Subgroups — #26032097

Order 11392.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152295	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152245	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151951	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151935	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151919	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151901	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151826	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151811	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151741	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151730	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151678	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151595	C₂	1	1	yes	maximal, characteristic, complemented
14	#102×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#466×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13083559	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
21	#15668005	—	2	1	no	—
22	#15668003	C₂²	1	1	yes	characteristic, complemented
23	#15667999	C₂²	1	1	yes	characteristic
24	#15667994	—	2	1	no	—
25	#15667993	—	2	1	no	—
26	#15667992	C₂²	1	1	yes	characteristic
27	#15667983	—	2	1	no	—
28	#15667983	—	2	1	no	—
29	#15667983	C₂²	1	1	yes	characteristic
30	#15667980	—	2	1	no	—
31	D₄×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
32	#15667974	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
34	#15667972	—	2	1	no	—
35	#15667971	C₂²	1	1	yes	characteristic, complemented
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×D₈₉	—	2	1	no	—
38	#15667945	C₂²	1	1	yes	characteristic, complemented
39	#15667943	C₂²	1	1	yes	characteristic, complemented
40	#15667941	C₂²	1	1	yes	characteristic, complemented
41	#15667937	C₂²	1	1	yes	characteristic, complemented
42	#15667937	C₂²	1	1	yes	characteristic, complemented
43	#15667935	C₂²	1	1	yes	characteristic, complemented
44	#15667934	C₂²	1	1	yes	characteristic
45	#15667934	C₂²	1	1	yes	characteristic
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic, complemented
50	#15667924	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
54	#121×C₈₉	C₂²	1	1	yes	characteristic
55	#121×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	#107×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₈₉	—	2	1	no	—
60	#13083574	—	2	2	no	—
61	#13083574	—	4	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
68	D₄×D₈₉	—	2	2	no	—
69	D₄×D₈₉	—	4	1	no	—
70	C₂²×Dic₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
75	#13083562	—	4	1	no	—
76	#13083559	—	2	2	no	—
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₈×D₈₉	—	4	1	no	—
80	C₄⋊Dic₈₉	—	2	1	no	—
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	C₂²⋊Dic₈₉	—	2	1	no	—
92	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	D₄	1	1	yes	characteristic
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₈₉	—	2	1	no	—
99	C₂×D₄×C₈₉	—	2	1	no	—
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
103	D₈×C₈₉	—	4	1	no	—
104	C₂×C₈×C₈₉	—	4	1	no	—
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₉	—	2	1	no	—
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₂³×C₈₉	—	2	1	no	—
112	C₂³×C₈₉	—	2	1	no	—
113	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₈₉	—	2	1	no	—
115	C₂²×D₈₉	—	4	2	no	—
116	C₂²×D₈₉	—	4	2	no	—
117	C₂²×D₈₉	—	8	1	no	—
118	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
119	D₄×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	2	1	no	—
121	D₄×C₈₉	—	2	1	no	—
122	D₄×C₈₉	—	2	1	no	—
123	D₄×C₈₉	—	8	1	no	—
124	D₄×C₈₉	—	8	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₈₉	—	2	2	no	—
132	C₂²⋊D₈₉	—	4	2	no	—
133	C₂²⋊D₈₉	—	4	2	no	—
134	C₂²⋊D₈₉	—	4	2	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	8	1	no	—
139	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
142	D₃₅₆	—	2	1	no	—
143	D₃₅₆	#41	1	2	yes	—
144	C₄×D₈₉	—	2	1	no	—
145	C₄×D₈₉	—	2	1	no	—
146	C₄×D₈₉	—	4	2	no	—
147	C₄×D₈₉	—	8	1	no	—
148	C₄×D₈₉	—	8	1	no	—
149	Dic₁₇₈	—	2	1	no	—
150	Dic₁₇₈	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	—	2	1	no	—
153	#10623494	—	2	1	no	—
154	#10623494	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	1	no	—
163	C₂²×C₈₉	—	4	1	no	—
164	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×D₈₉	—	2	1	no	—
166	C₂×D₈₉	—	2	1	no	—
167	C₂×D₈₉	—	2	2	no	—
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	4	1	no	—
170	C₂×D₈₉	—	8	4	no	—
171	C₂×D₈₉	#135	1	2	yes	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	4	1	no	—
174	C₄×C₈₉	—	4	1	no	—
175	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
176	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
177	Dic₈₉	—	2	1	no	—
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	—	8	1	no	—
185	C₂×C₈₉	—	8	1	no	—
186	C₂×C₈₉	#546	1	1	yes	characteristic
187	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
188	C₂×C₈₉	#576	1	1	yes	characteristic
189	D₈₉	—	2	2	no	—
190	D₈₉	—	4	2	no	—
191	#3297	—	89	1	no	maximal, Sylow
192	C₈₉	#3297	1	1	yes	characteristic, complemented, Sylow
193	#553	—	89	1	no	—
194	#546	—	89	1	no	—
195	#488	—	89	1	no	—
196	#484	—	89	1	no	—
197	#482	—	89	1	no	—
198	#478	—	89	1	no	—
199	#466	D₈₉	1	1	yes	characteristic, complemented
200	#465	—	89	1	no	—
201	#463	—	89	1	no	—
202	#449	—	89	1	no	—
203	#447	—	89	1	no	—
204	C₄×D₈	—	89	1	no	—
205	C₈×D₄	—	89	1	no	—
206	#416	—	89	1	no	—
207	C₂×#102	—	89	1	no	—
208	C₂×#41	—	89	1	no	—
209	C₂²×D₄	—	89	1	no	—
210	C₂×SD₁₆	—	89	1	no	—
211	C₂×D₈	—	89	1	no	—
212	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
213	C₂²×C₈	—	89	1	no	—
214	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
215	#126	—	178	1	no	—
216	#125	—	89	1	no	—
217	#124	—	89	1	no	—
218	#123	—	178	1	no	—
219	#123	—	178	1	no	—
220	C₂²⋊Q₈	—	89	1	no	—
221	#121	—	89	1	no	—
222	#121	—	178	1	no	—
223	#121	—	178	1	no	—
224	#121	C₂×D₈₉	1	1	yes	characteristic
225	#121	C₂×D₈₉	1	1	yes	characteristic, complemented
226	#120	—	178	1	no	—
227	C₄×D₄	—	89	1	no	—
228	C₄×D₄	—	89	1	no	—
229	C₄×D₄	—	89	1	no	—
230	C₄×D₄	—	178	1	no	—
231	#117	—	89	1	no	—
232	C₂×C₄⋊C₄	—	89	1	no	—
233	C₂×C₂²⋊C₄	—	178	1	no	—
234	#107	—	89	1	no	—
235	#107	C₂×D₈₉	1	1	yes	characteristic, complemented
236	#106	—	89	1	no	—
237	C₄⋊C₈	—	89	1	no	—
238	#103	—	89	1	no	—
239	#103	—	89	1	no	—
240	#103	—	89	1	no	—
241	#102	—	89	1	no	—
242	#102	—	89	1	no	—
243	#102	—	89	1	no	—
244	#102	—	89	1	no	—
245	#102	—	89	1	no	—
246	#102	C₂×D₈₉	1	1	yes	characteristic
247	#102	C₂×D₈₉	1	1	yes	characteristic, complemented
248	C₂²⋊C₈	—	89	1	no	—
249	C₂²⋊C₈	—	89	1	no	—
250	C₄×C₈	—	89	1	no	—
251	C₂⁴	—	178	1	no	—
252	#41	—	356	1	no	—
253	#41	—	178	2	no	—
254	C₂×Q₈	—	89	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	89	1	no	—
258	C₂×D₄	—	89	1	no	—
259	C₂×D₄	—	356	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	—	178	2	no	—
262	C₂×D₄	—	178	1	no	—
263	C₂×D₄	—	178	1	no	—
264	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
266	C₂²×C₄	—	89	1	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	—	178	1	no	—
269	C₂²×C₄	—	178	1	no	—
270	C₂²×C₄	—	178	1	no	—
271	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
272	SD₁₆	—	356	1	no	—
273	D₈	—	4	1	no	—
274	D₈	—	178	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	89	1	no	—
277	C₂×C₈	—	89	1	no	—
278	C₂×C₈	—	356	1	no	—
279	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	C₄⋊C₄	—	89	1	no	—
282	C₄⋊C₄	—	89	1	no	—
283	C₄⋊C₄	—	89	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	—	178	1	no	—
287	C₄⋊C₄	—	178	1	no	—
288	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
289	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
290	C₂²⋊C₄	—	2	1	no	—
291	C₂²⋊C₄	—	2	1	no	—
292	C₂²⋊C₄	—	89	1	no	—
293	C₂²⋊C₄	—	89	1	no	—
294	C₂²⋊C₄	—	178	1	no	—
295	C₂²⋊C₄	—	178	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	C₄²	—	89	1	no	—
302	C₄²	—	178	1	no	—
303	C₂³	—	2	1	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	89	1	no	—
306	C₂³	—	356	2	no	—
307	C₂³	—	356	2	no	—
308	C₂³	—	178	1	no	—
309	C₂³	—	712	1	no	—
310	C₂³	D₄×D₈₉	1	1	yes	characteristic
311	Q₈	—	178	1	no	—
312	Q₈	—	178	1	no	—
313	D₄	—	2	1	no	—
314	D₄	—	2	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	8	1	no	—
318	D₄	—	8	1	no	—
319	D₄	—	89	2	no	—
320	D₄	—	356	2	no	—
321	D₄	—	356	2	no	—
322	D₄	—	356	2	no	—
323	D₄	—	178	2	no	—
324	D₄	—	178	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	—	89	1	no	—
330	C₂×C₄	—	89	1	no	—
331	C₂×C₄	—	89	1	no	—
332	C₂×C₄	—	356	2	no	—
333	C₂×C₄	—	178	1	no	—
334	C₂×C₄	—	178	1	no	—
335	C₂×C₄	—	178	1	no	—
336	C₂×C₄	—	178	1	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	—	712	1	no	—
339	C₂×C₄	—	712	1	no	—
340	C₂×C₄	—	712	1	no	—
341	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
342	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	178	1	no	—
346	C₈	—	178	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	89	2	no	—
357	C₂²	—	356	1	no	—
358	C₂²	—	356	1	no	—
359	C₂²	—	178	2	no	—
360	C₂²	—	178	1	no	—
361	C₂²	—	178	1	no	—
362	C₂²	—	712	4	no	—
363	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
364	C₄	—	2	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	—	178	1	no	—
373	C₄	#41×D₈₉	1	1	yes	characteristic
374	C₄	#15668057	1	1	yes	characteristic
375	C₂	—	4	1	no	—
376	C₂	—	8	1	no	—
377	C₂	—	8	1	no	—
378	C₂	—	356	2	no	—
379	C₂	—	178	2	no	—
380	C₂	#19152237	1	1	yes	characteristic
381	C₂	#135×D₈₉	1	1	yes	characteristic
382	C₂	#19152399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#466×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#466	D₈₉
hypocenter	C₈₉	#3297
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

Complements

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

N	H ≅ G/N	type
1. #19152295	C₂	semidirect
2. #19152245	C₂	semidirect
3. #19151951	C₂	semidirect
4. #19151935	C₂	semidirect
5. #19151919	C₂	semidirect
6. #19151901	C₂	semidirect
7. #19151826	C₂	semidirect
8. #19151811	C₂	semidirect
9. #19151741	C₂	semidirect
10. #19151730	C₂	semidirect
11. #19151678	C₂	semidirect
12. C₈×C₂²⋊D₈₉	C₂	semidirect
13. #19151595	C₂	semidirect
14. #102×D₈₉	C₂	semidirect
15. #466×C₈₉	C₂	semidirect
18. C₂×#13083562	C₂²	semidirect
22. #15668003	C₂²	semidirect
31. D₄×Dic₈₉	C₂²	semidirect
35. #15667971	C₂²	semidirect
38. #15667945	C₂²	semidirect
39. #15667943	C₂²	semidirect
40. #15667941	C₂²	semidirect
41. #15667937	C₂²	semidirect
42. #15667937	C₂²	semidirect
43. #15667935	C₂²	semidirect
49. C₂²⋊#10623494	C₂²	semidirect
51. C₈×Dic₈₉	C₂²	semidirect
53. C₂²×C₈×C₈₉	C₂²	semidirect
55. #121×C₈₉	C₂²	semidirect
56. #107×C₈₉	C₂²	semidirect
58. #102×C₈₉	C₂²	semidirect
192. C₈₉	#3297	semidirect
199. #466	D₈₉	semidirect
214. C₂²×C₈	C₂×D₈₉	semidirect
225. #121	C₂×D₈₉	semidirect
235. #107	C₂×D₈₉	semidirect
247. #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
#19152295	2
#19152245	2
#19151951	2
#19151935	2
#19151919	2
#19151901	2
#19151826	2
#19151811	2
#19151741	2
#19151730	2
#19151678	2
C₈×C₂²⋊D₈₉	2
#19151595	2
#102×D₈₉	2
#466×C₈₉	2
#3297	89