Subgroups — #26032225

Order 11392.

394 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	#19152301	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152252	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151968	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151930	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151845	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151828	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151804	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151744	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151744	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151708	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151668	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151596	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151591	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	C₂²	1	1	yes	characteristic
17	C₂×#13083573	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	C₂×#13083557	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
23	#15668012	C₂²	1	1	yes	characteristic
24	#15668007	—	2	1	no	—
25	#15667997	—	2	1	no	—
26	#15667996	—	2	1	no	—
27	#15667996	—	2	1	no	—
28	#15667992	C₂²	1	1	yes	characteristic
29	#15667992	C₂²	1	1	yes	characteristic
30	#15667987	—	2	1	no	—
31	#15667987	C₂²	1	1	yes	characteristic
32	#15667987	C₂²	1	1	yes	characteristic
33	#15667986	—	2	1	no	—
34	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
35	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
36	#15667974	—	2	1	no	—
37	#15667970	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊Dic₈₉	—	2	1	no	—
39	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
40	#15667943	C₂²	1	1	yes	characteristic
41	#15667937	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667935	C₂²	1	1	yes	characteristic
44	#15667935	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667923	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	#13083573	—	2	2	no	—
60	#13083573	—	2	2	no	—
61	#13083573	—	4	1	no	—
62	#13083573	—	4	1	no	—
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	D₄	1	1	yes	characteristic
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊D₈₉	—	2	1	no	—
70	C₂×C₂²⋊D₈₉	—	2	1	no	—
71	C₂²×Dic₈₉	—	2	1	no	—
72	C₂²×Dic₈₉	—	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	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
77	#13083563	—	4	1	no	—
78	#13083563	—	4	1	no	—
79	#13083559	—	2	2	no	—
80	#13083557	—	4	1	no	—
81	C₂×#10623494	C₂³	1	1	yes	characteristic
82	C₂×#10623494	C₂³	1	1	yes	characteristic
83	C₈×D₈₉	—	4	1	no	—
84	C₄⋊Dic₈₉	—	2	1	no	—
85	C₄⋊Dic₈₉	—	2	1	no	—
86	C₄⋊Dic₈₉	—	2	1	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
89	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
90	#13083553	—	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	D₄	1	1	yes	characteristic
98	#13083551	D₄	1	1	yes	characteristic
99	C₄×Dic₈₉	—	2	1	no	—
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂³×C₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
115	D₄×C₈₉	—	2	1	no	—
116	D₄×C₈₉	#41	1	2	yes	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₈₉	—	4	2	no	—
125	C₂²⋊D₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	—	4	2	no	—
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	1	no	—
136	D₃₅₆	—	4	2	no	—
137	D₃₅₆	—	8	1	no	—
138	C₄×D₈₉	—	2	1	no	—
139	C₄×D₈₉	—	2	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	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	2	1	no	—
146	Dic₁₇₈	—	2	1	no	—
147	Dic₁₇₈	—	2	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	C₈×C₈₉	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	#10623494	—	2	1	no	—
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	1	no	—
159	C₂×D₈₉	—	2	1	no	—
160	C₂×D₈₉	—	2	1	no	—
161	C₂×D₈₉	—	2	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	C₂×#41	1	1	yes	characteristic
171	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	C₂×C₈₉	—	4	2	no	—
178	C₂×C₈₉	#546	1	1	yes	characteristic
179	C₂×C₈₉	#575	1	1	yes	characteristic
180	C₂×C₈₉	#577	1	1	yes	characteristic
181	D₈₉	—	4	1	no	—
182	D₈₉	—	4	1	no	—
183	D₈₉	—	8	1	no	—
184	#3307	—	89	1	no	maximal, Sylow
185	C₈₉	#3307	1	1	yes	characteristic, complemented, Sylow
186	#553	—	89	1	no	—
187	#547	—	89	1	no	—
188	#491	—	89	1	no	—
189	#485	—	89	1	no	—
190	#483	—	89	1	no	—
191	#478	D₈₉	1	1	yes	characteristic, complemented
192	#468	—	89	1	no	—
193	#465	—	89	1	no	—
194	#461	—	89	1	no	—
195	#449	—	89	1	no	—
196	#449	—	89	1	no	—
197	#442	—	89	1	no	—
198	#435	—	89	1	no	—
199	#417	—	89	1	no	—
200	#416	—	89	1	no	—
201	C₂×#41	—	89	1	no	—
202	C₂×#41	—	89	1	no	—
203	C₂×SD₁₆	—	89	1	no	—
204	C₂×SD₁₆	—	89	1	no	—
205	C₂×D₈	—	89	1	no	—
206	C₂×M₁₆	—	89	1	no	—
207	C₂²×C₈	—	89	1	no	—
208	C₄⋊Q₈	—	89	1	no	—
209	#126	—	178	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	C₂²⋊Q₈	—	89	1	no	—
215	C₂²⋊Q₈	—	89	1	no	—
216	#121	—	89	1	no	—
217	#121	—	89	1	no	—
218	#121	—	178	1	no	—
219	#121	—	178	1	no	—
220	C₄×D₄	—	89	1	no	—
221	C₄×D₄	—	89	1	no	—
222	C₄×D₄	—	178	1	no	—
223	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
224	#117	—	89	1	no	—
225	C₂×C₄⋊C₄	—	89	1	no	—
226	C₂×C₄⋊C₄	—	178	1	no	—
227	#107	C₂×D₈₉	1	1	yes	characteristic
228	#106	—	89	1	no	—
229	#106	C₂×D₈₉	1	1	yes	characteristic
230	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
231	#103	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	—	89	1	no	—
235	#103	—	89	1	no	—
236	#103	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	C₂×D₈₉	1	1	yes	characteristic
240	#102	C₂×D₈₉	1	1	yes	characteristic
241	C₂²⋊C₈	—	89	1	no	—
242	C₂²⋊C₈	—	89	1	no	—
243	#97	—	89	1	no	—
244	#41	—	356	1	no	—
245	#41	—	356	1	no	—
246	#41	—	178	2	no	—
247	#41	—	178	2	no	—
248	C₂×Q₈	—	89	1	no	—
249	C₂×Q₈	—	89	1	no	—
250	C₂×D₄	—	89	1	no	—
251	C₂×D₄	—	89	1	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂×D₄	—	178	1	no	—
254	C₂×D₄	—	178	1	no	—
255	C₂×D₄	—	178	1	no	—
256	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	89	1	no	—
259	C₂²×C₄	—	89	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	—	178	1	no	—
263	C₂²×C₄	—	178	1	no	—
264	SD₁₆	—	356	1	no	—
265	SD₁₆	—	356	1	no	—
266	D₈	—	178	2	no	—
267	M₁₆	—	356	1	no	—
268	C₂×C₈	—	89	1	no	—
269	C₂×C₈	—	89	1	no	—
270	C₂×C₈	—	356	1	no	—
271	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₄⋊C₄	—	89	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	—	178	1	no	—
282	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
284	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	89	1	no	—
287	C₂²⋊C₄	—	89	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₄²	—	178	1	no	—
295	C₄²	C₂²×D₈₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	89	1	no	—
298	C₂³	—	89	1	no	—
299	C₂³	—	178	1	no	—
300	Q₈	—	178	1	no	—
301	Q₈	—	178	1	no	—
302	Q₈	—	178	1	no	—
303	Q₈	—	178	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	356	1	no	—
306	D₄	—	356	2	no	—
307	D₄	—	356	2	no	—
308	D₄	—	356	2	no	—
309	D₄	—	178	1	no	—
310	D₄	—	178	1	no	—
311	D₄	—	712	1	no	—
312	D₄	#13083575	1	2	yes	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	89	1	no	—
318	C₂×C₄	—	89	1	no	—
319	C₂×C₄	—	356	2	no	—
320	C₂×C₄	—	356	2	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	178	1	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	—	712	1	no	—
329	C₂×C₄	—	712	1	no	—
330	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
331	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	178	1	no	—
336	C₈	—	178	1	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	356	1	no	—
341	C₂²	—	356	1	no	—
342	C₂²	—	356	1	no	—
343	C₂²	—	178	1	no	—
344	C₂²	—	178	1	no	—
345	C₂²	—	178	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	—	356	1	no	—
357	C₄	—	356	1	no	—
358	C₄	—	178	1	no	—
359	C₄	—	178	1	no	—
360	C₄	C₂×#13083575	1	1	yes	characteristic
361	C₄	#15668058	1	1	yes	characteristic
362	C₂	—	4	2	no	—
363	C₂	—	356	1	no	—
364	C₂	—	356	1	no	—
365	C₂	—	712	1	no	—
366	C₂	#19152239	1	1	yes	characteristic
367	C₂	#19152394	1	1	yes	characteristic
368	C₂	#19152405	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	#3307
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. #19152301	C₂	semidirect
2. #19152252	C₂	semidirect
3. #19151968	C₂	semidirect
4. #19151941	C₂	semidirect
5. #19151930	C₂	semidirect
6. #19151845	C₂	semidirect
7. #19151828	C₂	semidirect
8. #19151804	C₂	semidirect
9. #19151744	C₂	semidirect
10. #19151744	C₂	semidirect
11. #19151708	C₂	semidirect
12. #19151668	C₂	semidirect
13. #19151596	C₂	semidirect
14. #19151591	C₂	semidirect
15. #478×C₈₉	C₂	semidirect
185. C₈₉	#3307	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
#19152301	2
#19152252	2
#19151968	2
#19151941	2
#19151930	2
#19151845	2
#19151828	2
#19151804	2
#19151744	2
#19151708	2
#19151668	2
#19151596	2
#19151591	2
#478×C₈₉	2
#3307	89