Subgroups — C₂₃×#2337 #15669956

Order 2944.

540 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#391	C₂	1	2	yes	maximal, complemented
3	C₂₃×#388	C₂	1	2	yes	maximal, complemented
4	C₂₃×#388	C₂	1	2	yes	maximal, complemented
5	C₂₃×#386	C₂	1	2	yes	maximal
6	C₂₃×#385	C₂	1	2	yes	maximal, complemented
7	C₂₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#124	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	2	yes	—
28	C₂₃×#95	C₂²	1	1	yes	characteristic
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂⁴×C₂₃	C₂³	1	2	yes	—
33	C₂×Q₈×C₂₃	—	4	2	no	—
34	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
35	C₂×D₄×C₂₃	—	4	2	no	—
36	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	4	2	no	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₂₃	—	2	4	no	—
49	C₄⋊C₄×C₂₃	—	4	2	no	—
50	C₂²⋊C₄×C₂₃	—	2	4	no	—
51	C₂²⋊C₄×C₂₃	—	4	2	no	—
52	C₂²⋊C₄×C₂₃	—	4	2	no	—
53	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
54	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
55	C₄²×C₂₃	—	4	1	no	—
56	C₄²×C₂₃	—	4	1	no	—
57	C₄²×C₂₃	—	4	2	no	—
58	C₄²×C₂₃	C₂×C₄	1	4	yes	complemented
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	2	2	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
66	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
67	Q₈×C₂₃	—	4	4	no	—
68	D₄×C₂₃	—	4	4	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	1	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	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	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	—	4	8	no	—
87	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
91	C₂×C₄×C₂₃	#41	1	4	yes	—
92	#2337	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₂₃	—	2	4	no	—
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	—	4	2	no	—
99	C₂²×C₂₃	—	4	2	no	—
100	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
101	C₂²×C₂₃	C₂×#41	1	2	yes	—
102	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
103	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	2	4	no	—
108	C₄×C₂₃	—	2	2	no	—
109	C₄×C₂₃	—	2	4	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	—	4	2	no	—
113	C₂×#124	C₂×C₂₃	1	1	yes	characteristic
114	#391	C₂×C₂₃	1	2	yes	complemented
115	#388	C₂×C₂₃	1	2	yes	complemented
116	#388	C₂×C₂₃	1	2	yes	complemented
117	#386	C₂×C₂₃	1	2	yes	—
118	#385	C₂×C₂₃	1	2	yes	complemented
119	#383	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
121	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
122	C₂×C₂₃	—	4	4	no	—
123	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
124	C₂×C₂₃	#518	1	1	yes	characteristic
125	C₂×C₂₃	#520	1	1	yes	characteristic
126	C₂×C₂₃	#538	1	2	yes	—
127	C₂×C₂₃	#542	1	2	yes	—
128	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
129	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
131	#124	C₄×C₂₃	1	8	yes	complemented
132	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
136	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
137	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
139	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
141	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
143	#95	C₂²×C₂₃	1	2	yes	—
144	#95	C₂²×C₂₃	1	2	yes	—
145	#95	C₂²×C₂₃	1	2	yes	—
146	#95	C₂²×C₂₃	1	1	yes	characteristic
147	#95	C₂²×C₂₃	1	2	yes	—
148	#95	C₂²×C₂₃	1	1	yes	characteristic
149	#95	C₂²×C₂₃	1	2	yes	—
150	C₂₃	#2337	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁴	C₂³×C₂₃	1	2	yes	—
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	complemented
173	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	complemented
174	C₄²	—	4	1	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	2	no	—
177	C₄²	C₂×C₄×C₂₃	1	4	yes	complemented
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
185	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	D₄	—	4	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
207	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
208	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
209	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
210	C₂×C₄	#41×C₂₃	1	4	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
219	C₂²	C₂×#41×C₂₃	1	2	yes	—
220	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
221	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
222	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	4	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₂	—	4	4	no	—
232	C₂	C₄×#41×C₂₃	1	1	yes	characteristic
233	C₂	C₂₃×#518	1	1	yes	characteristic
234	C₂	C₂₃×#520	1	1	yes	characteristic
235	C₂	C₂₃×#538	1	2	yes	—
236	C₂	C₂₃×#542	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
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×#391	C₂	semidirect
3. C₂₃×#388	C₂	semidirect
4. C₂₃×#388	C₂	semidirect
6. C₂₃×#385	C₂	semidirect
7. C₂₃×#383	C₂	semidirect
8. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
9. C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
13. C₂₃×#124	C₄	semidirect
34. C₂×Q₈×C₂₃	C₂×C₄	semidirect
53. C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
54. C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
58. C₄²×C₂₃	C₂×C₄	semidirect
92. #2337	C₂₃	direct (N × H)
114. #391	C₂×C₂₃	semidirect
115. #388	C₂×C₂₃	semidirect
116. #388	C₂×C₂₃	semidirect
118. #385	C₂×C₂₃	semidirect
119. #383	C₂×C₂₃	semidirect
120. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
121. C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
131. #124	C₄×C₂₃	semidirect
150. C₂₃	#2337	direct (N × H)
153. C₂×Q₈	C₂×C₄×C₂₃	semidirect
172. C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
173. C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
177. 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×#124	2
C₂₃×#391	2
C₂₃×#388	2
C₂₃×#386	2
C₂₃×#385	2
C₂₃×#383	2
C₄×C₄⋊C₄×C₂₃	2
C₄×C₂²⋊C₄×C₂₃	2
#2337	23

Subgroups — C23×#2337 #15669956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2337 #15669956