Subgroups — #26030039

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152154	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13083575	C₂	1	1	yes	maximal, characteristic
3	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151899	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151891	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151887	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151881	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151877	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151635	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151633	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊C₈×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151593	C₂	1	1	yes	maximal, characteristic
14	#19151593	C₂	1	1	yes	maximal, characteristic
15	#479×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
19	#15668011	C₂²	1	1	yes	characteristic
20	#15668011	C₂²	1	1	yes	characteristic
21	#15668004	C₂²	1	1	yes	characteristic
22	#15667990	—	2	1	no	—
23	#15667990	—	2	1	no	—
24	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
25	#15667977	—	2	1	no	—
26	#15667977	—	2	1	no	—
27	C₄×D₃₅₆	—	2	1	no	—
28	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
29	#15667974	—	2	1	no	—
30	#15667974	C₂²	1	1	yes	characteristic
31	#15667970	—	2	1	no	—
32	#15667970	C₂²	1	1	yes	characteristic
33	#15667970	C₂²	1	1	yes	characteristic
34	#15667970	C₂²	1	1	yes	characteristic
35	C₄²×D₈₉	—	2	1	no	—
36	C₄²×D₈₉	C₂²	1	1	yes	characteristic
37	#15667946	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667934	C₂²	1	1	yes	characteristic
50	#15667932	C₂²	1	1	yes	characteristic
51	#15667932	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#13083575	—	2	2	no	—
60	#13083575	Q₈	1	4	yes	complemented
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
71	C₂×#10623494	C₂³	1	1	yes	characteristic
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₈×D₈₉	—	2	2	no	—
74	C₈×D₈₉	—	2	2	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083551	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	D₄	1	1	yes	characteristic
93	C₄×Dic₈₉	D₄	1	1	yes	characteristic
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₄²×C₈₉	—	2	1	no	—
105	C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²×D₈₉	—	2	1	no	—
107	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₈₉	—	2	1	no	—
109	Q₈×C₈₉	C₂×Q₈	1	2	yes	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	D₃₅₆	—	2	4	no	—
124	D₃₅₆	—	2	1	no	—
125	D₃₅₆	C₂×Q₈	1	2	yes	—
126	C₄×D₈₉	—	2	4	no	—
127	C₄×D₈₉	—	2	2	no	—
128	C₄×D₈₉	—	4	1	no	—
129	C₄×D₈₉	—	4	2	no	—
130	C₄×D₈₉	—	4	2	no	—
131	C₄×D₈₉	C₂×Q₈	1	2	yes	—
132	C₄×D₈₉	#41	1	2	yes	—
133	Dic₁₇₈	—	4	2	no	—
134	Dic₁₇₈	—	4	2	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	#10623494	—	2	1	no	—
138	#10623494	—	2	1	no	—
139	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₈₉	—	2	1	no	—
141	C₂×D₈₉	—	2	1	no	—
142	C₂×D₈₉	—	2	2	no	—
143	C₂×D₈₉	—	4	1	no	—
144	C₂×D₈₉	—	4	1	no	—
145	C₂×D₈₉	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	4	1	no	—
150	C₄×C₈₉	—	4	1	no	—
151	C₄×C₈₉	—	4	1	no	—
152	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	4	1	no	—
156	Dic₈₉	—	4	1	no	—
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	C₂²⋊Q₈	1	2	yes	—
159	Dic₈₉	#135	1	2	yes	—
160	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
162	C₂×C₈₉	#575	1	1	yes	characteristic
163	D₈₉	—	2	2	no	—
164	D₈₉	—	4	2	no	—
165	#3076	—	89	1	no	maximal, Sylow
166	C₈₉	#3076	1	1	yes	characteristic, complemented, Sylow
167	#533	—	89	1	no	—
168	C₄×#41	—	89	1	no	—
169	#479	—	89	1	no	—
170	#479	D₈₉	1	1	yes	characteristic, complemented
171	#478	—	89	1	no	—
172	#478	—	89	1	no	—
173	#477	—	89	1	no	—
174	#476	—	89	1	no	—
175	#475	—	89	1	no	—
176	#474	—	89	1	no	—
177	#427	—	89	1	no	—
178	#427	—	89	1	no	—
179	C₂×C₄⋊C₈	—	89	1	no	—
180	#416	—	89	1	no	—
181	#416	—	89	1	no	—
182	C₂×#41	—	89	1	no	—
183	C₂²×C₈	—	89	1	no	—
184	C₂²×C₈	—	89	1	no	—
185	C₄⋊Q₈	—	89	1	no	—
186	C₄⋊Q₈	—	89	1	no	—
187	#125	—	89	1	no	—
188	#125	C₂×D₈₉	1	1	yes	characteristic
189	C₂²⋊Q₈	—	178	1	no	—
190	C₂²⋊Q₈	—	178	1	no	—
191	C₄×Q₈	—	89	1	no	—
192	C₄×Q₈	—	178	1	no	—
193	C₄×Q₈	—	178	1	no	—
194	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
195	C₄×D₄	—	89	1	no	—
196	C₄×D₄	—	89	1	no	—
197	C₄×D₄	—	178	1	no	—
198	C₄×D₄	—	178	1	no	—
199	#117	—	89	1	no	—
200	#117	—	89	1	no	—
201	#117	—	89	1	no	—
202	#117	—	178	1	no	—
203	C₂×C₄²	—	89	1	no	—
204	C₂×C₄²	—	178	1	no	—
205	#107	—	89	1	no	—
206	#107	—	89	1	no	—
207	#107	—	89	1	no	—
208	#107	C₂×D₈₉	1	1	yes	characteristic
209	#106	—	89	1	no	—
210	#106	—	89	1	no	—
211	#106	—	89	1	no	—
212	#106	C₂×D₈₉	1	1	yes	characteristic
213	C₄⋊C₈	—	89	1	no	—
214	C₄⋊C₈	—	89	1	no	—
215	C₄⋊C₈	—	89	1	no	—
216	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
217	#103	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	C₂×D₈₉	1	1	yes	characteristic
220	#103	C₂×D₈₉	1	1	yes	characteristic
221	#102	—	89	1	no	—
222	#102	—	89	1	no	—
223	#102	—	89	1	no	—
224	#102	—	89	1	no	—
225	#41	—	89	4	no	—
226	#41	—	178	2	no	—
227	C₂×Q₈	—	178	1	no	—
228	C₂×Q₈	—	178	1	no	—
229	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×D₄	—	89	1	no	—
231	C₂×D₄	—	178	1	no	—
232	C₂²×C₄	—	89	1	no	—
233	C₂²×C₄	—	89	1	no	—
234	C₂²×C₄	—	89	1	no	—
235	C₂²×C₄	—	178	1	no	—
236	C₂²×C₄	—	178	1	no	—
237	C₂²×C₄	—	178	1	no	—
238	C₂×C₈	—	89	1	no	—
239	C₂×C₈	—	89	1	no	—
240	C₂×C₈	—	178	2	no	—
241	C₂×C₈	—	178	2	no	—
242	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	89	1	no	—
248	C₄⋊C₄	—	89	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	178	1	no	—
251	C₄⋊C₄	—	178	1	no	—
252	C₄⋊C₄	—	178	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	178	1	no	—
259	C₂²⋊C₄	—	178	1	no	—
260	C₂²⋊C₄	—	178	1	no	—
261	C₂²⋊C₄	—	178	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	89	1	no	—
265	C₄²	—	89	1	no	—
266	C₄²	—	89	1	no	—
267	C₄²	—	178	1	no	—
268	C₄²	—	178	1	no	—
269	C₄²	—	178	1	no	—
270	C₄²	—	178	1	no	—
271	C₄²	C₂²×D₈₉	1	1	yes	characteristic
272	C₂³	—	89	1	no	—
273	C₂³	—	178	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	356	2	no	—
276	Q₈	—	356	2	no	—
277	Q₈	Q₈×D₈₉	1	2	yes	—
278	D₄	—	89	2	no	—
279	D₄	—	178	4	no	—
280	D₄	—	178	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	89	1	no	—
286	C₂×C₄	—	89	1	no	—
287	C₂×C₄	—	89	2	no	—
288	C₂×C₄	—	89	2	no	—
289	C₂×C₄	—	89	1	no	—
290	C₂×C₄	—	356	2	no	—
291	C₂×C₄	—	356	1	no	—
292	C₂×C₄	—	356	2	no	—
293	C₂×C₄	—	178	2	no	—
294	C₂×C₄	—	178	4	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	—	178	1	no	—
297	C₂×C₄	—	178	1	no	—
298	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
299	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	178	1	no	—
304	C₈	—	178	1	no	—
305	C₂²	—	89	2	no	—
306	C₂²	—	356	1	no	—
307	C₂²	—	356	1	no	—
308	C₂²	—	178	2	no	—
309	C₂²	—	178	1	no	—
310	C₂²	—	178	1	no	—
311	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	89	2	no	—
319	C₄	—	89	2	no	—
320	C₄	—	356	1	no	—
321	C₄	—	356	1	no	—
322	C₄	—	356	1	no	—
323	C₄	—	178	1	no	—
324	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
325	C₄	#41×D₈₉	1	1	yes	characteristic
326	C₂	—	356	2	no	—
327	C₂	—	178	2	no	—
328	C₂	C₂²⋊Q₈×D₈₉	1	1	yes	characteristic
329	C₂	#135×D₈₉	1	1	yes	characteristic
330	C₂	#19152397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#479×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083575	C₂²
hypercenter	#479	D₈₉
hypocenter	C₈₉	#3076
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #19152154	C₂	semidirect
3. #19151905	C₂	semidirect
4. #19151899	C₂	semidirect
5. #19151895	C₂	semidirect
6. #19151891	C₂	semidirect
7. #19151887	C₂	semidirect
8. #19151881	C₂	semidirect
9. #19151877	C₂	semidirect
10. #19151635	C₂	semidirect
11. #19151633	C₂	semidirect
12. C₄⋊C₈×D₈₉	C₂	semidirect
15. #479×C₈₉	C₂	semidirect
60. #13083575	Q₈	semidirect
166. C₈₉	#3076	semidirect
170. #479	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152154	2
C₄×#13083575	2
#19151905	2
#19151899	2
#19151895	2
#19151891	2
#19151887	2
#19151881	2
#19151877	2
#19151635	2
#19151633	2
C₄⋊C₈×D₈₉	2
#19151593	2
#479×C₈₉	2
#3076	89