Subgroups — C₂₃×#2499 #15670118

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#397	C₂	1	4	yes	maximal
4	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#387	C₂	1	4	yes	maximal, complemented
7	C₂₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×C₂²⋊Q₈	—	2	4	no	—
13	C₂₃×#117	—	2	4	no	—
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	4	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
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	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	4	yes	—
27	C₂₃×#95	C₂²	1	4	yes	—
28	C₂₃×#95	C₂²	1	4	yes	—
29	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₂₃	—	2	2	no	—
31	C₂×Q₈×C₂₃	—	2	2	no	—
32	C₂×Q₈×C₂₃	—	2	2	no	—
33	C₂²×C₄×C₂₃	—	2	2	no	—
34	C₂²×C₄×C₂₃	—	2	2	no	—
35	C₂²×C₄×C₂₃	D₄	1	4	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	—	2	4	no	—
48	C₄⋊C₄×C₂₃	—	2	8	no	—
49	C₂²⋊C₄×C₂₃	—	2	4	no	—
50	C₂²⋊C₄×C₂₃	—	2	4	no	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₄²×C₂₃	—	2	4	no	—
53	C₄²×C₂₃	—	2	2	no	—
54	C₄²×C₂₃	—	2	2	no	—
55	C₂³×C₂₃	—	2	2	no	—
56	C₂³×C₂₃	—	2	1	no	—
57	C₂³×C₂₃	—	2	1	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	C₂×D₄	1	2	yes	—
60	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
61	Q₈×C₂₃	—	4	4	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	1	no	—
64	C₂×C₄×C₂₃	—	2	1	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	8	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	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	4	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₃	#41	1	4	yes	—
82	C₂×C₄×C₂₃	#41	1	2	yes	—
83	C₂×C₄×C₂₃	#41	1	2	yes	—
84	#2499	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₂₃	—	2	2	no	—
86	C₂²×C₂₃	—	2	2	no	—
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	—	4	2	no	—
89	C₂²×C₂₃	—	4	2	no	—
90	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
92	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
93	C₂²×C₂₃	C₂×#41	1	2	yes	—
94	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
95	C₄×C₂₃	—	2	2	no	—
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	4	2	no	—
98	C₄×C₂₃	—	4	2	no	—
99	C₄×C₂₃	—	4	2	no	—
100	C₄×C₂₃	—	4	2	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	#123	1	4	yes	—
103	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
104	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
105	#397	C₂×C₂₃	1	4	yes	—
106	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
107	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
108	#387	C₂×C₂₃	1	4	yes	complemented
109	#384	C₂×C₂₃	1	1	yes	characteristic, complemented
110	#382	C₂×C₂₃	1	1	yes	characteristic, complemented
111	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
112	C₂×C₂₃	—	4	2	no	—
113	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
114	C₂×C₂₃	#532	1	1	yes	characteristic
115	C₂×C₂₃	#536	1	1	yes	characteristic
116	C₂×C₂₃	#552	1	2	yes	—
117	C₂×C₂₃	#556	1	2	yes	—
118	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
120	C₂²⋊Q₈	—	2	4	no	—
121	#117	—	2	4	no	—
122	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
130	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
131	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
133	#95	C₂²×C₂₃	1	2	yes	—
134	#95	C₂²×C₂₃	1	4	yes	—
135	#95	C₂²×C₂₃	1	4	yes	—
136	#95	C₂²×C₂₃	1	4	yes	—
137	C₂₃	#2499	1	1	yes	characteristic, direct product factor, Sylow
138	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	D₄×C₂₃	1	4	yes	—
145	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
149	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
152	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	2	8	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	1	no	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	2	no	—
168	C₂³	C₂×D₄×C₂₃	1	2	yes	—
169	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
189	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
190	C₂×C₄	#41×C₂₃	1	4	yes	—
191	C₂×C₄	#41×C₂₃	1	2	yes	—
192	C₂×C₄	#41×C₂₃	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
199	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
200	C₂²	C₂×#41×C₂₃	1	2	yes	—
201	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
202	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	C₂₃×#123	1	4	yes	—
211	C₂	—	4	2	no	—
212	C₂	C₂×C₂₃×#123	1	1	yes	characteristic
213	C₂	C₂₃×#532	1	1	yes	characteristic
214	C₂	C₂₃×#536	1	1	yes	characteristic
215	C₂	C₂₃×#552	1	2	yes	—
216	C₂	C₂₃×#556	1	2	yes	—

	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
4. C₂₃×#391	C₂	semidirect
5. C₂₃×#391	C₂	semidirect
6. C₂₃×#387	C₂	semidirect
7. C₂₃×#384	C₂	semidirect
8. C₂₃×#382	C₂	semidirect
84. #2499	C₂₃	direct (N × H)
106. #391	C₂×C₂₃	semidirect
107. #391	C₂×C₂₃	semidirect
108. #387	C₂×C₂₃	semidirect
109. #384	C₂×C₂₃	semidirect
110. #382	C₂×C₂₃	semidirect
137. C₂₃	#2499	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₂²⋊Q₈	2
C₂×C₂₃×#117	2
C₂₃×#397	2
C₂₃×#391	2
C₂₃×#387	2
C₂₃×#384	2
C₂₃×#382	2
C₄×C₂²⋊C₄×C₂₃	2
#2499	23

Subgroups — C23×#2499 #15670118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2499 #15670118