Subgroups — #26027671

Order 11392.

400 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151537	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151536	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151531	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151524	C₂	1	1	yes	maximal, characteristic
5	#19151511	C₂	1	1	yes	maximal, characteristic
6	#19151503	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151502	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151490	C₂	1	1	yes	maximal, characteristic
9	#19151488	C₂	1	1	yes	maximal, characteristic
10	#19151469	C₂	1	1	yes	maximal, characteristic
11	#19151469	C₂	1	1	yes	maximal, characteristic
12	#19151465	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151465	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151460	C₂	1	1	yes	maximal, characteristic
15	#398×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	C₂×#13083553	C₂²	1	1	yes	characteristic
20	C₂×#13083553	C₂²	1	1	yes	characteristic
21	C₂×#13083551	C₂²	1	1	yes	characteristic
22	C₂×#13083551	C₂²	1	1	yes	characteristic
23	C₂×#13083551	C₂²	1	1	yes	characteristic
24	C₂×#13083551	C₂²	1	1	yes	characteristic
25	C₂×#13083551	C₂²	1	1	yes	characteristic
26	C₂×#13083551	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
29	#15667919	C₂²	1	1	yes	characteristic
30	#15667919	C₂²	1	1	yes	characteristic
31	#15667919	C₂²	1	1	yes	characteristic
32	#15667919	C₂²	1	1	yes	characteristic
33	#15667919	C₂²	1	1	yes	characteristic
34	#15667919	C₂²	1	1	yes	characteristic
35	#15667919	C₂²	1	1	yes	characteristic
36	#15667919	C₂²	1	1	yes	characteristic
37	#15667919	C₂²	1	1	yes	characteristic
38	#15667919	C₂²	1	1	yes	characteristic
39	#15667919	C₂²	1	1	yes	characteristic
40	#15667919	C₂²	1	1	yes	characteristic
41	#15667919	C₂²	1	1	yes	characteristic
42	#15667919	C₂²	1	1	yes	characteristic
43	#15667919	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
48	#95×C₈₉	C₂²	1	1	yes	characteristic
49	#95×C₈₉	C₂²	1	1	yes	characteristic
50	#95×C₈₉	C₂²	1	1	yes	characteristic
51	C₂³×D₈₉	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₇₈	—	2	2	no	—
53	C₂×Dic₁₇₈	—	2	2	no	—
54	C₂×Dic₁₇₈	—	4	1	no	—
55	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
60	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₈₉	—	2	2	no	—
63	C₂×C₄×D₈₉	—	4	1	no	—
64	C₂×C₄×D₈₉	—	4	1	no	—
65	C₄⋊Dic₈₉	—	4	1	no	—
66	#13083553	—	4	1	no	—
67	#13083553	—	4	1	no	—
68	#13083551	—	2	2	no	—
69	#13083551	—	2	2	no	—
70	#13083551	—	2	2	no	—
71	#13083551	—	2	2	no	—
72	#13083551	—	4	1	no	—
73	#13083551	—	4	1	no	—
74	C₄×Dic₈₉	—	2	2	no	—
75	C₄×Dic₈₉	—	2	2	no	—
76	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₈₉	—	4	1	no	—
84	C₄⋊C₄×C₈₉	—	4	1	no	—
85	C₄⋊C₄×C₈₉	—	4	1	no	—
86	C₄⋊C₄×C₈₉	—	4	1	no	—
87	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
88	C₂²×D₈₉	—	2	1	no	—
89	C₂²×D₈₉	—	2	1	no	—
90	C₂²×D₈₉	—	2	1	no	—
91	C₂²×D₈₉	—	2	1	no	—
92	C₂²×D₈₉	—	2	1	no	—
93	C₂²×D₈₉	—	2	1	no	—
94	C₂²×D₈₉	#41	1	2	yes	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
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	—	2	1	no	—
115	C₂×C₄×C₈₉	#41	1	2	yes	—
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
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	—	2	1	no	—
130	C₂×Dic₈₉	—	2	1	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	#41	1	2	yes	—
137	C₄×D₈₉	—	4	4	no	—
138	Dic₁₇₈	—	4	4	no	—
139	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
141	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×D₈₉	—	2	2	no	—
147	C₂×D₈₉	—	2	2	no	—
148	C₂×D₈₉	—	2	2	no	—
149	C₂×D₈₉	—	4	1	no	—
150	C₂×D₈₉	—	4	1	no	—
151	C₂×D₈₉	—	4	1	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₄×C₈₉	—	2	2	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	—	4	1	no	—
157	C₄×C₈₉	—	4	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	Dic₈₉	—	2	2	no	—
161	Dic₈₉	—	4	1	no	—
162	Dic₈₉	—	4	1	no	—
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	—	4	1	no	—
166	Dic₈₉	—	4	1	no	—
167	C₂×C₈₉	#538	1	1	yes	characteristic
168	C₂×C₈₉	#542	1	1	yes	characteristic
169	C₂×C₈₉	#542	1	1	yes	characteristic
170	C₂×C₈₉	#551	1	1	yes	characteristic
171	C₂×C₈₉	#552	1	1	yes	characteristic
172	C₂×C₈₉	#552	1	1	yes	characteristic
173	C₂×C₈₉	#563	1	1	yes	characteristic
174	D₈₉	—	4	2	no	—
175	#2764	—	89	1	no	maximal, Sylow
176	C₈₉	#2764	1	1	yes	characteristic, complemented, Sylow
177	#400	—	89	1	no	—
178	#400	—	89	1	no	—
179	#398	—	89	1	no	—
180	#398	D₈₉	1	1	yes	characteristic, complemented
181	#397	—	89	1	no	—
182	#393	—	89	1	no	—
183	#391	—	89	1	no	—
184	#391	—	89	1	no	—
185	#388	—	89	1	no	—
186	#388	—	89	1	no	—
187	#385	—	89	1	no	—
188	#385	—	89	1	no	—
189	#383	—	89	1	no	—
190	#383	—	89	1	no	—
191	#381	—	89	1	no	—
192	C₂²×Q₈	—	89	1	no	—
193	C₂³×C₄	—	89	1	no	—
194	C₂×C₄⋊C₄	—	89	1	no	—
195	C₂×C₄⋊C₄	—	89	1	no	—
196	C₂×C₄⋊C₄	—	89	1	no	—
197	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
198	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	—	89	1	no	—
202	C₂×C₂²⋊C₄	—	89	1	no	—
203	C₂×C₂²⋊C₄	—	89	1	no	—
204	C₂×C₂²⋊C₄	—	89	1	no	—
205	C₂×C₂²⋊C₄	—	89	1	no	—
206	C₂×C₂²⋊C₄	—	89	1	no	—
207	C₂×C₄²	—	89	1	no	—
208	C₂×C₄²	—	89	1	no	—
209	#95	—	89	1	no	—
210	#95	—	89	1	no	—
211	#95	—	89	1	no	—
212	#95	—	89	1	no	—
213	#95	—	89	1	no	—
214	#95	—	89	1	no	—
215	#95	—	89	1	no	—
216	#95	—	89	1	no	—
217	#95	—	89	1	no	—
218	#95	—	89	1	no	—
219	#95	—	89	1	no	—
220	#95	—	89	1	no	—
221	#95	—	89	1	no	—
222	#95	—	89	1	no	—
223	#95	—	89	1	no	—
224	#95	C₂×D₈₉	1	1	yes	characteristic
225	#95	C₂×D₈₉	1	1	yes	characteristic
226	#95	C₂×D₈₉	1	1	yes	characteristic
227	C₂⁴	—	89	1	no	—
228	C₂×Q₈	—	356	1	no	—
229	C₂×Q₈	—	178	2	no	—
230	C₂×Q₈	—	178	2	no	—
231	C₂²×C₄	—	89	1	no	—
232	C₂²×C₄	—	89	1	no	—
233	C₂²×C₄	—	89	1	no	—
234	C₂²×C₄	—	89	1	no	—
235	C₂²×C₄	—	89	1	no	—
236	C₂²×C₄	—	89	1	no	—
237	C₂²×C₄	—	89	1	no	—
238	C₂²×C₄	—	356	1	no	—
239	C₂²×C₄	—	356	1	no	—
240	C₂²×C₄	—	178	2	no	—
241	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	356	1	no	—
253	C₄⋊C₄	—	356	1	no	—
254	C₄⋊C₄	—	356	1	no	—
255	C₂²⋊C₄	—	356	1	no	—
256	C₂²⋊C₄	—	356	1	no	—
257	C₂²⋊C₄	—	178	2	no	—
258	C₂²⋊C₄	—	178	2	no	—
259	C₂²⋊C₄	—	178	2	no	—
260	C₂²⋊C₄	—	178	2	no	—
261	C₄²	—	178	2	no	—
262	C₄²	—	178	2	no	—
263	C₂³	—	89	2	no	—
264	C₂³	—	178	1	no	—
265	C₂³	—	178	1	no	—
266	C₂³	—	178	1	no	—
267	C₂³	—	178	1	no	—
268	C₂³	—	178	1	no	—
269	C₂³	—	178	1	no	—
270	C₂³	C₂³×D₈₉	1	1	yes	characteristic
271	Q₈	—	356	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	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	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	89	2	no	—
293	C₂×C₄	—	356	4	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	—	178	1	no	—
297	C₂×C₄	—	178	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	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	—	178	1	no	—
314	C₂×C₄	#13083575	1	2	yes	—
315	C₂²	—	356	1	no	—
316	C₂²	—	356	1	no	—
317	C₂²	—	356	1	no	—
318	C₂²	—	356	1	no	—
319	C₂²	—	178	2	no	—
320	C₂²	—	178	2	no	—
321	C₂²	—	178	2	no	—
322	C₂²	#41×D₈₉	1	1	yes	characteristic
323	C₂²	#41×D₈₉	1	1	yes	characteristic
324	C₂²	C₂×#13083575	1	1	yes	characteristic
325	C₂²	#15668057	1	1	yes	characteristic
326	C₂²	#15668057	1	1	yes	characteristic
327	C₂²	#15668059	1	1	yes	characteristic
328	C₂²	#15668060	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	356	1	no	—
337	C₄	—	356	1	no	—
338	C₄	—	356	1	no	—
339	C₄	—	356	1	no	—
340	C₄	—	356	1	no	—
341	C₄	—	356	1	no	—
342	C₄	—	178	2	no	—
343	C₂	—	356	2	no	—
344	C₂	#19152180	1	1	yes	characteristic
345	C₂	#19152216	1	1	yes	characteristic
346	C₂	#19152222	1	1	yes	characteristic
347	C₂	#19152275	1	1	yes	characteristic
348	C₂	#19152280	1	1	yes	characteristic
349	C₂	#19152286	1	1	yes	characteristic
350	C₂	#19152349	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#398×C₈₉	C₂
prime radical	C₂³×D₈₉	C₂³
hypercenter	#398	D₈₉
hypocenter	C₈₉	#2764
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #398

Complements

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

N	H ≅ G/N	type
1. #19151537	C₂	semidirect
2. #19151536	C₂	semidirect
3. #19151531	C₂	semidirect
6. #19151503	C₂	semidirect
7. #19151502	C₂	semidirect
12. #19151465	C₂	semidirect
13. #19151465	C₂	semidirect
15. #398×C₈₉	C₂	semidirect
176. C₈₉	#2764	semidirect
180. #398	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19151537	2
#19151536	2
#19151531	2
#19151524	2
#19151511	2
#19151503	2
#19151502	2
#19151490	2
#19151488	2
#19151469	2
#19151465	2
#19151460	2
#398×C₈₉	2
#2764	89