Subgroups — #15683246

Order 2944.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093045	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092900	C₂	1	1	yes	maximal, characteristic
3	#13092726	C₂	1	2	yes	maximal, complemented
4	#13092715	C₂	1	2	yes	maximal, complemented
5	#13092710	C₂	1	2	yes	maximal, complemented
6	#13092695	C₂	1	2	yes	maximal, complemented
7	#13092460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625000	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092443	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092418	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#416	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#117	C₂²	1	1	yes	characteristic
15	Q₈×Dic₂₃	C₂²	1	2	yes	—
16	D₄×Dic₂₃	—	2	2	no	—
17	D₄×Dic₂₃	C₂²	1	2	yes	—
18	#10625097	—	2	1	no	—
19	#10625097	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₂₃	—	2	1	no	—
21	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂×#67267	C₂²	1	1	yes	characteristic
24	#10625061	C₂²	1	2	yes	—
25	#10625059	C₂²	1	2	yes	—
26	#10625049	—	2	2	no	—
27	#10625041	—	2	1	no	—
28	#10625040	—	2	1	no	—
29	C₂₃×#103	C₂²	1	1	yes	characteristic
30	C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂₃×#102	C₂²	1	1	yes	characteristic
32	C₂₃×#102	C₂²	1	1	yes	characteristic
33	#10625017	C₂²	1	2	yes	—
34	#10625014	C₂²	1	2	yes	—
35	#10625000	C₂²	1	2	yes	—
36	#10625000	C₂²	1	2	yes	—
37	Dic₂₃⋊C₈	C₂²	1	2	yes	—
38	Dic₂₃⋊C₈	C₂²	1	2	yes	—
39	#10624990	C₂²	1	2	yes	—
40	#10624989	C₂²	1	2	yes	—
41	#41×C₂₃	—	2	2	no	—
42	#41×C₂₃	Q₈	1	4	yes	complemented
43	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₂₃	—	2	1	no	—
45	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₂₃	—	2	1	no	—
49	C₂²×Dic₂₃	—	2	1	no	—
50	C₂²×Dic₂₃	—	2	2	no	—
51	C₂×Dic₄₆	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²⋊C₄×C₂₃	—	2	1	no	—
58	C₄²×C₂₃	—	2	1	no	—
59	C₂²⋊Dic₂₃	—	2	2	no	—
60	C₂²⋊Dic₂₃	D₄	1	2	yes	—
61	C₄⋊Dic₂₃	—	2	1	no	—
62	C₄⋊Dic₂₃	—	2	1	no	—
63	C₄⋊Dic₂₃	—	2	2	no	—
64	C₄⋊Dic₂₃	—	2	2	no	—
65	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
66	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
67	C₄⋊Dic₂₃	C₂³	1	2	yes	—
68	C₄⋊Dic₂₃	C₂³	1	2	yes	—
69	Dic₂₃⋊C₄	—	2	2	no	—
70	Dic₂₃⋊C₄	—	2	2	no	—
71	C₄×Dic₂₃	—	2	2	no	—
72	C₄×Dic₂₃	C₂³	1	2	yes	—
73	#67267	—	2	2	no	—
74	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
75	C₂³×C₂₃	—	2	1	no	—
76	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₂₃	—	2	1	no	—
78	Q₈×C₂₃	C₂×Q₈	1	2	yes	—
79	D₄×C₂₃	—	2	4	no	—
80	D₄×C₂₃	—	2	1	no	—
81	D₄×C₂₃	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
88	C₂×C₄×C₂₃	#41	1	2	yes	—
89	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₂₃	—	2	2	no	—
91	C₂×Dic₂₃	—	2	2	no	—
92	C₂×Dic₂₃	—	8	2	no	—
93	C₂×Dic₂₃	—	8	1	no	—
94	C₂×Dic₂₃	—	8	1	no	—
95	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
96	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
97	Dic₄₆	—	8	2	no	—
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	—	2	1	no	—
100	C₂₃⋊C₈	—	2	2	no	—
101	#3083	—	23	1	no	maximal, Sylow
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	4	1	no	—
106	C₂²×C₂₃	—	4	1	no	—
107	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
108	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
113	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
114	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
115	Dic₂₃	—	2	2	no	—
116	Dic₂₃	—	2	2	no	—
117	Dic₂₃	—	4	2	no	—
118	Dic₂₃	—	4	2	no	—
119	#544	—	23	1	no	—
120	#520	—	23	1	no	—
121	#479	—	23	2	no	—
122	#478	—	23	2	no	—
123	#477	—	23	2	no	—
124	#474	—	23	2	no	—
125	#428	—	23	1	no	—
126	C₂×#107	—	23	1	no	—
127	#424	—	23	1	no	—
128	#417	—	23	1	no	—
129	#416	D₂₃	1	1	yes	characteristic
130	C₂×C₂₃	—	2	2	no	—
131	C₂×C₂₃	—	4	2	no	—
132	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₂₃	#576	1	1	yes	characteristic
134	C₂×C₂₃	#578	1	1	yes	characteristic
135	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
136	C₂×M₁₆	—	23	1	no	—
137	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
138	C₄⋊Q₈	—	23	2	no	—
139	C₄⋊Q₈	—	46	1	no	—
140	#125	—	23	2	no	—
141	#125	—	46	1	no	—
142	C₂²⋊Q₈	—	46	2	no	—
143	C₄×Q₈	—	23	2	no	—
144	C₄×D₄	—	23	2	no	—
145	C₄×D₄	—	46	2	no	—
146	#117	—	23	1	no	—
147	#117	—	46	1	no	—
148	#117	C₂×D₂₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	23	1	no	—
150	C₂×C₄⋊C₄	—	23	1	no	—
151	C₂×C₄⋊C₄	—	46	1	no	—
152	#107	—	23	2	no	—
153	#107	—	23	2	no	—
154	#107	—	23	2	no	—
155	#106	—	23	2	no	—
156	C₄⋊C₈	—	23	2	no	—
157	C₄⋊C₈	—	23	2	no	—
158	#103	—	23	2	no	—
159	#103	C₂×D₂₃	1	1	yes	characteristic
160	#103	C₂×D₂₃	1	1	yes	characteristic
161	#102	—	23	2	no	—
162	#102	C₂×D₂₃	1	1	yes	characteristic
163	#102	C₂×D₂₃	1	1	yes	characteristic
164	C₂₃	#3083	1	1	yes	characteristic, complemented, Sylow
165	#41	—	2	2	no	—
166	#41	Dic₄₆	1	4	yes	complemented
167	C₂×Q₈	—	46	2	no	—
168	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	46	2	no	—
173	C₂²×C₄	—	46	1	no	—
174	C₂²×C₄	—	46	1	no	—
175	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
176	M₁₆	—	46	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	23	2	no	—
179	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
181	C₄⋊C₄	—	23	1	no	—
182	C₄⋊C₄	—	23	1	no	—
183	C₄⋊C₄	—	23	2	no	—
184	C₄⋊C₄	—	23	2	no	—
185	C₄⋊C₄	—	46	2	no	—
186	C₄⋊C₄	—	46	2	no	—
187	C₄⋊C₄	—	46	2	no	—
188	C₄⋊C₄	—	46	2	no	—
189	C₄⋊C₄	—	46	1	no	—
190	C₄⋊C₄	—	46	1	no	—
191	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	23	2	no	—
195	C₂²⋊C₄	—	46	2	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	23	2	no	—
198	C₄²	—	46	2	no	—
199	C₂³	—	2	1	no	—
200	C₂³	D₄×D₂₃	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	184	2	no	—
203	Q₈	C₂×Dic₄₆	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₄₆	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	23	2	no	—
212	C₂×C₄	—	23	2	no	—
213	C₂×C₄	—	46	2	no	—
214	C₂×C₄	—	46	2	no	—
215	C₂×C₄	—	184	2	no	—
216	C₂×C₄	—	184	1	no	—
217	C₂×C₄	—	184	1	no	—
218	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
219	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
220	C₂×C₄	#67290	1	2	yes	—
221	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	46	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#10625049	1	2	yes	—
231	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	46	2	no	—
236	C₄	—	46	2	no	—
237	C₄	—	92	2	no	—
238	C₄	—	92	2	no	—
239	C₄	#10625049	1	2	yes	—
240	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
241	C₄	C₂×#67290	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#10625049	1	1	yes	characteristic
245	C₂	#13093219	1	1	yes	characteristic
246	C₂	#13093231	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#416	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#41×C₂₃	C₂²
hypercenter	#416	D₂₃
hypocenter	C₂₃	#3083
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. #13093045	C₂	semidirect
3. #13092726	C₂	semidirect
4. #13092715	C₂	semidirect
5. #13092710	C₂	semidirect
6. #13092695	C₂	semidirect
7. #13092460	C₂	semidirect
8. C₂×#10625000	C₂	semidirect
9. #13092443	C₂	semidirect
42. #41×C₂₃	Q₈	semidirect
164. C₂₃	#3083	semidirect
166. #41	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
#13093045	2
#13092900	2
#13092726	2
#13092715	2
#13092710	2
#13092695	2
#13092460	2
C₂×#10625000	2
#13092443	2
#13092418	2
C₂₃×#416	2
#3083	23