Subgroups — C₂₃×#2455 #15670074

Order 2944.

660 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	C₂×C₂₃×#123	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
3	C₂₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#394	C₂	1	2	yes	maximal, complemented
5	C₂₃×#392	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#389	C₂	1	2	yes	maximal, complemented
7	C₂₃×#388	C₂	1	2	yes	maximal, complemented
8	C₂₃×#385	C₂	1	2	yes	maximal, complemented
9	C₂₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#123	—	2	8	no	—
15	C₄×D₄×C₂₃	—	2	8	no	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	1	yes	characteristic
32	C₂₃×#95	C₂²	1	1	yes	characteristic
33	C₂⁴×C₂₃	C₂³	1	2	yes	—
34	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×D₄×C₂₃	—	2	4	no	—
36	C₂×D₄×C₂₃	—	2	4	no	—
37	C₂×D₄×C₂₃	—	2	4	no	—
38	C₂×D₄×C₂₃	—	2	4	no	—
39	C₂×D₄×C₂₃	—	4	1	no	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	—	2	4	no	—
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	4	1	no	—
44	C₂²×C₄×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
52	C₄⋊C₄×C₂₃	—	2	4	no	—
53	C₄⋊C₄×C₂₃	—	2	4	no	—
54	C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₄⋊C₄×C₂₃	—	2	4	no	—
56	C₄⋊C₄×C₂₃	—	4	1	no	—
57	C₂²⋊C₄×C₂₃	—	2	4	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂²⋊C₄×C₂₃	—	2	4	no	—
60	C₂²⋊C₄×C₂₃	—	2	4	no	—
61	C₂²⋊C₄×C₂₃	—	2	4	no	—
62	C₂²⋊C₄×C₂₃	—	4	1	no	—
63	C₂²⋊C₄×C₂₃	D₄	1	8	yes	complemented
64	C₄²×C₂₃	—	2	4	no	—
65	C₂³×C₂₃	—	2	2	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	—	2	2	no	—
71	C₂³×C₂₃	—	2	1	no	—
72	C₂³×C₂₃	—	2	2	no	—
73	C₂³×C₂₃	—	2	1	no	—
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	C₂×D₄	1	4	yes	—
76	C₂³×C₂₃	#41	1	2	yes	—
77	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
78	D₄×C₂₃	—	2	16	no	—
79	D₄×C₂₃	—	4	4	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	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	—	4	8	no	—
98	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
99	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
100	C₂×C₄×C₂₃	#41	1	4	yes	—
101	#2455	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₂₃	—	2	4	no	—
103	C₂²×C₂₃	—	2	4	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	4	no	—
106	C₂²×C₂₃	—	2	4	no	—
107	C₂²×C₂₃	—	4	2	no	—
108	C₂²×C₂₃	—	4	2	no	—
109	C₂²×C₂₃	—	4	1	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	—	4	2	no	—
113	C₂²×C₂₃	—	4	1	no	—
114	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
115	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
116	C₂²×C₂₃	C₂×#41	1	2	yes	—
117	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₄×C₂₃	—	2	4	no	—
120	C₄×C₂₃	—	2	4	no	—
121	C₄×C₂₃	—	2	4	no	—
122	C₄×C₂₃	—	4	2	no	—
123	C₄×C₂₃	—	4	2	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	—	4	1	no	—
126	C₂×#123	C₂×C₂₃	1	2	yes	complemented
127	C₂×C₄×D₄	C₂×C₂₃	1	2	yes	complemented
128	#399	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#394	C₂×C₂₃	1	2	yes	complemented
130	#392	C₂×C₂₃	1	1	yes	characteristic
131	#389	C₂×C₂₃	1	2	yes	complemented
132	#388	C₂×C₂₃	1	2	yes	complemented
133	#385	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#385	C₂×C₂₃	1	2	yes	complemented
135	C₂×C₂₃	—	4	4	no	—
136	C₂×C₂₃	—	4	2	no	—
137	C₂×C₂₃	#525	1	2	yes	—
138	C₂×C₂₃	#539	1	1	yes	characteristic
139	C₂×C₂₃	D₄²	1	1	yes	characteristic
140	C₂×C₂₃	#547	1	2	yes	—
141	C₂×C₂₃	#559	1	1	yes	characteristic
142	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
143	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
144	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
146	#123	—	2	8	no	—
147	C₄×D₄	—	2	8	no	—
148	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
160	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
161	#95	C₂²×C₂₃	1	2	yes	—
162	#95	C₂²×C₂₃	1	2	yes	—
163	#95	C₂²×C₂₃	1	1	yes	characteristic
164	#95	C₂²×C₂₃	1	1	yes	characteristic
165	C₂₃	#2455	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁴	C₂³×C₂₃	1	2	yes	—
167	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	2	4	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	4	1	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
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	2	yes	—
183	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₂²⋊C₄	—	2	4	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	4	1	no	—
196	C₂²⋊C₄	D₄×C₂₃	1	8	yes	complemented
197	C₄²	—	2	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	C₂×D₄×C₂₃	1	4	yes	—
209	C₂³	#41×C₂₃	1	2	yes	—
210	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
211	D₄	—	2	16	no	—
212	D₄	—	4	4	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	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	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	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
232	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
233	C₂×C₄	#41×C₂₃	1	4	yes	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	1	no	—
246	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
247	C₂²	C₂×#41×C₂₃	1	2	yes	—
248	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
249	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
250	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	4	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₂	—	4	4	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂₃×#525	1	2	yes	—
261	C₂	C₂₃×#539	1	1	yes	characteristic
262	C₂	D₄²×C₂₃	1	1	yes	characteristic
263	C₂	C₂₃×#547	1	2	yes	—
264	C₂	C₂₃×#559	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂₃×#392	C₂
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. C₂×C₂₃×#123	C₂	semidirect
2. C₂×C₄×D₄×C₂₃	C₂	semidirect
3. C₂₃×#399	C₂	semidirect
4. C₂₃×#394	C₂	semidirect
6. C₂₃×#389	C₂	semidirect
7. C₂₃×#388	C₂	semidirect
8. C₂₃×#385	C₂	semidirect
9. C₂₃×#385	C₂	semidirect
63. C₂²⋊C₄×C₂₃	D₄	semidirect
101. #2455	C₂₃	direct (N × H)
126. C₂×#123	C₂×C₂₃	semidirect
127. C₂×C₄×D₄	C₂×C₂₃	semidirect
128. #399	C₂×C₂₃	semidirect
129. #394	C₂×C₂₃	semidirect
131. #389	C₂×C₂₃	semidirect
132. #388	C₂×C₂₃	semidirect
133. #385	C₂×C₂₃	semidirect
134. #385	C₂×C₂₃	semidirect
165. C₂₃	#2455	direct (N × H)
196. C₂²⋊C₄	D₄×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×#123	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#399	2
C₂₃×#394	2
C₂₃×#392	2
C₂₃×#389	2
C₂₃×#388	2
C₂₃×#385	2
#2455	23

Subgroups — C23×#2455 #15670074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2455 #15670074