Subgroups — #26031364

Order 11392.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152222	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152173	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151990	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151982	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151960	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151823	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151816	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151807	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151802	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151763	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151755	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151746	C₂	1	1	yes	maximal, characteristic
13	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151657	C₂	1	1	yes	maximal, characteristic, complemented
15	#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₅₆	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×#13083563	C₂²	1	1	yes	characteristic
23	C₂×#13083561	C₂²	1	1	yes	characteristic
24	C₂×#13083561	C₂²	1	1	yes	characteristic
25	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
26	#15668010	C₂²	1	1	yes	characteristic
27	#15668007	—	2	1	no	—
28	#15668001	—	2	1	no	—
29	#15668001	C₂²	1	1	yes	characteristic
30	#15668000	C₂²	1	1	yes	characteristic
31	#15668000	C₂²	1	1	yes	characteristic
32	#15667997	—	2	1	no	—
33	#15667997	—	2	1	no	—
34	#15667997	—	2	1	no	—
35	#15667992	C₂²	1	1	yes	characteristic
36	#15667990	—	2	1	no	—
37	#15667990	—	2	1	no	—
38	#15667988	—	2	1	no	—
39	#15667987	C₂²	1	1	yes	characteristic
40	#15667977	C₂²	1	1	yes	characteristic
41	#15667974	C₂²	1	1	yes	characteristic
42	#15667969	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667932	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	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	#103×C₈₉	C₂²	1	1	yes	characteristic
55	#103×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083575	—	4	1	no	—
60	#13083575	—	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	Q₈×D₈₉	—	4	1	no	—
65	Q₈×D₈₉	—	4	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
69	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
74	Dic₃₅₆	—	4	1	no	—
75	#13083564	—	4	1	no	—
76	#13083564	—	4	1	no	—
77	#13083563	—	4	1	no	—
78	#13083563	—	4	1	no	—
79	#13083561	—	4	1	no	—
80	#13083561	—	4	1	no	—
81	D₇₁₂	—	4	1	no	—
82	C₂×#10623494	C₂³	1	1	yes	characteristic
83	C₂×#10623494	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₈₉	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	D₄	1	1	yes	characteristic
89	#13083553	D₄	1	1	yes	characteristic
90	#13083551	—	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	—	2	1	no	—
98	#13083551	—	2	1	no	—
99	#13083551	D₄	1	1	yes	characteristic
100	#13083551	D₄	1	1	yes	characteristic
101	C₄×Dic₈₉	—	2	1	no	—
102	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
103	C₂×Q₈×C₈₉	—	2	1	no	—
104	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
110	C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₄²×C₈₉	C₂³	1	1	yes	characteristic
114	C₂²×D₈₉	—	2	1	no	—
115	C₂²×D₈₉	—	2	1	no	—
116	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
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	—	8	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	—	2	1	no	—
126	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	2	1	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	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	2	1	no	—
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	—	2	1	no	—
138	D₃₅₆	—	4	2	no	—
139	D₃₅₆	—	8	1	no	—
140	C₄×D₈₉	—	2	1	no	—
141	C₄×D₈₉	—	2	1	no	—
142	C₄×D₈₉	—	4	2	no	—
143	C₄×D₈₉	—	4	2	no	—
144	C₄×D₈₉	—	8	1	no	—
145	Dic₁₇₈	—	2	1	no	—
146	Dic₁₇₈	—	2	1	no	—
147	Dic₁₇₈	—	2	1	no	—
148	Dic₁₇₈	—	2	1	no	—
149	Dic₁₇₈	—	4	2	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	#10623494	—	2	1	no	—
153	#10623494	—	2	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	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	—	4	1	no	—
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	4	1	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	—	4	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
171	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	2	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	C₂×C₈₉	#535	1	1	yes	characteristic
178	C₂×C₈₉	#577	1	1	yes	characteristic
179	C₂×C₈₉	#578	1	1	yes	characteristic
180	D₈₉	—	4	1	no	—
181	D₈₉	—	8	1	no	—
182	D₈₉	—	8	1	no	—
183	#3225	—	89	1	no	maximal, Sylow
184	C₈₉	#3225	1	1	yes	characteristic, complemented, Sylow
185	#542	—	89	1	no	—
186	#536	—	89	1	no	—
187	#497	—	89	1	no	—
188	#495	—	89	1	no	—
189	#489	—	89	1	no	—
190	#487	D₈₉	1	1	yes	characteristic, complemented
191	#464	—	89	1	no	—
192	#463	—	89	1	no	—
193	#462	—	89	1	no	—
194	#461	—	89	1	no	—
195	#452	—	89	1	no	—
196	#451	—	89	1	no	—
197	#449	—	89	1	no	—
198	#448	—	89	1	no	—
199	#433	—	89	1	no	—
200	C₂×#41	—	89	1	no	—
201	C₂²×Q₈	—	89	1	no	—
202	C₂×Q₁₆	—	89	1	no	—
203	C₂×Q₁₆	—	89	1	no	—
204	C₂×Q₁₆	—	89	1	no	—
205	C₂×SD₁₆	—	89	1	no	—
206	C₂×SD₁₆	—	89	1	no	—
207	C₂×SD₁₆	—	89	1	no	—
208	C₂×SD₁₆	—	89	1	no	—
209	C₂×D₈	—	89	1	no	—
210	C₄⋊Q₈	—	89	1	no	—
211	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
212	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
213	#126	—	178	1	no	—
214	#124	—	89	1	no	—
215	#124	—	89	1	no	—
216	#124	—	89	1	no	—
217	#124	—	178	1	no	—
218	#123	—	178	1	no	—
219	#123	—	178	1	no	—
220	#123	—	178	1	no	—
221	C₂²⋊Q₈	—	89	1	no	—
222	C₂²⋊Q₈	—	178	1	no	—
223	C₂²⋊Q₈	—	178	1	no	—
224	C₂²⋊Q₈	—	178	1	no	—
225	#121	—	89	1	no	—
226	C₄×Q₈	—	89	1	no	—
227	C₄×D₄	—	89	1	no	—
228	#117	—	89	1	no	—
229	C₄⋊C₈	—	89	1	no	—
230	C₄⋊C₈	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	C₂×D₈₉	1	1	yes	characteristic
234	#103	C₂×D₈₉	1	1	yes	characteristic
235	#103	C₂×D₈₉	1	1	yes	characteristic
236	#103	C₂×D₈₉	1	1	yes	characteristic
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	C₂²⋊C₈	—	89	1	no	—
240	C₂²⋊C₈	—	89	1	no	—
241	#97	—	89	1	no	—
242	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
243	#41	—	356	1	no	—
244	#41	—	356	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	89	1	no	—
247	C₂×Q₈	—	89	1	no	—
248	C₂×Q₈	—	356	1	no	—
249	C₂×Q₈	—	356	1	no	—
250	C₂×Q₈	—	178	1	no	—
251	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×D₄	—	89	1	no	—
254	C₂×D₄	—	89	1	no	—
255	C₂×D₄	—	178	1	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂²×C₄	—	89	1	no	—
258	C₂²×C₄	—	178	1	no	—
259	C₂²×C₄	—	178	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	Q₁₆	—	356	1	no	—
262	Q₁₆	—	356	1	no	—
263	Q₁₆	—	356	1	no	—
264	SD₁₆	—	356	1	no	—
265	SD₁₆	—	356	1	no	—
266	SD₁₆	—	356	1	no	—
267	SD₁₆	—	356	1	no	—
268	D₈	—	356	1	no	—
269	C₂×C₈	—	89	1	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	—	89	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	—	178	1	no	—
282	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
283	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
284	C₂²⋊C₄	—	89	1	no	—
285	C₂²⋊C₄	—	89	1	no	—
286	C₂²⋊C₄	—	178	1	no	—
287	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₂²⋊C₄	—	178	1	no	—
295	C₄²	—	89	1	no	—
296	C₄²	—	178	1	no	—
297	C₄²	C₂²×D₈₉	1	1	yes	characteristic
298	C₂³	—	89	1	no	—
299	C₂³	—	178	1	no	—
300	C₂³	—	178	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	8	1	no	—
306	Q₈	—	356	2	no	—
307	Q₈	—	178	1	no	—
308	Q₈	—	178	1	no	—
309	Q₈	—	178	1	no	—
310	Q₈	—	178	1	no	—
311	D₄	—	356	2	no	—
312	D₄	—	178	1	no	—
313	D₄	—	178	1	no	—
314	D₄	—	178	1	no	—
315	D₄	—	178	1	no	—
316	D₄	—	712	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	89	1	no	—
322	C₂×C₄	—	89	1	no	—
323	C₂×C₄	—	89	1	no	—
324	C₂×C₄	—	356	2	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	—	178	1	no	—
329	C₂×C₄	—	178	1	no	—
330	C₂×C₄	—	712	1	no	—
331	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
332	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	178	1	no	—
337	C₈	—	178	1	no	—
338	C₂²	—	356	1	no	—
339	C₂²	—	356	1	no	—
340	C₂²	—	356	1	no	—
341	C₂²	—	356	1	no	—
342	C₂²	—	356	1	no	—
343	C₂²	—	356	1	no	—
344	C₂²	—	178	1	no	—
345	C₂²	—	178	1	no	—
346	C₂²	—	178	1	no	—
347	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	356	1	no	—
355	C₄	—	356	1	no	—
356	C₄	—	178	1	no	—
357	C₄	—	178	1	no	—
358	C₄	—	178	1	no	—
359	C₄	#15668058	1	1	yes	characteristic
360	C₄	#15668058	1	1	yes	characteristic
361	C₂	—	356	1	no	—
362	C₂	—	712	1	no	—
363	C₂	—	712	1	no	—
364	C₂	#19152166	1	1	yes	characteristic
365	C₂	#19152404	1	1	yes	characteristic
366	C₂	#19152411	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#487×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19151746	C₂
hypercenter	#487	D₈₉
hypocenter	C₈₉	#3225
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

Complements

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

N	H ≅ G/N	type
1. #19152222	C₂	semidirect
2. #19152173	C₂	semidirect
3. #19151990	C₂	semidirect
4. #19151982	C₂	semidirect
5. #19151960	C₂	semidirect
6. #19151823	C₂	semidirect
7. #19151816	C₂	semidirect
8. #19151807	C₂	semidirect
9. #19151802	C₂	semidirect
10. #19151763	C₂	semidirect
11. #19151755	C₂	semidirect
13. #19151738	C₂	semidirect
14. #19151657	C₂	semidirect
15. #487×C₈₉	C₂	semidirect
184. C₈₉	#3225	semidirect
190. #487	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152222	2
#19152173	2
#19151990	2
#19151982	2
#19151960	2
#19151823	2
#19151816	2
#19151807	2
#19151802	2
#19151763	2
#19151755	2
#19151746	2
#19151738	2
#19151657	2
#487×C₈₉	2
#3225	89