Subgroups — #26032086

Order 11392.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152286	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152241	C₂	1	1	yes	maximal, characteristic
3	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151921	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151826	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151817	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151763	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151750	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13083563	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151593	C₂	1	1	yes	maximal, characteristic, complemented
14	#102×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#488×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	C₂×#13083562	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic, complemented
23	#15668007	—	2	1	no	—
24	#15668003	—	2	1	no	—
25	#15667997	—	2	1	no	—
26	#15667994	—	2	1	no	—
27	#15667992	C₂²	1	1	yes	characteristic
28	#15667992	C₂²	1	1	yes	characteristic, complemented
29	#15667988	C₂²	1	1	yes	characteristic
30	#15667985	—	2	1	no	—
31	#15667983	C₂²	1	1	yes	characteristic
32	#15667982	—	2	1	no	—
33	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
34	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
35	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
36	#15667970	—	2	1	no	—
37	#15667970	C₂²	1	1	yes	characteristic, complemented
38	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₄×D₈₉	—	2	1	no	—
40	#15667945	C₂²	1	1	yes	characteristic, complemented
41	#15667944	C₂²	1	1	yes	characteristic, complemented
42	#15667943	C₂²	1	1	yes	characteristic, complemented
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic, complemented
44	#15667937	C₂²	1	1	yes	characteristic, complemented
45	#15667936	C₂²	1	1	yes	characteristic, complemented
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic, complemented
52	#125×C₈₉	C₂²	1	1	yes	characteristic, complemented
53	#124×C₈₉	C₂²	1	1	yes	characteristic
54	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
55	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	#102×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	—	2	1	no	—
60	#13083575	—	2	2	no	—
61	#13083575	—	4	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	D₄	1	1	yes	characteristic
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	D₄×D₈₉	—	2	2	no	—
69	D₄×D₈₉	—	4	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
75	#13083564	—	4	1	no	—
76	#13083563	—	2	2	no	—
77	#13083562	—	4	1	no	—
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₈×D₈₉	—	4	1	no	—
81	C₈×D₈₉	—	4	1	no	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	—	2	1	no	—
84	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
85	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	—	2	1	no	—
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	D₄	1	1	yes	characteristic
97	#13083551	D₄	1	1	yes	characteristic
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₈₉	—	2	1	no	—
109	C₂²⋊C₄×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	—	4	2	no	—
114	C₂²×D₈₉	—	4	2	no	—
115	C₂²×D₈₉	—	8	1	no	—
116	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
118	Q₈×C₈₉	—	2	1	no	—
119	Q₈×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	2	1	no	—
121	D₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	4	2	no	—
129	C₂²⋊D₈₉	—	8	1	no	—
130	C₂×Dic₈₉	—	2	1	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	D₃₅₆	—	2	2	no	—
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	—	4	2	no	—
138	D₃₅₆	#41	1	2	yes	—
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	C₄×D₈₉	—	8	1	no	—
146	Dic₁₇₈	—	2	1	no	—
147	Dic₁₇₈	—	2	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	C₈×C₈₉	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	#10623494	—	2	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	—	4	1	no	—
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	2	no	—
160	C₂×D₈₉	—	2	1	no	—
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₂×D₈₉	—	8	4	no	—
165	C₂×D₈₉	#135	1	2	yes	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	—	4	1	no	—
171	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
172	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	2	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	Dic₈₉	—	4	1	no	—
178	C₂×C₈₉	—	8	1	no	—
179	C₂×C₈₉	#546	1	1	yes	characteristic
180	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
181	C₂×C₈₉	#577	1	1	yes	characteristic
182	D₈₉	—	2	2	no	—
183	D₈₉	—	4	1	no	—
184	D₈₉	—	4	2	no	—
185	#3296	—	89	1	no	maximal, Sylow
186	C₈₉	#3296	1	1	yes	characteristic, complemented, Sylow
187	#552	—	89	1	no	—
188	#546	—	89	1	no	—
189	#488	D₈₉	1	1	yes	characteristic, complemented
190	#483	—	89	1	no	—
191	#482	—	89	1	no	—
192	#478	—	89	1	no	—
193	#467	—	89	1	no	—
194	#465	—	89	1	no	—
195	#464	—	89	1	no	—
196	#452	—	89	1	no	—
197	#450	—	89	1	no	—
198	C₄×SD₁₆	—	89	1	no	—
199	C₈×D₄	—	89	1	no	—
200	#416	—	89	1	no	—
201	C₂×#102	—	89	1	no	—
202	C₂×#41	—	89	1	no	—
203	C₂²×D₄	—	89	1	no	—
204	C₂×Q₁₆	—	89	1	no	—
205	C₂×SD₁₆	—	89	1	no	—
206	C₂×SD₁₆	—	89	1	no	—
207	C₂²×C₈	—	89	1	no	—
208	C₂²×C₈	—	89	1	no	—
209	#126	—	178	1	no	—
210	#125	—	178	1	no	—
211	#125	C₂×D₈₉	1	1	yes	characteristic, complemented
212	#124	C₂×D₈₉	1	1	yes	characteristic
213	#123	—	178	1	no	—
214	#123	—	178	1	no	—
215	C₂²⋊Q₈	—	89	1	no	—
216	C₂²⋊Q₈	—	89	1	no	—
217	C₂²⋊Q₈	—	89	1	no	—
218	#121	—	89	1	no	—
219	#121	—	178	1	no	—
220	#120	—	178	1	no	—
221	C₄×Q₈	—	89	1	no	—
222	C₄×D₄	—	89	1	no	—
223	C₄×D₄	—	89	1	no	—
224	#117	—	89	1	no	—
225	#117	—	178	1	no	—
226	C₂×C₄⋊C₄	—	89	1	no	—
227	C₂×C₂²⋊C₄	—	178	1	no	—
228	#107	—	89	1	no	—
229	#106	—	89	1	no	—
230	#106	—	89	1	no	—
231	C₄⋊C₈	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
235	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
236	#102	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	—	89	1	no	—
240	#102	C₂×D₈₉	1	1	yes	characteristic
241	#102	C₂×D₈₉	1	1	yes	characteristic
242	C₂²⋊C₈	—	89	1	no	—
243	C₂²⋊C₈	—	89	1	no	—
244	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
245	C₂⁴	—	178	1	no	—
246	#41	—	356	1	no	—
247	#41	—	178	2	no	—
248	C₂×Q₈	—	89	1	no	—
249	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×D₄	—	89	1	no	—
251	C₂×D₄	—	89	1	no	—
252	C₂×D₄	—	356	1	no	—
253	C₂×D₄	—	178	1	no	—
254	C₂×D₄	—	178	1	no	—
255	C₂×D₄	—	178	2	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂²×C₄	—	89	1	no	—
259	C₂²×C₄	—	89	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	—	178	1	no	—
263	Q₁₆	—	356	1	no	—
264	SD₁₆	—	356	1	no	—
265	SD₁₆	—	178	2	no	—
266	C₂×C₈	—	89	1	no	—
267	C₂×C₈	—	89	1	no	—
268	C₂×C₈	—	356	1	no	—
269	C₂×C₈	—	356	1	no	—
270	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	89	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	—	178	1	no	—
282	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	89	1	no	—
287	C₂²⋊C₄	—	89	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₄²	—	178	1	no	—
295	C₄²	—	178	1	no	—
296	C₄²	C₂²×D₈₉	1	1	yes	characteristic
297	C₂³	—	2	1	no	—
298	C₂³	—	89	1	no	—
299	C₂³	—	89	1	no	—
300	C₂³	—	356	2	no	—
301	C₂³	—	356	2	no	—
302	C₂³	—	178	1	no	—
303	C₂³	—	712	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	178	1	no	—
307	Q₈	—	178	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	89	2	no	—
311	D₄	—	356	2	no	—
312	D₄	—	356	2	no	—
313	D₄	—	178	2	no	—
314	D₄	—	178	1	no	—
315	D₄	—	712	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	89	1	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	356	2	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	178	1	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	—	178	1	no	—
329	C₂×C₄	—	178	1	no	—
330	C₂×C₄	—	712	1	no	—
331	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
332	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	178	1	no	—
337	C₈	—	178	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	89	2	no	—
342	C₂²	—	356	1	no	—
343	C₂²	—	356	1	no	—
344	C₂²	—	178	1	no	—
345	C₂²	—	178	2	no	—
346	C₂²	—	178	1	no	—
347	C₂²	—	178	1	no	—
348	C₂²	—	178	1	no	—
349	C₂²	—	178	1	no	—
350	C₂²	—	712	4	no	—
351	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	356	1	no	—
358	C₄	—	356	1	no	—
359	C₄	—	356	1	no	—
360	C₄	—	178	1	no	—
361	C₄	—	178	1	no	—
362	C₄	#41×D₈₉	1	1	yes	characteristic
363	C₄	#15668058	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	356	2	no	—
366	C₂	—	356	1	no	—
367	C₂	—	178	2	no	—
368	C₂	#19152241	1	1	yes	characteristic
369	C₂	#135×D₈₉	1	1	yes	characteristic
370	C₂	#19152405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#488×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152241	C₂
hypercenter	#488	D₈₉
hypocenter	C₈₉	#3296
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #19152286	C₂	semidirect
3. #19151927	C₂	semidirect
4. #19151921	C₂	semidirect
5. #19151895	C₂	semidirect
6. #19151838	C₂	semidirect
7. #19151826	C₂	semidirect
8. #19151817	C₂	semidirect
9. #19151763	C₂	semidirect
10. #19151750	C₂	semidirect
11. C₄×#13083563	C₂	semidirect
12. C₈×D₃₅₆	C₂	semidirect
13. #19151593	C₂	semidirect
14. #102×D₈₉	C₂	semidirect
15. #488×C₈₉	C₂	semidirect
18. C₂×#13083564	C₂²	semidirect
22. C₂×C₈×D₈₉	C₂²	semidirect
28. #15667992	C₂²	semidirect
33. C₄×Dic₁₇₈	C₂²	semidirect
37. #15667970	C₂²	semidirect
40. #15667945	C₂²	semidirect
41. #15667944	C₂²	semidirect
42. #15667943	C₂²	semidirect
43. C₄⋊#10623494	C₂²	semidirect
44. #15667937	C₂²	semidirect
45. #15667936	C₂²	semidirect
51. #15667924	C₂²	semidirect
52. #125×C₈₉	C₂²	semidirect
54. #103×C₈₉	C₂²	semidirect
55. #103×C₈₉	C₂²	semidirect
58. C₄×C₈×C₈₉	C₂²	semidirect
186. C₈₉	#3296	semidirect
189. #488	D₈₉	semidirect
211. #125	C₂×D₈₉	semidirect
234. #103	C₂×D₈₉	semidirect
235. #103	C₂×D₈₉	semidirect
244. C₄×C₈	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152286	2
#19152241	2
#19151927	2
#19151921	2
#19151895	2
#19151838	2
#19151826	2
#19151817	2
#19151763	2
#19151750	2
C₄×#13083563	2
C₈×D₃₅₆	2
#19151593	2
#102×D₈₉	2
#488×C₈₉	2
#3296	89