Subgroups — C₂₃×#2343 #15669962

Order 2944.

532 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×#117	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#387	C₂	1	2	yes	maximal, complemented
3	C₂₃×#387	C₂	1	2	yes	maximal, complemented
4	C₂₃×#385	C₂	1	2	yes	maximal
5	C₂₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#381	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
10	C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#117	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
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	1	yes	characteristic
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂₃×#95	C₂²	1	2	yes	—
36	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	—	2	4	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
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	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	C₄⋊C₄×C₂₃	—	2	4	no	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
55	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
56	C₂²⋊C₄×C₂₃	—	2	4	no	—
57	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
58	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
59	C₄²×C₂₃	—	2	4	no	—
60	C₄²×C₂₃	—	4	2	no	—
61	C₄²×C₂₃	C₂×C₄	1	4	yes	complemented
62	C₄²×C₂₃	C₂×C₄	1	4	yes	complemented
63	C₂³×C₂₃	—	2	2	no	—
64	C₂³×C₂₃	—	2	1	no	—
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
67	C₂³×C₂₃	#41	1	4	yes	—
68	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₄×C₂₃	—	2	8	no	—
70	C₂×C₄×C₂₃	—	2	2	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	1	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	1	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	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	4	8	no	—
90	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₂₃	#41	1	4	yes	—
97	#2343	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	2	no	—
100	C₂²×C₂₃	—	2	4	no	—
101	C₂²×C₂₃	—	2	4	no	—
102	C₂²×C₂₃	#117	1	4	yes	—
103	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
104	C₂²×C₂₃	C₂×#41	1	2	yes	—
105	C₂²×C₂₃	C₂×#41	1	2	yes	—
106	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	2	2	no	—
114	C₄×C₂₃	—	2	4	no	—
115	C₄×C₂₃	—	4	2	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	—	4	2	no	—
118	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
119	#387	C₂×C₂₃	1	2	yes	complemented
120	#387	C₂×C₂₃	1	2	yes	complemented
121	#385	C₂×C₂₃	1	2	yes	—
122	#383	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#382	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#381	C₂×C₂₃	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
127	#376	C₂×C₂₃	1	1	yes	characteristic, complemented
128	C₂×#95	C₂×C₂₃	1	1	yes	characteristic
129	C₂×C₂₃	—	2	4	no	—
130	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
131	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
132	C₂×C₂₃	#520	1	1	yes	characteristic
133	C₂×C₂₃	#551	1	2	yes	—
134	C₂×C₂₃	#552	1	2	yes	—
135	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
136	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
137	#117	C₄×C₂₃	1	8	yes	complemented
138	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
146	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
147	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
149	#95	C₂²×C₂₃	1	1	yes	characteristic
150	#95	C₂²×C₂₃	1	1	yes	characteristic
151	#95	C₂²×C₂₃	1	2	yes	—
152	#95	C₂²×C₂₃	1	1	yes	characteristic
153	#95	C₂²×C₂₃	1	2	yes	—
154	#95	C₂²×C₂₃	1	2	yes	—
155	#95	C₂²×C₂₃	1	2	yes	—
156	#95	C₂²×C₂₃	1	1	yes	characteristic
157	#95	C₂²×C₂₃	1	2	yes	—
158	#95	C₂²×C₂₃	1	2	yes	—
159	C₂₃	#2343	1	1	yes	characteristic, direct product factor, Sylow
160	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
166	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	complemented
179	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	complemented
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
182	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
183	C₄²	—	2	4	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₂×C₄×C₂₃	1	4	yes	complemented
186	C₄²	C₂×C₄×C₂₃	1	4	yes	complemented
187	C₂³	—	2	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	1	no	—
190	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
191	C₂³	#41×C₂₃	1	4	yes	—
192	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	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	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
215	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
216	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
217	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
218	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
219	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
220	C₂×C₄	#41×C₂₃	1	4	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	C₂₃×#117	1	4	yes	—
226	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
227	C₂²	C₂×#41×C₂₃	1	2	yes	—
228	C₂²	C₂×#41×C₂₃	1	2	yes	—
229	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
230	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₂	—	2	4	no	—
242	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
243	C₂	C₄×#41×C₂₃	1	1	yes	characteristic
244	C₂	C₂₃×#520	1	1	yes	characteristic
245	C₂	C₂₃×#551	1	2	yes	—
246	C₂	C₂₃×#552	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#387	C₂	semidirect
3. C₂₃×#387	C₂	semidirect
5. C₂₃×#383	C₂	semidirect
6. C₂₃×#382	C₂	semidirect
8. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
10. C₂₃×#376	C₂	semidirect
14. C₂₃×#117	C₄	semidirect
54. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
55. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
61. C₄²×C₂₃	C₂×C₄	semidirect
62. C₄²×C₂₃	C₂×C₄	semidirect
97. #2343	C₂₃	direct (N × H)
119. #387	C₂×C₂₃	semidirect
120. #387	C₂×C₂₃	semidirect
122. #383	C₂×C₂₃	semidirect
123. #382	C₂×C₂₃	semidirect
125. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
127. #376	C₂×C₂₃	semidirect
137. #117	C₄×C₂₃	semidirect
159. C₂₃	#2343	direct (N × H)
178. C₄⋊C₄	C₂×C₄×C₂₃	semidirect
179. C₄⋊C₄	C₂×C₄×C₂₃	semidirect
185. C₄²	C₂×C₄×C₂₃	semidirect
186. C₄²	C₂×C₄×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×#117	2
C₂₃×#387	2
C₂₃×#385	2
C₂₃×#383	2
C₂₃×#382	2
C₂₃×#381	2
C₄×C₄⋊C₄×C₂₃	2
C₄×C₂²⋊C₄×C₂₃	2
C₂₃×#376	2
C₂×C₂₃×#95	2
#2343	23

Subgroups — C23×#2343 #15669962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2343 #15669962