Subgroups — #26031138

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	#19152178	C₂	1	1	yes	maximal, characteristic
2	#19152172	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151936	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151932	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151927	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151923	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151853	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151847	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151836	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151830	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151759	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151759	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151737	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151563	C₂	1	1	yes	maximal, characteristic, complemented
15	#451×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083574	C₂²	1	1	yes	characteristic
18	C₂×#13083561	C₂²	1	1	yes	characteristic
19	C₂×#13083561	C₂²	1	1	yes	characteristic
20	C₂×#13083557	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	#15668010	—	2	1	no	—
23	#15668001	—	2	1	no	—
24	#15667995	—	2	1	no	—
25	#15667995	—	2	1	no	—
26	#15667990	—	2	1	no	—
27	#15667988	C₂²	1	1	yes	characteristic
28	#15667988	C₂²	1	1	yes	characteristic
29	#15667988	C₂²	1	1	yes	characteristic
30	#15667988	C₂²	1	1	yes	characteristic
31	#15667988	C₂²	1	1	yes	characteristic
32	#15667987	—	2	1	no	—
33	#15667987	C₂²	1	1	yes	characteristic
34	#15667986	C₂²	1	1	yes	characteristic
35	#15667985	—	2	1	no	—
36	#15667985	—	2	1	no	—
37	#15667971	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
39	#15667946	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667944	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667936	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667934	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×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×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	#13083575	—	4	1	no	—
60	#13083575	—	4	1	no	—
61	#13083574	—	4	1	no	—
62	#13083574	—	4	1	no	—
63	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	D₄	1	1	yes	characteristic
67	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
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	#13083561	—	4	1	no	—
76	#13083561	—	4	1	no	—
77	#13083557	—	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	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
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	—	2	1	no	—
97	#13083551	—	2	1	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	—	4	1	no	—
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
107	Q₁₆×C₈₉	—	4	1	no	—
108	Q₁₆×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
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	Q₈×C₈₉	—	2	1	no	—
121	Q₈×C₈₉	—	2	1	no	—
122	Q₈×C₈₉	—	2	1	no	—
123	Q₈×C₈₉	—	2	1	no	—
124	Q₈×C₈₉	—	4	2	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	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₈₉	—	4	1	no	—
135	C₂²⋊D₈₉	—	4	1	no	—
136	C₂²⋊D₈₉	—	8	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	2	1	no	—
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	—	2	1	no	—
144	D₃₅₆	—	4	1	no	—
145	D₃₅₆	—	4	2	no	—
146	D₃₅₆	—	8	1	no	—
147	C₄×D₈₉	—	2	1	no	—
148	C₄×D₈₉	—	2	1	no	—
149	C₄×D₈₉	—	2	1	no	—
150	C₄×D₈₉	—	2	1	no	—
151	C₄×D₈₉	—	4	2	no	—
152	C₄×D₈₉	—	8	1	no	—
153	Dic₁₇₈	—	4	1	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	—	89	1	no	—
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	D₈₉	1	1	yes	characteristic, complemented
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	C₂×D₈₉	1	1	yes	characteristic
210	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
211	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
234	#107	—	89	1	no	—
235	#107	—	89	1	no	—
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	—	2	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	89	1	no	—
258	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
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	—	4	1	no	—
275	Q₁₆	—	4	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	—	89	1	no	—
286	C₄⋊C₄	—	89	1	no	—
287	C₄⋊C₄	—	89	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	—	178	1	no	—
293	C₄⋊C₄	—	178	1	no	—
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	—	2	1	no	—
312	Q₈	—	2	1	no	—
313	Q₈	—	2	1	no	—
314	Q₈	—	2	1	no	—
315	Q₈	—	4	2	no	—
316	Q₈	—	356	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	—	4	2	no	—
332	C₂×C₄	—	89	1	no	—
333	C₂×C₄	—	89	1	no	—
334	C₂×C₄	—	356	2	no	—
335	C₂×C₄	—	178	1	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	—	712	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	#15668057	1	1	yes	characteristic
373	C₄	#15668057	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	#19152158	1	1	yes	characteristic
379	C₂	#19152407	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	#451×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152178	C₂
hypercenter	#451	D₈₉
hypocenter	C₈₉	#3201
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
2. #19152172	C₂	semidirect
3. #19151936	C₂	semidirect
4. #19151932	C₂	semidirect
5. #19151927	C₂	semidirect
6. #19151923	C₂	semidirect
7. #19151853	C₂	semidirect
8. #19151847	C₂	semidirect
9. #19151836	C₂	semidirect
10. #19151830	C₂	semidirect
11. #19151759	C₂	semidirect
12. #19151759	C₂	semidirect
13. #19151737	C₂	semidirect
14. #19151563	C₂	semidirect
15. #451×C₈₉	C₂	semidirect
191. C₈₉	#3201	semidirect
204. #451	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152178	2
#19152172	2
#19151936	2
#19151932	2
#19151927	2
#19151923	2
#19151853	2
#19151847	2
#19151836	2
#19151830	2
#19151759	2
#19151737	2
#19151563	2
#451×C₈₉	2
#3201	89