Subgroups — C₂₃×#1441 #15669060

Order 2944.

564 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#95	C₂	1	2	yes	maximal
4	C₂×C₂₃×#95	C₂	1	2	yes	maximal
5	C₂×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
12	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
14	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₂×C₄⋊C₄×C₂₃	C₄	1	4	yes	complemented
16	C₂₃×#95	—	2	4	no	—
17	C₂₃×#95	—	2	2	no	—
18	C₂₃×#95	—	2	4	no	—
19	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	—	2	1	no	—
21	C₂²×C₄×C₂₃	—	2	1	no	—
22	C₂²×C₄×C₂₃	—	2	2	no	—
23	C₂²×C₄×C₂₃	—	2	1	no	—
24	C₂²×C₄×C₂₃	—	2	1	no	—
25	C₂²×C₄×C₂₃	—	2	2	no	—
26	C₂²×C₄×C₂₃	—	2	1	no	—
27	C₂²×C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄×C₂₃	—	2	2	no	—
29	C₂²×C₄×C₂₃	—	2	2	no	—
30	C₂²×C₄×C₂₃	—	2	2	no	—
31	C₂²×C₄×C₂₃	—	2	2	no	—
32	C₂²×C₄×C₂₃	—	2	2	no	—
33	C₂²×C₄×C₂₃	—	2	2	no	—
34	C₂²×C₄×C₂₃	—	2	2	no	—
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	2	no	—
38	C₂²×C₄×C₂₃	—	2	2	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	D₄	1	4	yes	—
46	C₂²×C₄×C₂₃	D₄	1	2	yes	—
47	C₂²×C₄×C₂₃	D₄	1	2	yes	—
48	C₂²×C₄×C₂₃	D₄	1	2	yes	—
49	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
50	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
51	C₄⋊C₄×C₂₃	—	2	8	no	—
52	C₄⋊C₄×C₂₃	—	2	8	no	—
53	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂³×C₂₃	C₂×D₄	1	2	yes	—
58	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
60	C₂³×C₂₃	#41	1	1	yes	characteristic
61	C₂³×C₂₃	#41	1	1	yes	characteristic
62	C₂³×C₂₃	#41	1	2	yes	—
63	C₂³×C₂₃	#41	1	2	yes	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	4	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	4	1	no	—
83	C₂×C₄×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	—	4	2	no	—
86	C₂×C₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	1	no	—
90	C₂×C₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
94	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	complemented
95	#1441	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₂₃	#117	1	1	yes	characteristic
99	C₂²×C₂₃	C₄×D₄	1	2	yes	—
100	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
101	C₂²×C₂₃	#120	1	1	yes	characteristic
102	C₂²×C₂₃	#121	1	1	yes	characteristic
103	C₂²×C₂₃	#121	1	1	yes	characteristic
104	C₂²×C₂₃	#121	1	2	yes	—
105	C₂²×C₂₃	#121	1	2	yes	—
106	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
109	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
110	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
111	C₂²×C₂₃	#123	1	1	yes	characteristic
112	C₂²×C₂₃	#123	1	2	yes	—
113	C₂²×C₂₃	#124	1	1	yes	characteristic
114	C₂²×C₂₃	#125	1	2	yes	—
115	C₂²×C₂₃	#125	1	2	yes	—
116	C₂²×C₂₃	#126	1	2	yes	—
117	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
118	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₄×C₂₃	—	2	4	no	—
120	C₄×C₂₃	—	2	4	no	—
121	C₄×C₂₃	—	4	2	no	—
122	C₄×C₂₃	—	4	4	no	—
123	C₄×C₂₃	—	4	4	no	—
124	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
125	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
126	C₂×#95	C₂×C₂₃	1	2	yes	—
127	C₂×#95	C₂×C₂₃	1	2	yes	—
128	C₂×#95	C₂×C₂₃	1	1	yes	characteristic
129	C₂×C₂₃	#380	1	1	yes	characteristic
130	C₂×C₂₃	#386	1	1	yes	characteristic
131	C₂×C₂₃	#387	1	2	yes	—
132	C₂×C₂₃	#389	1	2	yes	—
133	C₂×C₂₃	#391	1	1	yes	characteristic
134	C₂×C₂₃	#394	1	1	yes	characteristic
135	C₂×C₂₃	#395	1	1	yes	characteristic
136	C₂×C₂₃	#396	1	2	yes	—
137	C₂×C₂₃	#398	1	1	yes	characteristic
138	C₂×C₂₃	#398	1	2	yes	—
139	C₂×C₂₃	#400	1	1	yes	characteristic
140	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
142	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
143	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
144	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	C₄×C₂₃	1	4	yes	complemented
150	#95	—	2	4	no	—
151	#95	—	2	2	no	—
152	#95	—	2	4	no	—
153	C₂₃	#1441	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
178	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
179	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
180	C₂²×C₄	D₄×C₂₃	1	4	yes	—
181	C₂²×C₄	D₄×C₂₃	1	2	yes	—
182	C₂²×C₄	D₄×C₂₃	1	2	yes	—
183	C₂²×C₄	D₄×C₂₃	1	2	yes	—
184	C₂²×C₄	Q₈×C₂₃	1	4	yes	—
185	C₂²×C₄	Q₈×C₂₃	1	2	yes	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	8	no	—
188	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
189	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
190	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
191	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₂³	C₂×D₄×C₂₃	1	2	yes	—
193	C₂³	C₂×Q₈×C₂₃	1	1	yes	characteristic
194	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
195	C₂³	#41×C₂₃	1	1	yes	characteristic
196	C₂³	#41×C₂₃	1	1	yes	characteristic
197	C₂³	#41×C₂₃	1	2	yes	—
198	C₂³	#41×C₂₃	1	2	yes	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
229	C₂×C₄	C₄⋊C₄×C₂₃	1	4	yes	complemented
230	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂²	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂²	C₂₃×#117	1	1	yes	characteristic
233	C₂²	C₄×D₄×C₂₃	1	2	yes	—
234	C₂²	C₄×Q₈×C₂₃	1	2	yes	—
235	C₂²	C₂₃×#120	1	1	yes	characteristic
236	C₂²	C₂₃×#121	1	1	yes	characteristic
237	C₂²	C₂₃×#121	1	1	yes	characteristic
238	C₂²	C₂₃×#121	1	2	yes	—
239	C₂²	C₂₃×#121	1	2	yes	—
240	C₂²	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
241	C₂²	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
243	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
244	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
245	C₂²	C₂₃×#123	1	1	yes	characteristic
246	C₂²	C₂₃×#123	1	2	yes	—
247	C₂²	C₂₃×#124	1	1	yes	characteristic
248	C₂²	C₂₃×#125	1	2	yes	—
249	C₂²	C₂₃×#125	1	2	yes	—
250	C₂²	C₂₃×#126	1	2	yes	—
251	C₂²	C₂₃×C₄⋊Q₈	1	2	yes	—
252	C₂²	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	4	4	no	—
258	C₂	C₂₃×#380	1	1	yes	characteristic
259	C₂	C₂₃×#386	1	1	yes	characteristic
260	C₂	C₂₃×#387	1	2	yes	—
261	C₂	C₂₃×#389	1	2	yes	—
262	C₂	C₂₃×#391	1	1	yes	characteristic
263	C₂	C₂₃×#394	1	1	yes	characteristic
264	C₂	C₂₃×#395	1	1	yes	characteristic
265	C₂	C₂₃×#396	1	2	yes	—
266	C₂	C₂₃×#398	1	1	yes	characteristic
267	C₂	C₂₃×#398	1	2	yes	—
268	C₂	C₂₃×#400	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂³
commutator	C₂³	C₂²×C₄×C₂₃
Frattini, cosocle	C₂⁴	C₂³×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂×C₄⋊C₄×C₂₃	C₄	semidirect
94. C₂×C₄×C₂₃	C₄⋊C₄	semidirect
95. #1441	C₂₃	direct (N × H)
149. C₂×C₄⋊C₄	C₄×C₂₃	semidirect
153. C₂₃	#1441	direct (N × H)
229. C₂×C₄	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#95	2
#1441	23

Subgroups — C23×#1441 #15669060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1441 #15669060