Subgroups — #15682229

Order 2944.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625103	C₂	1	1	yes	maximal, characteristic
2	#13092897	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092527	C₂	1	2	yes	maximal, complemented
4	#13092517	C₂	1	2	yes	maximal, complemented
5	C₄×#67273	C₂	1	2	yes	maximal, complemented
6	C₄×#67272	C₂	1	2	yes	maximal, complemented
7	#13092460	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092411	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625014	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092373	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67288	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₂₃	—	2	1	no	—
15	C₄×D₄×C₂₃	—	2	1	no	—
16	C₄×D₄×C₂₃	C₂²	1	2	yes	—
17	C₄×D₄×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#117	C₂²	1	1	yes	characteristic
19	C₂₃×#117	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
21	C₂×#67273	C₂²	1	2	yes	—
22	#10625103	C₄	1	8	yes	complemented
23	C₂×#67272	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×#67267	C₂²	1	1	yes	characteristic
26	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
27	#10625063	—	2	1	no	—
28	C₄⋊C₄×D₂₃	—	2	1	no	—
29	(C₂²⋊D₂₃)⋊C₄	—	2	2	no	—
30	C₄×D₉₂	C₂²	1	2	yes	complemented
31	C₄×Dic₄₆	C₂²	1	2	yes	complemented
32	#10625014	C₂²	1	2	yes	—
33	#10625014	C₂²	1	2	yes	—
34	#10624992	C₂²	1	1	yes	characteristic, complemented
35	#10624992	C₂²	1	1	yes	characteristic, complemented
36	#10624991	C₂²	1	1	yes	characteristic, complemented
37	#10624991	C₂²	1	1	yes	characteristic, complemented
38	#10624990	C₂²	1	2	yes	complemented
39	#10624989	C₂²	1	2	yes	complemented
40	#10624985	C₂²	1	2	yes	complemented
41	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
42	C₂⁴×C₂₃	—	2	1	no	—
43	C₂×D₄×C₂₃	—	2	1	no	—
44	C₂×D₄×C₂₃	—	2	2	no	—
45	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
46	C₂×D₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₂₃	—	2	1	no	—
51	C₂²×Dic₂₃	—	2	1	no	—
52	#67288	—	2	2	no	—
53	#67288	C₂×C₄	1	4	yes	complemented
54	C₂×D₉₂	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₂₃	—	2	1	no	—
56	C₂×C₄×D₂₃	—	2	2	no	—
57	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₂₃	—	2	1	no	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₂²⋊C₄×C₂₃	—	2	2	no	—
65	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
66	C₄²×C₂₃	—	2	1	no	—
67	C₄²×C₂₃	C₂³	1	2	yes	—
68	#67273	C₂×C₄	1	8	yes	complemented
69	#67272	C₂×C₄	1	8	yes	complemented
70	(C₂×D₂₃)⋊C₄	—	2	2	no	—
71	C₄⋊Dic₂₃	—	2	1	no	—
72	C₄⋊Dic₂₃	C₂³	1	2	yes	—
73	Dic₂₃⋊C₄	—	2	2	no	—
74	C₄×Dic₂₃	—	2	2	no	—
75	#67267	C₂×C₄	1	4	yes	complemented
76	C₂×C₂₃⋊C₈	—	2	2	no	—
77	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	—	2	4	no	—
82	C₂³×C₂₃	—	8	1	no	—
83	C₂³×C₂₃	—	8	1	no	—
84	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₂₃	—	2	1	no	—
86	D₄×C₂₃	—	2	4	no	—
87	D₄×C₂₃	—	2	2	no	—
88	D₄×C₂₃	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	8	2	no	—
92	C₂×C₄×C₂₃	—	8	2	no	—
93	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#41	1	2	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₂₃	—	2	4	no	—
100	C₂×Dic₂₃	—	2	1	no	—
101	C₂×Dic₂₃	—	2	2	no	—
102	C₂×Dic₂₃	—	8	1	no	—
103	D₉₂	—	2	1	no	—
104	D₉₂	C₂²×C₄	1	2	yes	—
105	C₄×D₂₃	—	2	4	no	—
106	C₄×D₂₃	—	8	2	no	—
107	Dic₄₆	—	2	1	no	—
108	Dic₄₆	C₂²×C₄	1	2	yes	—
109	C₂₃⋊C₈	—	2	1	no	—
110	C₂₃⋊C₈	—	2	1	no	—
111	C₂₃⋊C₈	C₂²×C₄	1	4	yes	—
112	#2946	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	4	no	—
116	C₂²×C₂₃	—	4	2	no	—
117	C₂²×C₂₃	—	4	2	no	—
118	C₂²×C₂₃	—	8	4	no	—
119	C₂²×C₂₃	C₄×D₄	1	2	yes	—
120	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₂₃	—	2	2	no	—
122	C₂×D₂₃	—	4	1	no	—
123	C₂×D₂₃	—	4	1	no	—
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	C₄×D₄	1	2	yes	—
129	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
131	Dic₂₃	—	2	2	no	—
132	Dic₂₃	—	4	2	no	—
133	C₂×AGL(1,ℤ/8)	—	23	1	no	—
134	#520	—	23	1	no	—
135	#518	D₂₃	1	1	yes	characteristic, complemented
136	#442	—	23	2	no	—
137	#440	—	23	2	no	—
138	C₄×SD₁₆	—	23	2	no	—
139	C₄×D₈	—	23	2	no	—
140	#428	—	23	1	no	—
141	#416	—	23	1	no	—
142	C₂×#102	—	23	1	no	—
143	#405	—	23	1	no	—
144	C₂×C₂₃	—	2	2	no	—
145	C₂×C₂₃	—	4	4	no	—
146	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂×C₂₃	#576	1	1	yes	characteristic
148	C₂×C₂₃	#577	1	1	yes	characteristic
149	D₂₃	—	4	2	no	—
150	C₂×#41	—	23	1	no	—
151	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
152	AGL(1,ℤ/8)	—	23	8	no	—
153	C₂×SD₁₆	—	23	2	no	—
154	C₂×D₈	—	23	2	no	—
155	C₂×M₁₆	—	23	1	no	—
156	C₂²×C₈	—	23	1	no	—
157	C₄×Q₈	—	23	2	no	—
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	23	2	no	—
161	C₄×D₄	—	46	2	no	—
162	C₄×D₄	C₂×D₂₃	1	2	yes	—
163	C₄×D₄	C₂×D₂₃	1	2	yes	—
164	#117	—	46	1	no	—
165	#117	C₂×D₂₃	1	1	yes	characteristic
166	#117	C₂×D₂₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	23	1	no	—
168	C₂×C₄⋊C₄	—	46	1	no	—
169	C₂×C₂²⋊C₄	—	2	2	no	—
170	#107	—	23	2	no	—
171	#106	—	23	2	no	—
172	#103	—	23	1	no	—
173	#103	—	23	1	no	—
174	#102	—	23	2	no	—
175	#102	—	23	2	no	—
176	#102	—	23	1	no	—
177	#102	—	23	1	no	—
178	#97	—	23	2	no	—
179	C₄×C₈	—	23	2	no	—
180	C₂₃	#2946	1	1	yes	characteristic, complemented, Sylow
181	C₂⁴	—	2	1	no	—
182	#41	—	23	4	no	—
183	#41	—	46	2	no	—
184	C₂×Q₈	—	23	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	23	1	no	—
188	C₂×D₄	—	46	1	no	—
189	C₂×D₄	C₄×D₂₃	1	4	yes	complemented
190	C₂×D₄	C₂²×D₂₃	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	46	1	no	—
194	C₂²×C₄	—	46	1	no	—
195	C₂²×C₄	—	46	2	no	—
196	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
197	SD₁₆	—	23	8	no	—
198	D₈	—	23	8	no	—
199	M₁₆	—	23	4	no	—
200	C₂×C₈	—	23	2	no	—
201	C₂×C₈	—	23	1	no	—
202	C₂×C₈	—	23	1	no	—
203	C₂×C₈	—	46	2	no	—
204	C₄⋊C₄	—	23	2	no	—
205	C₄⋊C₄	—	46	1	no	—
206	C₄⋊C₄	—	46	2	no	—
207	C₄⋊C₄	C₂²⋊D₂₃	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₂₃	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	46	2	no	—
215	C₂²⋊C₄	C₂²⋊D₂₃	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	46	2	no	—
218	C₄²	C₂²×D₂₃	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	46	1	no	—
224	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
225	Q₈	—	23	2	no	—
226	Q₈	—	46	1	no	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	23	2	no	—
230	D₄	—	46	4	no	—
231	D₄	—	46	1	no	—
232	D₄	C₂×C₄×D₂₃	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	46	1	no	—
238	C₂×C₄	—	46	4	no	—
239	C₂×C₄	—	46	2	no	—
240	C₂×C₄	—	184	2	no	—
241	C₂×C₄	—	184	1	no	—
242	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
243	C₂×C₄	#67288	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
248	C₈	—	23	4	no	—
249	C₈	—	46	1	no	—
250	C₈	—	46	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	46	2	no	—
258	C₂²	—	92	1	no	—
259	C₂²	—	92	1	no	—
260	C₂²	C₄×C₂²⋊D₂₃	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	46	2	no	—
267	C₄	—	92	2	no	—
268	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
269	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
270	C₄	C₂×#67288	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	92	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic
275	C₂	#13093220	1	1	yes	characteristic
276	C₂	#13093228	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#518	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#10625103	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#518	D₂₃
hypocenter	C₂₃	#2946
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #13092897	C₂	semidirect
3. #13092527	C₂	semidirect
4. #13092517	C₂	semidirect
5. C₄×#67273	C₂	semidirect
6. C₄×#67272	C₂	semidirect
7. #13092460	C₂	semidirect
8. #13092411	C₂	semidirect
9. C₂×#10625014	C₂	semidirect
10. #13092373	C₂	semidirect
11. C₂₃×#518	C₂	semidirect
22. #10625103	C₄	semidirect
30. C₄×D₉₂	C₂²	semidirect
31. C₄×Dic₄₆	C₂²	semidirect
34. #10624992	C₂²	semidirect
35. #10624992	C₂²	semidirect
36. #10624991	C₂²	semidirect
37. #10624991	C₂²	semidirect
38. #10624990	C₂²	semidirect
39. #10624989	C₂²	semidirect
40. #10624985	C₂²	semidirect
41. C₄×C₂₃⋊C₈	C₂²	semidirect
45. C₂×D₄×C₂₃	C₂×C₄	semidirect
53. #67288	C₂×C₄	semidirect
68. #67273	C₂×C₄	semidirect
69. #67272	C₂×C₄	semidirect
75. #67267	C₂×C₄	semidirect
135. #518	D₂₃	semidirect
180. C₂₃	#2946	semidirect
189. C₂×D₄	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625103	2
#13092897	2
#13092527	2
#13092517	2
C₄×#67273	2
C₄×#67272	2
#13092460	2
#13092411	2
C₂×#10625014	2
#13092373	2
C₂₃×#518	2
#2946	23