Subgroups — #26029124

Order 11392.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668033	C₂	1	1	yes	maximal, characteristic
2	#19152080	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152077	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151699	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151697	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151690	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13083559	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151678	C₂	1	1	yes	maximal, characteristic, complemented
11	#107×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151598	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151596	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151555	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083573	C₂²	1	1	yes	characteristic
18	#15668033	C₄	1	8	yes	complemented
19	C₂×#13083564	C₂²	1	1	yes	characteristic
20	C₂×#13083563	C₂²	1	1	yes	characteristic
21	C₂×#13083559	C₂²	1	1	yes	characteristic
22	C₂×#13083557	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
26	#15667977	C₂²	1	1	yes	characteristic
27	C₄×D₃₅₆	—	2	1	no	—
28	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	#15667974	—	2	1	no	—
31	#15667974	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₉	—	2	1	no	—
33	#15667972	—	2	1	no	—
34	#15667969	—	2	1	no	—
35	#15667969	C₂²	1	1	yes	characteristic
36	#15667968	—	2	1	no	—
37	C₂×#13083553	—	2	1	no	—
38	C₄⋊C₄×D₈₉	—	2	1	no	—
39	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
41	#15667946	C₂²	1	1	yes	characteristic
42	#15667945	C₂²	1	1	yes	characteristic
43	#15667945	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₄×#10623494	C₂²	1	1	yes	characteristic
52	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
56	#107×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	#97×C₈₉	C₂²	1	1	yes	characteristic
60	#13083575	—	2	2	no	—
61	#13083575	C₂×C₄	1	4	yes	complemented
62	#13083573	—	2	2	no	—
63	#13083573	C₂×C₄	1	4	yes	complemented
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	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	—	2	1	no	—
74	C₂×C₄×D₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
76	#13083564	C₂×C₄	1	4	yes	complemented
77	#13083563	C₂×C₄	1	4	yes	complemented
78	#13083559	C₂×C₄	1	4	yes	complemented
79	#13083557	—	2	2	no	—
80	C₂×#10623494	C₂³	1	1	yes	characteristic
81	C₂×#10623494	C₂³	1	1	yes	characteristic
82	C₈×D₈₉	C₂×C₄	1	4	yes	complemented
83	C₄⋊Dic₈₉	—	2	1	no	—
84	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	—	2	1	no	—
90	#13083553	D₄	1	1	yes	characteristic
91	#13083553	D₄	1	1	yes	characteristic
92	C₂²⋊Dic₈₉	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	D₄	1	1	yes	characteristic
96	#13083551	D₄	1	1	yes	characteristic
97	C₄×Dic₈₉	—	2	1	no	—
98	C₄×Dic₈₉	—	2	1	no	—
99	C₄×Dic₈₉	—	2	1	no	—
100	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	SD₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	C₂³	1	1	yes	characteristic
113	C₂³×C₈₉	—	2	1	no	—
114	C₂²×D₈₉	—	2	1	no	—
115	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	C₂²×C₄	1	2	yes	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	C₂²×C₄	1	2	yes	—
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	—	8	1	no	—
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₈₉	—	2	4	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	4	no	—
130	C₂×Dic₈₉	—	2	1	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	8	1	no	—
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	D₃₅₆	—	2	4	no	—
137	D₃₅₆	—	2	1	no	—
138	D₃₅₆	C₂²×C₄	1	2	yes	—
139	C₄×D₈₉	—	2	4	no	—
140	C₄×D₈₉	—	8	1	no	—
141	C₄×D₈₉	—	8	1	no	—
142	C₄×D₈₉	—	8	1	no	—
143	C₄×D₈₉	—	8	1	no	—
144	C₄×D₈₉	C₂²×C₄	1	2	yes	—
145	C₄×D₈₉	#41	1	2	yes	—
146	Dic₁₇₈	—	2	1	no	—
147	Dic₁₇₈	C₂²×C₄	1	2	yes	—
148	C₈×C₈₉	—	2	1	no	—
149	C₈×C₈₉	C₂²×C₄	1	2	yes	—
150	#10623494	—	2	1	no	—
151	#10623494	C₂²×C₄	1	2	yes	—
152	C₂²×C₈₉	—	2	2	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	2	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	C₄×D₄	1	2	yes	—
162	C₄×C₈₉	—	2	2	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	4	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
168	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	—	2	1	no	—
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	—	4	1	no	—
174	Dic₈₉	C₄×D₄	1	2	yes	—
175	C₂×C₈₉	—	4	2	no	—
176	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₈₉	#576	1	1	yes	characteristic
178	C₂×C₈₉	#578	1	1	yes	characteristic
179	D₈₉	—	2	2	no	—
180	D₈₉	—	4	2	no	—
181	#2951	—	89	1	no	maximal, Sylow
182	C₈₉	#2951	1	1	yes	characteristic, complemented, Sylow
183	C₂×#135	—	89	1	no	—
184	#520	—	89	1	no	—
185	#520	—	89	1	no	—
186	#440	—	89	1	no	—
187	#440	—	89	1	no	—
188	#440	—	89	1	no	—
189	#440	D₈₉	1	1	yes	characteristic, complemented
190	C₄×Q₁₆	—	89	1	no	—
191	C₄×Q₁₆	—	89	1	no	—
192	C₄×D₈	—	89	1	no	—
193	C₄×D₈	—	89	1	no	—
194	C₂×#107	—	89	1	no	—
195	#417	—	89	1	no	—
196	#417	—	89	1	no	—
197	#405	—	89	1	no	—
198	C₂×#41	—	89	1	no	—
199	C₂×#41	—	89	1	no	—
200	#135	—	89	8	no	—
201	C₂×Q₁₆	—	89	1	no	—
202	C₂×SD₁₆	—	89	1	no	—
203	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
204	C₂×D₈	—	89	1	no	—
205	C₂×M₁₆	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×Q₈	—	89	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
211	C₄×D₄	—	89	1	no	—
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	—	89	1	no	—
214	C₄×D₄	—	178	1	no	—
215	C₄×D₄	—	178	1	no	—
216	C₄×D₄	—	178	1	no	—
217	C₄×D₄	—	178	1	no	—
218	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
219	#117	—	89	1	no	—
220	#117	—	178	1	no	—
221	#117	—	178	1	no	—
222	C₂×C₄⋊C₄	—	89	1	no	—
223	C₂×C₄⋊C₄	—	89	1	no	—
224	C₂×C₄⋊C₄	—	178	1	no	—
225	C₂×C₄⋊C₄	—	178	1	no	—
226	#107	—	89	1	no	—
227	#107	—	89	1	no	—
228	#107	—	89	1	no	—
229	#107	C₂×D₈₉	1	1	yes	characteristic
230	#103	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	C₂×D₈₉	1	1	yes	characteristic
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	#102	C₂×D₈₉	1	1	yes	characteristic
238	#97	—	89	1	no	—
239	#97	C₂×D₈₉	1	1	yes	characteristic
240	C₄×C₈	—	89	1	no	—
241	C₄×C₈	—	89	1	no	—
242	#41	—	89	4	no	—
243	#41	—	89	4	no	—
244	#41	—	178	2	no	—
245	#41	—	178	2	no	—
246	C₂×Q₈	—	89	1	no	—
247	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
248	C₂×D₄	—	89	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	C₂²×C₄	—	178	1	no	—
259	C₂²×C₄	—	178	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	Q₁₆	—	89	4	no	—
262	SD₁₆	—	89	4	no	—
263	SD₁₆	C₄×D₈₉	1	4	yes	complemented
264	D₈	—	89	4	no	—
265	M₁₆	—	178	2	no	—
266	C₂×C₈	—	89	1	no	—
267	C₂×C₈	—	89	4	no	—
268	C₂×C₈	—	89	1	no	—
269	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	89	1	no	—
274	C₄⋊C₄	—	89	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₄⋊C₄	—	178	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	C₂²×D₈₉	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	89	1	no	—
285	C₂²⋊C₄	—	89	1	no	—
286	C₂²⋊C₄	—	178	1	no	—
287	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	89	1	no	—
291	C₄²	—	178	1	no	—
292	C₄²	—	178	1	no	—
293	C₄²	—	178	1	no	—
294	C₄²	C₂²×D₈₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	89	1	no	—
297	C₂³	—	178	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	89	2	no	—
300	Q₈	—	178	1	no	—
301	Q₈	C₂×C₄×D₈₉	1	2	yes	—
302	D₄	—	2	1	no	—
303	D₄	—	89	2	no	—
304	D₄	—	178	4	no	—
305	D₄	—	178	4	no	—
306	D₄	—	178	1	no	—
307	D₄	C₂×C₄×D₈₉	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	89	1	no	—
313	C₂×C₄	—	89	2	no	—
314	C₂×C₄	—	89	2	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	89	1	no	—
317	C₂×C₄	—	178	4	no	—
318	C₂×C₄	—	178	4	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	712	1	no	—
323	C₂×C₄	—	712	1	no	—
324	C₂×C₄	—	712	1	no	—
325	C₂×C₄	—	712	1	no	—
326	C₂×C₄	—	712	1	no	—
327	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
328	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	89	2	no	—
332	C₈	—	178	1	no	—
333	C₈	C₂×C₄×D₈₉	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	89	2	no	—
338	C₂²	—	356	1	no	—
339	C₂²	—	356	1	no	—
340	C₂²	—	178	2	no	—
341	C₂²	—	178	1	no	—
342	C₂²	—	178	1	no	—
343	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	89	2	no	—
350	C₄	—	356	1	no	—
351	C₄	—	356	1	no	—
352	C₄	—	178	1	no	—
353	C₄	—	178	1	no	—
354	C₄	—	178	2	no	—
355	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
356	C₄	#41×D₈₉	1	1	yes	characteristic
357	C₂	—	4	2	no	—
358	C₂	—	356	2	no	—
359	C₂	—	178	2	no	—
360	C₂	C₄×D₄×D₈₉	1	1	yes	characteristic
361	C₂	#19152402	1	1	yes	characteristic
362	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#440×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#15668033	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#440	D₈₉
hypocenter	C₈₉	#2951
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #19152080	C₂	semidirect
3. #19152077	C₂	semidirect
4. #19151700	C₂	semidirect
5. #19151699	C₂	semidirect
6. #19151697	C₂	semidirect
7. C₄×#13083564	C₂	semidirect
8. #19151690	C₂	semidirect
9. C₄×#13083559	C₂	semidirect
10. #19151678	C₂	semidirect
11. #107×D₈₉	C₂	semidirect
12. #19151598	C₂	semidirect
13. #19151596	C₂	semidirect
14. #19151555	C₂	semidirect
15. #440×C₈₉	C₂	semidirect
18. #15668033	C₄	semidirect
61. #13083575	C₂×C₄	semidirect
63. #13083573	C₂×C₄	semidirect
76. #13083564	C₂×C₄	semidirect
77. #13083563	C₂×C₄	semidirect
78. #13083559	C₂×C₄	semidirect
82. C₈×D₈₉	C₂×C₄	semidirect
104. SD₁₆×C₈₉	C₂×C₄	semidirect
182. C₈₉	#2951	semidirect
189. #440	D₈₉	semidirect
263. SD₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668033	2
#19152080	2
#19152077	2
#19151700	2
#19151699	2
#19151697	2
C₄×#13083564	2
#19151690	2
C₄×#13083559	2
#19151678	2
#107×D₈₉	2
#19151598	2
#19151596	2
#19151555	2
#440×C₈₉	2
#2951	89