Subgroups — C₂₃×#2494 #15670113

Order 2944.

412 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#401	C₂	1	2	yes	maximal, complemented
2	C₂₃×#397	C₂	1	2	yes	maximal
3	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#388	C₂	1	2	yes	maximal
6	C₂₃×#387	C₂	1	2	yes	maximal, complemented
7	C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	1	yes	characteristic
27	C₂₃×#95	C₂²	1	1	yes	characteristic
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂₃×#95	C₂²	1	2	yes	—
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₂₃	—	2	4	no	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₂²⋊C₄×C₂₃	—	2	2	no	—
56	C₂²⋊C₄×C₂₃	—	2	4	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	4	2	no	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	—	2	2	no	—
61	C₄²×C₂₃	—	4	2	no	—
62	C₂³×C₂₃	—	2	1	no	—
63	C₂³×C₂₃	—	2	1	no	—
64	C₂³×C₂₃	—	2	1	no	—
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	—	2	1	no	—
67	C₂³×C₂₃	—	2	1	no	—
68	C₂³×C₂₃	#41	1	2	yes	—
69	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	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	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	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	#41	1	2	yes	—
98	C₂×C₄×C₂₃	#41	1	2	yes	—
99	C₂×C₄×C₂₃	#41	1	2	yes	—
100	C₂×C₄×C₂₃	#41	1	2	yes	—
101	#2494	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	4	1	no	—
106	C₂²×C₂₃	—	4	1	no	—
107	C₂²×C₂₃	—	4	1	no	—
108	C₂²×C₂₃	—	4	1	no	—
109	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
110	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
111	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
112	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
113	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₄×C₂₃	—	2	2	no	—
117	C₄×C₂₃	—	4	1	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	—	4	1	no	—
121	C₄×C₂₃	—	4	2	no	—
122	C₄×C₂₃	—	4	1	no	—
123	C₄×C₂₃	—	4	2	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	#126	1	4	yes	—
126	#401	C₂×C₂₃	1	2	yes	complemented
127	#397	C₂×C₂₃	1	2	yes	—
128	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#388	C₂×C₂₃	1	2	yes	—
131	#387	C₂×C₂₃	1	2	yes	complemented
132	#382	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#380	C₂×C₂₃	1	1	yes	characteristic
134	#380	C₂×C₂₃	1	1	yes	characteristic
135	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
137	C₂×C₂₃	—	4	2	no	—
138	C₂×C₂₃	C₂×#126	1	1	yes	characteristic
139	C₂×C₂₃	#529	1	1	yes	characteristic
140	C₂×C₂₃	#542	1	1	yes	characteristic
141	C₂×C₂₃	#552	1	1	yes	characteristic
142	C₂×C₂₃	#553	1	1	yes	characteristic
143	C₂×C₂₃	#553	1	1	yes	characteristic
144	C₂×C₂₃	#556	1	1	yes	characteristic
145	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
156	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
159	#95	C₂²×C₂₃	1	1	yes	characteristic
160	#95	C₂²×C₂₃	1	1	yes	characteristic
161	#95	C₂²×C₂₃	1	2	yes	—
162	#95	C₂²×C₂₃	1	2	yes	—
163	#95	C₂²×C₂₃	1	2	yes	—
164	#95	C₂²×C₂₃	1	2	yes	—
165	#95	C₂²×C₂₃	1	2	yes	—
166	#95	C₂²×C₂₃	1	2	yes	—
167	#95	C₂²×C₂₃	1	2	yes	—
168	C₂₃	#2494	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
178	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
181	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	1	no	—
198	C₂³	—	2	1	no	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	#41×C₂₃	1	2	yes	—
203	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	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	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	#41×C₂₃	1	2	yes	—
232	C₂×C₄	#41×C₂₃	1	2	yes	—
233	C₂×C₄	#41×C₂₃	1	2	yes	—
234	C₂×C₄	#41×C₂₃	1	2	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
243	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
244	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
245	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
246	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
247	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	C₂₃×#126	1	4	yes	—
259	C₂	—	4	2	no	—
260	C₂	C₂×C₂₃×#126	1	1	yes	characteristic
261	C₂	C₂₃×#529	1	1	yes	characteristic
262	C₂	C₂₃×#542	1	1	yes	characteristic
263	C₂	C₂₃×#552	1	1	yes	characteristic
264	C₂	C₂₃×#553	1	1	yes	characteristic
265	C₂	C₂₃×#553	1	1	yes	characteristic
266	C₂	C₂₃×#556	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#401	C₂	semidirect
3. C₂₃×#389	C₂	semidirect
4. C₂₃×#389	C₂	semidirect
6. C₂₃×#387	C₂	semidirect
7. C₂₃×#382	C₂	semidirect
10. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
101. #2494	C₂₃	direct (N × H)
126. #401	C₂×C₂₃	semidirect
128. #389	C₂×C₂₃	semidirect
129. #389	C₂×C₂₃	semidirect
131. #387	C₂×C₂₃	semidirect
132. #382	C₂×C₂₃	semidirect
135. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
168. C₂₃	#2494	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#401	2
C₂₃×#397	2
C₂₃×#389	2
C₂₃×#388	2
C₂₃×#387	2
C₂₃×#382	2
C₂₃×#380	2
C₄×C₄⋊C₄×C₂₃	2
C₄×C₂²⋊C₄×C₂₃	2
#2494	23

Subgroups — C23×#2494 #15670113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2494 #15670113