Subgroups — #26034855

Order 11392.

912 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	C₈₉⋊#564	C₂	1	2	yes	maximal, complemented
2	#19152348	C₂	1	2	yes	maximal, complemented
3	#19152326	C₂	1	2	yes	maximal, complemented
4	#19152305	C₂	1	2	yes	maximal, complemented
5	#19152288	C₂	1	2	yes	maximal, complemented
6	#19152288	C₂	1	2	yes	maximal, complemented
7	#19152285	C₂	1	4	yes	maximal, complemented
8	#19152227	C₂	1	2	yes	maximal, complemented
9	#19152226	C₂	1	2	yes	maximal, complemented
10	#19152210	C₂	1	2	yes	maximal, complemented
11	#19152205	C₂	1	4	yes	maximal, complemented
12	#19152193	C₂	1	2	yes	maximal
13	#19152155	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15668004	C₂	1	1	yes	maximal, characteristic, complemented
15	#529×C₈₉	C₂	1	1	yes	maximal, characteristic
16	#15668012	C₂²	1	1	yes	characteristic
17	#15668012	C₂²	1	1	yes	characteristic
18	#15668012	C₂²	1	2	yes	—
19	#15668011	C₂²	1	4	yes	—
20	#15668010	C₂²	1	4	yes	—
21	#15668005	C₂²	1	4	yes	—
22	#15668005	C₂²	1	4	yes	—
23	#15668005	C₂²	1	4	yes	—
24	#15668005	C₂²	1	4	yes	—
25	#15668004	C₂²	1	2	yes	—
26	#15668004	C₂²	1	2	yes	—
27	#15668004	C₂²	1	2	yes	—
28	#15668004	C₂²	1	2	yes	—
29	#15668004	C₂²	1	2	yes	—
30	#15668004	C₂²	1	2	yes	—
31	#15668003	C₂²	1	4	yes	—
32	#15668003	C₂²	1	4	yes	—
33	#15668002	C₂²	1	4	yes	—
34	#15668002	C₂²	1	4	yes	—
35	#15668002	C₂²	1	4	yes	—
36	#15668002	C₂²	1	4	yes	—
37	#15668002	C₂²	1	4	yes	—
38	#15668002	C₂²	1	4	yes	—
39	#15667993	C₂²	1	4	yes	—
40	#15667993	C₂²	1	4	yes	—
41	#15667991	C₂²	1	2	yes	—
42	#15667991	C₂²	1	2	yes	—
43	#15667991	C₂²	1	2	yes	—
44	#15667991	C₂²	1	2	yes	—
45	#15667989	C₂²	1	4	yes	—
46	#15667989	C₂²	1	4	yes	—
47	#15667989	C₂²	1	4	yes	—
48	#15667989	C₂²	1	4	yes	—
49	C₄×Dic₁₇₈	C₂²	1	2	yes	—
50	C₄×Dic₁₇₈	C₂²	1	2	yes	—
51	C₄×Dic₁₇₈	C₂²	1	2	yes	—
52	C₄×Dic₁₇₈	C₂²	1	2	yes	—
53	#15667977	C₂²	1	4	yes	—
54	D₄×Dic₈₉	C₂²	1	4	yes	—
55	#15667971	C₂²	1	2	yes	—
56	#15667971	C₂²	1	2	yes	—
57	#15667968	C₂²	1	4	yes	—
58	#15667968	C₂²	1	4	yes	—
59	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
60	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
61	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
62	C₂×#13083553	C₂²	1	2	yes	—
63	C₂×#13083553	C₂²	1	2	yes	—
64	#126×C₈₉	C₂²	1	2	yes	—
65	#125×C₈₉	C₂²	1	1	yes	characteristic
66	#124×C₈₉	C₂²	1	1	yes	characteristic
67	#123×C₈₉	C₂²	1	2	yes	—
68	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
69	#121×C₈₉	C₂²	1	1	yes	characteristic
70	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
71	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
72	C₄×D₄×C₈₉	C₂²	1	2	yes	—
73	#117×C₈₉	C₂²	1	2	yes	—
74	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
75	C₂×Dic₁₇₈	C₂³	1	2	yes	—
76	C₂×Dic₁₇₈	C₂³	1	4	yes	—
77	C₂×Dic₁₇₈	C₂³	1	2	yes	—
78	C₂²×Dic₈₉	C₂³	1	2	yes	—
79	C₂²×Dic₈₉	C₂³	1	4	yes	—
80	C₂²×Dic₈₉	C₂³	1	2	yes	—
81	C₄⋊Dic₈₉	C₂³	1	2	yes	—
82	C₄⋊Dic₈₉	C₂³	1	2	yes	—
83	C₄⋊Dic₈₉	C₂³	1	2	yes	—
84	C₄⋊Dic₈₉	C₂³	1	2	yes	—
85	C₄⋊Dic₈₉	C₂³	1	4	yes	—
86	C₄⋊Dic₈₉	C₂³	1	2	yes	—
87	C₄⋊Dic₈₉	C₂³	1	2	yes	—
88	C₄⋊Dic₈₉	C₂³	1	2	yes	—
89	C₄⋊Dic₈₉	C₂³	1	4	yes	—
90	C₄⋊Dic₈₉	C₂³	1	2	yes	—
91	#13083553	C₂³	1	4	yes	—
92	#13083553	C₂³	1	4	yes	—
93	#13083553	C₂³	1	4	yes	—
94	#13083553	C₂³	1	4	yes	—
95	#13083553	C₂³	1	4	yes	—
96	#13083553	C₂³	1	4	yes	—
97	#13083553	C₂³	1	4	yes	—
98	#13083553	C₂³	1	4	yes	—
99	#13083553	C₂³	1	4	yes	—
100	#13083553	C₂³	1	4	yes	—
101	#13083553	C₂³	1	4	yes	—
102	#13083553	C₂³	1	4	yes	—
103	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
104	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
105	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
106	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
107	C₄×Dic₈₉	C₂³	1	4	yes	—
108	C₄×Dic₈₉	C₂³	1	4	yes	—
109	C₄×Dic₈₉	C₂³	1	4	yes	—
110	C₄×Dic₈₉	C₂³	1	4	yes	—
111	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×D₄×C₈₉	C₂³	1	2	yes	—
113	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
116	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
118	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
119	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
120	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
121	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
122	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
123	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
124	C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
125	C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
126	C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
127	C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
128	C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
129	C₄²×C₈₉	C₂³	1	1	yes	characteristic
130	C₄²×C₈₉	C₂³	1	1	yes	characteristic
131	C₄²×C₈₉	C₂³	1	2	yes	—
132	C₄²×C₈₉	C₂³	1	1	yes	characteristic
133	C₄²×C₈₉	C₂³	1	1	yes	characteristic
134	C₂³×C₈₉	C₂⁴	1	2	yes	—
135	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
136	Q₈×C₈₉	—	4	1	no	—
137	D₄×C₈₉	—	2	4	no	—
138	D₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	4	no	—
142	C₂×C₄×C₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
149	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
150	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
151	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
152	C₂×Dic₈₉	—	2	4	no	—
153	C₂×Dic₈₉	—	4	4	no	—
154	C₂×Dic₈₉	—	4	2	no	—
155	C₂×Dic₈₉	C₂⁴	1	4	yes	—
156	C₂×Dic₈₉	C₂⁴	1	4	yes	—
157	C₂×Dic₈₉	C₂⁴	1	4	yes	—
158	C₂×Dic₈₉	C₂⁴	1	4	yes	—
159	Dic₁₇₈	—	2	4	no	—
160	Dic₁₇₈	—	4	2	no	—
161	Dic₁₇₈	—	4	4	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	2	no	—
165	C₂²×C₈₉	—	2	2	no	—
166	C₂²×C₈₉	—	2	2	no	—
167	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
168	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	2	2	no	—
173	C₄×C₈₉	—	2	2	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	2	2	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
178	Dic₈₉	—	2	4	no	—
179	Dic₈₉	—	2	4	no	—
180	Dic₈₉	—	2	4	no	—
181	Dic₈₉	—	2	4	no	—
182	C₂×C₈₉	—	2	2	no	—
183	C₂×C₈₉	—	4	2	no	—
184	C₂×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
185	C₂×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
186	C₂×C₈₉	#585	1	1	yes	characteristic
187	#3563	—	89	1	no	maximal, Sylow
188	C₈₉	#3563	1	1	yes	characteristic, complemented, Sylow
189	#564	—	89	2	no	—
190	#563	—	89	2	no	—
191	#557	—	89	2	no	—
192	#554	—	89	2	no	—
193	#552	—	89	4	no	—
194	#552	—	89	2	no	—
195	#552	—	89	2	no	—
196	#544	—	89	2	no	—
197	#544	—	89	2	no	—
198	#541	—	89	4	no	—
199	#541	—	89	2	no	—
200	#539	—	89	2	no	—
201	#533	—	89	1	no	—
202	#529	D₈₉	1	1	yes	characteristic
203	C₂×#125	—	89	1	no	—
204	C₄⋊Q₈	—	89	4	no	—
205	C₄⋊Q₈	—	89	4	no	—
206	C₄⋊Q₈	—	89	1	no	—
207	C₄⋊Q₈	—	89	1	no	—
208	C₄⋊Q₈	—	89	2	no	—
209	#126	—	89	4	no	—
210	#126	—	89	4	no	—
211	#126	—	89	4	no	—
212	#126	—	89	4	no	—
213	#126	C₂×D₈₉	1	2	yes	—
214	#125	—	89	4	no	—
215	#125	—	89	4	no	—
216	#125	—	89	4	no	—
217	#125	—	89	4	no	—
218	#125	—	89	4	no	—
219	#125	—	89	4	no	—
220	#125	—	89	4	no	—
221	#125	—	89	4	no	—
222	#125	—	89	2	no	—
223	#125	—	89	2	no	—
224	#125	—	89	2	no	—
225	#125	—	89	2	no	—
226	#125	—	89	2	no	—
227	#125	—	89	2	no	—
228	#125	C₂×D₈₉	1	1	yes	characteristic
229	#124	C₂×D₈₉	1	1	yes	characteristic
230	#123	—	89	4	no	—
231	#123	—	89	4	no	—
232	#123	C₂×D₈₉	1	2	yes	—
233	C₂²⋊Q₈	—	89	2	no	—
234	C₂²⋊Q₈	—	89	2	no	—
235	C₂²⋊Q₈	—	89	2	no	—
236	C₂²⋊Q₈	—	89	2	no	—
237	C₂²⋊Q₈	—	89	4	no	—
238	C₂²⋊Q₈	—	89	4	no	—
239	C₂²⋊Q₈	—	89	4	no	—
240	C₂²⋊Q₈	—	89	4	no	—
241	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
242	#121	C₂×D₈₉	1	1	yes	characteristic
243	C₄×Q₈	—	89	4	no	—
244	C₄×Q₈	—	89	2	no	—
245	C₄×Q₈	—	89	2	no	—
246	C₄×Q₈	—	89	2	no	—
247	C₄×Q₈	—	89	2	no	—
248	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
249	C₄×D₄	—	89	4	no	—
250	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
251	C₄×D₄	C₂×D₈₉	1	2	yes	—
252	#117	—	89	2	no	—
253	#117	—	89	2	no	—
254	#117	—	89	4	no	—
255	#117	—	89	4	no	—
256	#117	C₂×D₈₉	1	2	yes	—
257	C₂×C₄⋊C₄	—	89	2	no	—
258	C₂×C₄⋊C₄	—	89	2	no	—
259	C₂×C₄⋊C₄	—	89	1	no	—
260	C₂×C₄⋊C₄	—	89	1	no	—
261	C₂×C₄⋊C₄	—	89	2	no	—
262	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
263	C₂×Q₈	—	89	2	no	—
264	C₂×Q₈	—	89	4	no	—
265	C₂×Q₈	—	89	2	no	—
266	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×D₄	C₂²×D₈₉	1	2	yes	—
268	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
269	C₂²×C₄	—	89	2	no	—
270	C₂²×C₄	—	89	4	no	—
271	C₂²×C₄	—	89	2	no	—
272	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
273	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
274	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
275	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	—	89	2	no	—
277	C₄⋊C₄	—	89	2	no	—
278	C₄⋊C₄	—	89	2	no	—
279	C₄⋊C₄	—	89	4	no	—
280	C₄⋊C₄	—	89	2	no	—
281	C₄⋊C₄	—	89	4	no	—
282	C₄⋊C₄	—	89	4	no	—
283	C₄⋊C₄	—	89	4	no	—
284	C₄⋊C₄	—	89	4	no	—
285	C₄⋊C₄	—	89	2	no	—
286	C₄⋊C₄	—	89	4	no	—
287	C₄⋊C₄	—	89	2	no	—
288	C₄⋊C₄	—	89	4	no	—
289	C₄⋊C₄	—	89	4	no	—
290	C₄⋊C₄	—	89	2	no	—
291	C₄⋊C₄	—	89	4	no	—
292	C₄⋊C₄	—	89	4	no	—
293	C₄⋊C₄	—	89	4	no	—
294	C₄⋊C₄	—	89	4	no	—
295	C₄⋊C₄	—	89	4	no	—
296	C₄⋊C₄	—	89	2	no	—
297	C₄⋊C₄	—	89	4	no	—
298	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
299	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
300	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
301	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
302	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
303	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
304	C₂²⋊C₄	—	89	4	no	—
305	C₂²⋊C₄	—	89	4	no	—
306	C₂²⋊C₄	—	89	4	no	—
307	C₂²⋊C₄	—	89	4	no	—
308	C₂²⋊C₄	C₂²×D₈₉	1	2	yes	—
309	C₂²⋊C₄	C₂²×D₈₉	1	2	yes	—
310	C₂²⋊C₄	C₂²×D₈₉	1	2	yes	—
311	C₂²⋊C₄	C₂²×D₈₉	1	2	yes	—
312	C₂²⋊C₄	C₂²×D₈₉	1	2	yes	—
313	C₄²	—	89	4	no	—
314	C₄²	—	89	4	no	—
315	C₄²	—	89	4	no	—
316	C₄²	—	89	4	no	—
317	C₄²	C₂²×D₈₉	1	1	yes	characteristic
318	C₄²	C₂²×D₈₉	1	1	yes	characteristic
319	C₄²	C₂²×D₈₉	1	1	yes	characteristic
320	C₄²	C₂²×D₈₉	1	1	yes	characteristic
321	C₄²	C₂²×D₈₉	1	2	yes	—
322	C₂³	C₂³×D₈₉	1	2	yes	—
323	C₂³	C₂³×D₈₉	1	1	yes	characteristic
324	Q₈	—	4	1	no	—
325	Q₈	—	356	2	no	—
326	Q₈	—	356	4	no	—
327	Q₈	—	178	4	no	—
328	D₄	—	2	4	no	—
329	D₄	—	4	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	89	4	no	—
335	C₂×C₄	—	89	4	no	—
336	C₂×C₄	—	89	4	no	—
337	C₂×C₄	—	89	4	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	356	4	no	—
340	C₂×C₄	—	178	4	no	—
341	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
342	C₂×C₄	C₂³×D₈₉	1	2	yes	—
343	C₂×C₄	C₂³×D₈₉	1	2	yes	—
344	C₂×C₄	C₂³×D₈₉	1	2	yes	—
345	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
346	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
347	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
348	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
349	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	#15668059	1	2	yes	—
356	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	2	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	2	no	—
365	C₄	—	178	4	no	—
366	C₄	—	178	4	no	—
367	C₄	—	178	4	no	—
368	C₄	—	178	4	no	—
369	C₄	#15668059	1	2	yes	—
370	C₂	—	2	2	no	—
371	C₂	—	4	2	no	—
372	C₂	C₂×#15668059	1	1	yes	characteristic
373	C₂	C₂×#15668059	1	1	yes	characteristic
374	C₂	#19152434	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#529×C₈₉	C₂
prime radical	#121×C₈₉	C₂²
hypercenter	#529	D₈₉
hypocenter	C₈₉	#3563
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #529

Complements

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

N	H ≅ G/N	type
1. C₈₉⋊#564	C₂	semidirect
2. #19152348	C₂	semidirect
3. #19152326	C₂	semidirect
4. #19152305	C₂	semidirect
5. #19152288	C₂	semidirect
6. #19152288	C₂	semidirect
7. #19152285	C₂	semidirect
8. #19152227	C₂	semidirect
9. #19152226	C₂	semidirect
10. #19152210	C₂	semidirect
11. #19152205	C₂	semidirect
13. #19152155	C₂	semidirect
14. C₂×#15668004	C₂	semidirect
188. C₈₉	#3563	semidirect

Maximal subgroups

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

subgroup	index
C₈₉⋊#564	2
#19152348	2
#19152326	2
#19152305	2
#19152288	2
#19152285	2
#19152227	2
#19152226	2
#19152210	2
#19152205	2
#19152193	2
#19152155	2
C₂×#15668004	2
#529×C₈₉	2
#3563	89