Subgroups — #26030459

Order 11392.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152208	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152081	C₂	1	1	yes	maximal, characteristic
3	#19151931	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151929	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151903	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151900	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151801	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151760	C₂	1	1	yes	maximal, characteristic
10	#19151705	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151695	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151690	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151656	C₂	1	1	yes	maximal, characteristic, complemented
15	#452×C₈₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	#15668010	C₂²	1	1	yes	characteristic
19	#15668005	—	2	1	no	—
20	#15668005	—	2	1	no	—
21	#15668003	—	2	1	no	—
22	#15668003	C₂²	1	1	yes	characteristic
23	#15668002	—	2	1	no	—
24	#15667991	C₂²	1	1	yes	characteristic
25	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
26	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
27	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
28	#15667977	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₉	—	2	1	no	—
30	D₄×Dic₈₉	—	2	1	no	—
31	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	#15667971	—	2	1	no	—
34	#15667971	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₉	—	2	1	no	—
36	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667944	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₂²×Dic₈₉	—	2	1	no	—
61	C₂²×Dic₈₉	—	2	1	no	—
62	C₂²×Dic₈₉	—	2	1	no	—
63	#13083564	—	2	2	no	—
64	#13083562	—	2	2	no	—
65	C₂×#10623494	C₂³	1	1	yes	characteristic
66	C₂×#10623494	C₂³	1	1	yes	characteristic
67	C₄⋊Dic₈₉	—	2	1	no	—
68	C₄⋊Dic₈₉	—	2	1	no	—
69	C₄⋊Dic₈₉	—	2	1	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
73	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	C₂²⋊Dic₈₉	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	C₂²⋊Dic₈₉	—	2	1	no	—
85	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
93	#41×C₈₉	—	2	2	no	—
94	#41×C₈₉	—	2	2	no	—
95	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₉	—	2	1	no	—
98	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₉	—	2	2	no	—
102	SD₁₆×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	—	2	1	no	—
109	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₉	—	2	1	no	—
111	Q₈×C₈₉	—	2	1	no	—
112	Q₈×C₈₉	—	2	1	no	—
113	Q₈×C₈₉	#41	1	2	yes	—
114	D₄×C₈₉	—	2	4	no	—
115	D₄×C₈₉	—	2	1	no	—
116	D₄×C₈₉	#41	1	2	yes	—
117	C₂×C₄×C₈₉	—	2	4	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	8	1	no	—
130	C₂×Dic₈₉	—	8	1	no	—
131	C₂×Dic₈₉	—	8	1	no	—
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	Dic₁₇₈	—	2	1	no	—
137	Dic₁₇₈	—	2	1	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	#10623494	—	2	1	no	—
141	#10623494	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	2	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	—	4	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	2	1	no	—
157	Dic₈₉	—	2	1	no	—
158	Dic₈₉	—	2	1	no	—
159	Dic₈₉	—	4	1	no	—
160	Dic₈₉	—	4	1	no	—
161	Dic₈₉	—	4	1	no	—
162	Dic₈₉	—	4	1	no	—
163	C₂×C₈₉	—	4	1	no	—
164	C₂×C₈₉	—	4	2	no	—
165	C₂×C₈₉	#525	1	1	yes	characteristic
166	C₂×C₈₉	#577	1	1	yes	characteristic
167	C₂×C₈₉	#578	1	1	yes	characteristic
168	#3120	—	89	1	no	maximal, Sylow
169	C₈₉	#3120	1	1	yes	characteristic, complemented, Sylow
170	#541	—	89	1	no	—
171	#520	—	89	1	no	—
172	#484	—	89	1	no	—
173	#483	—	89	1	no	—
174	#479	—	89	1	no	—
175	#478	—	89	1	no	—
176	#462	—	89	1	no	—
177	#461	—	89	1	no	—
178	#452	—	89	1	no	—
179	#452	D₈₉	1	1	yes	characteristic
180	#441	—	89	1	no	—
181	#440	—	89	1	no	—
182	C₄×Q₁₆	—	89	1	no	—
183	C₄×SD₁₆	—	89	1	no	—
184	#433	—	89	1	no	—
185	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
186	C₂×Q₁₆	—	89	1	no	—
187	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
188	C₂×SD₁₆	—	89	1	no	—
189	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₄⋊Q₈	—	89	1	no	—
191	#126	—	178	1	no	—
192	#126	—	178	1	no	—
193	#125	—	89	1	no	—
194	#125	—	178	1	no	—
195	#125	—	178	1	no	—
196	C₂²⋊Q₈	—	89	1	no	—
197	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
198	C₄×Q₈	—	89	1	no	—
199	C₄×Q₈	—	89	1	no	—
200	C₄×Q₈	—	89	1	no	—
201	C₄×Q₈	—	89	1	no	—
202	C₄×D₄	—	89	1	no	—
203	C₄×D₄	—	89	1	no	—
204	C₄×D₄	—	178	1	no	—
205	C₄×D₄	—	178	1	no	—
206	#117	—	89	1	no	—
207	#117	—	178	1	no	—
208	C₂×C₄⋊C₄	—	89	1	no	—
209	C₂×C₄⋊C₄	—	178	1	no	—
210	#107	—	89	1	no	—
211	#107	—	89	1	no	—
212	#106	—	89	1	no	—
213	#106	—	89	1	no	—
214	C₄⋊C₈	—	89	1	no	—
215	C₄⋊C₈	—	89	1	no	—
216	#103	—	89	1	no	—
217	#103	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	—	89	1	no	—
220	#103	—	89	1	no	—
221	#103	C₂×D₈₉	1	1	yes	characteristic
222	#102	—	89	1	no	—
223	#102	C₂×D₈₉	1	1	yes	characteristic
224	C₂²⋊C₈	—	89	1	no	—
225	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
226	#97	—	89	1	no	—
227	C₄×C₈	—	89	1	no	—
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	C₂×Q₈	—	89	1	no	—
231	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	178	1	no	—
237	C₂²×C₄	—	178	1	no	—
238	C₂²×C₄	—	178	1	no	—
239	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	178	2	no	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	178	2	no	—
244	C₂×C₈	—	89	1	no	—
245	C₂×C₈	—	89	1	no	—
246	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	89	1	no	—
251	C₄⋊C₄	—	89	1	no	—
252	C₄⋊C₄	—	89	1	no	—
253	C₄⋊C₄	—	89	1	no	—
254	C₄⋊C₄	—	178	1	no	—
255	C₄⋊C₄	—	178	1	no	—
256	C₄⋊C₄	—	178	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	—	178	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	—	178	1	no	—
264	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	89	1	no	—
267	C₂²⋊C₄	—	89	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₂²⋊C₄	—	178	1	no	—
271	C₄²	—	89	1	no	—
272	C₄²	—	89	1	no	—
273	C₄²	—	178	1	no	—
274	C₄²	—	178	1	no	—
275	C₄²	—	178	1	no	—
276	C₄²	—	178	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	D₄×D₈₉	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	178	1	no	—
283	Q₈	—	178	1	no	—
284	Q₈	#13083573	1	2	yes	—
285	D₄	—	2	4	no	—
286	D₄	—	2	1	no	—
287	D₄	#13083573	1	2	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	89	1	no	—
296	C₂×C₄	—	89	1	no	—
297	C₂×C₄	—	89	1	no	—
298	C₂×C₄	—	178	1	no	—
299	C₂×C₄	—	178	1	no	—
300	C₂×C₄	—	178	1	no	—
301	C₂×C₄	—	178	1	no	—
302	C₂×C₄	—	712	1	no	—
303	C₂×C₄	—	712	1	no	—
304	C₂×C₄	—	712	1	no	—
305	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	178	1	no	—
310	C₈	—	178	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	356	1	no	—
323	C₄	—	356	1	no	—
324	C₄	—	356	1	no	—
325	C₄	—	356	1	no	—
326	C₄	—	178	1	no	—
327	C₄	—	178	1	no	—
328	C₄	—	178	1	no	—
329	C₄	—	178	1	no	—
330	C₄	C₂×#13083573	1	1	yes	characteristic
331	C₄	C₂×#13083573	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	4	2	no	—
334	C₂	#19152104	1	1	yes	characteristic
335	C₂	#19152405	1	1	yes	characteristic
336	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#452×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#41×C₈₉	C₂²
hypercenter	#452	D₈₉
hypocenter	C₈₉	#3120
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #19152208	C₂	semidirect
3. #19151931	C₂	semidirect
4. #19151929	C₂	semidirect
5. #19151903	C₂	semidirect
6. #19151900	C₂	semidirect
7. #19151806	C₂	semidirect
8. #19151801	C₂	semidirect
10. #19151705	C₂	semidirect
11. #19151695	C₂	semidirect
12. #19151690	C₂	semidirect
13. SD₁₆×Dic₈₉	C₂	semidirect
14. #19151656	C₂	semidirect
169. C₈₉	#3120	semidirect

Maximal subgroups

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

subgroup	index
#19152208	2
#19152081	2
#19151931	2
#19151929	2
#19151903	2
#19151900	2
#19151806	2
#19151801	2
#19151760	2
#19151705	2
#19151695	2
#19151690	2
SD₁₆×Dic₈₉	2
#19151656	2
#452×C₈₉	2
#3120	89