Subgroups — #26030163

Order 11392.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152105	C₂	1	1	yes	maximal, characteristic
2	#117×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151938	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151937	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151932	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151921	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151920	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151915	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151910	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151636	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151634	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151594	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15667924	C₂	1	1	yes	maximal, characteristic, complemented
15	#416×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
17	C₂²×#10623494	C₂²	1	1	yes	characteristic
18	#15667993	—	2	1	no	—
19	#15667988	C₂²	1	1	yes	characteristic
20	#15667988	C₂²	1	1	yes	characteristic
21	#15667983	C₂²	1	1	yes	characteristic
22	#15667983	C₂²	1	1	yes	characteristic
23	#15667980	—	2	1	no	—
24	C₄×C₂²⋊D₈₉	—	2	1	no	—
25	C₄×C₂²⋊D₈₉	—	2	1	no	—
26	#15667971	C₂²	1	1	yes	characteristic
27	#15667971	C₂²	1	1	yes	characteristic
28	#15667970	C₂²	1	1	yes	characteristic
29	#15667970	C₂²	1	1	yes	characteristic
30	#15667969	—	2	1	no	—
31	#15667968	—	2	1	no	—
32	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×D₈₉	—	2	1	no	—
35	C₄²×D₈₉	—	2	1	no	—
36	#15667946	C₂²	1	1	yes	characteristic, complemented
37	#15667946	C₂²	1	1	yes	characteristic, complemented
38	#15667945	C₂²	1	1	yes	characteristic, complemented
39	#15667945	C₂²	1	1	yes	characteristic, complemented
40	#15667944	C₂²	1	1	yes	characteristic, complemented
41	#15667944	C₂²	1	1	yes	characteristic, complemented
42	#15667943	C₂²	1	1	yes	characteristic, complemented
43	#15667943	C₂²	1	1	yes	characteristic, complemented
44	#15667936	C₂²	1	1	yes	characteristic, complemented
45	#15667936	C₂²	1	1	yes	characteristic, complemented
46	#15667933	C₂²	1	1	yes	characteristic, complemented
47	#15667933	C₂²	1	1	yes	characteristic, complemented
48	#15667924	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
54	#117×C₈₉	C₂²	1	1	yes	characteristic
55	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₈₉	C₂²	1	1	yes	characteristic, complemented
58	#102×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₈₉	—	2	1	no	—
61	C₂×C₂²⋊D₈₉	—	2	1	no	—
62	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
63	C₂×C₄×D₈₉	—	2	2	no	—
64	C₂×C₄×D₈₉	—	2	2	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₂×#10623494	C₂³	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	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	#41×C₈₉	—	2	2	no	—
95	#41×C₈₉	—	2	2	no	—
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	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	—	2	1	no	—
109	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₈₉	—	2	2	no	—
111	C₂²×D₈₉	—	2	2	no	—
112	C₂²×D₈₉	—	4	1	no	—
113	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	D₄×C₈₉	—	2	1	no	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	—	4	2	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	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#41	1	2	yes	—
126	C₂×C₄×C₈₉	#41	1	2	yes	—
127	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	4	2	no	—
129	C₂²⋊D₈₉	—	4	2	no	—
130	C₂×Dic₈₉	—	2	2	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	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₄×D₈₉	—	2	2	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	1	no	—
140	C₄×D₈₉	—	2	1	no	—
141	C₄×D₈₉	—	2	1	no	—
142	C₄×D₈₉	—	2	1	no	—
143	C₄×D₈₉	—	4	2	no	—
144	C₄×D₈₉	—	4	2	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	#10623494	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	#123	1	2	yes	—
155	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	1	no	—
158	C₂×D₈₉	—	2	1	no	—
159	C₂×D₈₉	—	2	1	no	—
160	C₂×D₈₉	—	2	1	no	—
161	C₂×D₈₉	—	4	4	no	—
162	C₂×D₈₉	#135	1	2	yes	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	—	4	1	no	—
165	C₄×C₈₉	—	4	1	no	—
166	C₄×C₈₉	#123	1	2	yes	—
167	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	2	1	no	—
171	Dic₈₉	—	4	1	no	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	—	4	1	no	—
174	Dic₈₉	—	4	1	no	—
175	C₂×C₈₉	—	2	2	no	—
176	C₂×C₈₉	—	8	1	no	—
177	C₂×C₈₉	C₂×#123	1	1	yes	characteristic
178	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
179	C₂×C₈₉	#575	1	1	yes	characteristic
180	D₈₉	—	2	2	no	—
181	D₈₉	—	4	1	no	—
182	#3091	—	89	1	no	maximal, Sylow
183	C₈₉	#3091	1	1	yes	characteristic, complemented, Sylow
184	#525	—	89	1	no	—
185	C₂×#117	—	89	1	no	—
186	#485	—	89	1	no	—
187	#485	—	89	1	no	—
188	#484	—	89	1	no	—
189	#483	—	89	1	no	—
190	#482	—	89	1	no	—
191	#482	—	89	1	no	—
192	#481	—	89	1	no	—
193	#480	—	89	1	no	—
194	#427	—	89	1	no	—
195	#427	—	89	1	no	—
196	#416	—	89	1	no	—
197	#416	D₈₉	1	1	yes	characteristic, complemented
198	C₂×C₂²⋊C₈	—	89	1	no	—
199	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
200	C₂³×C₄	—	89	1	no	—
201	C₂²×C₈	—	89	1	no	—
202	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
203	#123	—	178	1	no	—
204	C₂²⋊Q₈	—	89	1	no	—
205	C₂²⋊Q₈	—	89	1	no	—
206	#121	—	89	1	no	—
207	#121	—	89	1	no	—
208	#120	—	178	1	no	—
209	C₄×D₄	—	178	1	no	—
210	C₄×D₄	—	178	1	no	—
211	#117	—	89	1	no	—
212	#117	—	89	1	no	—
213	#117	—	89	1	no	—
214	#117	—	89	1	no	—
215	#117	—	178	1	no	—
216	#117	—	178	1	no	—
217	#117	C₂×D₈₉	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	89	1	no	—
219	C₂×C₄⋊C₄	—	89	1	no	—
220	C₂×C₂²⋊C₄	—	178	1	no	—
221	C₂×C₄²	—	178	1	no	—
222	#107	—	89	1	no	—
223	#107	—	89	1	no	—
224	#107	—	89	1	no	—
225	#107	—	89	1	no	—
226	#106	—	89	1	no	—
227	#106	—	89	1	no	—
228	#106	—	89	1	no	—
229	#106	—	89	1	no	—
230	#103	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
233	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	C₂×D₈₉	1	1	yes	characteristic, complemented
237	#102	C₂×D₈₉	1	1	yes	characteristic, complemented
238	C₂²⋊C₈	—	89	1	no	—
239	C₂²⋊C₈	—	89	1	no	—
240	C₂²⋊C₈	—	89	1	no	—
241	C₂²⋊C₈	—	89	1	no	—
242	C₂⁴	—	89	1	no	—
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	89	1	no	—
252	C₂²×C₄	—	89	1	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	C₂²×C₄	—	178	2	no	—
259	C₂²×C₄	—	178	2	no	—
260	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₄⋊C₄	—	89	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	89	1	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₂²⋊C₄	—	178	1	no	—
284	C₂²⋊C₄	—	178	1	no	—
285	C₂²⋊C₄	—	178	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	178	1	no	—
288	C₄²	—	178	1	no	—
289	C₄²	—	178	1	no	—
290	C₄²	—	178	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	89	1	no	—
293	C₂³	—	89	1	no	—
294	C₂³	—	356	1	no	—
295	C₂³	—	178	2	no	—
296	C₂³	—	178	2	no	—
297	C₂³	D₄×D₈₉	1	1	yes	characteristic
298	Q₈	—	2	1	no	—
299	Q₈	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	2	no	—
303	D₄	—	356	2	no	—
304	D₄	—	356	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	89	1	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	356	2	no	—
312	C₂×C₄	—	356	2	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	2	no	—
318	C₂×C₄	—	178	2	no	—
319	C₂×C₄	—	178	2	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	D₄×D₈₉	1	1	yes	characteristic
325	C₂×C₄	#13083573	1	2	yes	—
326	C₂×C₄	#13083574	1	2	yes	—
327	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	178	1	no	—
331	C₈	—	178	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	89	2	no	—
338	C₂²	—	356	4	no	—
339	C₂²	—	178	1	no	—
340	C₂²	—	178	1	no	—
341	C₂²	—	178	1	no	—
342	C₂²	—	178	1	no	—
343	C₂²	—	178	1	no	—
344	C₂²	#15667994	1	2	yes	—
345	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	356	1	no	—
350	C₄	—	356	1	no	—
351	C₄	—	356	1	no	—
352	C₄	—	356	1	no	—
353	C₄	—	178	1	no	—
354	C₄	—	178	1	no	—
355	C₄	#15667994	1	2	yes	—
356	C₄	C₂×#13083573	1	1	yes	characteristic
357	C₄	C₂×#13083574	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	8	1	no	—
360	C₂	—	356	1	no	—
361	C₂	—	178	2	no	—
362	C₂	C₂×#15667994	1	1	yes	characteristic
363	C₂	#135×D₈₉	1	1	yes	characteristic
364	C₂	#19152395	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#416×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152105	C₂
hypercenter	#416	D₈₉
hypocenter	C₈₉	#3091
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #117×D₈₉	C₂	semidirect
3. #19151938	C₂	semidirect
4. #19151937	C₂	semidirect
5. #19151932	C₂	semidirect
6. #19151927	C₂	semidirect
7. #19151921	C₂	semidirect
8. #19151920	C₂	semidirect
9. #19151915	C₂	semidirect
10. #19151910	C₂	semidirect
11. #19151636	C₂	semidirect
12. #19151634	C₂	semidirect
13. #19151594	C₂	semidirect
14. C₂×#15667924	C₂	semidirect
15. #416×C₈₉	C₂	semidirect
36. #15667946	C₂²	semidirect
37. #15667946	C₂²	semidirect
38. #15667945	C₂²	semidirect
39. #15667945	C₂²	semidirect
40. #15667944	C₂²	semidirect
41. #15667944	C₂²	semidirect
42. #15667943	C₂²	semidirect
43. #15667943	C₂²	semidirect
44. #15667936	C₂²	semidirect
45. #15667936	C₂²	semidirect
46. #15667933	C₂²	semidirect
47. #15667933	C₂²	semidirect
55. #103×C₈₉	C₂²	semidirect
56. #103×C₈₉	C₂²	semidirect
57. #102×C₈₉	C₂²	semidirect
58. #102×C₈₉	C₂²	semidirect
183. C₈₉	#3091	semidirect
197. #416	D₈₉	semidirect
232. #103	C₂×D₈₉	semidirect
233. #103	C₂×D₈₉	semidirect
236. #102	C₂×D₈₉	semidirect
237. #102	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152105	2
#117×D₈₉	2
#19151938	2
#19151937	2
#19151932	2
#19151927	2
#19151921	2
#19151920	2
#19151915	2
#19151910	2
#19151636	2
#19151634	2
#19151594	2
C₂×#15667924	2
#416×C₈₉	2
#3091	89