Subgroups — C₂₃×#2339 #15669958

Order 2944.

928 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
3	C₂₃×#391	C₂	1	2	yes	maximal, complemented
4	C₂₃×#386	C₂	1	4	yes	maximal, complemented
5	C₂₃×#381	C₂	1	2	yes	maximal, complemented
6	C₂₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#95	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₃	—	2	8	no	—
10	C₂×#41×C₂₃	C₄	1	8	yes	complemented
11	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂²	1	4	yes	—
15	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
16	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₂₃	—	2	8	no	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
23	C₂₃×#95	C₂²	1	1	yes	characteristic
24	C₂₃×#95	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	4	yes	—
26	C₂₃×#95	C₂²	1	1	yes	characteristic
27	C₂₃×#95	C₂²	1	4	yes	—
28	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
29	C₂⁴×C₂₃	C₂³	1	2	yes	—
30	#41×C₂₃	—	2	32	no	—
31	C₂×Q₈×C₂₃	—	2	4	no	—
32	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
33	C₂×D₄×C₂₃	—	2	8	no	—
34	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
36	C₂×D₄×C₂₃	D₄	1	8	yes	—
37	C₂²×C₄×C₂₃	—	2	8	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	—	2	8	no	—
40	C₂²×C₄×C₂₃	—	2	8	no	—
41	C₂²×C₄×C₂₃	—	4	4	no	—
42	C₂²×C₄×C₂₃	C₂×C₄	1	8	yes	complemented
43	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	complemented
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₂²⋊C₄×C₂₃	—	2	8	no	—
52	C₂²⋊C₄×C₂₃	—	2	4	no	—
53	C₂²⋊C₄×C₂₃	—	4	2	no	—
54	C₄²×C₂₃	—	2	4	no	—
55	C₂³×C₂₃	—	2	4	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	—	2	2	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	1	no	—
61	C₂³×C₂₃	—	2	1	no	—
62	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂³×C₂₃	C₂×D₄	1	8	yes	—
65	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
66	Q₈×C₂₃	—	2	8	no	—
67	D₄×C₂₃	—	2	16	no	—
68	D₄×C₂₃	C₂²⋊C₄	1	16	yes	complemented
69	C₂×C₄×C₂₃	—	2	16	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	8	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	4	16	no	—
82	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₂₃	#41	1	4	yes	complemented
87	#2339	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₂₃	—	2	4	no	—
89	C₂²×C₂₃	—	2	4	no	—
90	C₂²×C₂₃	—	2	8	no	—
91	C₂²×C₂₃	—	2	8	no	—
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	—	2	2	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
98	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
100	C₂²×C₂₃	C₂×#41	1	2	yes	—
101	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	—	4	4	no	—
107	C₄×C₂₃	—	4	2	no	—
108	C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂²×#41	C₂×C₂₃	1	1	yes	characteristic
110	C₂×C₄×D₄	C₂×C₂₃	1	2	yes	complemented
111	#391	C₂×C₂₃	1	2	yes	complemented
112	#386	C₂×C₂₃	1	4	yes	complemented
113	#381	C₂×C₂₃	1	2	yes	complemented
114	#380	C₂×C₂₃	1	1	yes	characteristic, complemented
115	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×#95	C₂×C₂₃	1	2	yes	complemented
117	C₂×C₂₃	—	2	8	no	—
118	C₂×C₂₃	—	4	2	no	—
119	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
121	C₂×C₂₃	#520	1	1	yes	characteristic
122	C₂×C₂₃	#546	1	2	yes	—
123	C₂×C₂₃	#547	1	2	yes	—
124	C₂×#41	—	2	8	no	—
125	C₂×#41	C₄×C₂₃	1	8	yes	complemented
126	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
128	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
129	C₂³×C₄	C₂²×C₂₃	1	4	yes	—
130	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
131	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
132	C₄×D₄	—	2	8	no	—
133	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
137	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
138	#95	C₂²×C₂₃	1	1	yes	characteristic
139	#95	C₂²×C₂₃	1	2	yes	—
140	#95	C₂²×C₂₃	1	4	yes	—
141	#95	C₂²×C₂₃	1	1	yes	characteristic
142	#95	C₂²×C₂₃	1	4	yes	—
143	C₂₃	#2339	1	1	yes	characteristic, direct product factor, Sylow
144	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
145	C₂⁴	C₂³×C₂₃	1	2	yes	—
146	#41	—	2	32	no	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
149	C₂×D₄	—	2	8	no	—
150	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
151	C₂×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
152	C₂×D₄	D₄×C₂₃	1	8	yes	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	C₂×C₄×C₂₃	1	8	yes	complemented
159	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	complemented
160	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
162	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
166	C₄⋊C₄	—	2	4	no	—
167	C₂²⋊C₄	—	2	8	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₂²⋊C₄	—	4	2	no	—
170	C₄²	—	2	4	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
179	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
180	C₂³	C₂×D₄×C₂₃	1	8	yes	—
181	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
182	Q₈	—	2	8	no	—
183	D₄	—	2	16	no	—
184	D₄	C₂²⋊C₄×C₂₃	1	16	yes	complemented
185	C₂×C₄	—	2	16	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	16	no	—
198	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
199	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
200	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
201	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
202	C₂×C₄	#41×C₂₃	1	4	yes	complemented
203	C₂²	—	2	8	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	8	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
213	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
214	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
215	C₂²	C₂×#41×C₂₃	1	2	yes	—
216	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
217	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	4	no	—
220	C₄	—	2	4	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	4	no	—
223	C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
224	C₂	—	2	8	no	—
225	C₂	—	4	2	no	—
226	C₂	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
227	C₂	C₄×#41×C₂₃	1	1	yes	characteristic
228	C₂	C₂₃×#520	1	1	yes	characteristic
229	C₂	C₂₃×#546	1	2	yes	—
230	C₂	C₂₃×#547	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×#41×C₂₃	C₂
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. C₂×C₄×D₄×C₂₃	C₂	semidirect
3. C₂₃×#391	C₂	semidirect
4. C₂₃×#386	C₂	semidirect
5. C₂₃×#381	C₂	semidirect
6. C₂₃×#380	C₂	semidirect
7. C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
8. C₂×C₂₃×#95	C₂	semidirect
10. C₂×#41×C₂₃	C₄	semidirect
32. C₂×Q₈×C₂₃	C₂×C₄	semidirect
35. C₂×D₄×C₂₃	C₂×C₄	semidirect
42. C₂²×C₄×C₂₃	C₂×C₄	semidirect
43. C₂²×C₄×C₂₃	C₂×C₄	semidirect
68. D₄×C₂₃	C₂²⋊C₄	semidirect
86. C₂×C₄×C₂₃	#41	semidirect
87. #2339	C₂₃	direct (N × H)
110. C₂×C₄×D₄	C₂×C₂₃	semidirect
111. #391	C₂×C₂₃	semidirect
112. #386	C₂×C₂₃	semidirect
113. #381	C₂×C₂₃	semidirect
114. #380	C₂×C₂₃	semidirect
115. C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
116. C₂×#95	C₂×C₂₃	semidirect
125. C₂×#41	C₄×C₂₃	semidirect
143. C₂₃	#2339	direct (N × H)
148. C₂×Q₈	C₂×C₄×C₂₃	semidirect
151. C₂×D₄	C₂×C₄×C₂₃	semidirect
158. C₂²×C₄	C₂×C₄×C₂₃	semidirect
159. C₂²×C₄	C₂×C₄×C₂₃	semidirect
184. D₄	C₂²⋊C₄×C₂₃	semidirect
202. C₂×C₄	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₃	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#391	2
C₂₃×#386	2
C₂₃×#381	2
C₂₃×#380	2
C₄×C₂²⋊C₄×C₂₃	2
C₂×C₂₃×#95	2
#2339	23

Subgroups — C23×#2339 #15669958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2339 #15669958