Subgroups — C₂₃×#1880 #15669499

Order 2944.

374 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	C₂²×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×SD₁₆×C₂₃	—	2	2	no	—
11	C₂×SD₁₆×C₂₃	—	2	2	no	—
12	C₂×SD₁₆×C₂₃	C₄	1	4	yes	complemented
13	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₂₃	—	2	2	no	—
16	C₄×Q₈×C₂₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
22	C₂×C₄²×C₂₃	—	2	1	no	—
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#103	—	2	2	no	—
25	C₂₃×#102	—	2	2	no	—
26	C₂⁴×C₂₃	—	2	1	no	—
27	C₂×Q₈×C₂₃	—	2	1	no	—
28	C₂×Q₈×C₂₃	—	2	1	no	—
29	C₂×Q₈×C₂₃	—	2	1	no	—
30	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₂₃	D₄	1	2	yes	—
32	C₂×Q₈×C₂₃	D₄	1	2	yes	—
33	C₂×D₄×C₂₃	—	2	1	no	—
34	C₂×D₄×C₂₃	—	2	1	no	—
35	C₂×D₄×C₂₃	—	2	1	no	—
36	C₂×D₄×C₂₃	—	2	1	no	—
37	C₂×D₄×C₂₃	—	2	1	no	—
38	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₃	—	2	1	no	—
40	C₂²×C₄×C₂₃	—	2	1	no	—
41	C₂²×C₄×C₂₃	—	2	1	no	—
42	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	SD₁₆×C₂₃	—	2	8	no	—
46	C₂×C₈×C₂₃	—	2	1	no	—
47	C₂×C₈×C₂₃	—	2	1	no	—
48	C₂×C₈×C₂₃	—	2	1	no	—
49	C₂×C₈×C₂₃	—	2	1	no	—
50	C₂×C₈×C₂₃	—	2	1	no	—
51	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₂₃	—	2	4	no	—
53	C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
57	C₂²⋊C₄×C₂₃	—	4	2	no	—
58	C₂²⋊C₄×C₂₃	—	4	2	no	—
59	C₄²×C₂₃	—	2	4	no	—
60	C₄²×C₂₃	—	2	1	no	—
61	C₄²×C₂₃	—	2	1	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	4	1	no	—
64	C₂³×C₂₃	—	4	1	no	—
65	C₂³×C₂₃	—	4	1	no	—
66	C₂³×C₂₃	—	4	1	no	—
67	C₂³×C₂₃	—	4	1	no	—
68	C₂³×C₂₃	—	4	1	no	—
69	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₂₃	—	2	2	no	—
71	Q₈×C₂₃	—	2	2	no	—
72	Q₈×C₂₃	—	2	2	no	—
73	Q₈×C₂₃	C₂²⋊C₄	1	4	yes	complemented
74	D₄×C₂₃	—	2	2	no	—
75	D₄×C₂₃	—	2	2	no	—
76	D₄×C₂₃	—	4	1	no	—
77	D₄×C₂₃	—	4	1	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	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	1	no	—
91	C₂×C₄×C₂₃	—	4	1	no	—
92	C₂×C₄×C₂₃	SD₁₆	1	2	yes	complemented
93	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
99	C₈×C₂₃	—	2	2	no	—
100	C₈×C₂₃	—	4	1	no	—
101	#1880	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	—	4	2	no	—
105	C₂²×C₂₃	—	8	1	no	—
106	C₂²×C₂₃	—	8	1	no	—
107	C₂²×C₂₃	—	8	1	no	—
108	C₂²×C₂₃	—	8	1	no	—
109	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
110	C₂²×C₂₃	#120	1	1	yes	characteristic
111	C₂²×C₂₃	#121	1	1	yes	characteristic
112	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²×C₂₃	#135	1	1	yes	characteristic
114	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂²×C₂₃	#137	1	1	yes	characteristic
116	C₄×C₂₃	—	2	4	no	—
117	C₄×C₂₃	—	2	2	no	—
118	C₄×C₂₃	—	4	1	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	—	8	1	no	—
121	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
123	C₄×C₂₃	#121	1	1	yes	characteristic
124	C₄×C₂₃	#123	1	1	yes	characteristic
125	C₂²×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
127	C₂×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
128	C₂×#102	C₂×C₂₃	1	1	yes	characteristic
129	#390	C₂×C₂₃	1	1	yes	characteristic
130	#340	C₂×C₂₃	1	1	yes	characteristic, complemented
131	#336	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂×C₂₃	—	8	2	no	—
133	C₂×C₂₃	#386	1	1	yes	characteristic
134	C₂×C₂₃	C₄×SD₁₆	1	1	yes	characteristic
135	C₂×C₂₃	#440	1	1	yes	characteristic
136	C₂×C₂₃	#448	1	1	yes	characteristic
137	C₂×C₂₃	#449	1	1	yes	characteristic
138	C₂×C₂₃	#460	1	1	yes	characteristic
139	C₂×C₂₃	#464	1	1	yes	characteristic
140	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
141	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×SD₁₆	—	2	2	no	—
143	C₂×SD₁₆	—	2	2	no	—
144	C₂×SD₁₆	C₄×C₂₃	1	4	yes	complemented
145	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	—	2	1	no	—
153	C₂×C₂²⋊C₄	—	2	1	no	—
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
156	#103	—	2	2	no	—
157	#102	—	2	2	no	—
158	C₂₃	#1880	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁴	—	2	1	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	C₂×C₄×C₂₃	1	2	yes	—
164	C₂×Q₈	D₄×C₂₃	1	2	yes	—
165	C₂×Q₈	D₄×C₂₃	1	2	yes	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	SD₁₆	—	2	8	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	4	1	no	—
197	C₂³	—	4	1	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	—	4	1	no	—
201	C₂³	—	4	1	no	—
202	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
203	Q₈	—	2	2	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	2	no	—
206	Q₈	C₂²⋊C₄×C₂₃	1	4	yes	complemented
207	D₄	—	2	2	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	1	no	—
210	D₄	—	4	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	SD₁₆×C₂₃	1	2	yes	complemented
226	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
231	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	8	1	no	—
238	C₂²	—	8	1	no	—
239	C₂²	—	8	1	no	—
240	C₂²	—	8	1	no	—
241	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
242	C₂²	C₂₃×#120	1	1	yes	characteristic
243	C₂²	C₂₃×#121	1	1	yes	characteristic
244	C₂²	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
245	C₂²	C₂₃×#135	1	1	yes	characteristic
246	C₂²	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂²	C₂₃×#137	1	1	yes	characteristic
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	8	1	no	—
253	C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
254	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
255	C₄	C₂₃×#121	1	1	yes	characteristic
256	C₄	C₂₃×#123	1	1	yes	characteristic
257	C₂	—	8	2	no	—
258	C₂	C₂₃×#386	1	1	yes	characteristic
259	C₂	C₄×SD₁₆×C₂₃	1	1	yes	characteristic
260	C₂	C₂₃×#440	1	1	yes	characteristic
261	C₂	C₂₃×#448	1	1	yes	characteristic
262	C₂	C₂₃×#449	1	1	yes	characteristic
263	C₂	C₂₃×#460	1	1	yes	characteristic
264	C₂	C₂₃×#464	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
2. C₂×C₄×Q₈×C₂₃	C₂	semidirect
3. C₂×C₂₃×#103	C₂	semidirect
6. C₂₃×#340	C₂	semidirect
7. C₂₃×#336	C₂	semidirect
12. C₂×SD₁₆×C₂₃	C₄	semidirect
73. Q₈×C₂₃	C₂²⋊C₄	semidirect
92. C₂×C₄×C₂₃	SD₁₆	semidirect
101. #1880	C₂₃	direct (N × H)
126. C₂×C₄×Q₈	C₂×C₂₃	semidirect
127. C₂×#103	C₂×C₂₃	semidirect
130. #340	C₂×C₂₃	semidirect
131. #336	C₂×C₂₃	semidirect
144. C₂×SD₁₆	C₄×C₂₃	semidirect
158. C₂₃	#1880	direct (N × H)
206. Q₈	C₂²⋊C₄×C₂₃	semidirect
225. C₂×C₄	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SD₁₆×C₂₃	2
C₂×C₄×Q₈×C₂₃	2
C₂×C₂₃×#103	2
C₂×C₂₃×#102	2
C₂₃×#390	2
C₂₃×#340	2
C₂₃×#336	2
#1880	23

Subgroups — C23×#1880 #15669499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1880 #15669499