Subgroups — #26026421

Order 11392.

424 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151539	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151539	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151521	C₂	1	1	yes	maximal, characteristic
4	#19151521	C₂	1	1	yes	maximal, characteristic
5	#19151519	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151517	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151510	C₂	1	1	yes	maximal, characteristic
8	#19151505	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151502	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151483	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151480	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151459	C₂	1	1	yes	maximal, characteristic
13	#19151456	C₂	1	1	yes	maximal, characteristic
14	C₂²⋊C₄×Dic₈₉	C₂	1	1	yes	maximal, characteristic
15	#393×C₈₉	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₂³×Dic₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083553	C₂²	1	1	yes	characteristic
21	C₂×#13083553	C₂²	1	1	yes	characteristic
22	C₂×#13083553	C₂²	1	1	yes	characteristic
23	C₂×#13083553	C₂²	1	1	yes	characteristic
24	C₂×#13083553	C₂²	1	1	yes	characteristic
25	C₂×#13083553	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₈₉	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	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊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	C₂×Dic₁₇₈	—	2	2	no	—
52	C₂×Dic₁₇₈	—	4	1	no	—
53	C₂×Dic₁₇₈	—	4	1	no	—
54	C₂²×Dic₈₉	—	4	1	no	—
55	C₂²×Dic₈₉	—	4	1	no	—
56	C₂²×Dic₈₉	Q₈	1	4	yes	complemented
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	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	4	1	no	—
67	#13083553	—	2	2	no	—
68	#13083553	—	4	1	no	—
69	#13083553	—	4	1	no	—
70	#13083553	—	4	1	no	—
71	#13083553	—	4	1	no	—
72	#13083553	—	4	1	no	—
73	C₂²⋊Dic₈₉	—	2	2	no	—
74	C₂²⋊Dic₈₉	—	4	1	no	—
75	C₂²⋊Dic₈₉	—	4	1	no	—
76	C₄×Dic₈₉	—	4	1	no	—
77	C₄×Dic₈₉	—	4	1	no	—
78	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₈₉	—	2	2	no	—
80	C₂×Q₈×C₈₉	—	4	1	no	—
81	C₂×Q₈×C₈₉	—	4	1	no	—
82	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
88	C₂²⋊C₄×C₈₉	—	2	2	no	—
89	C₂²⋊C₄×C₈₉	—	4	1	no	—
90	C₂²⋊C₄×C₈₉	—	4	1	no	—
91	C₂³×C₈₉	—	2	1	no	—
92	C₂³×C₈₉	—	2	1	no	—
93	C₂³×C₈₉	—	2	1	no	—
94	C₂³×C₈₉	—	2	1	no	—
95	C₂³×C₈₉	—	2	1	no	—
96	C₂³×C₈₉	—	2	1	no	—
97	C₂³×C₈₉	C₂×Q₈	1	2	yes	—
98	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
99	Q₈×C₈₉	—	4	4	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	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	#41	1	2	yes	—
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	—	2	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	4	4	no	—
140	C₂×Dic₈₉	C₂×Q₈	1	2	yes	—
141	C₂×Dic₈₉	C₂×Q₈	1	2	yes	—
142	C₂×Dic₈₉	#41	1	2	yes	—
143	Dic₁₇₈	—	4	4	no	—
144	C₂²×C₈₉	—	2	2	no	—
145	C₂²×C₈₉	—	2	2	no	—
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	—	4	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
152	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	Dic₈₉	—	2	2	no	—
165	Dic₈₉	—	2	2	no	—
166	Dic₈₉	—	2	2	no	—
167	Dic₈₉	—	4	1	no	—
168	Dic₈₉	—	4	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	—	4	1	no	—
172	C₂×C₈₉	—	4	2	no	—
173	C₂×C₈₉	#533	1	1	yes	characteristic
174	C₂×C₈₉	#538	1	1	yes	characteristic
175	C₂×C₈₉	#538	1	1	yes	characteristic
176	C₂×C₈₉	#542	1	1	yes	characteristic
177	C₂×C₈₉	#542	1	1	yes	characteristic
178	C₂×C₈₉	#543	1	1	yes	characteristic
179	C₂×C₈₉	#544	1	1	yes	characteristic
180	#2665	—	89	1	no	maximal, Sylow
181	C₈₉	#2665	1	1	yes	characteristic, complemented, Sylow
182	#400	—	89	1	no	—
183	#400	—	89	1	no	—
184	#396	—	89	1	no	—
185	#396	—	89	1	no	—
186	#395	—	89	1	no	—
187	#395	—	89	1	no	—
188	#393	—	89	1	no	—
189	#393	D₈₉	1	1	yes	characteristic
190	#391	—	89	1	no	—
191	#391	—	89	1	no	—
192	#387	—	89	1	no	—
193	#387	—	89	1	no	—
194	#381	—	89	1	no	—
195	#380	—	89	1	no	—
196	C₄×C₂²⋊C₄	—	89	1	no	—
197	C₂²×Q₈	—	89	1	no	—
198	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
199	C₂³×C₄	—	89	1	no	—
200	C₂×C₄⋊C₄	—	89	1	no	—
201	C₂×C₄⋊C₄	—	89	1	no	—
202	C₂×C₄⋊C₄	—	89	1	no	—
203	C₂×C₄⋊C₄	—	89	1	no	—
204	C₂×C₄⋊C₄	—	89	1	no	—
205	C₂×C₄⋊C₄	—	89	1	no	—
206	C₂×C₄⋊C₄	—	89	1	no	—
207	C₂×C₄⋊C₄	—	89	1	no	—
208	C₂×C₂²⋊C₄	—	89	1	no	—
209	C₂×C₂²⋊C₄	—	89	1	no	—
210	C₂×C₂²⋊C₄	—	89	1	no	—
211	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
214	C₂×C₄²	—	89	1	no	—
215	C₂×C₄²	—	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	—	89	1	no	—
225	#95	—	89	1	no	—
226	#95	—	89	1	no	—
227	#95	—	89	1	no	—
228	#95	—	89	1	no	—
229	#95	C₂×D₈₉	1	1	yes	characteristic
230	#95	C₂×D₈₉	1	1	yes	characteristic
231	#95	C₂×D₈₉	1	1	yes	characteristic
232	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	356	1	no	—
237	C₂×Q₈	—	356	1	no	—
238	C₂×Q₈	—	178	2	no	—
239	C₂²×C₄	—	89	4	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	—	89	1	no	—
242	C₂²×C₄	—	89	1	no	—
243	C₂²×C₄	—	89	1	no	—
244	C₂²×C₄	—	89	1	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	89	1	no	—
248	C₂²×C₄	—	356	1	no	—
249	C₂²×C₄	—	356	1	no	—
250	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	—	356	1	no	—
257	C₄⋊C₄	—	356	1	no	—
258	C₄⋊C₄	—	356	1	no	—
259	C₄⋊C₄	—	356	1	no	—
260	C₄⋊C₄	—	356	1	no	—
261	C₄⋊C₄	—	356	1	no	—
262	C₄⋊C₄	—	178	2	no	—
263	C₄⋊C₄	—	178	2	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	356	1	no	—
268	C₂²⋊C₄	—	356	1	no	—
269	C₂²⋊C₄	—	178	2	no	—
270	C₄²	—	356	1	no	—
271	C₄²	—	356	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	Q₈×D₈₉	1	2	yes	—
279	C₂³	C₂³×D₈₉	1	1	yes	characteristic
280	Q₈	—	4	4	no	—
281	Q₈	—	356	4	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	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	89	2	no	—
300	C₂×C₄	—	89	2	no	—
301	C₂×C₄	—	89	2	no	—
302	C₂×C₄	—	356	4	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	—	178	1	no	—
315	C₂×C₄	—	178	1	no	—
316	C₂×C₄	—	178	1	no	—
317	C₂×C₄	—	178	1	no	—
318	C₂×C₄	—	178	1	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	#13083573	1	2	yes	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
333	C₂²	#41×D₈₉	1	1	yes	characteristic
334	C₂²	C₂×#13083573	1	1	yes	characteristic
335	C₂²	#15668057	1	1	yes	characteristic
336	C₂²	#15668057	1	1	yes	characteristic
337	C₂²	#15668058	1	1	yes	characteristic
338	C₂²	#15668059	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	356	1	no	—
346	C₄	—	356	1	no	—
347	C₄	—	356	1	no	—
348	C₄	—	356	1	no	—
349	C₄	—	356	1	no	—
350	C₄	—	178	2	no	—
351	C₄	—	178	2	no	—
352	C₄	—	178	2	no	—
353	C₂	—	4	2	no	—
354	C₂	#19152150	1	1	yes	characteristic
355	C₂	#19152182	1	1	yes	characteristic
356	C₂	#19152183	1	1	yes	characteristic
357	C₂	#19152212	1	1	yes	characteristic
358	C₂	#19152217	1	1	yes	characteristic
359	C₂	#19152223	1	1	yes	characteristic
360	C₂	#19152227	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#393×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂³
hypercenter	#393	D₈₉
hypocenter	C₈₉	#2665
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #393

Complements

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

N	H ≅ G/N	type
1. #19151539	C₂	semidirect
2. #19151539	C₂	semidirect
5. #19151519	C₂	semidirect
6. #19151517	C₂	semidirect
8. #19151505	C₂	semidirect
9. #19151502	C₂	semidirect
10. #19151483	C₂	semidirect
11. #19151480	C₂	semidirect
56. C₂²×Dic₈₉	Q₈	semidirect
181. C₈₉	#2665	semidirect

Maximal subgroups

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

subgroup	index
#19151539	2
#19151521	2
#19151519	2
#19151517	2
#19151510	2
#19151505	2
#19151502	2
#19151483	2
#19151480	2
#19151459	2
#19151456	2
C₂²⋊C₄×Dic₈₉	2
#393×C₈₉	2
#2665	89