Subgroups — #26032594

Order 11392.

420 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152249	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152248	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152002	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151913	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151912	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151910	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151877	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151876	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151833	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151832	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151678	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151661	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#15667945	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#480×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	—	2	1	no	—
17	C₂×#13083573	—	2	1	no	—
18	C₂×#13083559	C₂²	1	1	yes	characteristic
19	C₂²×#10623494	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
21	#15668011	C₂²	1	1	yes	characteristic
22	#15668011	C₂²	1	1	yes	characteristic
23	#15667997	—	2	1	no	—
24	#15667995	—	2	1	no	—
25	#15667994	—	2	1	no	—
26	#15667993	—	2	1	no	—
27	#15667990	—	2	1	no	—
28	#15667989	—	2	1	no	—
29	#15667987	C₂²	1	1	yes	characteristic
30	#15667986	C₂²	1	1	yes	characteristic
31	#15667983	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	#15667974	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	#15667946	C₂²	1	1	yes	characteristic
39	#15667946	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667945	C₂²	1	1	yes	characteristic
42	#15667945	C₂²	1	1	yes	characteristic
43	#15667945	C₂²	1	1	yes	characteristic
44	#15667945	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667934	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	#15667924	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	#121×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	#102×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	4	no	—
60	#13083573	—	4	4	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
74	#13083559	—	2	2	no	—
75	C₂×#10623494	—	2	2	no	—
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₈×D₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	D₄	1	1	yes	characteristic
95	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₉	—	2	1	no	—
97	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₂³×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
111	D₄×C₈₉	—	2	1	no	—
112	D₄×C₈₉	—	2	1	no	—
113	D₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	4	2	no	—
123	C₂²⋊D₈₉	—	4	2	no	—
124	C₂²⋊D₈₉	D₈	1	4	yes	complemented
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	4	2	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	4	2	no	—
139	C₄×D₈₉	—	4	2	no	—
140	C₄×D₈₉	—	4	2	no	—
141	Dic₁₇₈	—	4	2	no	—
142	Dic₁₇₈	—	4	2	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	—	2	1	no	—
145	#10623494	—	2	1	no	—
146	#10623494	#41	1	2	yes	complemented
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	C₂×D₈	1	2	yes	—
153	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₂×D₈₉	—	4	1	no	—
158	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	C₂×D₈	1	2	yes	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₈₉	—	2	1	no	—
167	Dic₈₉	—	2	1	no	—
168	Dic₈₉	—	4	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	C₂×D₈	1	2	yes	—
172	C₂×C₈₉	—	2	2	no	—
173	C₂×C₈₉	—	8	1	no	—
174	C₂×C₈₉	#547	1	1	yes	characteristic
175	C₂×C₈₉	C₂²×D₈	1	1	yes	characteristic
176	C₂×C₈₉	#578	1	1	yes	characteristic
177	D₈₉	—	2	2	no	—
178	D₈₉	—	8	1	no	—
179	#3334	—	89	1	no	maximal, Sylow
180	C₈₉	#3334	1	1	yes	characteristic, complemented, Sylow
181	#547	—	89	1	no	—
182	#547	—	89	1	no	—
183	#500	—	89	1	no	—
184	#480	—	89	1	no	—
185	#480	—	89	1	no	—
186	#480	—	89	1	no	—
187	#480	D₈₉	1	1	yes	characteristic, complemented
188	#474	—	89	1	no	—
189	#474	—	89	1	no	—
190	#466	—	89	1	no	—
191	#466	—	89	1	no	—
192	C₄×D₈	—	89	1	no	—
193	C₈×D₄	—	89	1	no	—
194	C₂×#107	—	89	1	no	—
195	C₂×#107	—	89	1	no	—
196	C₂×#41	—	178	1	no	—
197	C₂×#41	—	178	1	no	—
198	C₂×D₈	—	89	1	no	—
199	C₂²×C₈	—	89	1	no	—
200	C₂²×C₈	—	89	1	no	—
201	C₄⋊Q₈	—	89	1	no	—
202	C₄⋊Q₈	—	89	1	no	—
203	#123	—	178	1	no	—
204	#123	—	178	1	no	—
205	#123	—	178	1	no	—
206	#123	—	178	1	no	—
207	C₂²⋊Q₈	—	178	1	no	—
208	C₂²⋊Q₈	—	178	1	no	—
209	#121	—	89	1	no	—
210	#121	—	89	1	no	—
211	#121	—	89	1	no	—
212	#121	C₂×D₈₉	1	1	yes	characteristic
213	C₄×D₄	—	89	1	no	—
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	89	1	no	—
216	C₂×C₄⋊C₄	—	89	1	no	—
217	C₂×C₄⋊C₄	—	89	1	no	—
218	C₂×C₄⋊C₄	—	89	1	no	—
219	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
220	#107	—	89	1	no	—
221	#107	—	89	1	no	—
222	#107	—	89	1	no	—
223	#107	—	89	1	no	—
224	#107	—	89	1	no	—
225	#107	—	89	1	no	—
226	#107	—	89	1	no	—
227	#107	C₂×D₈₉	1	1	yes	characteristic
228	#107	C₂×D₈₉	1	1	yes	characteristic
229	C₄⋊C₈	—	89	1	no	—
230	#102	—	89	1	no	—
231	#102	—	89	1	no	—
232	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#102	C₂×D₈₉	1	1	yes	characteristic
235	#102	C₂×D₈₉	1	1	yes	characteristic
236	C₂²⋊C₈	—	89	1	no	—
237	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
238	C₄×C₈	—	89	1	no	—
239	#41	—	356	4	no	—
240	#41	—	356	4	no	—
241	C₂×Q₈	—	178	1	no	—
242	C₂×Q₈	—	178	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	89	1	no	—
245	C₂×D₄	—	89	1	no	—
246	C₂×D₄	—	178	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	89	1	no	—
252	C₂²×C₄	—	178	1	no	—
253	C₂²×C₄	—	178	1	no	—
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
258	D₈	—	178	2	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	89	1	no	—
267	C₄⋊C₄	—	89	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	89	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	89	1	no	—
280	C₂²⋊C₄	—	89	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₂²⋊C₄	—	178	1	no	—
284	C₂²⋊C₄	—	178	1	no	—
285	C₂²⋊C₄	—	178	1	no	—
286	C₄²	—	89	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	89	1	no	—
289	C₂³	—	178	1	no	—
290	C₂³	D₄×D₈₉	1	1	yes	characteristic
291	Q₈	—	356	2	no	—
292	Q₈	—	356	2	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	4	2	no	—
296	D₄	—	89	4	no	—
297	D₄	—	356	2	no	—
298	D₄	—	356	2	no	—
299	D₄	—	356	2	no	—
300	D₄	—	178	1	no	—
301	D₄	—	178	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	89	1	no	—
309	C₂×C₄	—	89	1	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	356	2	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	356	2	no	—
316	C₂×C₄	—	178	1	no	—
317	C₂×C₄	—	178	1	no	—
318	C₂×C₄	—	178	1	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	89	2	no	—
326	C₈	—	178	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	89	2	no	—
333	C₂²	—	356	1	no	—
334	C₂²	—	356	1	no	—
335	C₂²	—	356	1	no	—
336	C₂²	—	178	1	no	—
337	C₂²	—	178	1	no	—
338	C₂²	D₈×D₈₉	1	2	yes	—
339	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	89	2	no	—
345	C₄	—	356	1	no	—
346	C₄	—	356	1	no	—
347	C₄	—	356	1	no	—
348	C₄	—	178	1	no	—
349	C₄	—	178	1	no	—
350	C₄	#41×D₈₉	1	1	yes	characteristic
351	C₄	#15668059	1	1	yes	characteristic
352	C₂	—	2	2	no	—
353	C₂	—	8	1	no	—
354	C₂	—	178	2	no	—
355	C₂	—	712	1	no	—
356	C₂	#19152246	1	1	yes	characteristic
357	C₂	C₂×D₈×D₈₉	1	1	yes	characteristic
358	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#480×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#480	D₈₉
hypocenter	C₈₉	#3334
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #480

Complements

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

N	H ≅ G/N	type
1. #19152249	C₂	semidirect
2. #19152248	C₂	semidirect
3. #19152002	C₂	semidirect
4. #19151913	C₂	semidirect
5. #19151912	C₂	semidirect
6. #19151910	C₂	semidirect
7. #19151877	C₂	semidirect
8. #19151876	C₂	semidirect
9. #19151833	C₂	semidirect
10. #19151832	C₂	semidirect
11. #19151678	C₂	semidirect
12. #19151661	C₂	semidirect
13. C₂×#15667945	C₂	semidirect
14. #107×D₈₉	C₂	semidirect
15. #480×C₈₉	C₂	semidirect
124. C₂²⋊D₈₉	D₈	semidirect
146. #10623494	#41	semidirect
180. C₈₉	#3334	semidirect
187. #480	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152249	2
#19152248	2
#19152002	2
#19151913	2
#19151912	2
#19151910	2
#19151877	2
#19151876	2
#19151833	2
#19151832	2
#19151678	2
#19151661	2
C₂×#15667945	2
#107×D₈₉	2
#480×C₈₉	2
#3334	89