Subgroups — #26031331

Order 11392.

364 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152208	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152172	C₂	1	1	yes	maximal, characteristic
3	#19151991	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151976	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151965	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151954	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151823	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151822	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151762	C₂	1	1	yes	maximal, characteristic
12	#19151758	C₂	1	1	yes	maximal, characteristic
13	#19151754	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151656	C₂	1	1	yes	maximal, characteristic, complemented
15	#451×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083564	C₂²	1	1	yes	characteristic
20	C₂×#13083564	C₂²	1	1	yes	characteristic
21	C₂×#13083563	C₂²	1	1	yes	characteristic
22	C₂×#13083561	C₂²	1	1	yes	characteristic
23	#15668010	C₂²	1	1	yes	characteristic
24	#15668010	C₂²	1	1	yes	characteristic
25	#15668010	C₂²	1	1	yes	characteristic
26	#15668005	—	2	1	no	—
27	#15668005	—	2	1	no	—
28	#15668003	—	2	1	no	—
29	#15668003	C₂²	1	1	yes	characteristic
30	#15668002	—	2	1	no	—
31	#15668001	C₂²	1	1	yes	characteristic
32	#15668001	C₂²	1	1	yes	characteristic
33	#15667995	—	2	1	no	—
34	#15667995	—	2	1	no	—
35	#15667991	C₂²	1	1	yes	characteristic
36	#15667988	—	2	1	no	—
37	#15667988	—	2	1	no	—
38	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
39	#15667977	C₂²	1	1	yes	characteristic
40	#15667971	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	1	no	—
60	#13083574	—	4	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	Dic₃₅₆	—	4	1	no	—
67	#13083564	—	4	1	no	—
68	#13083564	—	4	1	no	—
69	#13083564	—	4	1	no	—
70	#13083563	—	4	1	no	—
71	#13083561	—	4	1	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	D₄	1	1	yes	characteristic
77	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
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	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
98	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	—	4	1	no	—
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
107	Q₁₆×C₈₉	—	4	1	no	—
108	Q₁₆×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂²⋊C₄×C₈₉	—	2	1	no	—
114	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	—	2	1	no	—
118	Q₈×C₈₉	—	2	1	no	—
119	Q₈×C₈₉	—	2	1	no	—
120	Q₈×C₈₉	—	2	1	no	—
121	Q₈×C₈₉	—	2	1	no	—
122	Q₈×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₈₉	—	4	2	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
140	D₃₅₆	—	2	1	no	—
141	D₃₅₆	—	2	1	no	—
142	C₄×D₈₉	—	4	2	no	—
143	Dic₁₇₈	—	2	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	2	1	no	—
146	Dic₁₇₈	—	2	1	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	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	—	2	1	no	—
154	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₂×D₈₉	—	4	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
164	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₈₉	—	2	1	no	—
166	Dic₈₉	—	2	1	no	—
167	Dic₈₉	—	2	1	no	—
168	Dic₈₉	—	2	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	—	4	1	no	—
172	C₂×C₈₉	—	4	1	no	—
173	C₂×C₈₉	#535	1	1	yes	characteristic
174	C₂×C₈₉	#578	1	1	yes	characteristic
175	C₂×C₈₉	#578	1	1	yes	characteristic
176	D₈₉	—	8	1	no	—
177	#3223	—	89	1	no	maximal, Sylow
178	C₈₉	#3223	1	1	yes	characteristic, complemented, Sylow
179	#541	—	89	1	no	—
180	#536	—	89	1	no	—
181	#497	—	89	1	no	—
182	#494	—	89	1	no	—
183	#491	—	89	1	no	—
184	#488	—	89	1	no	—
185	#464	—	89	1	no	—
186	#464	—	89	1	no	—
187	#462	—	89	1	no	—
188	#462	—	89	1	no	—
189	#452	—	89	1	no	—
190	#452	—	89	1	no	—
191	#451	—	89	1	no	—
192	#451	D₈₉	1	1	yes	characteristic, complemented
193	#433	—	89	1	no	—
194	C₂×#41	—	89	1	no	—
195	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
196	C₂×Q₁₆	—	89	1	no	—
197	C₂×Q₁₆	—	89	1	no	—
198	C₂×Q₁₆	—	89	1	no	—
199	C₂×Q₁₆	—	89	1	no	—
200	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
201	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
202	C₂×SD₁₆	—	89	1	no	—
203	C₂×SD₁₆	—	89	1	no	—
204	C₄⋊Q₈	—	89	1	no	—
205	C₄⋊Q₈	—	89	1	no	—
206	C₄⋊Q₈	—	89	1	no	—
207	#126	—	178	1	no	—
208	#126	—	178	1	no	—
209	#125	—	89	1	no	—
210	#125	—	178	1	no	—
211	#125	—	178	1	no	—
212	#124	—	89	1	no	—
213	#124	—	89	1	no	—
214	#123	—	178	1	no	—
215	#123	—	178	1	no	—
216	C₂²⋊Q₈	—	89	1	no	—
217	C₂²⋊Q₈	—	178	1	no	—
218	C₂²⋊Q₈	—	178	1	no	—
219	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
220	C₄×Q₈	—	89	1	no	—
221	C₄×Q₈	—	89	1	no	—
222	#117	—	89	1	no	—
223	C₄⋊C₈	—	89	1	no	—
224	C₄⋊C₈	—	89	1	no	—
225	#103	—	89	1	no	—
226	#103	—	89	1	no	—
227	#103	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	C₂×D₈₉	1	1	yes	characteristic
230	#103	C₂×D₈₉	1	1	yes	characteristic
231	#102	—	89	1	no	—
232	#102	—	89	1	no	—
233	C₂²⋊C₈	—	89	1	no	—
234	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
235	#97	—	89	1	no	—
236	C₄×C₈	—	89	1	no	—
237	#41	—	356	1	no	—
238	#41	—	356	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	89	1	no	—
243	C₂×Q₈	—	89	1	no	—
244	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×D₄	—	89	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
252	Q₁₆	—	4	1	no	—
253	Q₁₆	—	4	1	no	—
254	Q₁₆	—	356	1	no	—
255	Q₁₆	—	356	1	no	—
256	Q₁₆	—	356	1	no	—
257	Q₁₆	—	356	1	no	—
258	SD₁₆	—	356	1	no	—
259	SD₁₆	—	356	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	89	1	no	—
262	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	89	1	no	—
267	C₄⋊C₄	—	89	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	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	—	178	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
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	C₄²	—	89	1	no	—
289	C₄²	—	89	1	no	—
290	C₄²	—	178	1	no	—
291	C₄²	—	178	1	no	—
292	C₂³	—	178	1	no	—
293	C₂³	D₄×D₈₉	1	1	yes	characteristic
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	178	1	no	—
302	Q₈	—	178	1	no	—
303	Q₈	—	178	1	no	—
304	Q₈	—	178	1	no	—
305	D₄	—	356	2	no	—
306	D₄	—	178	1	no	—
307	D₄	—	178	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	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	89	1	no	—
317	C₂×C₄	—	89	1	no	—
318	C₂×C₄	—	89	1	no	—
319	C₂×C₄	—	356	2	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	178	1	no	—
328	C₈	—	178	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	356	1	no	—
333	C₂²	—	356	1	no	—
334	C₂²	—	356	1	no	—
335	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	356	1	no	—
342	C₄	—	356	1	no	—
343	C₄	—	356	1	no	—
344	C₄	—	178	1	no	—
345	C₄	—	178	1	no	—
346	C₄	—	178	1	no	—
347	C₄	—	178	1	no	—
348	C₄	#15668057	1	1	yes	characteristic
349	C₄	#15668057	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	712	1	no	—
352	C₂	#19152163	1	1	yes	characteristic
353	C₂	#19152411	1	1	yes	characteristic
354	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	#451×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083574	C₂²
hypercenter	#451	D₈₉
hypocenter	C₈₉	#3223
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
1. #19152208	C₂	semidirect
3. #19151991	C₂	semidirect
4. #19151976	C₂	semidirect
5. #19151965	C₂	semidirect
6. #19151954	C₂	semidirect
7. #19151823	C₂	semidirect
8. #19151822	C₂	semidirect
9. #19151807	C₂	semidirect
10. #19151806	C₂	semidirect
13. #19151754	C₂	semidirect
14. #19151656	C₂	semidirect
15. #451×C₈₉	C₂	semidirect
178. C₈₉	#3223	semidirect
192. #451	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152208	2
#19152172	2
#19151991	2
#19151976	2
#19151965	2
#19151954	2
#19151823	2
#19151822	2
#19151807	2
#19151806	2
#19151762	2
#19151758	2
#19151754	2
#19151656	2
#451×C₈₉	2
#3223	89