Subgroups — #26032248

Order 11392.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152260	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152253	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151958	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151935	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151913	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151893	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151843	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151833	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151813	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151742	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151735	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151671	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151598	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083574	—	2	1	no	—
18	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	C₂×#13083557	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
23	#15668012	C₂²	1	1	yes	characteristic
24	#15668000	—	2	1	no	—
25	#15668000	C₂²	1	1	yes	characteristic
26	#15667994	—	2	1	no	—
27	#15667994	—	2	1	no	—
28	#15667992	—	2	1	no	—
29	#15667992	C₂²	1	1	yes	characteristic
30	#15667991	—	2	1	no	—
31	#15667987	—	2	1	no	—
32	#15667987	C₂²	1	1	yes	characteristic
33	#15667983	C₂²	1	1	yes	characteristic
34	#15667983	C₂²	1	1	yes	characteristic
35	C₄×D₃₅₆	—	2	1	no	—
36	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667945	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	#15667936	C₂²	1	1	yes	characteristic
46	#15667935	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	#13083575	—	2	2	no	—
60	#13083575	—	4	1	no	—
61	#13083574	—	4	4	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	Q₈×D₈₉	—	2	2	no	—
66	Q₈×D₈₉	—	4	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	—	2	1	no	—
69	C₂×D₃₅₆	D₄	1	1	yes	characteristic
70	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊D₈₉	—	2	1	no	—
72	C₂×C₂²⋊D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	—	2	1	no	—
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
79	#13083562	—	4	1	no	—
80	#13083559	—	4	1	no	—
81	#13083557	—	4	1	no	—
82	C₂×#10623494	C₂³	1	1	yes	characteristic
83	C₂×#10623494	C₂³	1	1	yes	characteristic
84	C₈×D₈₉	—	4	1	no	—
85	C₄⋊Dic₈₉	—	2	1	no	—
86	C₄⋊Dic₈₉	—	2	1	no	—
87	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
88	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
90	#13083553	—	2	1	no	—
91	#13083553	—	2	1	no	—
92	C₂²⋊Dic₈₉	—	2	1	no	—
93	C₂²⋊Dic₈₉	—	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	D₄	1	1	yes	characteristic
99	#13083551	D₄	1	1	yes	characteristic
100	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	—	2	1	no	—
103	SD₁₆×C₈₉	—	2	2	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	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	C₂³×C₈₉	—	2	1	no	—
113	C₂²×D₈₉	—	2	1	no	—
114	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	#41	1	2	yes	complemented
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	4	2	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	C₂²⋊D₈₉	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	D₃₅₆	—	2	2	no	—
135	D₃₅₆	—	2	1	no	—
136	D₃₅₆	—	2	1	no	—
137	D₃₅₆	—	4	2	no	—
138	D₃₅₆	—	4	2	no	—
139	C₄×D₈₉	—	2	1	no	—
140	C₄×D₈₉	—	2	1	no	—
141	C₄×D₈₉	—	2	1	no	—
142	C₄×D₈₉	—	2	1	no	—
143	C₄×D₈₉	—	4	2	no	—
144	C₄×D₈₉	—	4	2	no	—
145	C₄×D₈₉	—	4	2	no	—
146	C₄×D₈₉	—	4	2	no	—
147	Dic₁₇₈	—	2	1	no	—
148	Dic₁₇₈	—	2	1	no	—
149	Dic₁₇₈	—	4	2	no	—
150	Dic₁₇₈	—	4	2	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	—	2	1	no	—
153	#10623494	—	2	1	no	—
154	#10623494	—	2	1	no	—
155	C₂²×C₈₉	—	4	1	no	—
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	—	4	1	no	—
158	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₈₉	—	2	1	no	—
160	C₂×D₈₉	—	2	1	no	—
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	2	1	no	—
164	C₂×D₈₉	—	4	1	no	—
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₂×D₈₉	AGL(1,ℤ/8)	1	2	yes	—
168	C₄×C₈₉	—	2	2	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
174	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
175	Dic₈₉	—	2	1	no	—
176	Dic₈₉	—	2	1	no	—
177	Dic₈₉	—	4	1	no	—
178	Dic₈₉	—	4	1	no	—
179	Dic₈₉	—	4	1	no	—
180	C₂×C₈₉	—	8	1	no	—
181	C₂×C₈₉	#546	1	1	yes	characteristic
182	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂×C₈₉	#578	1	1	yes	characteristic
184	D₈₉	—	2	2	no	—
185	D₈₉	—	4	1	no	—
186	D₈₉	—	8	1	no	—
187	#3308	—	89	1	no	maximal, Sylow
188	C₈₉	#3308	1	1	yes	characteristic, complemented, Sylow
189	#548	—	89	1	no	—
190	#547	—	89	1	no	—
191	#489	—	89	1	no	—
192	#484	—	89	1	no	—
193	#480	—	89	1	no	—
194	#477	—	89	1	no	—
195	#468	—	89	1	no	—
196	#466	—	89	1	no	—
197	#463	—	89	1	no	—
198	#449	—	89	1	no	—
199	#448	—	89	1	no	—
200	#440	D₈₉	1	1	yes	characteristic, complemented
201	#435	—	89	1	no	—
202	#417	—	89	1	no	—
203	C₂×#103	—	89	1	no	—
204	C₂×#41	—	89	1	no	—
205	C₂×#41	—	178	1	no	—
206	C₂²×Q₈	—	89	1	no	—
207	C₂×SD₁₆	—	89	1	no	—
208	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
209	C₂×D₈	—	89	1	no	—
210	C₂×M₁₆	—	89	1	no	—
211	C₂²×C₈	—	89	1	no	—
212	C₄⋊Q₈	—	89	1	no	—
213	#124	—	89	1	no	—
214	#124	—	178	1	no	—
215	#123	—	178	1	no	—
216	#123	—	178	1	no	—
217	C₂²⋊Q₈	—	89	1	no	—
218	C₂²⋊Q₈	—	178	1	no	—
219	C₂²⋊Q₈	—	178	1	no	—
220	#121	—	89	1	no	—
221	#121	—	89	1	no	—
222	#121	—	89	1	no	—
223	#121	—	178	1	no	—
224	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
225	C₄×D₄	—	89	1	no	—
226	C₄×D₄	—	178	1	no	—
227	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
228	C₂×C₄⋊C₄	—	89	1	no	—
229	C₂×C₄⋊C₄	—	89	1	no	—
230	#107	—	89	1	no	—
231	#107	—	89	1	no	—
232	#107	C₂×D₈₉	1	1	yes	characteristic
233	C₄⋊C₈	—	89	1	no	—
234	#103	—	89	1	no	—
235	#103	—	89	1	no	—
236	#103	—	89	1	no	—
237	#103	—	89	1	no	—
238	#103	—	89	1	no	—
239	#103	C₂×D₈₉	1	1	yes	characteristic
240	#102	—	89	1	no	—
241	#102	—	89	1	no	—
242	#102	—	89	1	no	—
243	#102	C₂×D₈₉	1	1	yes	characteristic
244	C₂²⋊C₈	—	89	1	no	—
245	C₂²⋊C₈	—	89	1	no	—
246	#97	C₂×D₈₉	1	1	yes	characteristic
247	#41	—	356	4	no	—
248	#41	—	356	1	no	—
249	#41	—	178	2	no	—
250	C₂×Q₈	—	89	1	no	—
251	C₂×Q₈	—	356	1	no	—
252	C₂×Q₈	—	178	1	no	—
253	C₂×Q₈	—	178	2	no	—
254	C₂×Q₈	—	178	1	no	—
255	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×D₄	—	89	1	no	—
257	C₂×D₄	—	89	1	no	—
258	C₂×D₄	—	178	1	no	—
259	C₂×D₄	—	178	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	—	178	1	no	—
262	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	89	1	no	—
265	C₂²×C₄	—	89	1	no	—
266	C₂²×C₄	—	178	1	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	—	178	1	no	—
269	C₂²×C₄	—	178	1	no	—
270	SD₁₆	—	2	2	no	—
271	SD₁₆	—	356	1	no	—
272	D₈	—	356	1	no	—
273	M₁₆	—	356	1	no	—
274	C₂×C₈	—	89	1	no	—
275	C₂×C₈	—	89	1	no	—
276	C₂×C₈	—	356	1	no	—
277	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
278	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	89	1	no	—
281	C₄⋊C₄	—	89	1	no	—
282	C₄⋊C₄	—	89	1	no	—
283	C₄⋊C₄	—	178	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	—	178	1	no	—
287	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
288	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
289	C₂²⋊C₄	—	2	1	no	—
290	C₂²⋊C₄	—	89	1	no	—
291	C₂²⋊C₄	—	89	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₂²⋊C₄	—	178	1	no	—
295	C₂²⋊C₄	—	178	1	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₂²⋊C₄	—	178	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	C₂²×D₈₉	1	1	yes	characteristic
300	C₂³	—	2	1	no	—
301	C₂³	—	89	1	no	—
302	C₂³	—	89	1	no	—
303	C₂³	—	178	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	—	356	2	no	—
306	Q₈	—	356	2	no	—
307	Q₈	—	178	1	no	—
308	Q₈	—	178	1	no	—
309	Q₈	#13083574	1	2	yes	complemented
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	356	2	no	—
313	D₄	—	356	2	no	—
314	D₄	—	356	2	no	—
315	D₄	—	356	2	no	—
316	D₄	—	178	2	no	—
317	D₄	—	178	1	no	—
318	D₄	—	178	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	89	1	no	—
324	C₂×C₄	—	89	1	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	356	2	no	—
328	C₂×C₄	—	356	2	no	—
329	C₂×C₄	—	178	1	no	—
330	C₂×C₄	—	178	1	no	—
331	C₂×C₄	—	178	1	no	—
332	C₂×C₄	—	178	1	no	—
333	C₂×C₄	—	178	1	no	—
334	C₂×C₄	—	178	1	no	—
335	C₂×C₄	—	178	1	no	—
336	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
337	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	178	1	no	—
342	C₈	—	178	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	89	2	no	—
347	C₂²	—	356	1	no	—
348	C₂²	—	356	1	no	—
349	C₂²	—	356	1	no	—
350	C₂²	—	178	1	no	—
351	C₂²	—	178	1	no	—
352	C₂²	—	178	1	no	—
353	C₂²	—	178	1	no	—
354	C₂²	—	178	1	no	—
355	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	356	1	no	—
362	C₄	—	356	1	no	—
363	C₄	—	356	1	no	—
364	C₄	—	178	1	no	—
365	C₄	—	178	1	no	—
366	C₄	C₂×#13083574	1	1	yes	characteristic
367	C₄	#15668058	1	1	yes	characteristic
368	C₂	—	8	1	no	—
369	C₂	—	356	1	no	—
370	C₂	—	178	2	no	—
371	C₂	—	712	1	no	—
372	C₂	#19152244	1	1	yes	characteristic
373	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
374	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	#440×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#440	D₈₉
hypocenter	C₈₉	#3308
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #19152260	C₂	semidirect
2. #19152253	C₂	semidirect
3. #19151958	C₂	semidirect
4. #19151935	C₂	semidirect
5. #19151913	C₂	semidirect
6. #19151893	C₂	semidirect
7. #19151843	C₂	semidirect
8. #19151833	C₂	semidirect
9. #19151813	C₂	semidirect
10. #19151742	C₂	semidirect
11. #19151735	C₂	semidirect
12. #19151671	C₂	semidirect
13. #19151598	C₂	semidirect
14. #103×D₈₉	C₂	semidirect
15. #440×C₈₉	C₂	semidirect
117. Q₈×C₈₉	#41	semidirect
188. C₈₉	#3308	semidirect
200. #440	D₈₉	semidirect
309. Q₈	#13083574	semidirect

Maximal subgroups

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

subgroup	index
#19152260	2
#19152253	2
#19151958	2
#19151935	2
#19151913	2
#19151893	2
#19151843	2
#19151833	2
#19151813	2
#19151742	2
#19151735	2
#19151671	2
#19151598	2
#103×D₈₉	2
#440×C₈₉	2
#3308	89