Subgroups — #26031690

Order 11392.

370 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152204	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152199	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151901	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151897	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151852	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151850	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151835	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151833	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151794	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151793	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151792	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151792	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151623	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
17	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
18	C₂×#13083559	C₂²	1	1	yes	characteristic
19	C₂×#13083559	C₂²	1	1	yes	characteristic
20	C₂×#13083557	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	#15668009	C₂²	1	1	yes	characteristic
23	#15668008	C₂²	1	1	yes	characteristic
24	#15668003	C₂²	1	1	yes	characteristic
25	#15667997	—	2	1	no	—
26	#15667997	—	2	1	no	—
27	#15667995	—	2	1	no	—
28	#15667994	—	2	1	no	—
29	#15667987	—	2	1	no	—
30	#15667987	—	2	1	no	—
31	#15667987	C₂²	1	1	yes	characteristic
32	#15667987	C₂²	1	1	yes	characteristic
33	#15667987	C₂²	1	1	yes	characteristic
34	#15667986	—	2	1	no	—
35	#15667985	—	2	1	no	—
36	#15667983	C₂²	1	1	yes	characteristic
37	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
38	#15667974	C₂²	1	1	yes	characteristic
39	#15667974	C₂²	1	1	yes	characteristic
40	#15667969	C₂²	1	1	yes	characteristic
41	#15667945	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	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	#15667932	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667932	C₂²	1	1	yes	characteristic
51	#15667932	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×D₃₅₆	—	2	1	no	—
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	—	2	1	no	—
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₂²⋊D₈₉	—	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	D₇₁₂	—	4	1	no	—
75	D₇₁₂	—	4	1	no	—
76	#13083559	—	4	1	no	—
77	#13083559	—	4	1	no	—
78	#13083557	—	4	1	no	—
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₂×#10623494	C₂³	1	1	yes	characteristic
81	C₈×D₈₉	—	4	1	no	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	#13083553	—	2	1	no	—
85	#13083553	D₄	1	1	yes	characteristic
86	#13083553	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	D₄	1	1	yes	characteristic
95	#13083551	D₄	1	1	yes	characteristic
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	1	no	—
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	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	1	no	—
111	C₂²×D₈₉	—	2	1	no	—
112	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
113	D₄×C₈₉	—	2	1	no	—
114	D₄×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	—	8	1	no	—
119	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	8	1	no	—
123	C₂²⋊D₈₉	—	8	1	no	—
124	C₂²⋊D₈₉	—	8	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	D₃₅₆	—	2	1	no	—
130	D₃₅₆	—	2	1	no	—
131	D₃₅₆	—	2	1	no	—
132	D₃₅₆	—	2	1	no	—
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	8	1	no	—
136	D₃₅₆	—	8	1	no	—
137	D₃₅₆	—	8	1	no	—
138	D₃₅₆	—	8	1	no	—
139	D₃₅₆	—	8	1	no	—
140	C₄×D₈₉	—	2	1	no	—
141	C₄×D₈₉	—	2	1	no	—
142	C₄×D₈₉	—	8	1	no	—
143	C₄×D₈₉	—	8	1	no	—
144	C₄×D₈₉	—	8	1	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	#10623494	—	2	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₈₉	—	2	1	no	—
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	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	—	4	1	no	—
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₂×D₈₉	—	4	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
171	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	2	1	no	—
175	Dic₈₉	—	4	1	no	—
176	C₂×C₈₉	—	8	1	no	—
177	C₂×C₈₉	#537	1	1	yes	characteristic
178	C₂×C₈₉	#576	1	1	yes	characteristic
179	C₂×C₈₉	#576	1	1	yes	characteristic
180	D₈₉	—	4	1	no	—
181	D₈₉	—	8	1	no	—
182	D₈₉	—	8	1	no	—
183	D₈₉	—	8	1	no	—
184	#3259	—	89	1	no	maximal, Sylow
185	C₈₉	#3259	1	1	yes	characteristic, complemented, Sylow
186	#540	—	89	1	no	—
187	#540	—	89	1	no	—
188	#478	—	89	1	no	—
189	#478	—	89	1	no	—
190	#478	—	89	1	no	—
191	#478	D₈₉	1	1	yes	characteristic, complemented
192	#469	—	89	1	no	—
193	#469	—	89	1	no	—
194	#466	—	89	1	no	—
195	#466	—	89	1	no	—
196	#459	—	89	1	no	—
197	#459	—	89	1	no	—
198	#459	—	89	1	no	—
199	#459	—	89	1	no	—
200	#424	—	89	1	no	—
201	C₂×D₈	—	89	1	no	—
202	C₂×D₈	—	89	1	no	—
203	C₂×D₈	—	89	1	no	—
204	C₂×D₈	—	89	1	no	—
205	C₂×M₁₆	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	#127	—	89	1	no	—
208	#127	—	89	1	no	—
209	#125	—	89	1	no	—
210	#125	C₂×D₈₉	1	1	yes	characteristic
211	#123	—	178	1	no	—
212	#123	—	178	1	no	—
213	#123	—	178	1	no	—
214	#123	—	178	1	no	—
215	#121	—	89	1	no	—
216	#121	—	89	1	no	—
217	#121	—	89	1	no	—
218	#121	—	89	1	no	—
219	#121	—	178	1	no	—
220	#121	—	178	1	no	—
221	#121	—	178	1	no	—
222	#121	—	178	1	no	—
223	C₄×D₄	—	89	1	no	—
224	C₄×D₄	—	89	1	no	—
225	C₄×D₄	—	89	1	no	—
226	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
227	#117	—	89	1	no	—
228	#107	—	89	1	no	—
229	#107	C₂×D₈₉	1	1	yes	characteristic
230	#106	—	89	1	no	—
231	#106	C₂×D₈₉	1	1	yes	characteristic
232	C₄⋊C₈	—	89	1	no	—
233	C₄⋊C₈	—	89	1	no	—
234	C₄⋊C₈	—	89	1	no	—
235	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
236	#102	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	—	89	1	no	—
240	#102	—	89	1	no	—
241	#102	—	89	1	no	—
242	#102	C₂×D₈₉	1	1	yes	characteristic
243	#102	C₂×D₈₉	1	1	yes	characteristic
244	C₂×D₄	—	89	1	no	—
245	C₂×D₄	—	89	1	no	—
246	C₂×D₄	—	89	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂×D₄	—	178	1	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂×D₄	—	178	1	no	—
254	C₂×D₄	—	178	1	no	—
255	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	89	1	no	—
258	C₂²×C₄	—	178	1	no	—
259	C₂²×C₄	—	178	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	D₈	—	356	1	no	—
262	D₈	—	356	1	no	—
263	D₈	—	356	1	no	—
264	D₈	—	356	1	no	—
265	M₁₆	—	356	1	no	—
266	C₂×C₈	—	89	1	no	—
267	C₂×C₈	—	89	1	no	—
268	C₂×C₈	—	356	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	—	2	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	C₂²×D₈₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
280	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	89	1	no	—
283	C₂²⋊C₄	—	89	1	no	—
284	C₂²⋊C₄	—	178	1	no	—
285	C₂²⋊C₄	—	178	1	no	—
286	C₂²⋊C₄	—	178	1	no	—
287	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₄²	—	89	1	no	—
292	C₄²	C₂²×D₈₉	1	1	yes	characteristic
293	C₂³	—	2	1	no	—
294	C₂³	—	89	1	no	—
295	C₂³	—	178	1	no	—
296	C₂³	—	178	1	no	—
297	C₂³	—	178	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	178	1	no	—
301	D₄	—	178	1	no	—
302	D₄	—	178	1	no	—
303	D₄	—	178	1	no	—
304	D₄	—	178	1	no	—
305	D₄	—	178	1	no	—
306	D₄	—	712	1	no	—
307	D₄	—	712	1	no	—
308	D₄	—	712	1	no	—
309	D₄	—	712	1	no	—
310	D₄	—	712	1	no	—
311	D₄	—	712	1	no	—
312	D₄	—	712	1	no	—
313	D₄	—	712	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	8	1	no	—
318	C₂×C₄	—	89	1	no	—
319	C₂×C₄	—	89	1	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	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	—	2	1	no	—
332	C₈	—	178	1	no	—
333	C₈	—	178	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	356	1	no	—
338	C₂²	—	356	1	no	—
339	C₂²	—	356	1	no	—
340	C₂²	—	356	1	no	—
341	C₂²	—	356	1	no	—
342	C₂²	—	356	1	no	—
343	C₂²	—	356	1	no	—
344	C₂²	—	356	1	no	—
345	C₂²	—	356	1	no	—
346	C₂²	—	178	1	no	—
347	C₂²	—	178	1	no	—
348	C₂²	—	178	1	no	—
349	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	356	1	no	—
356	C₄	—	178	1	no	—
357	C₄	—	178	1	no	—
358	C₄	—	178	1	no	—
359	C₄	#15668058	1	1	yes	characteristic
360	C₄	#15668060	1	1	yes	characteristic
361	C₂	—	8	1	no	—
362	C₂	—	356	1	no	—
363	C₂	—	712	1	no	—
364	C₂	—	712	1	no	—
365	C₂	—	712	1	no	—
366	C₂	#19152178	1	1	yes	characteristic
367	C₂	#19152400	1	1	yes	characteristic
368	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	#478×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#478	D₈₉
hypocenter	C₈₉	#3259
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #19152204	C₂	semidirect
2. #19152199	C₂	semidirect
3. #19151901	C₂	semidirect
4. #19151897	C₂	semidirect
5. #19151895	C₂	semidirect
6. #19151852	C₂	semidirect
7. #19151850	C₂	semidirect
8. #19151835	C₂	semidirect
9. #19151833	C₂	semidirect
10. #19151794	C₂	semidirect
11. #19151793	C₂	semidirect
12. #19151792	C₂	semidirect
13. #19151792	C₂	semidirect
14. #19151623	C₂	semidirect
15. #478×C₈₉	C₂	semidirect
185. C₈₉	#3259	semidirect
191. #478	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152204	2
#19152199	2
#19151901	2
#19151897	2
#19151895	2
#19151852	2
#19151850	2
#19151835	2
#19151833	2
#19151794	2
#19151793	2
#19151792	2
#19151623	2
#478×C₈₉	2
#3259	89