Subgroups — #26034149

Order 11392.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152333	C₂	1	4	yes	maximal, complemented
2	#19152320	C₂	1	4	yes	maximal, complemented
3	#19152272	C₂	1	8	yes	maximal, complemented
4	#19152220	C₂	1	4	yes	maximal, complemented
5	#19152172	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152167	C₂	1	1	yes	maximal, characteristic
7	#19152151	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152135	C₂	1	4	yes	maximal, complemented
9	C₂×#15668001	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152072	C₂	1	1	yes	maximal, characteristic, complemented
11	#19152066	C₂	1	1	yes	maximal, characteristic, complemented
12	#519×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13083574	C₂²	1	1	yes	characteristic
14	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
15	#15668011	C₂²	1	2	yes	complemented
16	#15668010	C₂²	1	2	yes	complemented
17	#15668009	C₂²	1	2	yes	—
18	#15668007	C₂²	1	8	yes	complemented
19	#15668006	C₂²	1	8	yes	complemented
20	#15668003	C₂²	1	2	yes	complemented
21	#15668001	C₂²	1	2	yes	—
22	#15668001	C₂²	1	2	yes	—
23	#15668001	C₂²	1	4	yes	—
24	#15668001	C₂²	1	2	yes	complemented
25	#15668000	C₂²	1	2	yes	—
26	#15668000	C₂²	1	8	yes	complemented
27	#15667997	C₂²	1	8	yes	complemented
28	#15667997	C₂²	1	8	yes	complemented
29	#15667995	C₂²	1	4	yes	—
30	#15667990	C₂²	1	8	yes	complemented
31	#15667989	C₂²	1	4	yes	—
32	#15667988	C₂²	1	4	yes	complemented
33	#15667987	C₂²	1	4	yes	—
34	#15667982	C₂²	1	4	yes	—
35	C₄×Dic₁₇₈	C₂²	1	4	yes	complemented
36	#15667977	C₂²	1	2	yes	complemented
37	C₄×D₃₅₆	C₂²	1	4	yes	—
38	C₄×D₃₅₆	C₂²	1	8	yes	complemented
39	#15667974	C₂²	1	4	yes	—
40	#15667974	C₂²	1	2	yes	complemented
41	#15667972	C₂²	1	4	yes	—
42	#15667971	C₂²	1	1	yes	characteristic
43	#15667969	C₂²	1	8	yes	complemented
44	#15667969	C₂²	1	2	yes	complemented
45	C₂×#13083551	C₂²	1	4	yes	—
46	C₂²⋊C₄×D₈₉	C₂²	1	4	yes	—
47	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
48	C₄²×D₈₉	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
50	C₄×Q₈×C₈₉	C₂²	1	4	yes	complemented
51	C₄×Q₈×C₈₉	C₂²	1	4	yes	complemented
52	#117×C₈₉	C₂²	1	1	yes	characteristic
53	#117×C₈₉	C₂²	1	4	yes	—
54	#117×C₈₉	C₂²	1	1	yes	characteristic
55	C₂³×D₈₉	C₂³	1	2	yes	—
56	#13083574	—	2	4	no	—
57	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₇₈	C₂³	1	4	yes	—
59	C₂×D₃₅₆	—	2	4	no	—
60	C₂×D₃₅₆	C₂³	1	2	yes	—
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	2	yes	—
63	C₂×D₃₅₆	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₈₉	C₂³	1	2	yes	—
65	C₂²×Dic₈₉	C₂³	1	2	yes	—
66	C₂×C₄×D₈₉	C₂³	1	2	yes	—
67	C₂×C₄×D₈₉	C₂³	1	8	yes	—
68	C₂×C₄×D₈₉	C₂³	1	4	yes	—
69	C₄⋊Dic₈₉	C₂³	1	2	yes	—
70	C₄⋊Dic₈₉	C₂³	1	4	yes	—
71	#13083553	C₂³	1	4	yes	—
72	#13083553	C₂³	1	4	yes	—
73	#13083553	C₂³	1	8	yes	—
74	#13083553	C₂³	1	4	yes	—
75	C₂²⋊Dic₈₉	C₂³	1	2	yes	—
76	#13083551	C₂³	1	4	yes	—
77	#13083551	C₂³	1	8	yes	—
78	#13083551	C₂³	1	8	yes	—
79	#13083551	C₂³	1	8	yes	—
80	#13083551	C₂³	1	8	yes	—
81	#13083551	C₂³	1	8	yes	—
82	C₄×Dic₈₉	C₂³	1	4	yes	—
83	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
84	C₄×Dic₈₉	C₂³	1	2	yes	—
85	C₄×Dic₈₉	C₂³	1	2	yes	—
86	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₈₉	—	2	4	no	—
88	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
89	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
90	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
91	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
93	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
94	C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
95	C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
96	C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
97	C₄²×C₈₉	C₂³	1	2	yes	—
98	C₄²×C₈₉	C₂³	1	8	yes	—
99	C₄²×C₈₉	C₂³	1	2	yes	—
100	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₂²×D₈₉	—	2	4	no	—
102	C₂²×D₈₉	—	4	4	no	—
103	C₂²×D₈₉	C₂⁴	1	4	yes	—
104	C₂²×D₈₉	C₂⁴	1	2	yes	—
105	Q₈×C₈₉	—	2	4	no	—
106	Q₈×C₈₉	—	2	4	no	—
107	C₂×C₄×C₈₉	—	2	4	no	—
108	C₂×C₄×C₈₉	#41	1	4	yes	—
109	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
110	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
111	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
112	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₈₉	—	2	4	no	—
115	C₂×Dic₈₉	—	4	2	no	—
116	C₂×Dic₈₉	C₂⁴	1	2	yes	—
117	C₂×Dic₈₉	C₂⁴	1	2	yes	—
118	C₂×Dic₈₉	C₂⁴	1	2	yes	—
119	C₂×Dic₈₉	C₂⁴	1	4	yes	—
120	D₃₅₆	—	2	4	no	—
121	D₃₅₆	—	2	4	no	—
122	D₃₅₆	—	2	2	no	—
123	D₃₅₆	—	4	8	no	—
124	C₄×D₈₉	—	2	4	no	—
125	C₄×D₈₉	—	4	8	no	—
126	C₄×D₈₉	—	4	4	no	—
127	Dic₁₇₈	—	2	2	no	—
128	Dic₁₇₈	—	4	4	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	C₂×#41	1	2	yes	—
131	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
132	C₂×D₈₉	—	2	2	no	—
133	C₂×D₈₉	—	2	4	no	—
134	C₂×D₈₉	—	2	4	no	—
135	C₂×D₈₉	—	2	8	no	—
136	C₂×D₈₉	—	8	4	no	—
137	C₄×C₈₉	—	2	4	no	—
138	C₄×C₈₉	—	2	4	no	—
139	C₄×C₈₉	—	2	4	no	—
140	C₄×C₈₉	C₂×#41	1	2	yes	—
141	C₄×C₈₉	C₂×#41	1	2	yes	—
142	Dic₈₉	—	2	2	no	—
143	Dic₈₉	—	2	2	no	—
144	Dic₈₉	—	2	4	no	—
145	Dic₈₉	—	2	2	no	—
146	C₂×C₈₉	—	2	2	no	—
147	C₂×C₈₉	C₂²×#41	1	1	yes	characteristic
148	C₂×C₈₉	#585	1	2	yes	—
149	D₈₉	—	4	2	no	—
150	D₈₉	—	4	4	no	—
151	#3510	—	89	1	no	maximal, Sylow
152	C₈₉	#3510	1	1	yes	characteristic, complemented, Sylow
153	#559	—	89	4	no	—
154	#556	—	89	4	no	—
155	#551	—	89	8	no	—
156	#542	—	89	4	no	—
157	#536	—	89	1	no	—
158	#535	—	89	1	no	—
159	#533	—	89	1	no	—
160	#529	—	89	4	no	—
161	C₂×#124	—	89	1	no	—
162	#519	D₈₉	1	1	yes	characteristic, complemented
163	#518	—	89	1	no	—
164	C₄×#41	—	89	1	no	—
165	C₂×#41	—	89	1	no	—
166	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
167	C₂²×D₄	—	89	1	no	—
168	C₄⋊Q₈	—	89	2	no	—
169	C₄⋊Q₈	—	89	2	no	—
170	#127	—	89	2	no	—
171	#126	—	89	8	no	—
172	#126	—	89	8	no	—
173	#125	—	89	2	no	—
174	#124	—	89	2	no	—
175	#124	—	89	2	no	—
176	#124	—	89	4	no	—
177	#124	—	89	2	no	—
178	#124	—	89	2	no	—
179	#124	—	89	8	no	—
180	#123	—	89	4	no	—
181	#123	—	89	8	no	—
182	#123	—	89	8	no	—
183	C₂²⋊Q₈	—	89	4	no	—
184	C₂²⋊Q₈	—	89	8	no	—
185	C₂²⋊Q₈	—	89	4	no	—
186	#121	—	89	4	no	—
187	#120	—	89	4	no	—
188	C₄×Q₈	—	89	2	no	—
189	C₄×Q₈	—	89	4	no	—
190	C₄×Q₈	C₂×D₈₉	1	4	yes	complemented
191	C₄×Q₈	C₂×D₈₉	1	4	yes	complemented
192	C₄×D₄	—	89	2	no	—
193	C₄×D₄	—	89	4	no	—
194	C₄×D₄	—	89	4	no	—
195	C₄×D₄	—	89	4	no	—
196	C₄×D₄	—	89	8	no	—
197	#117	—	89	1	no	—
198	#117	—	89	2	no	—
199	#117	—	89	8	no	—
200	#117	C₂×D₈₉	1	1	yes	characteristic
201	#117	C₂×D₈₉	1	1	yes	characteristic
202	#117	C₂×D₈₉	1	4	yes	—
203	C₂×C₂²⋊C₄	—	89	4	no	—
204	C₂×C₂²⋊C₄	—	89	4	no	—
205	C₂×C₄²	—	89	1	no	—
206	C₂×C₄²	—	89	2	no	—
207	C₂⁴	—	89	2	no	—
208	#41	—	178	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	89	4	no	—
211	C₂×Q₈	—	89	1	no	—
212	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
213	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
214	C₂×D₄	—	89	2	no	—
215	C₂×D₄	—	89	8	no	—
216	C₂×D₄	—	89	2	no	—
217	C₂×D₄	—	89	2	no	—
218	C₂×D₄	—	89	1	no	—
219	C₂×D₄	—	178	4	no	—
220	C₂²×C₄	—	89	2	no	—
221	C₂²×C₄	—	89	4	no	—
222	C₂²×C₄	—	89	8	no	—
223	C₂²×C₄	—	89	2	no	—
224	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
225	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
226	C₄⋊C₄	—	89	4	no	—
227	C₄⋊C₄	—	89	2	no	—
228	C₄⋊C₄	—	89	4	no	—
229	C₄⋊C₄	—	89	8	no	—
230	C₄⋊C₄	—	89	4	no	—
231	C₄⋊C₄	—	89	4	no	—
232	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
233	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
234	C₄⋊C₄	C₂²×D₈₉	1	8	yes	—
235	C₂²⋊C₄	—	89	2	no	—
236	C₂²⋊C₄	—	89	8	no	—
237	C₂²⋊C₄	—	89	8	no	—
238	C₂²⋊C₄	—	89	8	no	—
239	C₂²⋊C₄	—	89	4	no	—
240	C₂²⋊C₄	—	89	8	no	—
241	C₂²⋊C₄	—	89	8	no	—
242	C₂²⋊C₄	C₂²×D₈₉	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₈₉	1	2	yes	—
244	C₄²	—	89	1	no	—
245	C₄²	—	89	4	no	—
246	C₄²	—	89	1	no	—
247	C₄²	—	89	2	no	—
248	C₄²	—	89	2	no	—
249	C₄²	C₂²×D₈₉	1	2	yes	—
250	C₄²	C₂²×D₈₉	1	8	yes	—
251	C₄²	C₂²×D₈₉	1	2	yes	—
252	C₂³	—	89	4	no	—
253	C₂³	—	89	2	no	—
254	C₂³	—	356	4	no	—
255	C₂³	—	178	4	no	—
256	C₂³	C₂³×D₈₉	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	356	4	no	—
260	Q₈	—	178	2	no	—
261	D₄	—	356	8	no	—
262	D₄	—	178	4	no	—
263	D₄	—	178	4	no	—
264	D₄	—	178	4	no	—
265	D₄	—	178	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	89	2	no	—
268	C₂×C₄	—	89	2	no	—
269	C₂×C₄	—	89	2	no	—
270	C₂×C₄	—	89	4	no	—
271	C₂×C₄	—	356	4	no	—
272	C₂×C₄	—	356	8	no	—
273	C₂×C₄	—	356	2	no	—
274	C₂×C₄	—	178	4	no	—
275	C₂×C₄	#13083574	1	4	yes	—
276	C₂×C₄	C₂³×D₈₉	1	4	yes	—
277	C₂×C₄	C₂³×D₈₉	1	4	yes	—
278	C₂×C₄	C₂³×D₈₉	1	4	yes	—
279	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	178	2	no	—
283	C₂²	—	178	4	no	—
284	C₂²	—	178	4	no	—
285	C₂²	—	178	8	no	—
286	C₂²	—	712	4	no	—
287	C₂²	C₂×#13083574	1	2	yes	—
288	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	178	2	no	—
293	C₄	—	178	2	no	—
294	C₄	—	178	4	no	—
295	C₄	—	178	2	no	—
296	C₄	C₂×#13083574	1	2	yes	—
297	C₄	C₂×#13083574	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	356	2	no	—
300	C₂	—	356	4	no	—
301	C₂	C₂²×#13083574	1	1	yes	characteristic
302	C₂	#19152435	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#519×C₈₉	C₂
prime radical	#19152167	C₂
hypercenter	#519	D₈₉
hypocenter	C₈₉	#3510
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #519

Complements

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

N	H ≅ G/N	type
1. #19152333	C₂	semidirect
2. #19152320	C₂	semidirect
3. #19152272	C₂	semidirect
4. #19152220	C₂	semidirect
5. #19152172	C₂	semidirect
7. #19152151	C₂	semidirect
8. #19152135	C₂	semidirect
9. C₂×#15668001	C₂	semidirect
10. #19152072	C₂	semidirect
11. #19152066	C₂	semidirect
12. #519×C₈₉	C₂	semidirect
15. #15668011	C₂²	semidirect
16. #15668010	C₂²	semidirect
18. #15668007	C₂²	semidirect
19. #15668006	C₂²	semidirect
20. #15668003	C₂²	semidirect
24. #15668001	C₂²	semidirect
26. #15668000	C₂²	semidirect
27. #15667997	C₂²	semidirect
28. #15667997	C₂²	semidirect
30. #15667990	C₂²	semidirect
32. #15667988	C₂²	semidirect
35. C₄×Dic₁₇₈	C₂²	semidirect
36. #15667977	C₂²	semidirect
38. C₄×D₃₅₆	C₂²	semidirect
40. #15667974	C₂²	semidirect
43. #15667969	C₂²	semidirect
44. #15667969	C₂²	semidirect
48. C₄²×D₈₉	C₂²	semidirect
50. C₄×Q₈×C₈₉	C₂²	semidirect
51. C₄×Q₈×C₈₉	C₂²	semidirect
152. C₈₉	#3510	semidirect
162. #519	D₈₉	semidirect
190. C₄×Q₈	C₂×D₈₉	semidirect
191. C₄×Q₈	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152333	2
#19152320	2
#19152272	2
#19152220	2
#19152172	2
#19152167	2
#19152151	2
#19152135	2
C₂×#15668001	2
#19152072	2
#19152066	2
#519×C₈₉	2
#3510	89