Subgroups — #26023280

Order 11392.

576 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15667990	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667969	C₂	1	1	yes	maximal, characteristic
4	#19151502	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151493	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151490	C₂	1	1	yes	maximal, characteristic
7	#19151489	C₂	1	1	yes	maximal, characteristic
8	#19151489	C₂	1	1	yes	maximal, characteristic
9	#19151488	C₂	1	1	yes	maximal, characteristic
10	#19151484	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151482	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083553	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151449	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151445	C₂	1	1	yes	maximal, characteristic
15	C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
18	#15667990	C₄	1	8	yes	complemented
19	C₄×Dic₁₇₈	—	2	4	no	—
20	#15667969	—	2	4	no	—
21	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
22	C₂×#13083553	C₂²	1	1	yes	characteristic
23	C₂×#13083553	C₂²	1	1	yes	characteristic
24	C₂×#13083553	C₂²	1	1	yes	characteristic
25	C₂×#13083553	C₂²	1	1	yes	characteristic
26	C₂×#13083553	C₂²	1	1	yes	characteristic
27	C₂×#13083553	C₂²	1	1	yes	characteristic
28	C₂×#13083551	C₂²	1	1	yes	characteristic
29	C₂×#13083551	C₂²	1	1	yes	characteristic
30	C₂×#13083551	C₂²	1	1	yes	characteristic
31	C₂×#13083551	C₂²	1	1	yes	characteristic
32	C₂×#13083551	C₂²	1	1	yes	characteristic
33	C₂×#13083551	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
39	#15667919	C₂²	1	1	yes	characteristic
40	#15667919	C₂²	1	1	yes	characteristic
41	#15667919	C₂²	1	1	yes	characteristic
42	#15667919	C₂²	1	1	yes	characteristic
43	#15667919	C₂²	1	1	yes	characteristic
44	#15667919	C₂²	1	1	yes	characteristic
45	#15667919	C₂²	1	1	yes	characteristic
46	#15667919	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
52	#95×C₈₉	C₂²	1	1	yes	characteristic
53	#95×C₈₉	C₂²	1	1	yes	characteristic
54	C₂³×D₈₉	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₇₈	—	2	2	no	—
56	C₂×Dic₁₇₈	—	2	2	no	—
57	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	—
58	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
60	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₈₉	—	2	2	no	—
66	C₂×C₄×D₈₉	—	2	2	no	—
67	C₂×C₄×D₈₉	C₂×C₄	1	4	yes	—
68	C₄⋊Dic₈₉	—	2	2	no	—
69	#13083553	—	2	2	no	—
70	#13083553	—	2	2	no	—
71	#13083553	—	2	2	no	—
72	#13083553	C₂×C₄	1	4	yes	—
73	#13083553	C₂×C₄	1	4	yes	—
74	#13083553	D₄	1	4	yes	—
75	#13083551	—	2	2	no	—
76	#13083551	—	2	2	no	—
77	#13083551	—	2	2	no	—
78	#13083551	—	2	2	no	—
79	#13083551	C₂×C₄	1	4	yes	—
80	#13083551	C₂×C₄	1	4	yes	—
81	C₄×Dic₈₉	—	2	2	no	—
82	C₄×Dic₈₉	—	2	2	no	—
83	C₄×Dic₈₉	—	2	2	no	—
84	C₄×Dic₈₉	—	2	2	no	—
85	C₄×Dic₈₉	—	2	2	no	—
86	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₈₉	—	2	2	no	—
94	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
95	C₄²×C₈₉	—	2	2	no	—
96	C₄²×C₈₉	—	2	2	no	—
97	C₄²×C₈₉	—	2	2	no	—
98	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
99	C₂²×D₈₉	—	2	1	no	—
100	C₂²×D₈₉	—	2	1	no	—
101	C₂²×D₈₉	—	2	1	no	—
102	C₂²×D₈₉	—	2	1	no	—
103	C₂²×D₈₉	—	2	1	no	—
104	C₂²×D₈₉	—	2	1	no	—
105	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
122	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₈₉	#41	1	2	yes	—
126	C₂×C₄×C₈₉	#41	1	2	yes	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	2	1	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	2	1	no	—
140	C₂×Dic₈₉	—	2	1	no	—
141	C₂×Dic₈₉	—	2	1	no	—
142	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
143	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
144	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
145	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
146	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
147	C₂×Dic₈₉	#41	1	2	yes	—
148	C₄×D₈₉	—	2	8	no	—
149	Dic₁₇₈	—	2	8	no	—
150	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
151	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
155	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×D₈₉	—	2	2	no	—
158	C₂×D₈₉	—	2	2	no	—
159	C₂×D₈₉	—	2	2	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	2	no	—
165	C₄×C₈₉	—	2	2	no	—
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	—	2	2	no	—
169	C₄×C₈₉	—	2	2	no	—
170	C₄×C₈₉	—	4	1	no	—
171	Dic₈₉	—	2	2	no	—
172	Dic₈₉	—	2	2	no	—
173	Dic₈₉	—	2	2	no	—
174	Dic₈₉	—	4	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	—	4	1	no	—
177	Dic₈₉	C₄×D₄	1	4	yes	—
178	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂×C₈₉	C₄×#41	1	1	yes	characteristic
180	C₂×C₈₉	#519	1	1	yes	characteristic
181	C₂×C₈₉	#547	1	1	yes	characteristic
182	C₂×C₈₉	#548	1	1	yes	characteristic
183	C₂×C₈₉	#552	1	1	yes	characteristic
184	C₂×C₈₉	#556	1	1	yes	characteristic
185	D₈₉	—	4	2	no	—
186	#2362	—	89	1	no	maximal, Sylow
187	C₈₉	#2362	1	1	yes	characteristic, complemented, Sylow
188	C₂×C₂²⋊Q₈	—	89	1	no	—
189	C₂×C₄×Q₈	—	89	1	no	—
190	C₂×#117	—	89	1	no	—
191	#391	—	89	1	no	—
192	#389	—	89	1	no	—
193	#388	—	89	1	no	—
194	#388	—	89	1	no	—
195	#388	—	89	1	no	—
196	#388	—	89	1	no	—
197	#387	—	89	1	no	—
198	#387	—	89	1	no	—
199	C₄×C₄⋊C₄	—	89	1	no	—
200	C₄×C₄⋊C₄	—	89	1	no	—
201	C₄×C₄⋊C₄	D₈₉	1	1	yes	characteristic, complemented
202	C₄×C₂²⋊C₄	—	89	1	no	—
203	C₂²×Q₈	—	89	1	no	—
204	C₂³×C₄	—	89	1	no	—
205	C₂²⋊Q₈	—	89	8	no	—
206	C₄×Q₈	—	178	4	no	—
207	#117	—	178	4	no	—
208	C₂×C₄⋊C₄	—	89	1	no	—
209	C₂×C₄⋊C₄	—	89	1	no	—
210	C₂×C₄⋊C₄	—	89	1	no	—
211	C₂×C₄⋊C₄	—	89	1	no	—
212	C₂×C₄⋊C₄	—	89	1	no	—
213	C₂×C₄⋊C₄	—	89	1	no	—
214	C₂×C₄⋊C₄	—	89	1	no	—
215	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	—	89	1	no	—
218	C₂×C₂²⋊C₄	—	89	1	no	—
219	C₂×C₂²⋊C₄	—	89	1	no	—
220	C₂×C₂²⋊C₄	—	89	1	no	—
221	C₂×C₂²⋊C₄	—	89	1	no	—
222	C₂×C₂²⋊C₄	—	89	1	no	—
223	C₂×C₄²	—	89	1	no	—
224	C₂×C₄²	—	89	1	no	—
225	C₂×C₄²	—	89	1	no	—
226	C₂×C₄²	—	89	1	no	—
227	C₂×C₄²	—	89	1	no	—
228	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
229	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
230	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
231	#95	—	89	1	no	—
232	#95	—	89	1	no	—
233	#95	—	89	1	no	—
234	#95	—	89	1	no	—
235	#95	—	89	1	no	—
236	#95	—	89	1	no	—
237	#95	—	89	1	no	—
238	#95	—	89	1	no	—
239	#95	C₂×D₈₉	1	1	yes	characteristic
240	#95	C₂×D₈₉	1	1	yes	characteristic
241	C₂⁴	—	89	1	no	—
242	C₂×Q₈	—	89	4	no	—
243	C₂×Q₈	—	178	2	no	—
244	C₂×Q₈	—	178	2	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	89	1	no	—
248	C₂²×C₄	—	89	1	no	—
249	C₂²×C₄	—	89	4	no	—
250	C₂²×C₄	—	89	1	no	—
251	C₂²×C₄	—	89	1	no	—
252	C₂²×C₄	—	89	1	no	—
253	C₂²×C₄	—	178	2	no	—
254	C₂²×C₄	—	178	2	no	—
255	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
260	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	89	4	no	—
264	C₄⋊C₄	—	89	4	no	—
265	C₄⋊C₄	—	89	4	no	—
266	C₄⋊C₄	—	178	2	no	—
267	C₄⋊C₄	—	178	2	no	—
268	C₄⋊C₄	—	178	2	no	—
269	C₄⋊C₄	—	178	2	no	—
270	C₄⋊C₄	C₄×D₈₉	1	4	yes	—
271	C₂²⋊C₄	—	89	4	no	—
272	C₂²⋊C₄	—	89	4	no	—
273	C₂²⋊C₄	—	178	2	no	—
274	C₂²⋊C₄	—	178	2	no	—
275	C₂²⋊C₄	—	178	2	no	—
276	C₂²⋊C₄	—	178	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	178	2	no	—
281	C₄²	—	178	2	no	—
282	C₄²	—	178	2	no	—
283	C₄²	—	178	2	no	—
284	C₄²	—	178	2	no	—
285	C₂³	—	89	2	no	—
286	C₂³	—	178	1	no	—
287	C₂³	—	178	1	no	—
288	C₂³	—	178	1	no	—
289	C₂³	—	178	1	no	—
290	C₂³	—	178	1	no	—
291	C₂³	—	178	1	no	—
292	C₂³	C₂³×D₈₉	1	1	yes	characteristic
293	Q₈	—	178	8	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	89	2	no	—
310	C₂×C₄	—	89	2	no	—
311	C₂×C₄	—	89	2	no	—
312	C₂×C₄	—	89	2	no	—
313	C₂×C₄	—	89	2	no	—
314	C₂×C₄	—	89	2	no	—
315	C₂×C₄	—	178	8	no	—
316	C₂×C₄	—	178	1	no	—
317	C₂×C₄	—	178	1	no	—
318	C₂×C₄	—	178	1	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	178	1	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	—	178	1	no	—
331	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
332	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
333	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
334	C₂×C₄	D₄×D₈₉	1	2	yes	—
335	C₂×C₄	#13083574	1	2	yes	—
336	C₂×C₄	#13083574	1	2	yes	—
337	C₂²	—	356	1	no	—
338	C₂²	—	356	1	no	—
339	C₂²	—	356	1	no	—
340	C₂²	—	356	1	no	—
341	C₂²	—	178	2	no	—
342	C₂²	—	178	2	no	—
343	C₂²	—	178	2	no	—
344	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
345	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
346	C₂²	#41×D₈₉	1	1	yes	characteristic
347	C₂²	C₂×#13083574	1	1	yes	characteristic
348	C₂²	C₂×#13083574	1	1	yes	characteristic
349	C₂²	#15668059	1	1	yes	characteristic
350	C₂²	#15668060	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	89	4	no	—
359	C₄	—	356	1	no	—
360	C₄	—	356	1	no	—
361	C₄	—	356	1	no	—
362	C₄	—	178	2	no	—
363	C₄	—	178	2	no	—
364	C₄	—	178	2	no	—
365	C₂	—	356	2	no	—
366	C₂	C₄×D₄×D₈₉	1	1	yes	characteristic
367	C₂	C₄×#13083574	1	1	yes	characteristic
368	C₂	#19152074	1	1	yes	characteristic
369	C₂	#19152249	1	1	yes	characteristic
370	C₂	#19152257	1	1	yes	characteristic
371	C₂	#19152281	1	1	yes	characteristic
372	C₂	#19152320	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₈₉	C₂
socle, cosocle	C₂³×C₈₉	C₂⁴
prime radical	C₂³×D₈₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#2362
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₁₇₈	C₂	semidirect
4. #19151502	C₂	semidirect
5. #19151493	C₂	semidirect
10. #19151484	C₂	semidirect
11. #19151482	C₂	semidirect
12. C₄×#13083553	C₂	semidirect
13. #19151449	C₂	semidirect
15. C₄×C₄⋊C₄×C₈₉	C₂	semidirect
18. #15667990	C₄	semidirect
187. C₈₉	#2362	semidirect
201. C₄×C₄⋊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×#15667990	2
C₂×C₄×Dic₁₇₈	2
C₂×#15667969	2
#19151502	2
#19151493	2
#19151490	2
#19151489	2
#19151488	2
#19151484	2
#19151482	2
C₄×#13083553	2
#19151449	2
#19151445	2
C₄×C₄⋊C₄×C₈₉	2
#2362	89