Subgroups — C₂₃×#8657 #19156283

Order 5888.

418 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1834	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1803	C₂	1	2	yes	maximal
4	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1296	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#446	—	2	2	no	—
8	C₂₃×#446	C₄	1	4	yes	complemented
9	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#103	—	2	2	no	—
13	C₂×C₂₃×#97	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#384	C₂²	1	2	yes	—
16	C₂₃×#340	—	2	1	no	—
17	C₂₃×#340	—	2	1	no	—
18	C₂₃×#340	—	2	2	no	—
19	C₂₃×#334	—	2	2	no	—
20	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	—	2	1	no	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₂₃	—	2	1	no	—
26	C₄×Q₈×C₂₃	—	2	1	no	—
27	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₄×Q₈×C₂₃	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	—	2	1	no	—
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×C₄⋊C₈	—	2	4	no	—
37	C₂₃×C₄⋊C₈	—	2	1	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₂₃×#103	—	4	4	no	—
42	C₂₃×#97	—	2	4	no	—
43	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₂₃	D₄	1	2	yes	—
45	C₂×Q₈×C₂₃	—	2	1	no	—
46	C₂×Q₈×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	2	2	no	—
48	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	—	4	2	no	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	4	2	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	—	4	2	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
64	C₄⋊C₄×C₂₃	—	2	2	no	—
65	C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	—	4	2	no	—
68	C₄⋊C₄×C₂₃	—	4	2	no	—
69	C₄⋊C₄×C₂₃	—	4	2	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	8	2	no	—
72	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₂₃	—	2	2	no	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₂₃	#41	1	1	yes	characteristic
79	#8657	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	#121	1	1	yes	characteristic
81	Q₈×C₂₃	—	2	4	no	—
82	Q₈×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	4	1	no	—
87	C₂×C₄×C₂₃	—	4	1	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	8	2	no	—
90	C₂×C₄×C₂₃	—	8	2	no	—
91	C₂×C₄×C₂₃	—	8	2	no	—
92	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#135	1	2	yes	—
99	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄×C₂₃	#137	1	2	yes	—
101	C₂×C₄×C₂₃	#137	1	2	yes	—
102	C₈×C₂₃	—	2	2	no	—
103	C₈×C₂₃	—	2	2	no	—
104	C₈×C₂₃	—	4	2	no	—
105	C₈×C₂₃	—	8	2	no	—
106	C₂×#446	C₂×C₂₃	1	1	yes	characteristic
107	#1834	C₂×C₂₃	1	1	yes	characteristic
108	#1803	C₂×C₂₃	1	2	yes	—
109	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
110	#1296	C₂×C₂₃	1	2	yes	—
111	C₂²×C₂₃	#380	1	1	yes	characteristic
112	C₂²×C₂₃	#464	1	2	yes	—
113	C₂²×C₂₃	#469	1	1	yes	characteristic
114	C₂²×C₂₃	#470	1	1	yes	characteristic
115	C₂²×C₂₃	#471	1	2	yes	—
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	—	16	2	no	—
121	C₄×C₂₃	#416	1	1	yes	characteristic
122	C₄×C₂₃	#419	1	1	yes	characteristic
123	C₄×C₂₃	#428	1	2	yes	—
124	C₄×C₂₃	#468	1	2	yes	—
125	C₄×C₂₃	#479	1	2	yes	—
126	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
127	#446	—	2	2	no	—
128	#446	C₄×C₂₃	1	4	yes	complemented
129	C₂×C₄⋊C₈	—	2	1	no	—
130	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×#103	—	2	2	no	—
133	C₂×#97	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
135	#384	C₂²×C₂₃	1	2	yes	—
136	#340	—	2	1	no	—
137	#340	—	2	1	no	—
138	#340	—	2	2	no	—
139	#334	—	2	2	no	—
140	C₂×C₂₃	#1702	1	2	yes	—
141	C₂×C₂₃	#1704	1	2	yes	—
142	C₂×C₂₃	#1805	1	1	yes	characteristic
143	C₂×C₂₃	#1809	1	1	yes	characteristic
144	C₂×C₂₃	#1815	1	1	yes	characteristic
145	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
153	C₄×Q₈	Q₈×C₂₃	1	2	yes	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
166	#103	—	4	4	no	—
167	#97	—	2	4	no	—
168	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
169	C₄×C₈	D₄×C₂₃	1	2	yes	—
170	C₂₃	#8657	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
189	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	8	2	no	—
198	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
204	C₄²	#41×C₂₃	1	1	yes	characteristic
205	C₂³	C₂₃×#121	1	1	yes	characteristic
206	Q₈	—	2	4	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
220	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
221	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
222	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
223	C₂×C₄	C₂₃×#135	1	2	yes	—
224	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
225	C₂×C₄	C₂₃×#137	1	2	yes	—
226	C₂×C₄	C₂₃×#137	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	4	2	no	—
230	C₈	—	8	2	no	—
231	C₂²	C₂₃×#380	1	1	yes	characteristic
232	C₂²	C₂₃×#464	1	2	yes	—
233	C₂²	C₂₃×#469	1	1	yes	characteristic
234	C₂²	C₂₃×#470	1	1	yes	characteristic
235	C₂²	C₂₃×#471	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	16	2	no	—
241	C₄	C₂₃×#416	1	1	yes	characteristic
242	C₄	C₂₃×#419	1	1	yes	characteristic
243	C₄	C₂₃×#428	1	2	yes	—
244	C₄	C₂₃×#468	1	2	yes	—
245	C₄	C₂₃×#479	1	2	yes	—
246	C₂	C₂₃×#1702	1	2	yes	—
247	C₂	C₂₃×#1704	1	2	yes	—
248	C₂	C₂₃×#1805	1	1	yes	characteristic
249	C₂	C₂₃×#1809	1	1	yes	characteristic
250	C₂	C₂₃×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#121
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
8. C₂₃×#446	C₄	semidirect
79. #8657	C₂₃	direct (N × H)
128. #446	C₄×C₂₃	semidirect
170. C₂₃	#8657	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×#446	2
C₂₃×#1834	2
C₂₃×#1803	2
C₂×C₂₃×#334	2
C₂₃×#1296	2
#8657	23

Subgroups — C23×#8657 #19156283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8657 #19156283