Subgroups — #26032854

Order 11392.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152296	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152292	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151998	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151924	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151922	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151913	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151911	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151901	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151897	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151852	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151851	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151680	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151674	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151639	C₂	1	1	yes	maximal, characteristic, complemented
15	#428×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
17	C₂×#13083557	C₂²	1	1	yes	characteristic
18	#15668007	—	2	1	no	—
19	#15668006	—	2	1	no	—
20	#15668003	C₂²	1	1	yes	characteristic
21	#15668003	C₂²	1	1	yes	characteristic
22	#15667997	—	2	1	no	—
23	#15667994	—	2	1	no	—
24	#15667987	C₂²	1	1	yes	characteristic
25	#15667987	C₂²	1	1	yes	characteristic
26	#15667986	—	2	1	no	—
27	#15667986	C₂²	1	1	yes	characteristic
28	#15667986	C₂²	1	1	yes	characteristic
29	#15667985	—	2	1	no	—
30	C₄×D₃₅₆	—	2	1	no	—
31	#15667974	C₂²	1	1	yes	characteristic
32	#15667974	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	—	2	1	no	—
34	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
35	#15667970	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667934	C₂²	1	1	yes	characteristic
44	#15667934	C₂²	1	1	yes	characteristic
45	#15667932	C₂²	1	1	yes	characteristic
46	#15667932	C₂²	1	1	yes	characteristic
47	#15667932	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	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	#117×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#107×C₈₉	C₂²	1	1	yes	characteristic
57	#106×C₈₉	C₂²	1	1	yes	characteristic
58	#106×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×D₃₅₆	—	2	1	no	—
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
70	D₇₁₂	—	2	2	no	—
71	#13083557	—	2	2	no	—
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	—	2	1	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
83	#13083551	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	D₄	1	1	yes	characteristic
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
95	M₁₆×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₄²×C₈₉	—	2	1	no	—
105	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₈₉	—	2	1	no	—
107	C₂²×D₈₉	—	2	1	no	—
108	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	8	1	no	—
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₈₉	—	4	1	no	—
119	C₂²⋊D₈₉	—	4	2	no	—
120	C₂²⋊D₈₉	—	8	1	no	—
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
126	D₃₅₆	—	2	1	no	—
127	D₃₅₆	—	2	1	no	—
128	D₃₅₆	—	2	1	no	—
129	D₃₅₆	—	2	1	no	—
130	D₃₅₆	—	4	2	no	—
131	D₃₅₆	—	8	1	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	8	1	no	—
135	C₄×D₈₉	—	8	1	no	—
136	C₄×D₈₉	—	8	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	#41	1	2	yes	—
139	#10623494	—	2	1	no	—
140	#10623494	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₈₉	—	2	1	no	—
146	C₂×D₈₉	—	2	1	no	—
147	C₂×D₈₉	—	2	1	no	—
148	C₂×D₈₉	—	4	1	no	—
149	C₂×D₈₉	—	4	1	no	—
150	C₂×D₈₉	—	4	1	no	—
151	C₂×D₈₉	—	4	1	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	—	4	1	no	—
157	C₄×C₈₉	—	4	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	—	4	1	no	—
166	C₂×C₈₉	—	4	1	no	—
167	C₂×C₈₉	#547	1	1	yes	characteristic
168	C₂×C₈₉	#575	1	1	yes	characteristic
169	C₂×C₈₉	#576	1	1	yes	characteristic
170	D₈₉	—	4	1	no	—
171	D₈₉	—	8	1	no	—
172	D₈₉	—	8	1	no	—
173	#3355	—	89	1	no	maximal, Sylow
174	C₈₉	#3355	1	1	yes	characteristic, complemented, Sylow
175	#553	—	89	1	no	—
176	#553	—	89	1	no	—
177	#499	—	89	1	no	—
178	#482	—	89	1	no	—
179	#482	—	89	1	no	—
180	#480	—	89	1	no	—
181	#480	—	89	1	no	—
182	#478	—	89	1	no	—
183	#478	—	89	1	no	—
184	#469	—	89	1	no	—
185	#469	—	89	1	no	—
186	C₄×D₈	—	89	1	no	—
187	#436	—	89	1	no	—
188	#428	—	89	1	no	—
189	#428	D₈₉	1	1	yes	characteristic, complemented
190	C₂×D₈	—	89	1	no	—
191	C₂×M₁₆	—	89	1	no	—
192	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
193	#126	—	178	1	no	—
194	#126	—	178	1	no	—
195	#125	—	89	1	no	—
196	#125	—	89	1	no	—
197	#123	—	178	1	no	—
198	#123	—	178	1	no	—
199	#121	—	89	1	no	—
200	#121	—	89	1	no	—
201	#121	—	89	1	no	—
202	#121	—	89	1	no	—
203	#121	—	178	1	no	—
204	#121	—	178	1	no	—
205	C₄×D₄	—	89	1	no	—
206	C₄×D₄	—	89	1	no	—
207	C₄×D₄	—	89	1	no	—
208	C₄×D₄	—	178	1	no	—
209	C₄×D₄	—	178	1	no	—
210	#117	—	89	1	no	—
211	#117	C₂×D₈₉	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	89	1	no	—
213	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
214	#107	—	89	1	no	—
215	#107	—	89	1	no	—
216	#107	—	89	1	no	—
217	#107	C₂×D₈₉	1	1	yes	characteristic
218	#107	C₂×D₈₉	1	1	yes	characteristic
219	#106	—	89	1	no	—
220	#106	—	89	1	no	—
221	#106	C₂×D₈₉	1	1	yes	characteristic
222	#106	C₂×D₈₉	1	1	yes	characteristic
223	C₄⋊C₈	—	89	1	no	—
224	#102	—	89	1	no	—
225	#102	—	89	1	no	—
226	#102	—	89	1	no	—
227	#102	—	89	1	no	—
228	#102	—	89	1	no	—
229	#102	—	89	1	no	—
230	C₂²⋊C₈	—	89	1	no	—
231	C₂²⋊C₈	—	89	1	no	—
232	C₄×C₈	—	89	1	no	—
233	C₂×D₄	—	89	1	no	—
234	C₂×D₄	—	89	1	no	—
235	C₂×D₄	—	89	1	no	—
236	C₂×D₄	—	178	1	no	—
237	C₂×D₄	—	178	1	no	—
238	C₂×D₄	—	178	1	no	—
239	C₂×D₄	—	178	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	89	1	no	—
242	C₂²×C₄	—	178	1	no	—
243	C₂²×C₄	—	178	1	no	—
244	C₂²×C₄	—	178	1	no	—
245	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
246	D₈	—	178	2	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	178	2	no	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	—	89	1	no	—
251	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	89	1	no	—
255	C₄⋊C₄	—	89	1	no	—
256	C₄⋊C₄	—	89	1	no	—
257	C₄⋊C₄	—	178	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	—	178	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	89	1	no	—
268	C₂²⋊C₄	—	89	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₂²⋊C₄	—	178	1	no	—
271	C₂²⋊C₄	—	178	1	no	—
272	C₂²⋊C₄	—	178	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	178	1	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	89	1	no	—
278	C₄²	—	178	1	no	—
279	C₂³	—	89	1	no	—
280	C₂³	—	178	1	no	—
281	C₂³	—	178	1	no	—
282	C₂³	D₄×D₈₉	1	1	yes	characteristic
283	D₄	—	356	2	no	—
284	D₄	—	356	2	no	—
285	D₄	—	356	1	no	—
286	D₄	—	178	1	no	—
287	D₄	—	178	1	no	—
288	D₄	—	178	1	no	—
289	D₄	—	178	1	no	—
290	D₄	—	712	1	no	—
291	D₄	—	712	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	712	1	no	—
307	C₂×C₄	—	712	1	no	—
308	C₂×C₄	—	712	1	no	—
309	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	178	1	no	—
313	C₈	—	178	1	no	—
314	C₈	#13083574	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	356	1	no	—
319	C₂²	—	356	1	no	—
320	C₂²	—	356	1	no	—
321	C₂²	—	356	1	no	—
322	C₂²	—	356	1	no	—
323	C₂²	—	356	1	no	—
324	C₂²	—	178	1	no	—
325	C₂²	—	178	1	no	—
326	C₂²	—	178	1	no	—
327	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	356	1	no	—
334	C₄	—	356	1	no	—
335	C₄	—	178	1	no	—
336	C₄	—	178	1	no	—
337	C₄	—	178	1	no	—
338	C₄	C₂×#13083574	1	1	yes	characteristic
339	C₄	#15668060	1	1	yes	characteristic
340	C₂	—	4	1	no	—
341	C₂	—	356	1	no	—
342	C₂	—	712	1	no	—
343	C₂	—	712	1	no	—
344	C₂	#19152249	1	1	yes	characteristic
345	C₂	#19152397	1	1	yes	characteristic
346	C₂	#19152402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#428×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#428	D₈₉
hypocenter	C₈₉	#3355
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
1. #19152296	C₂	semidirect
2. #19152292	C₂	semidirect
3. #19151998	C₂	semidirect
4. #19151924	C₂	semidirect
5. #19151922	C₂	semidirect
6. #19151913	C₂	semidirect
7. #19151911	C₂	semidirect
8. #19151901	C₂	semidirect
9. #19151897	C₂	semidirect
10. #19151852	C₂	semidirect
11. #19151851	C₂	semidirect
12. #19151680	C₂	semidirect
13. #19151674	C₂	semidirect
14. #19151639	C₂	semidirect
15. #428×C₈₉	C₂	semidirect
174. C₈₉	#3355	semidirect
189. #428	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152296	2
#19152292	2
#19151998	2
#19151924	2
#19151922	2
#19151913	2
#19151911	2
#19151901	2
#19151897	2
#19151852	2
#19151851	2
#19151680	2
#19151674	2
#19151639	2
#428×C₈₉	2
#3355	89