Subgroups — #15682089

Order 2944.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092924	C₂	1	1	yes	maximal, characteristic
3	#13092486	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₂₃	C₂	1	4	yes	maximal, complemented
5	#13092445	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092384	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092380	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092372	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#131	—	2	4	no	—
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	2	yes	complemented
15	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂²	1	2	yes	complemented
17	#10625117	C₄	1	2	yes	—
18	#10625109	C₄	1	2	yes	—
19	#10625108	C₄	1	2	yes	—
20	#10625106	C₄	1	2	yes	—
21	C₄×C₂²⋊D₂₃	C₂²	1	4	yes	—
22	#10625097	C₂²	1	1	yes	characteristic
23	C₂×#67267	C₂²	1	1	yes	characteristic
24	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₂₃	—	2	4	no	—
26	C₂×C₈×D₂₃	C₂²	1	2	yes	—
27	C₂×C₈×D₂₃	C₂²	1	2	yes	—
28	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
29	(C₂×D₂₃)⋊C₈	C₂²	1	4	yes	—
30	#10624997	C₂²	1	2	yes	complemented
31	Dic₂₃⋊C₈	C₂²	1	4	yes	complemented
32	C₈×Dic₂₃	C₂²	1	2	yes	complemented
33	C₂³×D₂₃	C₂×C₄	1	1	yes	characteristic
34	#41×C₂₃	—	2	2	no	—
35	#41×C₂₃	—	2	2	no	—
36	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₃	C₂×C₄	1	4	yes	—
42	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₂₃	—	2	2	no	—
44	C₂×C₄×D₂₃	—	2	2	no	—
45	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₃	C₂³	1	2	yes	—
47	M₁₆×C₂₃	—	2	2	no	—
48	M₁₆×C₂₃	—	2	4	no	—
49	C₂×C₈×C₂₃	—	2	4	no	—
50	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
51	C₂×C₈×C₂₃	C₂³	1	2	yes	—
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
56	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
58	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₂₃	C₂³	1	2	yes	—
60	#67267	—	2	2	no	—
61	C₂×C₂₃⋊C₈	—	2	2	no	—
62	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
63	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
64	C₈×D₂₃	—	2	4	no	—
65	C₈×D₂₃	—	2	4	no	—
66	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
67	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₂₃	—	2	2	no	—
69	C₂²×D₂₃	—	2	4	no	—
70	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
71	Q₈×C₂₃	—	2	2	no	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃	#41	1	2	yes	—
80	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₂₃	—	2	8	no	—
82	C₂×Dic₂₃	—	2	2	no	—
83	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
84	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
85	C₄×D₂₃	—	2	4	no	—
86	C₄×D₂₃	—	2	2	no	—
87	C₄×D₂₃	—	2	2	no	—
88	C₈×C₂₃	—	2	2	no	—
89	C₈×C₂₃	C₂×D₄	1	2	yes	—
90	C₈×C₂₃	C₂×D₄	1	2	yes	—
91	C₂₃⋊C₈	—	2	2	no	—
92	C₂₃⋊C₈	—	2	2	no	—
93	#2929	—	23	1	no	maximal, Sylow
94	C₂²×C₂₃	—	2	2	no	—
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	2	4	no	—
97	C₂²×C₂₃	C₄×D₄	1	2	yes	—
98	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×D₂₃	—	2	8	no	—
100	C₂×D₂₃	—	2	2	no	—
101	C₂×D₂₃	—	2	2	no	—
102	C₂×D₂₃	#131	1	4	yes	complemented
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	C₄×D₄	1	2	yes	—
105	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
107	Dic₂₃	—	2	2	no	—
108	Dic₂₃	—	2	4	no	—
109	C₂×#131	D₂₃	1	1	yes	characteristic, complemented
110	C₂³×C₈	—	23	1	no	—
111	#525	—	23	1	no	—
112	#435	—	23	4	no	—
113	C₈×D₄	—	23	4	no	—
114	#424	—	23	1	no	—
115	#408	—	23	1	no	—
116	#407	—	23	1	no	—
117	#405	—	23	1	no	—
118	C₂×C₂₃	—	2	2	no	—
119	C₂×C₂₃	—	4	2	no	—
120	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂×C₂₃	C₂×#131	1	2	yes	—
122	D₂₃	—	2	4	no	—
123	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
124	C₂³×C₄	—	23	1	no	—
125	#131	—	2	4	no	—
126	C₂×M₁₆	—	23	1	no	—
127	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
128	C₂×M₁₆	C₂×D₂₃	1	2	yes	complemented
129	C₂²×C₈	—	23	1	no	—
130	C₂²×C₈	—	23	2	no	—
131	C₂²×C₈	—	23	2	no	—
132	C₂²×C₈	—	46	4	no	—
133	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₂₃	1	2	yes	complemented
135	#123	—	23	2	no	—
136	C₂²⋊Q₈	—	23	2	no	—
137	#121	—	23	2	no	—
138	#120	—	23	2	no	—
139	C₄×D₄	—	23	4	no	—
140	#117	—	23	1	no	—
141	C₄⋊C₈	—	23	4	no	—
142	C₂²⋊C₈	—	23	4	no	—
143	C₂²⋊C₈	—	23	4	no	—
144	#97	—	23	2	no	—
145	C₄×C₈	—	23	2	no	—
146	C₂₃	#2929	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	—	23	1	no	—
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
151	C₂×D₄	—	23	4	no	—
152	C₂×D₄	C₄×D₂₃	1	2	yes	—
153	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
154	C₂²×C₄	—	23	1	no	—
155	C₂²×C₄	—	23	2	no	—
156	C₂²×C₄	—	23	2	no	—
157	C₂²×C₄	—	46	2	no	—
158	C₂²×C₄	—	46	2	no	—
159	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	4	no	—
163	M₁₆	—	46	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	23	2	no	—
166	C₂×C₈	—	23	2	no	—
167	C₂×C₈	—	46	2	no	—
168	C₂×C₈	—	46	4	no	—
169	C₂×C₈	—	46	4	no	—
170	C₂×C₈	C₂²⋊D₂₃	1	4	yes	complemented
171	C₂×C₈	C₂²×D₂₃	1	2	yes	—
172	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
174	C₄⋊C₄	—	23	2	no	—
175	C₄⋊C₄	—	23	4	no	—
176	C₂²⋊C₄	—	23	2	no	—
177	C₂²⋊C₄	—	23	4	no	—
178	C₂²⋊C₄	—	23	4	no	—
179	C₄²	—	23	2	no	—
180	C₂³	—	23	2	no	—
181	C₂³	—	46	4	no	—
182	C₂³	—	46	2	no	—
183	C₂³	C₂×C₄×D₂₃	1	2	yes	—
184	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	46	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	23	4	no	—
192	C₂×C₄	—	23	2	no	—
193	C₂×C₄	—	46	2	no	—
194	C₂×C₄	—	46	4	no	—
195	C₂×C₄	—	46	2	no	—
196	C₂×C₄	—	46	2	no	—
197	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
199	C₂×C₄	#67288	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
201	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	46	2	no	—
204	C₈	—	46	2	no	—
205	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
206	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	23	4	no	—
211	C₂²	—	46	8	no	—
212	C₂²	—	46	2	no	—
213	C₂²	—	46	2	no	—
214	C₂²	C₄×C₂²⋊D₂₃	1	2	yes	—
215	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	46	2	no	—
218	C₄	—	46	4	no	—
219	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
220	C₄	C₂×#67288	1	1	yes	characteristic
221	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	46	4	no	—
225	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic
226	C₂	#131×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#131	C₂
prime radical	#13092924	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	#2929
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₂₃	C₂	semidirect
3. #13092486	C₂	semidirect
4. C₈×C₂²⋊D₂₃	C₂	semidirect
5. #13092445	C₂	semidirect
6. #13092384	C₂	semidirect
7. #13092380	C₂	semidirect
8. #13092372	C₂	semidirect
9. C₂×C₂₃×#131	C₂	semidirect
14. C₂×M₁₆×C₂₃	C₂²	semidirect
16. C₂²×C₈×C₂₃	C₂²	semidirect
28. C₂²⋊(C₂₃⋊C₈)	C₂²	semidirect
30. #10624997	C₂²	semidirect
31. Dic₂₃⋊C₈	C₂²	semidirect
32. C₈×Dic₂₃	C₂²	semidirect
50. C₂×C₈×C₂₃	D₄	semidirect
102. C₂×D₂₃	#131	semidirect
109. C₂×#131	D₂₃	semidirect
128. C₂×M₁₆	C₂×D₂₃	semidirect
134. C₂²×C₈	C₂×D₂₃	semidirect
146. C₂₃	#2929	semidirect
170. C₂×C₈	C₂²⋊D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₂₃	2
#13092924	2
#13092486	2
C₈×C₂²⋊D₂₃	2
#13092445	2
#13092384	2
#13092380	2
#13092372	2
C₂×C₂₃×#131	2
#2929	23