Subgroups — #26032420

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 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
2	Q₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151968	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151942	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151935	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	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151696	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151668	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151605	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#479×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×#13083557	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
23	#15668012	—	2	1	no	—
24	#15668012	C₂²	1	1	yes	characteristic
25	#15668007	—	2	1	no	—
26	#15667997	—	2	1	no	—
27	#15667992	—	2	1	no	—
28	#15667992	—	2	1	no	—
29	#15667992	C₂²	1	1	yes	characteristic
30	#15667992	C₂²	1	1	yes	characteristic
31	#15667987	—	2	1	no	—
32	#15667987	C₂²	1	1	yes	characteristic
33	#15667987	C₂²	1	1	yes	characteristic
34	C₄×D₃₅₆	—	2	1	no	—
35	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
36	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
37	#15667974	—	2	1	no	—
38	#15667970	C₂²	1	1	yes	characteristic
39	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667937	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667936	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	#15667935	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×Q₈×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	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	Q₈×D₈₉	—	2	2	no	—
64	Q₈×D₈₉	—	2	2	no	—
65	Q₈×D₈₉	—	4	1	no	—
66	Q₈×D₈₉	—	4	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	—	2	1	no	—
69	C₂×D₃₅₆	D₄	1	1	yes	characteristic
70	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
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	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
77	#13083563	—	4	1	no	—
78	#13083563	—	4	1	no	—
79	#13083561	—	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×Q₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	—	2	1	no	—
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	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂²×D₈₉	—	2	1	no	—
112	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×C₈₉	#41	1	2	yes	—
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	—	2	1	no	—
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	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	D₃₅₆	—	2	2	no	—
129	D₃₅₆	—	2	1	no	—
130	D₃₅₆	—	2	1	no	—
131	D₃₅₆	—	4	2	no	—
132	D₃₅₆	—	8	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	2	1	no	—
135	C₄×D₈₉	—	2	1	no	—
136	C₄×D₈₉	—	2	1	no	—
137	C₄×D₈₉	—	4	2	no	—
138	C₄×D₈₉	—	4	2	no	—
139	C₄×D₈₉	—	8	1	no	—
140	C₄×D₈₉	—	8	1	no	—
141	Dic₁₇₈	—	2	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	—	2	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	4	2	no	—
146	Dic₁₇₈	—	4	2	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	#10623494	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	4	1	no	—
158	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	#137	1	2	yes	—
161	C₄×C₈₉	—	2	2	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	4	1	no	—
165	C₄×C₈₉	—	4	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
168	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	2	1	no	—
171	Dic₈₉	—	4	1	no	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	—	4	1	no	—
174	C₂×C₈₉	#546	1	1	yes	characteristic
175	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
176	C₂×C₈₉	#577	1	1	yes	characteristic
177	D₈₉	—	2	2	no	—
178	D₈₉	—	4	1	no	—
179	D₈₉	—	8	1	no	—
180	#3320	—	89	1	no	maximal, Sylow
181	C₈₉	#3320	1	1	yes	characteristic, complemented, Sylow
182	#553	—	89	1	no	—
183	D₄×Q₈	—	89	1	no	—
184	#491	—	89	1	no	—
185	#485	—	89	1	no	—
186	#484	—	89	1	no	—
187	#479	D₈₉	1	1	yes	characteristic, complemented
188	#468	—	89	1	no	—
189	#465	—	89	1	no	—
190	#461	—	89	1	no	—
191	#448	—	89	1	no	—
192	#448	—	89	1	no	—
193	#440	—	89	1	no	—
194	#435	—	89	1	no	—
195	#419	—	89	1	no	—
196	C₂×#103	—	89	1	no	—
197	C₂²×Q₈	—	89	1	no	—
198	C₂²×Q₈	—	89	1	no	—
199	C₂×SD₁₆	—	89	1	no	—
200	C₂×SD₁₆	—	89	1	no	—
201	C₂×SD₁₆	—	89	1	no	—
202	C₂×M₁₆	—	89	1	no	—
203	C₂²×C₈	—	89	1	no	—
204	C₄⋊Q₈	—	89	1	no	—
205	C₄⋊Q₈	—	178	1	no	—
206	#126	—	178	1	no	—
207	#125	C₂×D₈₉	1	1	yes	characteristic
208	#123	—	178	1	no	—
209	C₂²⋊Q₈	—	89	1	no	—
210	C₂²⋊Q₈	—	89	1	no	—
211	C₂²⋊Q₈	—	178	1	no	—
212	C₂²⋊Q₈	—	178	1	no	—
213	#121	—	89	1	no	—
214	#121	—	89	1	no	—
215	#121	—	178	1	no	—
216	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
217	C₄×D₄	—	89	1	no	—
218	C₄×D₄	—	89	1	no	—
219	C₄×D₄	—	178	1	no	—
220	C₄×D₄	—	178	1	no	—
221	#117	—	89	1	no	—
222	C₂×C₄⋊C₄	—	89	1	no	—
223	#107	—	89	1	no	—
224	#107	C₂×D₈₉	1	1	yes	characteristic
225	#106	C₂×D₈₉	1	1	yes	characteristic
226	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
227	#103	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	C₂×D₈₉	1	1	yes	characteristic
235	#103	C₂×D₈₉	1	1	yes	characteristic
236	#102	—	89	1	no	—
237	C₂²⋊C₈	—	89	1	no	—
238	C₂²⋊C₈	—	89	1	no	—
239	#97	—	89	1	no	—
240	C₂×Q₈	—	89	1	no	—
241	C₂×Q₈	—	89	1	no	—
242	C₂×Q₈	—	356	1	no	—
243	C₂×Q₈	—	356	1	no	—
244	C₂×Q₈	—	178	1	no	—
245	C₂×Q₈	—	178	1	no	—
246	C₂×Q₈	—	178	2	no	—
247	C₂×Q₈	—	178	2	no	—
248	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×D₄	—	89	1	no	—
250	C₂×D₄	—	89	1	no	—
251	C₂×D₄	—	178	1	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	89	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	SD₁₆	—	356	1	no	—
260	SD₁₆	—	356	1	no	—
261	SD₁₆	—	178	2	no	—
262	M₁₆	—	356	1	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	356	1	no	—
266	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	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	—	89	1	no	—
282	C₂²⋊C₄	—	89	1	no	—
283	C₂²⋊C₄	—	178	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	C₄²	—	2	1	no	—
290	C₄²	—	178	1	no	—
291	C₄²	C₂²×D₈₉	1	1	yes	characteristic
292	C₂³	—	89	1	no	—
293	C₂³	—	89	1	no	—
294	C₂³	—	178	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	356	2	no	—
297	Q₈	—	356	2	no	—
298	Q₈	—	178	1	no	—
299	Q₈	—	178	1	no	—
300	Q₈	—	178	1	no	—
301	Q₈	—	178	1	no	—
302	Q₈	#13083575	1	2	yes	—
303	D₄	—	356	2	no	—
304	D₄	—	178	2	no	—
305	D₄	—	178	1	no	—
306	D₄	—	178	1	no	—
307	D₄	—	712	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	89	1	no	—
313	C₂×C₄	—	89	1	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	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	712	1	no	—
324	C₂×C₄	—	712	1	no	—
325	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
326	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	178	1	no	—
331	C₈	—	178	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	—	178	1	no	—
339	C₂²	—	178	1	no	—
340	C₂²	—	178	1	no	—
341	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	356	1	no	—
349	C₄	—	356	1	no	—
350	C₄	—	356	1	no	—
351	C₄	—	178	1	no	—
352	C₄	—	178	1	no	—
353	C₄	C₂×#13083575	1	1	yes	characteristic
354	C₄	#15668058	1	1	yes	characteristic
355	C₂	—	356	1	no	—
356	C₂	—	178	2	no	—
357	C₂	—	712	1	no	—
358	C₂	#19152239	1	1	yes	characteristic
359	C₂	#137×D₈₉	1	1	yes	characteristic
360	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	#479×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152301	C₂
hypercenter	#479	D₈₉
hypocenter	C₈₉	#3320
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
2. Q₈×D₃₅₆	C₂	semidirect
3. #19151968	C₂	semidirect
4. #19151942	C₂	semidirect
5. #19151935	C₂	semidirect
6. #19151845	C₂	semidirect
7. #19151828	C₂	semidirect
8. #19151804	C₂	semidirect
9. #19151738	C₂	semidirect
10. #19151738	C₂	semidirect
11. #19151696	C₂	semidirect
12. #19151668	C₂	semidirect
13. #19151605	C₂	semidirect
14. #103×D₈₉	C₂	semidirect
15. #479×C₈₉	C₂	semidirect
181. C₈₉	#3320	semidirect
187. #479	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
Q₈×D₃₅₆	2
#19151968	2
#19151942	2
#19151935	2
#19151845	2
#19151828	2
#19151804	2
#19151738	2
#19151696	2
#19151668	2
#19151605	2
#103×D₈₉	2
#479×C₈₉	2
#3320	89