Subgroups — #26031309

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152201	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152172	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151984	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151969	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151965	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151954	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151816	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151815	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151802	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151801	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151745	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151741	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151737	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151656	C₂	1	1	yes	maximal, characteristic, complemented
15	#448×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083562	C₂²	1	1	yes	characteristic
19	C₂×#13083561	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
22	C₂×#13083559	C₂²	1	1	yes	characteristic
23	#15668010	C₂²	1	1	yes	characteristic
24	#15668010	C₂²	1	1	yes	characteristic
25	#15668008	C₂²	1	1	yes	characteristic
26	#15668003	C₂²	1	1	yes	characteristic
27	#15668001	C₂²	1	1	yes	characteristic
28	#15668001	C₂²	1	1	yes	characteristic
29	#15667996	—	2	1	no	—
30	#15667995	—	2	1	no	—
31	#15667995	—	2	1	no	—
32	#15667994	—	2	1	no	—
33	#15667988	—	2	1	no	—
34	#15667988	—	2	1	no	—
35	#15667986	C₂²	1	1	yes	characteristic
36	#15667984	—	2	1	no	—
37	#15667983	—	2	1	no	—
38	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
39	#15667974	C₂²	1	1	yes	characteristic
40	#15667971	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667941	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	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
55	#121×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	1	no	—
60	#13083574	—	4	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	2	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	—	2	1	no	—
71	#13083563	—	4	1	no	—
72	#13083562	—	4	1	no	—
73	#13083561	—	4	1	no	—
74	#13083561	—	4	1	no	—
75	D₇₁₂	—	4	1	no	—
76	#13083559	—	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	D₄	1	1	yes	characteristic
81	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
90	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
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	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
98	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	—	4	1	no	—
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×D₄×C₈₉	—	2	1	no	—
105	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	—	2	1	no	—
107	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
108	SD₁₆×C₈₉	—	4	1	no	—
109	SD₁₆×C₈₉	—	4	1	no	—
110	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂²⋊C₄×C₈₉	—	2	1	no	—
114	C₂³×C₈₉	—	2	1	no	—
115	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×D₈₉	—	2	1	no	—
117	C₂²×D₈₉	—	2	1	no	—
118	Q₈×C₈₉	—	2	1	no	—
119	Q₈×C₈₉	—	2	1	no	—
120	Q₈×C₈₉	—	2	1	no	—
121	Q₈×C₈₉	—	2	1	no	—
122	Q₈×C₈₉	—	4	2	no	—
123	D₄×C₈₉	—	2	1	no	—
124	D₄×C₈₉	—	2	1	no	—
125	D₄×C₈₉	—	8	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₈₉	—	4	2	no	—
135	C₂²⋊D₈₉	—	8	1	no	—
136	C₂²⋊D₈₉	—	8	1	no	—
137	C₂²⋊D₈₉	—	8	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	2	1	no	—
140	C₂×Dic₈₉	—	8	1	no	—
141	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
144	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
145	D₃₅₆	—	2	1	no	—
146	D₃₅₆	—	2	1	no	—
147	D₃₅₆	—	2	1	no	—
148	D₃₅₆	—	2	1	no	—
149	C₄×D₈₉	—	4	2	no	—
150	C₄×D₈₉	—	8	1	no	—
151	Dic₁₇₈	—	2	1	no	—
152	Dic₁₇₈	—	2	1	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	—	2	1	no	—
155	#10623494	—	2	1	no	—
156	#10623494	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	1	no	—
163	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₈₉	—	4	1	no	—
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₂×D₈₉	—	4	1	no	—
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	4	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	4	1	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	—	4	1	no	—
174	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
175	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
176	Dic₈₉	—	2	1	no	—
177	Dic₈₉	—	2	1	no	—
178	Dic₈₉	—	2	1	no	—
179	Dic₈₉	—	2	1	no	—
180	Dic₈₉	—	4	1	no	—
181	Dic₈₉	—	4	1	no	—
182	C₂×C₈₉	—	4	1	no	—
183	C₂×C₈₉	—	8	1	no	—
184	C₂×C₈₉	#535	1	1	yes	characteristic
185	C₂×C₈₉	#577	1	1	yes	characteristic
186	C₂×C₈₉	#577	1	1	yes	characteristic
187	D₈₉	—	8	1	no	—
188	D₈₉	—	8	1	no	—
189	#3221	—	89	1	no	maximal, Sylow
190	C₈₉	#3221	1	1	yes	characteristic, complemented, Sylow
191	#540	—	89	1	no	—
192	#536	—	89	1	no	—
193	#496	—	89	1	no	—
194	#492	—	89	1	no	—
195	#491	—	89	1	no	—
196	#488	—	89	1	no	—
197	#463	—	89	1	no	—
198	#463	—	89	1	no	—
199	#461	—	89	1	no	—
200	#461	—	89	1	no	—
201	#449	—	89	1	no	—
202	#449	—	89	1	no	—
203	#448	—	89	1	no	—
204	#448	D₈₉	1	1	yes	characteristic, complemented
205	#433	—	89	1	no	—
206	C₂×#41	—	89	1	no	—
207	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
208	C₂×SD₁₆	—	89	1	no	—
209	C₂×SD₁₆	—	89	1	no	—
210	C₂×SD₁₆	—	89	1	no	—
211	C₂×SD₁₆	—	89	1	no	—
212	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
213	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
214	C₂×D₈	—	89	1	no	—
215	C₂×D₈	—	89	1	no	—
216	C₄⋊Q₈	—	89	1	no	—
217	C₄⋊Q₈	—	89	1	no	—
218	#127	—	89	1	no	—
219	#125	—	89	1	no	—
220	#124	—	89	1	no	—
221	#124	—	89	1	no	—
222	#123	—	178	1	no	—
223	#123	—	178	1	no	—
224	#123	—	178	1	no	—
225	#123	—	178	1	no	—
226	C₂²⋊Q₈	—	178	1	no	—
227	C₂²⋊Q₈	—	178	1	no	—
228	#121	—	89	1	no	—
229	#121	—	178	1	no	—
230	#121	—	178	1	no	—
231	#121	C₂×D₈₉	1	1	yes	characteristic
232	C₄×D₄	—	89	1	no	—
233	C₄×D₄	—	89	1	no	—
234	#117	—	89	1	no	—
235	C₄⋊C₈	—	89	1	no	—
236	C₄⋊C₈	—	89	1	no	—
237	#103	—	89	1	no	—
238	#103	—	89	1	no	—
239	#103	—	89	1	no	—
240	#103	—	89	1	no	—
241	#103	C₂×D₈₉	1	1	yes	characteristic
242	#103	C₂×D₈₉	1	1	yes	characteristic
243	#102	—	89	1	no	—
244	#102	—	89	1	no	—
245	C₂²⋊C₈	—	89	1	no	—
246	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
247	#97	—	89	1	no	—
248	C₄×C₈	—	89	1	no	—
249	#41	—	356	1	no	—
250	#41	—	356	1	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	89	1	no	—
255	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	89	1	no	—
259	C₂×D₄	—	89	1	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	—	178	1	no	—
262	C₂×D₄	—	178	1	no	—
263	C₂×D₄	—	178	1	no	—
264	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	178	1	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	—	178	1	no	—
269	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
270	SD₁₆	—	4	1	no	—
271	SD₁₆	—	4	1	no	—
272	SD₁₆	—	356	1	no	—
273	SD₁₆	—	356	1	no	—
274	SD₁₆	—	356	1	no	—
275	SD₁₆	—	356	1	no	—
276	D₈	—	356	1	no	—
277	D₈	—	356	1	no	—
278	C₂×C₈	—	89	1	no	—
279	C₂×C₈	—	89	1	no	—
280	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
282	C₄⋊C₄	—	89	1	no	—
283	C₄⋊C₄	—	89	1	no	—
284	C₄⋊C₄	—	89	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	—	178	1	no	—
287	C₄⋊C₄	—	178	1	no	—
288	C₄⋊C₄	—	178	1	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₄⋊C₄	—	178	1	no	—
291	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	89	1	no	—
294	C₂²⋊C₄	—	89	1	no	—
295	C₂²⋊C₄	—	178	1	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₂²⋊C₄	—	178	1	no	—
298	C₂²⋊C₄	—	178	1	no	—
299	C₂²⋊C₄	—	178	1	no	—
300	C₂²⋊C₄	—	178	1	no	—
301	C₂²⋊C₄	—	178	1	no	—
302	C₄²	—	89	1	no	—
303	C₄²	—	89	1	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	178	1	no	—
306	C₂³	—	178	1	no	—
307	C₂³	D₄×D₈₉	1	1	yes	characteristic
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	178	1	no	—
314	Q₈	—	178	1	no	—
315	D₄	—	2	1	no	—
316	D₄	—	2	1	no	—
317	D₄	—	8	1	no	—
318	D₄	—	356	2	no	—
319	D₄	—	178	1	no	—
320	D₄	—	178	1	no	—
321	D₄	—	178	1	no	—
322	D₄	—	178	1	no	—
323	D₄	—	712	1	no	—
324	D₄	—	712	1	no	—
325	D₄	—	712	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	89	1	no	—
333	C₂×C₄	—	89	1	no	—
334	C₂×C₄	—	89	1	no	—
335	C₂×C₄	—	89	1	no	—
336	C₂×C₄	—	356	2	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	—	178	1	no	—
339	C₂×C₄	—	712	1	no	—
340	C₂×C₄	—	712	1	no	—
341	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
342	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	178	1	no	—
346	C₈	—	178	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	356	1	no	—
354	C₂²	—	356	1	no	—
355	C₂²	—	356	1	no	—
356	C₂²	—	356	1	no	—
357	C₂²	—	356	1	no	—
358	C₂²	—	356	1	no	—
359	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	4	1	no	—
362	C₄	—	4	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	356	1	no	—
365	C₄	—	356	1	no	—
366	C₄	—	178	1	no	—
367	C₄	—	178	1	no	—
368	C₄	—	178	1	no	—
369	C₄	—	178	1	no	—
370	C₄	#15668057	1	1	yes	characteristic
371	C₄	#15668057	1	1	yes	characteristic
372	C₂	—	4	1	no	—
373	C₂	—	8	1	no	—
374	C₂	—	712	1	no	—
375	C₂	—	712	1	no	—
376	C₂	#19152163	1	1	yes	characteristic
377	C₂	#19152404	1	1	yes	characteristic
378	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	#448×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#448	D₈₉
hypocenter	C₈₉	#3221
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #448

Complements

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

N	H ≅ G/N	type
1. #19152201	C₂	semidirect
2. #19152172	C₂	semidirect
3. #19151984	C₂	semidirect
4. #19151969	C₂	semidirect
5. #19151965	C₂	semidirect
6. #19151954	C₂	semidirect
7. #19151816	C₂	semidirect
8. #19151815	C₂	semidirect
9. #19151802	C₂	semidirect
10. #19151801	C₂	semidirect
11. #19151745	C₂	semidirect
12. #19151741	C₂	semidirect
13. #19151737	C₂	semidirect
14. #19151656	C₂	semidirect
15. #448×C₈₉	C₂	semidirect
190. C₈₉	#3221	semidirect
204. #448	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152201	2
#19152172	2
#19151984	2
#19151969	2
#19151965	2
#19151954	2
#19151816	2
#19151815	2
#19151802	2
#19151801	2
#19151745	2
#19151741	2
#19151737	2
#19151656	2
#448×C₈₉	2
#3221	89