Subgroups — #26033317

Order 11392.

368 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	#19152330	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152322	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151984	C₂	1	1	yes	maximal, characteristic
4	#19151983	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151822	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151822	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151815	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151810	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151800	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151796	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151711	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151705	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083561	C₂²	1	1	yes	characteristic
17	C₂×#13083561	C₂²	1	1	yes	characteristic
18	C₂×#13083561	C₂²	1	1	yes	characteristic
19	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	#15668008	C₂²	1	1	yes	characteristic
22	#15668007	—	2	1	no	—
23	#15668007	—	2	1	no	—
24	#15668001	C₂²	1	1	yes	characteristic
25	#15668001	C₂²	1	1	yes	characteristic
26	#15668001	C₂²	1	1	yes	characteristic
27	#15667998	—	2	1	no	—
28	#15667996	—	2	1	no	—
29	#15667994	—	2	1	no	—
30	#15667992	—	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×D₃₅₆	C₂²	1	1	yes	characteristic
36	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
37	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
38	#15667972	—	2	1	no	—
39	#15667970	—	2	1	no	—
40	#15667944	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	#15667936	C₂²	1	1	yes	characteristic
45	#15667936	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667934	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×D₃₅₆	C₂³	1	1	yes	characteristic
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	—	2	1	no	—
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	#13083561	—	4	1	no	—
68	#13083561	—	4	1	no	—
69	#13083561	—	4	1	no	—
70	D₇₁₂	—	2	2	no	—
71	#13083559	—	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	—	2	1	no	—
77	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	#13083553	D₄	1	1	yes	characteristic
81	#13083553	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₈₉	—	2	1	no	—
83	#13083551	—	2	1	no	—
84	#13083551	—	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	#13083551	—	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	C₂²×D₈₉	—	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	Q₈×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	2	1	no	—
121	D₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	4	2	no	—
129	C₂²⋊D₈₉	—	8	1	no	—
130	C₂×Dic₈₉	—	2	1	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	8	1	no	—
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	D₃₅₆	—	2	1	no	—
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	—	2	1	no	—
140	D₃₅₆	—	2	1	no	—
141	C₄×D₈₉	—	4	2	no	—
142	C₄×D₈₉	—	8	1	no	—
143	Dic₁₇₈	—	4	1	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	#41	1	2	yes	—
146	#10623494	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	C₂²×C₈₉	—	4	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₂×D₈₉	—	4	1	no	—
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	—	2	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	C₂×#41	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	C₂×C₈₉	—	8	1	no	—
172	C₂×C₈₉	#550	1	1	yes	characteristic
173	C₂×C₈₉	#575	1	1	yes	characteristic
174	C₂×C₈₉	#577	1	1	yes	characteristic
175	D₈₉	—	8	1	no	—
176	D₈₉	—	8	1	no	—
177	#3401	—	89	1	no	maximal, Sylow
178	C₈₉	#3401	1	1	yes	characteristic, complemented, Sylow
179	#559	—	89	1	no	—
180	#556	—	89	1	no	—
181	#497	D₈₉	1	1	yes	characteristic, complemented
182	#496	—	89	1	no	—
183	#495	—	89	1	no	—
184	#464	—	89	1	no	—
185	#464	—	89	1	no	—
186	#463	—	89	1	no	—
187	#463	—	89	1	no	—
188	#461	—	89	1	no	—
189	#460	—	89	1	no	—
190	#446	—	89	1	no	—
191	#442	—	89	1	no	—
192	#441	—	89	1	no	—
193	C₄×SD₁₆	—	89	1	no	—
194	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
195	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂×SD₁₆	—	89	1	no	—
197	C₂×SD₁₆	—	89	1	no	—
198	C₂×SD₁₆	—	89	1	no	—
199	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
200	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
201	C₂×D₈	—	89	1	no	—
202	C₂×D₈	—	89	1	no	—
203	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
204	#127	—	89	1	no	—
205	#126	—	178	1	no	—
206	#126	—	178	1	no	—
207	#124	—	89	1	no	—
208	#124	—	89	1	no	—
209	#124	—	89	1	no	—
210	#124	—	178	1	no	—
211	#124	C₂×D₈₉	1	1	yes	characteristic
212	#123	—	178	1	no	—
213	#123	—	178	1	no	—
214	C₂²⋊Q₈	—	178	1	no	—
215	C₄×Q₈	—	89	1	no	—
216	C₄×Q₈	—	89	1	no	—
217	C₄×Q₈	—	89	1	no	—
218	C₄×Q₈	—	89	1	no	—
219	C₄×D₄	—	89	1	no	—
220	C₄×D₄	—	89	1	no	—
221	C₄×D₄	—	89	1	no	—
222	C₄×D₄	—	178	1	no	—
223	#117	—	178	1	no	—
224	#106	—	89	1	no	—
225	C₄⋊C₈	—	89	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	—	89	1	no	—
231	#102	—	89	1	no	—
232	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#97	—	89	1	no	—
235	#97	C₂×D₈₉	1	1	yes	characteristic
236	C₄×C₈	—	89	1	no	—
237	C₂×Q₈	—	89	1	no	—
238	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₂×D₄	—	89	1	no	—
242	C₂×D₄	—	89	1	no	—
243	C₂×D₄	—	178	1	no	—
244	C₂×D₄	—	178	1	no	—
245	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	—	178	1	no	—
247	C₂²×C₄	—	178	1	no	—
248	C₂²×C₄	—	178	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	4	1	no	—
251	SD₁₆	—	2	2	no	—
252	SD₁₆	—	4	1	no	—
253	SD₁₆	—	356	1	no	—
254	SD₁₆	—	356	1	no	—
255	SD₁₆	—	356	1	no	—
256	D₈	—	356	1	no	—
257	D₈	—	178	2	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	89	1	no	—
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	89	1	no	—
267	C₄⋊C₄	—	89	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₄²	—	89	1	no	—
284	C₄²	—	89	1	no	—
285	C₄²	—	178	1	no	—
286	C₄²	—	178	1	no	—
287	C₄²	—	178	1	no	—
288	C₄²	C₂²×D₈₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	178	1	no	—
291	C₂³	—	178	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	356	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	356	2	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	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	89	1	no	—
312	C₂×C₄	—	89	1	no	—
313	C₂×C₄	—	89	1	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	178	1	no	—
316	C₂×C₄	—	178	1	no	—
317	C₂×C₄	—	712	1	no	—
318	C₂×C₄	—	712	1	no	—
319	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
320	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	178	1	no	—
324	C₈	—	178	1	no	—
325	C₈	#13083573	1	2	yes	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	356	1	no	—
330	C₂²	—	356	1	no	—
331	C₂²	—	356	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	—	2	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	—	178	1	no	—
344	C₄	—	178	1	no	—
345	C₄	—	178	1	no	—
346	C₄	—	178	1	no	—
347	C₄	C₂×#13083573	1	1	yes	characteristic
348	C₄	#15668057	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	712	1	no	—
351	C₂	—	712	1	no	—
352	C₂	#19152268	1	1	yes	characteristic
353	C₂	#19152397	1	1	yes	characteristic
354	C₂	#19152407	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	#19151984	C₂
hypercenter	#497	D₈₉
hypocenter	C₈₉	#3401
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. #19152330	C₂	semidirect
2. #19152322	C₂	semidirect
4. #19151983	C₂	semidirect
5. #19151822	C₂	semidirect
6. #19151822	C₂	semidirect
7. #19151815	C₂	semidirect
8. #19151810	C₂	semidirect
9. #19151800	C₂	semidirect
10. #19151796	C₂	semidirect
11. #19151726	C₂	semidirect
12. #19151711	C₂	semidirect
13. #19151705	C₂	semidirect
14. SD₁₆×Dic₈₉	C₂	semidirect
15. #497×C₈₉	C₂	semidirect
178. C₈₉	#3401	semidirect
181. #497	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152330	2
#19152322	2
#19151984	2
#19151983	2
#19151822	2
#19151815	2
#19151810	2
#19151800	2
#19151796	2
#19151726	2
#19151711	2
#19151705	2
SD₁₆×Dic₈₉	2
#497×C₈₉	2
#3401	89