Subgroups — C₂₃×#10276 #19157902

Order 5888.

574 conjugacy classes of subgroups.

The subgroup lattice has 248 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	C₂×C₂₃×#475	C₂	1	2	yes	maximal
3	C₂₃×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#332	C₂	1	2	yes	maximal
5	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₂₃	—	2	2	no	—
7	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#494	—	2	4	no	—
10	C₂₃×#475	—	2	4	no	—
11	C₂₃×#475	—	2	4	no	—
12	C₂×C₂₃×#106	—	2	2	no	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#103	—	2	2	no	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	2	no	—
18	C₂₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₂₃	—	2	2	no	—
20	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂×Q₁₆×C₂₃	—	2	8	no	—
22	C₂×Q₁₆×C₂₃	—	4	4	no	—
23	C₂×Q₁₆×C₂₃	—	4	4	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	D₄	1	2	yes	—
27	C₂₃×C₄⋊Q₈	—	2	2	no	—
28	C₂₃×C₄⋊Q₈	—	2	2	no	—
29	C₂₃×C₄⋊Q₈	—	2	2	no	—
30	C₂₃×C₄⋊Q₈	D₄	1	4	yes	—
31	C₄×Q₈×C₂₃	—	2	2	no	—
32	C₄×Q₈×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	—	2	1	no	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#106	—	2	8	no	—
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
43	C₂₃×#103	—	2	8	no	—
44	C₄×C₈×C₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	—	2	1	no	—
46	C₂×Q₈×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	2	4	no	—
48	C₂×Q₈×C₂₃	—	2	2	no	—
49	C₂×Q₈×C₂₃	—	4	2	no	—
50	C₂×Q₈×C₂₃	—	4	4	no	—
51	C₂×Q₈×C₂₃	—	4	4	no	—
52	C₂×Q₈×C₂₃	Q₁₆	1	2	yes	complemented
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	—	4	2	no	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
59	Q₁₆×C₂₃	—	4	16	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	4	1	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
68	C₄⋊C₄×C₂₃	—	2	4	no	—
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₄⋊C₄×C₂₃	—	2	4	no	—
71	C₄⋊C₄×C₂₃	—	4	2	no	—
72	C₄⋊C₄×C₂₃	—	4	2	no	—
73	C₄⋊C₄×C₂₃	—	4	2	no	—
74	C₄²×C₂₃	—	4	1	no	—
75	C₄²×C₂₃	—	4	1	no	—
76	C₄²×C₂₃	C₂×D₄	1	2	yes	—
77	C₄²×C₂₃	C₂×D₄	1	2	yes	—
78	#10276	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₃	#121	1	1	yes	characteristic
80	Q₈×C₂₃	—	2	4	no	—
81	Q₈×C₂₃	—	4	2	no	—
82	Q₈×C₂₃	—	4	8	no	—
83	Q₈×C₂₃	—	8	2	no	—
84	Q₈×C₂₃	—	8	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	4	1	no	—
88	C₂×C₄×C₂₃	—	4	1	no	—
89	C₂×C₄×C₂₃	—	4	1	no	—
90	C₂×C₄×C₂₃	—	4	4	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	8	4	no	—
93	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#120	1	2	yes	—
95	C₂×C₄×C₂₃	#121	1	2	yes	—
96	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
100	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₄×C₂₃	#137	1	2	yes	—
102	C₈×C₂₃	—	2	4	no	—
103	C₈×C₂₃	—	4	4	no	—
104	C₈×C₂₃	—	8	1	no	—
105	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
106	C₂×#475	C₂×C₂₃	1	2	yes	—
107	#1984	C₂×C₂₃	1	1	yes	characteristic
108	C₂×#332	C₂×C₂₃	1	2	yes	—
109	#1277	C₂×C₂₃	1	1	yes	characteristic
110	C₂²×C₂₃	#392	1	1	yes	characteristic
111	C₂²×C₂₃	#460	1	2	yes	—
112	C₂²×C₂₃	#467	1	1	yes	characteristic
113	C₂²×C₂₃	#470	1	1	yes	characteristic
114	C₂²×C₂₃	#473	1	2	yes	—
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	—	8	4	no	—
118	C₄×C₂₃	—	8	1	no	—
119	C₄×C₂₃	#450	1	2	yes	—
120	C₄×C₂₃	#451	1	2	yes	—
121	C₄×C₂₃	#452	1	2	yes	—
122	C₄×C₂₃	#492	1	1	yes	characteristic
123	C₄×C₂₃	#496	1	1	yes	characteristic
124	C₂²×Q₁₆	—	2	2	no	—
125	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
126	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
127	#494	—	2	4	no	—
128	#475	—	2	4	no	—
129	#475	—	2	4	no	—
130	C₂×#106	—	2	2	no	—
131	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
133	C₂×#103	—	2	2	no	—
134	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
135	#336	—	2	2	no	—
136	#332	—	2	4	no	—
137	C₂×C₂₃	#1670	1	2	yes	—
138	C₂×C₂₃	#1690	1	2	yes	—
139	C₂×C₂₃	#2001	1	1	yes	characteristic
140	C₂×C₂₃	#2014	1	1	yes	characteristic
141	C₂×C₂₃	#2020	1	1	yes	characteristic
142	C₂²×Q₈	—	2	2	no	—
143	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
144	C₂×Q₁₆	—	2	8	no	—
145	C₂×Q₁₆	—	4	4	no	—
146	C₂×Q₁₆	—	4	4	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	D₄×C₂₃	1	2	yes	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	D₄×C₂₃	1	4	yes	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
162	#106	—	2	8	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
166	#103	—	2	8	no	—
167	C₄×C₈	—	2	2	no	—
168	C₂₃	#10276	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	Q₁₆×C₂₃	1	2	yes	complemented
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
183	Q₁₆	—	4	16	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×D₄×C₂₃	1	2	yes	—
201	C₄²	C₂×D₄×C₂₃	1	2	yes	—
202	C₂³	C₂₃×#121	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	4	8	no	—
206	Q₈	—	8	2	no	—
207	Q₈	—	8	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
217	C₂×C₄	C₂₃×#120	1	2	yes	—
218	C₂×C₄	C₂₃×#121	1	2	yes	—
219	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
220	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
221	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
223	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
224	C₂×C₄	C₂₃×#137	1	2	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	4	4	no	—
227	C₈	—	8	1	no	—
228	C₂²	C₂₃×#392	1	1	yes	characteristic
229	C₂²	C₂₃×#460	1	2	yes	—
230	C₂²	C₂₃×#467	1	1	yes	characteristic
231	C₂²	C₂₃×#470	1	1	yes	characteristic
232	C₂²	C₂₃×#473	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	4	no	—
236	C₄	—	8	1	no	—
237	C₄	C₂₃×#450	1	2	yes	—
238	C₄	C₂₃×#451	1	2	yes	—
239	C₄	C₂₃×#452	1	2	yes	—
240	C₄	C₂₃×#492	1	1	yes	characteristic
241	C₄	C₂₃×#496	1	1	yes	characteristic
242	C₂	C₂₃×#1670	1	2	yes	—
243	C₂	C₂₃×#1690	1	2	yes	—
244	C₂	C₂₃×#2001	1	1	yes	characteristic
245	C₂	C₂₃×#2014	1	1	yes	characteristic
246	C₂	C₂₃×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂×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
52. C₂×Q₈×C₂₃	Q₁₆	semidirect
78. #10276	C₂₃	direct (N × H)
168. C₂₃	#10276	direct (N × H)
176. C₂×Q₈	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
C₂×C₂₃×#475	2
C₂₃×#1984	2
C₂×C₂₃×#332	2
C₂₃×#1277	2
#10276	23

Subgroups — C23×#10276 #19157902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10276 #19157902