Subgroups — #15682210

Order 2944.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625120	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625064	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092526	C₂	1	2	yes	maximal, complemented
5	#13092520	C₂	1	2	yes	maximal, complemented
6	#13092502	C₂	1	2	yes	maximal, complemented
7	#13092497	C₂	1	2	yes	maximal, complemented
8	#13092457	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092423	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×D₉₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	—	2	1	no	—
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	#10625120	C₄	1	8	yes	complemented
18	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
19	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₂×#67274	C₂²	1	2	yes	—
21	D₄×Dic₂₃	—	2	1	no	—
22	D₄×Dic₂₃	—	2	1	no	—
23	D₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
24	D₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
25	C₂×#67272	C₂²	1	2	yes	—
26	C₂×(C₂×D₂₃)⋊C₄	—	2	1	no	—
27	#10625097	—	2	1	no	—
28	#10625097	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₂₃	—	2	1	no	—
30	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
31	C₂×#67267	C₂²	1	1	yes	characteristic
32	#10625064	—	2	1	no	—
33	#10625064	—	2	1	no	—
34	#10625064	C₂²	1	2	yes	—
35	#10625064	C₂²	1	2	yes	—
36	C₂₃×#103	C₂²	1	2	yes	complemented
37	C₂₃×#102	C₂²	1	2	yes	complemented
38	#10625003	C₂²	1	2	yes	—
39	#10625003	C₂²	1	2	yes	—
40	#10624997	C₂²	1	2	yes	complemented
41	C₈×Dic₂₃	C₂²	1	2	yes	complemented
42	#10624990	C₂²	1	2	yes	complemented
43	#10624989	C₂²	1	2	yes	complemented
44	C₂³×D₂₃	—	2	1	no	—
45	#41×C₂₃	—	2	2	no	—
46	#41×C₂₃	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×Dic₂₃	—	2	1	no	—
54	C₂²×Dic₂₃	—	2	1	no	—
55	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
56	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
57	C₂×D₉₂	—	2	1	no	—
58	C₂×D₉₂	—	2	2	no	—
59	C₂×D₉₂	C₂×C₄	1	4	yes	complemented
60	C₂×D₉₂	C₂³	1	2	yes	—
61	C₂×C₄×D₂₃	—	2	2	no	—
62	C₂×C₄×D₂₃	—	2	2	no	—
63	C₂×C₄×D₂₃	—	4	4	no	—
64	M₁₆×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
68	C₂²⋊Dic₂₃	—	2	1	no	—
69	C₂²⋊Dic₂₃	—	2	1	no	—
70	C₂²⋊Dic₂₃	—	2	1	no	—
71	#67274	C₂×C₄	1	8	yes	complemented
72	#67272	C₂×C₄	1	8	yes	complemented
73	(C₂×D₂₃)⋊C₄	—	2	2	no	—
74	(C₂×D₂₃)⋊C₄	—	2	2	no	—
75	C₄⋊Dic₂₃	—	2	1	no	—
76	C₄⋊Dic₂₃	—	2	1	no	—
77	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
79	C₄×Dic₂₃	—	2	1	no	—
80	C₄×Dic₂₃	—	2	1	no	—
81	C₄×Dic₂₃	—	2	1	no	—
82	C₄×Dic₂₃	D₄	1	1	yes	characteristic
83	C₄×Dic₂₃	D₄	1	1	yes	characteristic
84	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
85	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
86	#67267	C₂×C₄	1	4	yes	complemented
87	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
88	C₂³×C₂₃	—	2	1	no	—
89	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₂₃	—	2	4	no	—
91	C₂²×D₂₃	—	2	2	no	—
92	C₂²×D₂₃	—	4	2	no	—
93	C₂²×D₂₃	—	4	2	no	—
94	Q₈×C₂₃	—	2	1	no	—
95	Q₈×C₂₃	C₂²×C₄	1	2	yes	complemented
96	D₄×C₂₃	—	2	4	no	—
97	D₄×C₂₃	—	2	1	no	—
98	D₄×C₂₃	C₂²×C₄	1	2	yes	complemented
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	#41	1	2	yes	—
106	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₂₃	—	2	2	no	—
108	C₂×Dic₂₃	—	2	1	no	—
109	C₂×Dic₂₃	—	2	1	no	—
110	C₂×Dic₂₃	—	4	1	no	—
111	C₂×Dic₂₃	—	4	2	no	—
112	C₂×Dic₂₃	—	4	2	no	—
113	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
117	D₉₂	—	2	4	no	—
118	D₉₂	—	2	2	no	—
119	D₉₂	C₂²×C₄	1	4	yes	—
120	C₄×D₂₃	—	4	4	no	—
121	C₄×D₂₃	—	4	4	no	—
122	C₈×C₂₃	—	2	2	no	—
123	C₂₃⋊C₈	C₂²×C₄	1	4	yes	complemented
124	C₄×AGL(1,ℤ/8)	—	23	1	no	maximal, Sylow
125	C₂²×C₂₃	—	2	1	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	4	1	no	—
129	C₂²×C₂₃	—	4	1	no	—
130	C₂²×C₂₃	C₄×D₄	1	2	yes	—
131	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₂₃	—	2	4	no	—
133	C₂×D₂₃	—	4	8	no	—
134	C₂×D₂₃	—	4	2	no	—
135	C₂×D₂₃	—	4	2	no	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	4	2	no	—
138	C₄×C₂₃	C₄×D₄	1	2	yes	—
139	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
141	Dic₂₃	—	2	1	no	—
142	Dic₂₃	—	2	1	no	—
143	Dic₂₃	—	2	1	no	—
144	Dic₂₃	—	4	1	no	—
145	Dic₂₃	—	4	1	no	—
146	Dic₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
147	C₂×AGL(1,ℤ/8)	—	23	1	no	—
148	C₄×#41	—	23	1	no	—
149	C₂×C₄×D₄	—	23	1	no	—
150	#442	—	23	2	no	—
151	#440	—	23	2	no	—
152	C₄×SD₁₆	—	23	2	no	—
153	C₄×D₈	—	23	2	no	—
154	#428	—	23	1	no	—
155	#418	—	23	1	no	—
156	#417	D₂₃	1	1	yes	characteristic, complemented
157	C₄×M₁₆	—	23	1	no	—
158	C₂×C₂₃	—	2	2	no	—
159	C₂×C₂₃	—	4	2	no	—
160	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₂₃	#575	1	1	yes	characteristic
163	D₂₃	—	4	4	no	—
164	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
165	C₂²×D₄	—	23	1	no	—
166	C₂³×C₄	—	46	1	no	—
167	AGL(1,ℤ/8)	—	23	8	no	—
168	C₂×SD₁₆	—	23	2	no	—
169	C₂×D₈	—	23	2	no	—
170	C₂×M₁₆	—	23	1	no	—
171	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
172	C₄×Q₈	—	23	1	no	—
173	C₄×Q₈	—	23	1	no	—
174	C₄×D₄	—	23	1	no	—
175	C₄×D₄	—	23	1	no	—
176	C₄×D₄	—	23	2	no	—
177	C₄×D₄	—	23	2	no	—
178	C₄×D₄	—	46	1	no	—
179	C₄×D₄	—	46	1	no	—
180	C₄×D₄	—	46	1	no	—
181	C₄×D₄	—	46	1	no	—
182	#117	—	23	1	no	—
183	#117	—	46	1	no	—
184	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	46	1	no	—
186	C₂×C₄²	—	23	1	no	—
187	C₂×C₄²	—	46	1	no	—
188	#107	—	23	2	no	—
189	#106	—	23	2	no	—
190	#103	C₂×D₂₃	1	2	yes	complemented
191	#102	—	23	2	no	—
192	#102	—	23	2	no	—
193	#102	C₂×D₂₃	1	2	yes	complemented
194	#97	—	23	2	no	—
195	C₄×C₈	—	23	2	no	—
196	C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
197	C₂⁴	—	46	1	no	—
198	#41	—	2	2	no	—
199	#41	C₄×D₂₃	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	23	4	no	—
203	C₂×D₄	—	23	2	no	—
204	C₂×D₄	—	46	1	no	—
205	C₂×D₄	—	46	2	no	—
206	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	23	1	no	—
210	C₂²×C₄	—	23	1	no	—
211	C₂²×C₄	—	46	2	no	—
212	C₂²×C₄	—	46	2	no	—
213	C₂²×C₄	—	46	1	no	—
214	C₂²×C₄	—	46	1	no	—
215	C₂²×C₄	—	92	4	no	—
216	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
217	SD₁₆	—	23	8	no	—
218	D₈	—	23	8	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	23	4	no	—
221	C₂×C₈	—	23	2	no	—
222	C₂×C₈	C₂²×D₂₃	1	2	yes	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	23	1	no	—
225	C₄⋊C₄	—	23	1	no	—
226	C₄⋊C₄	—	46	1	no	—
227	C₄⋊C₄	—	46	1	no	—
228	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
229	C₂²⋊C₄	—	46	2	no	—
230	C₂²⋊C₄	—	46	2	no	—
231	C₂²⋊C₄	—	46	1	no	—
232	C₂²⋊C₄	—	46	1	no	—
233	C₂²⋊C₄	—	46	1	no	—
234	C₄²	—	23	1	no	—
235	C₄²	—	23	1	no	—
236	C₄²	—	23	1	no	—
237	C₄²	—	23	1	no	—
238	C₄²	—	46	1	no	—
239	C₄²	—	46	1	no	—
240	C₄²	—	46	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	46	4	no	—
243	C₂³	—	46	2	no	—
244	C₂³	—	92	2	no	—
245	C₂³	—	92	2	no	—
246	C₂³	D₄×D₂₃	1	1	yes	characteristic
247	Q₈	—	2	1	no	—
248	Q₈	C₂×C₄×D₂₃	1	2	yes	complemented
249	D₄	—	2	4	no	—
250	D₄	—	2	1	no	—
251	D₄	—	23	4	no	—
252	D₄	—	46	4	no	—
253	D₄	—	46	2	no	—
254	D₄	C₂×C₄×D₂₃	1	2	yes	complemented
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	23	1	no	—
260	C₂×C₄	—	23	1	no	—
261	C₂×C₄	—	23	1	no	—
262	C₂×C₄	—	23	1	no	—
263	C₂×C₄	—	46	2	no	—
264	C₂×C₄	—	46	1	no	—
265	C₂×C₄	—	46	1	no	—
266	C₂×C₄	—	92	4	no	—
267	C₂×C₄	—	92	4	no	—
268	C₂×C₄	—	92	2	no	—
269	C₂×C₄	—	92	1	no	—
270	C₂×C₄	—	92	2	no	—
271	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
272	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
273	C₂×C₄	#67290	1	2	yes	—
274	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	23	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	46	4	no	—
283	C₂²	—	92	8	no	—
284	C₂²	—	92	2	no	—
285	C₂²	—	92	2	no	—
286	C₂²	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
287	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	23	2	no	—
291	C₄	—	46	1	no	—
292	C₄	—	46	1	no	—
293	C₄	—	46	1	no	—
294	C₄	—	92	1	no	—
295	C₄	—	92	1	no	—
296	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
297	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
298	C₄	C₂×#67290	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	2	no	—
301	C₂	—	92	4	no	—
302	C₂	C₂×(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₂₃	1	1	yes	characteristic
304	C₂	#13093216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#417	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#10625120	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#417	D₂₃
hypocenter	C₂₃	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. #41×Dic₂₃	C₂	semidirect
3. C₂×#10625064	C₂	semidirect
4. #13092526	C₂	semidirect
5. #13092520	C₂	semidirect
6. #13092502	C₂	semidirect
7. #13092497	C₂	semidirect
8. #13092457	C₂	semidirect
9. #13092423	C₂	semidirect
10. M₁₆×Dic₂₃	C₂	semidirect
11. C₂₃×#417	C₂	semidirect
17. #10625120	C₄	semidirect
18. Q₈×Dic₂₃	C₂²	semidirect
19. Q₈×Dic₂₃	C₂²	semidirect
23. D₄×Dic₂₃	C₂²	semidirect
24. D₄×Dic₂₃	C₂²	semidirect
36. C₂₃×#103	C₂²	semidirect
37. C₂₃×#102	C₂²	semidirect
40. #10624997	C₂²	semidirect
41. C₈×Dic₂₃	C₂²	semidirect
42. #10624990	C₂²	semidirect
43. #10624989	C₂²	semidirect
46. #41×C₂₃	C₂×C₄	semidirect
59. C₂×D₉₂	C₂×C₄	semidirect
71. #67274	C₂×C₄	semidirect
72. #67272	C₂×C₄	semidirect
86. #67267	C₂×C₄	semidirect
95. Q₈×C₂₃	C₂²×C₄	semidirect
98. D₄×C₂₃	C₂²×C₄	semidirect
123. C₂₃⋊C₈	C₂²×C₄	semidirect
146. Dic₂₃	AGL(1,ℤ/8)	semidirect
156. #417	D₂₃	semidirect
190. #103	C₂×D₂₃	semidirect
193. #102	C₂×D₂₃	semidirect
196. C₂₃	C₄×AGL(1,ℤ/8)	semidirect
199. #41	C₄×D₂₃	semidirect
248. Q₈	C₂×C₄×D₂₃	semidirect
254. D₄	C₂×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×#10625120	2
#41×Dic₂₃	2
C₂×#10625064	2
#13092526	2
#13092520	2
#13092502	2
#13092497	2
#13092457	2
#13092423	2
M₁₆×Dic₂₃	2
C₂₃×#417	2
C₄×AGL(1,ℤ/8)	23