2	no	—
142	C₄×C₄⋊C₄	—	2	2	no	—
143	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
144	#340	—	2	2	no	—
145	#340	—	2	2	no	—
146	#333	—	2	2	no	—
147	C₂×C₂₃	#1632	1	2	yes	—
148	C₂×C₂₃	#1633	1	1	yes	characteristic
149	C₂×C₂₃	#1634	1	1	yes	characteristic
150	C₂×C₂₃	#1878	1	2	yes	—
151	C₂×C₂₃	#1887	1	1	yes	characteristic
152	C₂²×Q₈	—	2	1	no	—
153	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
154	C₂×Q₁₆	—	2	8	no	—
155	C₂×Q₁₆	—	4	4	no	—
156	C₂×Q₁₆	—	4	4	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₄⋊Q₈	—	2	1	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
163	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
164	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
169	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
177	#95	—	2	4	no	—
178	#95	—	2	2	no	—
179	C₂₃	#9066	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
194	Q₁₆	—	4	16	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	8	2	no	—
209	C₄⋊C₄	Q₁₆×C₂₃	1	4	yes	complemented
210	C₄²	—	4	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	8	2	no	—
213	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₄²	#41×C₂₃	1	2	yes	—
216	C₂³	C₂₃×#120	1	1	yes	characteristic
217	Q₈	—	4	4	no	—
218	Q₈	—	4	4	no	—
219	Q₈	—	8	2	no	—
220	Q₈	—	8	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
237	C₂×C₄	C₂₃×#121	1	2	yes	—
238	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
239	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
240	C₂×C₄	C₂₃×#135	1	2	yes	—
241	C₂×C₄	C₂₃×#137	1	2	yes	—
242	C₂×C₄	C₂₃×#137	1	2	yes	—
243	C₈	—	2	4	no	—
244	C₈	—	8	2	no	—
245	C₂²	C₂₃×#386	1	1	yes	characteristic
246	C₂²	C₂₃×#451	1	2	yes	—
247	C₂²	C₂₃×#452	1	2	yes	—
248	C₂²	C₂₃×#453	1	1	yes	characteristic
249	C₂²	C₂₃×#453	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	—	8	2	no	—
256	C₄	C₄×Q₁₆×C₂₃	1	2	yes	—
257	C₄	C₂₃×#441	1	2	yes	—
258	C₄	C₂₃×#462	1	2	yes	—
259	C₄	C₂₃×#464	1	2	yes	—
260	C₂	C₂₃×#1632	1	2	yes	—
261	C₂	C₂₃×#1633	1	1	yes	characteristic
262	C₂	C₂₃×#1634	1	1	yes	characteristic
263	C₂	C₂₃×#1878	1	2	yes	—
264	C₂	C₂₃×#1887	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#120
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂₃×#494	C₄	semidirect
76. C₄⋊C₄×C₂₃	Q₁₆	semidirect
83. #9066	C₂₃	direct (N × H)
137. #494	C₄×C₂₃	semidirect
179. C₂₃	#9066	direct (N × H)
209. C₄⋊C₄	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#494	2
C₂₃×#2369	2
C₂₃×#1987	2
C₂×C₂₃×#333	2
C₂₃×#1279	2
#9066	23

Subgroups — C23×#9066 #19156692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9066 #19156692