Subgroups — #26031139

Order 11392.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152179	C₂	1	1	yes	maximal, characteristic
2	#19152171	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151928	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151925	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151855	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151845	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151840	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151830	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151762	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151758	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151755	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151563	C₂	1	1	yes	maximal, characteristic, complemented
15	#484×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×#13083574	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	C₂×#13083564	C₂²	1	1	yes	characteristic
21	C₂×#13083563	C₂²	1	1	yes	characteristic
22	C₂×#13083561	C₂²	1	1	yes	characteristic
23	C₂×#13083557	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
25	#15668011	—	2	1	no	—
26	#15667998	—	2	1	no	—
27	#15667997	—	2	1	no	—
28	#15667994	—	2	1	no	—
29	#15667992	—	2	1	no	—
30	#15667992	C₂²	1	1	yes	characteristic
31	#15667992	C₂²	1	1	yes	characteristic
32	#15667992	C₂²	1	1	yes	characteristic
33	#15667991	C₂²	1	1	yes	characteristic
34	#15667988	C₂²	1	1	yes	characteristic
35	#15667987	—	2	1	no	—
36	#15667987	C₂²	1	1	yes	characteristic
37	#15667987	C₂²	1	1	yes	characteristic
38	#15667986	—	2	1	no	—
39	#15667986	—	2	1	no	—
40	#15667970	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667935	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	1	no	—
60	#13083574	—	4	1	no	—
61	#13083574	—	4	1	no	—
62	#13083574	—	4	1	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
67	Q₈×D₈₉	—	4	1	no	—
68	Q₈×D₈₉	—	4	1	no	—
69	C₂×D₃₅₆	—	2	1	no	—
70	C₂×D₃₅₆	—	2	1	no	—
71	C₂×D₃₅₆	D₄	1	1	yes	characteristic
72	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊D₈₉	—	2	1	no	—
74	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
76	C₂×C₄×D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
80	Dic₃₅₆	—	4	1	no	—
81	#13083564	—	4	1	no	—
82	#13083563	—	4	1	no	—
83	#13083561	—	4	1	no	—
84	#13083557	—	4	1	no	—
85	C₂×#10623494	C₂³	1	1	yes	characteristic
86	C₂×#10623494	C₂³	1	1	yes	characteristic
87	C₈×D₈₉	—	4	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	#13083553	—	2	1	no	—
93	#13083553	—	2	1	no	—
94	C₂²⋊Dic₈₉	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	—	2	1	no	—
97	#13083551	—	2	1	no	—
98	#13083551	—	2	1	no	—
99	#13083551	—	2	1	no	—
100	#13083551	—	2	1	no	—
101	#13083551	—	2	1	no	—
102	C₄×Dic₈₉	—	2	1	no	—
103	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	—	2	1	no	—
105	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂²⋊C₄×C₈₉	—	2	1	no	—
114	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	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	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×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	—	2	1	no	—
126	C₂×C₄×C₈₉	—	8	1	no	—
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₈₉	—	4	1	no	—
130	C₂²⋊D₈₉	—	4	1	no	—
131	C₂²⋊D₈₉	—	4	2	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	D₃₅₆	—	2	1	no	—
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	—	4	1	no	—
140	D₃₅₆	—	8	1	no	—
141	D₃₅₆	—	8	1	no	—
142	C₄×D₈₉	—	2	1	no	—
143	C₄×D₈₉	—	2	1	no	—
144	C₄×D₈₉	—	2	1	no	—
145	C₄×D₈₉	—	2	1	no	—
146	C₄×D₈₉	—	4	2	no	—
147	C₄×D₈₉	—	4	2	no	—
148	Dic₁₇₈	—	2	1	no	—
149	Dic₁₇₈	—	2	1	no	—
150	Dic₁₇₈	—	2	1	no	—
151	Dic₁₇₈	—	2	1	no	—
152	Dic₁₇₈	—	4	1	no	—
153	Dic₁₇₈	—	4	2	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	—	2	1	no	—
156	#10623494	—	2	1	no	—
157	#10623494	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	2	1	no	—
164	C₂×D₈₉	—	2	1	no	—
165	C₂×D₈₉	—	2	1	no	—
166	C₂×D₈₉	—	2	1	no	—
167	C₂×D₈₉	—	2	1	no	—
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	4	1	no	—
170	C₂×D₈₉	—	4	1	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	—	4	1	no	—
174	C₄×C₈₉	—	4	1	no	—
175	C₄×C₈₉	—	4	1	no	—
176	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
177	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
178	Dic₈₉	—	2	1	no	—
179	Dic₈₉	—	2	1	no	—
180	Dic₈₉	—	4	1	no	—
181	Dic₈₉	—	4	1	no	—
182	Dic₈₉	—	4	1	no	—
183	C₂×C₈₉	—	4	1	no	—
184	C₂×C₈₉	#534	1	1	yes	characteristic
185	C₂×C₈₉	#577	1	1	yes	characteristic
186	C₂×C₈₉	#578	1	1	yes	characteristic
187	D₈₉	—	4	1	no	—
188	D₈₉	—	4	1	no	—
189	D₈₉	—	8	1	no	—
190	#3201	—	89	1	no	maximal, Sylow
191	C₈₉	#3201	1	1	yes	characteristic, complemented, Sylow
192	#537	—	89	1	no	—
193	#536	—	89	1	no	—
194	#485	—	89	1	no	—
195	#484	D₈₉	1	1	yes	characteristic, complemented
196	#483	—	89	1	no	—
197	#482	—	89	1	no	—
198	#470	—	89	1	no	—
199	#468	—	89	1	no	—
200	#467	—	89	1	no	—
201	#465	—	89	1	no	—
202	#452	—	89	1	no	—
203	#452	—	89	1	no	—
204	#451	—	89	1	no	—
205	#448	—	89	1	no	—
206	#408	—	89	1	no	—
207	C₂×#41	—	89	1	no	—
208	C₂×#41	—	89	1	no	—
209	C₂²×Q₈	—	89	1	no	—
210	C₂×Q₁₆	—	89	1	no	—
211	C₂×Q₁₆	—	89	1	no	—
212	C₂×SD₁₆	—	89	1	no	—
213	C₂×SD₁₆	—	89	1	no	—
214	C₂×M₁₆	—	89	1	no	—
215	C₂²×C₈	—	89	1	no	—
216	C₄⋊Q₈	—	178	1	no	—
217	#124	—	178	1	no	—
218	#123	—	178	1	no	—
219	#123	—	178	1	no	—
220	C₂²⋊Q₈	—	89	1	no	—
221	C₂²⋊Q₈	—	89	1	no	—
222	C₂²⋊Q₈	—	89	1	no	—
223	C₂²⋊Q₈	—	89	1	no	—
224	C₂²⋊Q₈	—	89	1	no	—
225	C₂²⋊Q₈	—	178	1	no	—
226	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
227	#121	—	89	1	no	—
228	#121	—	89	1	no	—
229	#121	—	178	1	no	—
230	#121	—	178	1	no	—
231	#121	—	178	1	no	—
232	#117	—	89	1	no	—
233	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
234	#107	C₂×D₈₉	1	1	yes	characteristic
235	#107	C₂×D₈₉	1	1	yes	characteristic
236	#106	—	89	1	no	—
237	#106	—	89	1	no	—
238	#103	—	89	1	no	—
239	#103	—	89	1	no	—
240	#103	—	89	1	no	—
241	#103	—	89	1	no	—
242	#103	C₂×D₈₉	1	1	yes	characteristic
243	#103	C₂×D₈₉	1	1	yes	characteristic
244	#102	—	89	1	no	—
245	#102	—	89	1	no	—
246	C₂²⋊C₈	—	89	1	no	—
247	C₂²⋊C₈	—	89	1	no	—
248	C₂²⋊C₈	—	89	1	no	—
249	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
250	#41	—	356	1	no	—
251	#41	—	356	1	no	—
252	#41	—	356	1	no	—
253	#41	—	356	1	no	—
254	C₂×Q₈	—	89	1	no	—
255	C₂×Q₈	—	89	1	no	—
256	C₂×Q₈	—	89	1	no	—
257	C₂×Q₈	—	356	1	no	—
258	C₂×Q₈	—	356	1	no	—
259	C₂×Q₈	—	178	1	no	—
260	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₂×D₄	—	89	1	no	—
262	C₂×D₄	—	89	1	no	—
263	C₂×D₄	—	89	1	no	—
264	C₂×D₄	—	89	1	no	—
265	C₂×D₄	—	178	1	no	—
266	C₂×D₄	—	178	1	no	—
267	C₂×D₄	—	178	1	no	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	89	1	no	—
270	C₂²×C₄	—	89	1	no	—
271	C₂²×C₄	—	178	1	no	—
272	C₂²×C₄	—	178	1	no	—
273	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
274	Q₁₆	—	356	1	no	—
275	Q₁₆	—	356	1	no	—
276	SD₁₆	—	356	1	no	—
277	SD₁₆	—	356	1	no	—
278	M₁₆	—	356	1	no	—
279	C₂×C₈	—	89	1	no	—
280	C₂×C₈	—	89	1	no	—
281	C₂×C₈	—	356	1	no	—
282	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
283	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	89	1	no	—
287	C₄⋊C₄	—	178	1	no	—
288	C₄⋊C₄	—	178	1	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₄⋊C₄	—	178	1	no	—
291	C₄⋊C₄	—	178	1	no	—
292	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
293	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
294	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
295	C₂²⋊C₄	—	2	1	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₂²⋊C₄	—	178	1	no	—
298	C₂²⋊C₄	—	178	1	no	—
299	C₂²⋊C₄	—	178	1	no	—
300	C₂²⋊C₄	—	178	1	no	—
301	C₂²⋊C₄	—	178	1	no	—
302	C₂²⋊C₄	—	178	1	no	—
303	C₂²⋊C₄	—	178	1	no	—
304	C₄²	—	178	1	no	—
305	C₂³	—	89	1	no	—
306	C₂³	—	89	1	no	—
307	C₂³	—	178	1	no	—
308	C₂³	D₄×D₈₉	1	1	yes	characteristic
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	356	1	no	—
312	Q₈	—	356	2	no	—
313	Q₈	—	178	1	no	—
314	Q₈	—	178	1	no	—
315	Q₈	—	178	1	no	—
316	Q₈	—	178	1	no	—
317	D₄	—	356	1	no	—
318	D₄	—	356	2	no	—
319	D₄	—	356	1	no	—
320	D₄	—	356	1	no	—
321	D₄	—	178	1	no	—
322	D₄	—	178	1	no	—
323	D₄	—	712	1	no	—
324	D₄	—	712	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	8	1	no	—
332	C₂×C₄	—	89	1	no	—
333	C₂×C₄	—	89	1	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	356	2	no	—
336	C₂×C₄	—	178	1	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	—	178	1	no	—
339	C₂×C₄	—	178	1	no	—
340	C₂×C₄	—	178	1	no	—
341	C₂×C₄	—	178	1	no	—
342	C₂×C₄	—	178	1	no	—
343	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
344	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	178	1	no	—
348	C₈	—	178	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	356	1	no	—
353	C₂²	—	356	1	no	—
354	C₂²	—	356	1	no	—
355	C₂²	—	178	1	no	—
356	C₂²	—	178	1	no	—
357	C₂²	—	178	1	no	—
358	C₂²	—	178	1	no	—
359	C₂²	—	178	1	no	—
360	C₂²	—	178	1	no	—
361	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	4	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	356	1	no	—
368	C₄	—	356	1	no	—
369	C₄	—	356	1	no	—
370	C₄	—	178	1	no	—
371	C₄	—	178	1	no	—
372	C₄	#15668058	1	1	yes	characteristic
373	C₄	#15668058	1	1	yes	characteristic
374	C₂	—	4	1	no	—
375	C₂	—	356	1	no	—
376	C₂	—	356	1	no	—
377	C₂	—	712	1	no	—
378	C₂	#19152159	1	1	yes	characteristic
379	C₂	#19152408	1	1	yes	characteristic
380	C₂	#19152411	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#484×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152179	C₂
hypercenter	#484	D₈₉
hypocenter	C₈₉	#3201
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
2. #19152171	C₂	semidirect
3. #19151941	C₂	semidirect
4. #19151928	C₂	semidirect
5. #19151925	C₂	semidirect
6. #19151855	C₂	semidirect
7. #19151845	C₂	semidirect
8. #19151840	C₂	semidirect
9. #19151830	C₂	semidirect
10. #19151762	C₂	semidirect
11. #19151758	C₂	semidirect
12. #19151755	C₂	semidirect
13. #19151738	C₂	semidirect
14. #19151563	C₂	semidirect
15. #484×C₈₉	C₂	semidirect
191. C₈₉	#3201	semidirect
195. #484	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152179	2
#19152171	2
#19151941	2
#19151928	2
#19151925	2
#19151855	2
#19151845	2
#19151840	2
#19151830	2
#19151762	2
#19151758	2
#19151755	2
#19151738	2
#19151563	2
#484×C₈₉	2
#3201	89