Subgroups — #26031332

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	#19152206	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152168	C₂	1	1	yes	maximal, characteristic
3	#19151989	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	#19151951	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151817	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151809	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151808	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151761	C₂	1	1	yes	maximal, characteristic
11	#19151757	C₂	1	1	yes	maximal, characteristic
12	#19151755	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151658	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	C₂×#13083564	C₂²	1	1	yes	characteristic
21	C₂×#13083564	C₂²	1	1	yes	characteristic
22	C₂×#13083564	C₂²	1	1	yes	characteristic
23	C₂×#13083562	C₂²	1	1	yes	characteristic
24	#15668012	C₂²	1	1	yes	characteristic
25	#15668010	C₂²	1	1	yes	characteristic
26	#15668010	C₂²	1	1	yes	characteristic
27	#15668007	—	2	1	no	—
28	#15668006	—	2	1	no	—
29	#15668004	—	2	1	no	—
30	#15668003	C₂²	1	1	yes	characteristic
31	#15668002	—	2	1	no	—
32	#15667999	C₂²	1	1	yes	characteristic
33	#15667994	—	2	1	no	—
34	#15667994	—	2	1	no	—
35	#15667992	C₂²	1	1	yes	characteristic
36	#15667989	—	2	1	no	—
37	#15667989	—	2	1	no	—
38	#15667988	C₂²	1	1	yes	characteristic
39	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
40	#15667969	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667936	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	#15667924	C₂²	1	1	yes	characteristic
49	#15667924	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×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	#124×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	4	1	no	—
60	#13083573	—	4	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	Q₈×D₈₉	—	4	1	no	—
66	Q₈×D₈₉	—	4	1	no	—
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
71	Dic₃₅₆	—	4	1	no	—
72	Dic₃₅₆	—	4	1	no	—
73	#13083564	—	4	1	no	—
74	#13083564	—	4	1	no	—
75	#13083564	—	4	1	no	—
76	#13083562	—	4	1	no	—
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	—	2	1	no	—
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	—	2	1	no	—
90	#13083553	—	2	1	no	—
91	#13083553	D₄	1	1	yes	characteristic
92	#13083553	D₄	1	1	yes	characteristic
93	C₂²⋊Dic₈₉	—	2	1	no	—
94	C₂²⋊Dic₈₉	—	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	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
104	Q₁₆×C₈₉	—	4	1	no	—
105	SD₁₆×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₂²⋊C₄×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	—	2	1	no	—
113	C₄²×C₈₉	C₂³	1	1	yes	characteristic
114	C₂³×C₈₉	—	2	1	no	—
115	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
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	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	2	1	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	4	2	no	—
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	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	2	1	no	—
139	C₄×D₈₉	—	4	2	no	—
140	Dic₁₇₈	—	2	1	no	—
141	Dic₁₇₈	—	2	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	—	2	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	2	1	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	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	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	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	—	4	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	—	4	1	no	—
172	C₂×C₈₉	—	8	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	—	4	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	D₈₉	1	1	yes	characteristic, complemented
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	—	89	1	no	—
193	#433	—	89	1	no	—
194	C₂×#41	—	89	1	no	—
195	C₂²×Q₈	—	89	1	no	—
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	—	89	1	no	—
201	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
202	C₂×SD₁₆	—	89	1	no	—
203	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
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	C₂×D₈₉	1	1	yes	characteristic
214	#123	—	178	1	no	—
215	#123	—	178	1	no	—
216	C₂²⋊Q₈	—	89	1	no	—
217	C₂²⋊Q₈	—	89	1	no	—
218	C₂²⋊Q₈	—	178	1	no	—
219	C₂²⋊Q₈	—	178	1	no	—
220	C₄×Q₈	—	89	1	no	—
221	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
222	#117	—	89	1	no	—
223	C₄⋊C₈	—	89	1	no	—
224	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
225	#103	—	89	1	no	—
226	#103	—	89	1	no	—
227	#103	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	C₂×D₈₉	1	1	yes	characteristic
231	#102	—	89	1	no	—
232	#102	C₂×D₈₉	1	1	yes	characteristic
233	C₂²⋊C₈	—	89	1	no	—
234	C₂²⋊C₈	—	89	1	no	—
235	#97	—	89	1	no	—
236	C₄×C₈	—	89	1	no	—
237	#41	—	356	1	no	—
238	#41	—	356	1	no	—
239	C₂×Q₈	—	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	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×D₄	—	178	1	no	—
248	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
249	C₂²×C₄	—	89	1	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	—	178	1	no	—
252	Q₁₆	—	4	1	no	—
253	Q₁₆	—	356	1	no	—
254	Q₁₆	—	356	1	no	—
255	Q₁₆	—	356	1	no	—
256	Q₁₆	—	356	1	no	—
257	Q₁₆	—	356	1	no	—
258	SD₁₆	—	4	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	—	2	1	no	—
282	C₂²⋊C₄	—	89	1	no	—
283	C₂²⋊C₄	—	89	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	—	2	1	no	—
289	C₄²	—	89	1	no	—
290	C₄²	—	178	1	no	—
291	C₄²	C₂²×D₈₉	1	1	yes	characteristic
292	C₂³	—	2	1	no	—
293	C₂³	—	89	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	2	no	—
299	Q₈	—	178	1	no	—
300	Q₈	—	178	1	no	—
301	Q₈	—	178	1	no	—
302	Q₈	—	178	1	no	—
303	Q₈	—	178	1	no	—
304	Q₈	—	178	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	356	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	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	—	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	D₄×D₈₉	1	1	yes	characteristic
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	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	178	1	no	—
333	C₂²	—	178	1	no	—
334	C₂²	—	178	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	—	356	1	no	—
344	C₄	—	356	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	#15668058	1	1	yes	characteristic
350	C₂	—	8	1	no	—
351	C₂	—	356	1	no	—
352	C₂	#19152164	1	1	yes	characteristic
353	C₂	#19152412	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	#464×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083573	C₂²
hypercenter	#464	D₈₉
hypocenter	C₈₉	#3223
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #19152206	C₂	semidirect
3. #19151989	C₂	semidirect
4. #19151976	C₂	semidirect
5. #19151965	C₂	semidirect
6. #19151951	C₂	semidirect
7. #19151817	C₂	semidirect
8. #19151809	C₂	semidirect
9. #19151808	C₂	semidirect
12. #19151755	C₂	semidirect
13. #19151752	C₂	semidirect
14. #19151658	C₂	semidirect
15. #464×C₈₉	C₂	semidirect
178. C₈₉	#3223	semidirect
186. #464	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152206	2
#19152168	2
#19151989	2
#19151976	2
#19151965	2
#19151951	2
#19151817	2
#19151809	2
#19151808	2
#19151761	2
#19151757	2
#19151755	2
#19151752	2
#19151658	2
#464×C₈₉	2
#3223	89