Subgroups — #26033328

Order 11392.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152322	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152317	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151991	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151979	C₂	1	1	yes	maximal, characteristic
5	#19151822	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151818	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151815	C₂	1	1	yes	maximal, characteristic
8	#19151810	C₂	1	1	yes	maximal, characteristic
9	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151805	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151699	C₂	1	1	yes	maximal, characteristic, complemented
14	Q₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083564	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083561	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	#15668010	C₂²	1	1	yes	characteristic
22	#15668007	—	2	1	no	—
23	#15668007	—	2	1	no	—
24	#15668005	—	2	1	no	—
25	#15668005	—	2	1	no	—
26	#15668001	C₂²	1	1	yes	characteristic
27	#15668001	C₂²	1	1	yes	characteristic
28	#15667999	C₂²	1	1	yes	characteristic
29	#15667992	—	2	1	no	—
30	#15667989	—	2	1	no	—
31	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
32	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
33	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
36	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
38	#15667970	—	2	1	no	—
39	#15667968	—	2	1	no	—
40	#15667946	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667933	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
57	#124×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂²×Dic₈₉	—	2	1	no	—
63	C₂×C₄×D₈₉	—	2	1	no	—
64	#13083564	—	4	1	no	—
65	#13083564	—	4	1	no	—
66	#13083563	—	2	2	no	—
67	#13083561	—	4	1	no	—
68	#13083559	—	4	1	no	—
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
75	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	D₄	1	1	yes	characteristic
81	#13083553	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₈₉	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	C₂²⋊Dic₈₉	—	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	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
95	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
100	Q₁₆×C₈₉	—	2	2	no	—
101	Q₁₆×C₈₉	—	4	1	no	—
102	SD₁₆×C₈₉	—	2	2	no	—
103	SD₁₆×C₈₉	—	4	1	no	—
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₂²⋊C₄×C₈₉	—	2	1	no	—
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	Q₈×C₈₉	—	2	1	no	—
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	—	2	1	no	—
118	Q₈×C₈₉	—	2	1	no	—
119	D₄×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	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	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	D₃₅₆	—	2	1	no	—
136	D₃₅₆	—	2	1	no	—
137	C₄×D₈₉	—	4	2	no	—
138	Dic₁₇₈	—	2	1	no	—
139	Dic₁₇₈	—	2	1	no	—
140	Dic₁₇₈	—	4	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	#41	1	2	yes	—
143	#10623494	—	2	1	no	—
144	#10623494	—	2	1	no	—
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₈₉	—	4	1	no	—
150	C₂×D₈₉	—	4	1	no	—
151	C₂×D₈₉	—	4	1	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	—	4	1	no	—
157	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
159	Dic₈₉	—	2	1	no	—
160	Dic₈₉	—	2	1	no	—
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	—	4	1	no	—
166	C₂×C₈₉	—	8	1	no	—
167	C₂×C₈₉	#550	1	1	yes	characteristic
168	C₂×C₈₉	#575	1	1	yes	characteristic
169	C₂×C₈₉	#578	1	1	yes	characteristic
170	D₈₉	—	8	1	no	—
171	#3402	—	89	1	no	maximal, Sylow
172	C₈₉	#3402	1	1	yes	characteristic, complemented, Sylow
173	#556	—	89	1	no	—
174	#556	—	89	1	no	—
175	#497	—	89	1	no	—
176	#497	D₈₉	1	1	yes	characteristic, complemented
177	#495	—	89	1	no	—
178	#464	—	89	1	no	—
179	#464	—	89	1	no	—
180	#463	—	89	1	no	—
181	#463	—	89	1	no	—
182	#462	—	89	1	no	—
183	#462	—	89	1	no	—
184	#446	—	89	1	no	—
185	#440	—	89	1	no	—
186	#440	—	89	1	no	—
187	C₄×Q₁₆	—	89	1	no	—
188	C₂×Q₁₆	—	89	1	no	—
189	C₂×Q₁₆	—	89	1	no	—
190	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
191	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
192	C₂×SD₁₆	—	89	1	no	—
193	C₂×SD₁₆	—	89	1	no	—
194	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
195	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂×D₈	—	89	1	no	—
197	C₄⋊Q₈	—	89	1	no	—
198	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
199	#126	—	178	1	no	—
200	#126	—	178	1	no	—
201	#126	—	178	1	no	—
202	#126	—	178	1	no	—
203	#124	—	89	1	no	—
204	#124	—	89	1	no	—
205	#124	—	89	1	no	—
206	#124	C₂×D₈₉	1	1	yes	characteristic
207	C₂²⋊Q₈	—	178	1	no	—
208	C₂²⋊Q₈	—	178	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×Q₈	—	89	1	no	—
211	C₄×Q₈	—	89	1	no	—
212	C₄×Q₈	—	89	1	no	—
213	C₄×Q₈	—	89	1	no	—
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	89	1	no	—
216	#117	—	178	1	no	—
217	#117	—	178	1	no	—
218	#107	—	89	1	no	—
219	C₄⋊C₈	—	89	1	no	—
220	C₄⋊C₈	—	89	1	no	—
221	C₄⋊C₈	—	89	1	no	—
222	#103	—	89	1	no	—
223	#103	—	89	1	no	—
224	#103	—	89	1	no	—
225	#103	—	89	1	no	—
226	#102	—	89	1	no	—
227	#102	—	89	1	no	—
228	#97	—	89	1	no	—
229	#97	C₂×D₈₉	1	1	yes	characteristic
230	C₄×C₈	—	89	1	no	—
231	C₂×Q₈	—	89	1	no	—
232	C₂×Q₈	—	89	1	no	—
233	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×D₄	—	89	1	no	—
237	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
238	C₂²×C₄	—	178	1	no	—
239	C₂²×C₄	—	178	1	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	4	1	no	—
242	Q₁₆	—	356	1	no	—
243	Q₁₆	—	356	1	no	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	4	1	no	—
246	SD₁₆	—	356	1	no	—
247	SD₁₆	—	178	2	no	—
248	D₈	—	356	1	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	—	2	1	no	—
255	C₄⋊C₄	—	89	1	no	—
256	C₄⋊C₄	—	89	1	no	—
257	C₄⋊C₄	—	89	1	no	—
258	C₄⋊C₄	—	89	1	no	—
259	C₄⋊C₄	—	89	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	—	178	1	no	—
264	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	178	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	—	89	1	no	—
277	C₄²	—	89	1	no	—
278	C₄²	—	178	1	no	—
279	C₄²	—	178	1	no	—
280	C₄²	—	178	1	no	—
281	C₄²	—	178	1	no	—
282	C₄²	C₂²×D₈₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	178	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	356	1	no	—
292	Q₈	—	178	1	no	—
293	Q₈	—	178	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	178	1	no	—
297	D₄	—	178	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	89	1	no	—
303	C₂×C₄	—	89	1	no	—
304	C₂×C₄	—	89	1	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
311	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	178	1	no	—
315	C₈	—	178	1	no	—
316	C₈	#13083573	1	2	yes	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	356	1	no	—
321	C₂²	—	356	1	no	—
322	C₂²	—	356	1	no	—
323	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	356	1	no	—
330	C₄	—	356	1	no	—
331	C₄	—	356	1	no	—
332	C₄	—	178	1	no	—
333	C₄	—	178	1	no	—
334	C₄	—	178	1	no	—
335	C₄	—	178	1	no	—
336	C₄	C₂×#13083573	1	1	yes	characteristic
337	C₄	#15668057	1	1	yes	characteristic
338	C₂	—	8	1	no	—
339	C₂	—	712	1	no	—
340	C₂	#19152268	1	1	yes	characteristic
341	C₂	#19152397	1	1	yes	characteristic
342	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	#497×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083559	C₂²
hypercenter	#497	D₈₉
hypocenter	C₈₉	#3402
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. #19152322	C₂	semidirect
2. #19152317	C₂	semidirect
3. #19151991	C₂	semidirect
5. #19151822	C₂	semidirect
6. #19151818	C₂	semidirect
9. #19151806	C₂	semidirect
10. #19151805	C₂	semidirect
11. #19151726	C₂	semidirect
12. #19151700	C₂	semidirect
13. #19151699	C₂	semidirect
14. Q₁₆×Dic₈₉	C₂	semidirect
15. #497×C₈₉	C₂	semidirect
172. C₈₉	#3402	semidirect
176. #497	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152322	2
#19152317	2
#19151991	2
#19151979	2
#19151822	2
#19151818	2
#19151815	2
#19151810	2
#19151806	2
#19151805	2
#19151726	2
#19151700	2
#19151699	2
Q₁₆×Dic₈₉	2
#497×C₈₉	2
#3402	89