Subgroups — C₂₃×#10274 #19157900

Order 5888.

626 conjugacy classes of subgroups.

The subgroup lattice has 274 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, complemented
2	C₂×C₂₃×#474	C₂	1	2	yes	maximal
3	C₂₃×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#332	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₂₃	—	2	2	no	—
7	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#494	—	2	4	no	—
10	C₂₃×#474	—	2	4	no	—
11	C₂₃×#474	—	2	4	no	—
12	C₂×C₂₃×#107	—	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×#102	—	2	2	no	—
16	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#332	—	2	4	no	—
19	C₂²×Q₈×C₂₃	—	2	2	no	—
20	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	—	2	1	no	—
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	2	no	—
26	C₂²×C₈×C₂₃	—	2	1	no	—
27	C₂²×C₈×C₂₃	D₄	1	2	yes	—
28	C₂₃×C₄⋊Q₈	—	2	2	no	—
29	C₂₃×C₄⋊Q₈	—	2	2	no	—
30	C₂₃×C₄⋊Q₈	—	2	2	no	—
31	C₂₃×C₄⋊Q₈	D₄	1	4	yes	—
32	C₄×D₄×C₂₃	—	2	2	no	—
33	C₄×D₄×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	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#107	—	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×#102	—	2	8	no	—
44	C₂₃×C₂²⋊C₈	—	4	4	no	—
45	C₄×C₈×C₂₃	—	2	2	no	—
46	C₂⁴×C₂₃	—	2	1	no	—
47	C₂×Q₈×C₂₃	—	2	4	no	—
48	C₂×Q₈×C₂₃	—	4	2	no	—
49	C₂×Q₈×C₂₃	—	4	4	no	—
50	C₂×Q₈×C₂₃	—	4	4	no	—
51	C₂×D₄×C₂₃	—	2	2	no	—
52	C₂×D₄×C₂₃	—	2	1	no	—
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	Q₁₆	1	2	yes	complemented
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	—	4	2	no	—
57	C₂²×C₄×C₂₃	—	4	1	no	—
58	C₂²×C₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	—	4	2	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
64	Q₁₆×C₂₃	—	4	16	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	4	no	—
68	C₂×C₈×C₂₃	—	4	2	no	—
69	C₂×C₈×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	4	1	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	D₈	1	4	yes	—
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	4	2	no	—
77	C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₂²⋊C₄×C₂₃	—	4	2	no	—
79	C₄²×C₂₃	C₂×D₄	1	2	yes	—
80	C₄²×C₂₃	C₂×D₄	1	2	yes	—
81	#10274	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	4	1	no	—
84	C₂³×C₂₃	—	4	2	no	—
85	C₂³×C₂₃	—	4	1	no	—
86	C₂³×C₂₃	—	4	2	no	—
87	C₂³×C₂₃	#121	1	1	yes	characteristic
88	Q₈×C₂₃	—	4	8	no	—
89	Q₈×C₂₃	—	8	2	no	—
90	Q₈×C₂₃	—	8	2	no	—
91	D₄×C₂₃	—	2	4	no	—
92	D₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	4	1	no	—
95	C₂×C₄×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	—	8	4	no	—
98	C₂×C₄×C₂₃	—	8	2	no	—
99	C₂×C₄×C₂₃	—	8	1	no	—
100	C₂×C₄×C₂₃	—	8	1	no	—
101	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
102	C₂×C₄×C₂₃	#120	1	2	yes	—
103	C₂×C₄×C₂₃	#121	1	2	yes	—
104	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
108	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×C₄×C₂₃	#137	1	2	yes	—
110	C₈×C₂₃	—	2	4	no	—
111	C₈×C₂₃	—	4	4	no	—
112	C₈×C₂₃	—	8	1	no	—
113	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
114	C₂×#474	C₂×C₂₃	1	2	yes	—
115	#1954	C₂×C₂₃	1	1	yes	characteristic
116	C₂×#332	C₂×C₂₃	1	2	yes	complemented
117	#1277	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₂²×C₂₃	—	4	4	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	8	2	no	—
121	C₂²×C₂₃	—	8	1	no	—
122	C₂²×C₂₃	—	8	1	no	—
123	C₂²×C₂₃	#392	1	1	yes	characteristic
124	C₂²×C₂₃	#459	1	2	yes	—
125	C₂²×C₂₃	#467	1	1	yes	characteristic
126	C₂²×C₂₃	#470	1	1	yes	characteristic
127	C₂²×C₂₃	#473	1	2	yes	—
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	—	8	1	no	—
130	C₄×C₂₃	—	8	4	no	—
131	C₄×C₂₃	#447	1	2	yes	—
132	C₄×C₂₃	#451	1	2	yes	—
133	C₄×C₂₃	#452	1	2	yes	—
134	C₄×C₂₃	#493	1	1	yes	characteristic
135	C₄×C₂₃	#496	1	1	yes	characteristic
136	C₂²×Q₁₆	—	2	2	no	—
137	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
138	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
139	#494	—	2	4	no	—
140	#474	—	2	4	no	—
141	#474	—	2	4	no	—
142	C₂×#107	—	2	2	no	—
143	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
145	C₂×#102	—	2	2	no	—
146	C₂×C₂²⋊C₈	—	2	2	no	—
147	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	#332	—	2	4	no	—
149	C₂×C₂₃	—	8	2	no	—
150	C₂×C₂₃	#1669	1	2	yes	—
151	C₂×C₂₃	#1689	1	2	yes	—
152	C₂×C₂₃	#2001	1	1	yes	characteristic
153	C₂×C₂₃	#2011	1	1	yes	characteristic
154	C₂×C₂₃	#2020	1	1	yes	characteristic
155	C₂²×Q₈	—	2	2	no	—
156	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
157	C₂³×C₄	—	2	1	no	—
158	C₂×Q₁₆	—	2	8	no	—
159	C₂×Q₁₆	—	4	4	no	—
160	C₂×Q₁₆	—	4	4	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	D₄×C₂₃	1	2	yes	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	D₄×C₂₃	1	4	yes	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
175	#107	—	2	8	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
179	#102	—	2	8	no	—
180	C₂²⋊C₈	—	4	4	no	—
181	C₄×C₈	—	2	2	no	—
182	C₂₃	#10274	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	—	2	1	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	Q₁₆×C₂₃	1	2	yes	complemented
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
201	Q₁₆	—	4	16	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	D₈×C₂₃	1	4	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₄²	C₂×D₄×C₂₃	1	2	yes	—
217	C₄²	C₂×D₄×C₂₃	1	2	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	4	1	no	—
220	C₂³	—	4	2	no	—
221	C₂³	—	4	1	no	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂₃×#121	1	1	yes	characteristic
224	Q₈	—	4	8	no	—
225	Q₈	—	8	2	no	—
226	Q₈	—	8	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	4	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
238	C₂×C₄	C₂₃×#120	1	2	yes	—
239	C₂×C₄	C₂₃×#121	1	2	yes	—
240	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
241	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
242	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
244	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
245	C₂×C₄	C₂₃×#137	1	2	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	4	4	no	—
248	C₈	—	8	1	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	1	no	—
253	C₂²	—	8	1	no	—
254	C₂²	C₂₃×#392	1	1	yes	characteristic
255	C₂²	C₂₃×#459	1	2	yes	—
256	C₂²	C₂₃×#467	1	1	yes	characteristic
257	C₂²	C₂₃×#470	1	1	yes	characteristic
258	C₂²	C₂₃×#473	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	8	1	no	—
261	C₄	—	8	4	no	—
262	C₄	C₂₃×#447	1	2	yes	—
263	C₄	C₂₃×#451	1	2	yes	—
264	C₄	C₂₃×#452	1	2	yes	—
265	C₄	C₂₃×#493	1	1	yes	characteristic
266	C₄	C₂₃×#496	1	1	yes	characteristic
267	C₂	—	8	2	no	—
268	C₂	C₂₃×#1669	1	2	yes	—
269	C₂	C₂₃×#1689	1	2	yes	—
270	C₂	C₂₃×#2001	1	1	yes	characteristic
271	C₂	C₂₃×#2011	1	1	yes	characteristic
272	C₂	C₂₃×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
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
1. C₂×C₂₃×#494	C₂	semidirect
4. C₂×C₂₃×#332	C₂	semidirect
5. C₂₃×#1277	C₂	semidirect
54. C₂×D₄×C₂₃	Q₁₆	semidirect
81. #10274	C₂₃	direct (N × H)
113. C₂×#494	C₂×C₂₃	semidirect
116. C₂×#332	C₂×C₂₃	semidirect
117. #1277	C₂×C₂₃	semidirect
182. C₂₃	#10274	direct (N × H)
191. C₂×D₄	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×#474	2
C₂₃×#1954	2
C₂×C₂₃×#332	2
C₂₃×#1277	2
#10274	23

Subgroups — C23×#10274 #19157900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10274 #19157900