Subgroups — #26029108

Order 11392.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#19152074	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152070	C₂	1	1	yes	maximal, characteristic
4	#19151700	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151699	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151697	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151689	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13083561	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13083562	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151683	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151635	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151605	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151601	C₂	1	1	yes	maximal, characteristic
14	#19151555	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	SD₁₆×D₈₉	C₄	1	8	yes	complemented
22	C₂×#13083557	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
25	C₄×Dic₁₇₈	—	2	1	no	—
26	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
27	#15667977	—	2	1	no	—
28	#15667977	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
30	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
31	#15667974	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₉	—	2	1	no	—
33	#15667972	—	2	1	no	—
34	#15667970	—	2	1	no	—
35	#15667970	C₂²	1	1	yes	characteristic
36	#15667970	C₂²	1	1	yes	characteristic
37	#15667969	—	2	1	no	—
38	#15667969	C₂²	1	1	yes	characteristic
39	C₂×#13083551	—	2	1	no	—
40	C₂²⋊C₄×D₈₉	—	2	1	no	—
41	#15667946	C₂²	1	1	yes	characteristic, complemented
42	#15667943	C₂²	1	1	yes	characteristic, complemented
43	#15667943	C₂²	1	1	yes	characteristic, complemented
44	#15667937	C₂²	1	1	yes	characteristic, complemented
45	#15667936	C₂²	1	1	yes	characteristic, complemented
46	#15667935	C₂²	1	1	yes	characteristic, complemented
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667932	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic, complemented
51	C₄×#10623494	C₂²	1	1	yes	characteristic, complemented
52	C₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
53	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
55	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
56	#107×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	#97×C₈₉	C₂²	1	1	yes	characteristic, complemented
60	C₂³×D₈₉	—	2	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	Q₈×D₈₉	—	2	2	no	—
64	Q₈×D₈₉	C₂×C₄	1	4	yes	complemented
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	D₄×D₈₉	—	2	2	no	—
68	D₄×D₈₉	C₂×C₄	1	4	yes	—
69	C₂²×Dic₈₉	—	2	1	no	—
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	#13083563	C₂×C₄	1	4	yes	complemented
75	#13083562	C₂×C₄	1	4	yes	complemented
76	#13083561	C₂×C₄	1	4	yes	complemented
77	#13083557	—	2	2	no	—
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₈×D₈₉	C₂×C₄	1	4	yes	complemented
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	#13083553	D₄	1	1	yes	characteristic
87	#13083553	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	D₄	1	1	yes	characteristic
94	#13083551	D₄	1	1	yes	characteristic
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	—	2	1	no	—
98	C₄×Dic₈₉	—	2	1	no	—
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×D₄×C₈₉	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	SD₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₄²×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	C₂³	1	1	yes	characteristic
113	C₂³×C₈₉	—	2	1	no	—
114	C₂²×D₈₉	—	2	1	no	—
115	C₂²×D₈₉	—	2	4	no	—
116	C₂²×D₈₉	—	8	1	no	—
117	C₂²×D₈₉	—	8	1	no	—
118	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
119	Q₈×C₈₉	—	2	1	no	—
120	Q₈×C₈₉	C₂²×C₄	1	2	yes	—
121	D₄×C₈₉	—	2	1	no	—
122	D₄×C₈₉	C₂²×C₄	1	2	yes	—
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	—	8	1	no	—
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	2	4	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	8	1	no	—
135	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	C₂²×C₄	1	2	yes	—
140	C₄×D₈₉	—	2	4	no	—
141	C₄×D₈₉	—	8	1	no	—
142	C₄×D₈₉	—	8	1	no	—
143	C₄×D₈₉	C₂²×C₄	1	2	yes	—
144	C₄×D₈₉	#41	1	2	yes	—
145	Dic₁₇₈	—	2	1	no	—
146	Dic₁₇₈	—	2	4	no	—
147	Dic₁₇₈	C₂²×C₄	1	2	yes	—
148	C₈×C₈₉	—	2	1	no	—
149	C₈×C₈₉	C₂²×C₄	1	2	yes	—
150	#10623494	—	2	1	no	—
151	#10623494	C₂²×C₄	1	2	yes	—
152	C₂²×C₈₉	—	2	2	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	—	4	1	no	—
155	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	1	no	—
158	C₂×D₈₉	—	2	2	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	—	8	4	no	—
162	C₂×D₈₉	C₄×D₄	1	2	yes	—
163	C₄×C₈₉	—	2	2	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	C₂³×C₄	1	1	yes	characteristic
169	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
170	Dic₈₉	—	2	2	no	—
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	4	1	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	C₄×D₄	1	2	yes	—
176	C₂×C₈₉	—	4	2	no	—
177	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₈₉	#577	1	1	yes	characteristic
179	C₂×C₈₉	#577	1	1	yes	characteristic
180	D₈₉	—	2	2	no	—
181	D₈₉	—	4	2	no	—
182	#2949	—	89	1	no	maximal, Sylow
183	C₈₉	#2949	1	1	yes	characteristic, complemented, Sylow
184	C₂²×SD₁₆	—	89	1	no	—
185	#519	—	89	1	no	—
186	#518	—	89	1	no	—
187	#440	—	89	1	no	—
188	#440	—	89	1	no	—
189	#440	—	89	1	no	—
190	#440	D₈₉	1	1	yes	characteristic, complemented
191	C₄×SD₁₆	—	89	1	no	—
192	C₄×SD₁₆	—	89	1	no	—
193	C₄×SD₁₆	—	89	1	no	—
194	C₄×SD₁₆	—	89	1	no	—
195	#427	—	89	1	no	—
196	#419	—	89	1	no	—
197	#418	—	89	1	no	—
198	#405	—	89	1	no	—
199	C₂²×Q₈	—	89	1	no	—
200	C₂²×D₄	—	89	1	no	—
201	C₂×SD₁₆	—	89	1	no	—
202	C₂×SD₁₆	—	89	1	no	—
203	C₂×SD₁₆	—	89	8	no	—
204	C₂×SD₁₆	—	89	1	no	—
205	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
206	C₂×M₁₆	—	89	1	no	—
207	C₂²×C₈	—	89	1	no	—
208	C₄×Q₈	—	89	1	no	—
209	C₄×Q₈	—	89	1	no	—
210	C₄×Q₈	—	89	1	no	—
211	C₄×Q₈	—	178	1	no	—
212	C₄×Q₈	—	178	1	no	—
213	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	89	1	no	—
216	C₄×D₄	—	89	1	no	—
217	C₄×D₄	—	178	1	no	—
218	C₄×D₄	—	178	1	no	—
219	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
220	#117	—	89	1	no	—
221	#117	—	89	1	no	—
222	#117	—	89	1	no	—
223	#117	—	178	1	no	—
224	#117	—	178	1	no	—
225	C₂×C₂²⋊C₄	—	178	1	no	—
226	C₂×C₂²⋊C₄	—	178	1	no	—
227	#107	—	89	1	no	—
228	#107	C₂×D₈₉	1	1	yes	characteristic, complemented
229	#106	—	89	1	no	—
230	#106	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	C₂×D₈₉	1	1	yes	characteristic
239	#97	—	89	1	no	—
240	#97	C₂×D₈₉	1	1	yes	characteristic, complemented
241	C₄×C₈	—	89	1	no	—
242	C₄×C₈	—	89	1	no	—
243	C₂⁴	—	178	1	no	—
244	C₂×Q₈	—	89	4	no	—
245	C₂×Q₈	—	89	1	no	—
246	C₂×Q₈	—	178	1	no	—
247	C₂×Q₈	—	178	2	no	—
248	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×D₄	—	89	4	no	—
250	C₂×D₄	—	89	1	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	C₂²×C₄	—	178	1	no	—
259	C₂²×C₄	—	178	1	no	—
260	SD₁₆	—	89	4	no	—
261	SD₁₆	—	89	4	no	—
262	SD₁₆	—	89	4	no	—
263	SD₁₆	C₄×D₈₉	1	4	yes	complemented
264	M₁₆	—	178	2	no	—
265	C₂×C₈	—	89	1	no	—
266	C₂×C₈	—	89	4	no	—
267	C₂×C₈	—	89	1	no	—
268	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	89	1	no	—
274	C₄⋊C₄	—	89	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	89	1	no	—
282	C₂²⋊C₄	—	89	1	no	—
283	C₂²⋊C₄	—	178	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	—	178	1	no	—
292	C₄²	—	178	1	no	—
293	C₄²	—	178	1	no	—
294	C₄²	—	178	1	no	—
295	C₄²	C₂²×D₈₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	89	1	no	—
298	C₂³	—	178	4	no	—
299	C₂³	—	178	1	no	—
300	C₂³	—	712	1	no	—
301	C₂³	—	712	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	89	2	no	—
304	Q₈	—	178	4	no	—
305	Q₈	—	178	1	no	—
306	Q₈	C₂×C₄×D₈₉	1	2	yes	—
307	D₄	—	2	1	no	—
308	D₄	—	89	2	no	—
309	D₄	—	178	4	no	—
310	D₄	—	178	1	no	—
311	D₄	C₂×C₄×D₈₉	1	2	yes	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	89	1	no	—
317	C₂×C₄	—	89	2	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	89	1	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	178	4	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	712	1	no	—
326	C₂×C₄	—	712	1	no	—
327	C₂×C₄	—	712	1	no	—
328	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	89	2	no	—
333	C₈	—	178	1	no	—
334	C₈	C₂×C₄×D₈₉	1	2	yes	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	89	2	no	—
339	C₂²	—	356	1	no	—
340	C₂²	—	356	1	no	—
341	C₂²	—	178	2	no	—
342	C₂²	—	178	1	no	—
343	C₂²	—	178	1	no	—
344	C₂²	—	712	4	no	—
345	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	89	2	no	—
352	C₄	—	356	1	no	—
353	C₄	—	356	1	no	—
354	C₄	—	178	1	no	—
355	C₄	—	178	1	no	—
356	C₄	—	178	2	no	—
357	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
358	C₄	#41×D₈₉	1	1	yes	characteristic
359	C₂	—	4	2	no	—
360	C₂	—	356	2	no	—
361	C₂	—	178	2	no	—
362	C₂	C₄×D₄×D₈₉	1	1	yes	characteristic
363	C₂	#19152407	1	1	yes	characteristic
364	C₂	#19152408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#440×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₄×D₈₉	C₂²
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#440	D₈₉
hypocenter	C₈₉	#2949
LCS	G ▷ 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
2. #19152074	C₂	semidirect
4. #19151700	C₂	semidirect
5. #19151699	C₂	semidirect
6. #19151697	C₂	semidirect
7. #19151689	C₂	semidirect
8. C₄×#13083561	C₂	semidirect
9. C₄×#13083562	C₂	semidirect
10. #19151683	C₂	semidirect
11. #19151635	C₂	semidirect
12. #19151605	C₂	semidirect
14. #19151555	C₂	semidirect
15. #440×C₈₉	C₂	semidirect
21. SD₁₆×D₈₉	C₄	semidirect
24. Q₈×Dic₈₉	C₂²	semidirect
26. C₄×Dic₁₇₈	C₂²	semidirect
28. #15667977	C₂²	semidirect
41. #15667946	C₂²	semidirect
42. #15667943	C₂²	semidirect
43. #15667943	C₂²	semidirect
44. #15667937	C₂²	semidirect
45. #15667936	C₂²	semidirect
46. #15667935	C₂²	semidirect
50. #15667922	C₂²	semidirect
51. C₄×#10623494	C₂²	semidirect
52. C₈×Dic₈₉	C₂²	semidirect
54. C₄×Q₈×C₈₉	C₂²	semidirect
56. #107×C₈₉	C₂²	semidirect
57. #103×C₈₉	C₂²	semidirect
59. #97×C₈₉	C₂²	semidirect
64. Q₈×D₈₉	C₂×C₄	semidirect
74. #13083563	C₂×C₄	semidirect
75. #13083562	C₂×C₄	semidirect
76. #13083561	C₂×C₄	semidirect
80. C₈×D₈₉	C₂×C₄	semidirect
104. SD₁₆×C₈₉	C₂×C₄	semidirect
183. C₈₉	#2949	semidirect
190. #440	D₈₉	semidirect
213. C₄×Q₈	C₂×D₈₉	semidirect
228. #107	C₂×D₈₉	semidirect
234. #103	C₂×D₈₉	semidirect
240. #97	C₂×D₈₉	semidirect
263. SD₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₁₆×D₈₉	2
#19152074	2
#19152070	2
#19151700	2
#19151699	2
#19151697	2
#19151689	2
C₄×#13083561	2
C₄×#13083562	2
#19151683	2
#19151635	2
#19151605	2
#19151601	2
#19151555	2
#440×C₈₉	2
#2949	89