Subgroups — #26026565

Order 11392.

544 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668012	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15668011	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667992	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667990	C₂	1	1	yes	maximal, characteristic
5	#19151535	C₂	1	1	yes	maximal, characteristic
6	#19151530	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151528	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151519	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151519	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151511	C₂	1	1	yes	maximal, characteristic
11	#19151490	C₂	1	1	yes	maximal, characteristic
12	#19151489	C₂	1	1	yes	maximal, characteristic
13	#19151483	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151472	C₂	1	1	yes	maximal, characteristic
15	#387×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₈₉	C₂²	1	1	yes	characteristic
20	#15668012	—	2	4	no	—
21	#15668011	—	2	4	no	—
22	#15667992	—	2	4	no	—
23	#15667990	—	2	4	no	—
24	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
31	C₂×#13083553	C₂²	1	1	yes	characteristic
32	C₂×#13083553	C₂²	1	1	yes	characteristic
33	C₂×#13083553	C₂²	1	1	yes	characteristic
34	C₂×#13083553	C₂²	1	1	yes	characteristic
35	C₂×#13083551	C₂²	1	1	yes	characteristic
36	C₂×#13083551	C₂²	1	1	yes	characteristic
37	C₂×#13083551	C₂²	1	1	yes	characteristic
38	C₂×#13083551	C₂²	1	1	yes	characteristic
39	C₂×#13083551	C₂²	1	1	yes	characteristic
40	C₂×#13083551	C₂²	1	1	yes	characteristic
41	C₂×C₄×Dic₈₉	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	#15667919	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
51	#95×C₈₉	C₂²	1	1	yes	characteristic
52	#95×C₈₉	C₂²	1	1	yes	characteristic
53	#95×C₈₉	C₂²	1	1	yes	characteristic
54	#95×C₈₉	C₂²	1	1	yes	characteristic
55	C₂³×D₈₉	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₇₈	—	2	2	no	—
57	C₂×Dic₁₇₈	—	2	2	no	—
58	C₂×Dic₁₇₈	—	2	2	no	—
59	C₂×Dic₁₇₈	—	2	2	no	—
60	C₂×Dic₁₇₈	—	2	2	no	—
61	C₂×Dic₁₇₈	—	2	2	no	—
62	C₂×Dic₁₇₈	—	4	1	no	—
63	C₂×Dic₁₇₈	—	4	1	no	—
64	C₂×Dic₁₇₈	—	4	1	no	—
65	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
66	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
67	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
68	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
69	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
71	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₈₉	—	2	2	no	—
73	C₂×C₄×D₈₉	—	2	2	no	—
74	C₂×C₄×D₈₉	—	4	1	no	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	—	2	2	no	—
80	C₄⋊Dic₈₉	—	4	1	no	—
81	C₄⋊Dic₈₉	D₄	1	4	yes	—
82	#13083553	—	2	2	no	—
83	#13083553	—	2	2	no	—
84	#13083553	—	2	2	no	—
85	#13083553	—	2	2	no	—
86	#13083551	—	2	2	no	—
87	#13083551	—	2	2	no	—
88	#13083551	—	2	2	no	—
89	#13083551	—	2	2	no	—
90	#13083551	—	4	1	no	—
91	#13083551	Q₈	1	4	yes	complemented
92	C₄×Dic₈₉	—	2	2	no	—
93	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	—	2	2	no	—
101	C₄⋊C₄×C₈₉	D₄	1	4	yes	—
102	C₄²×C₈₉	—	2	2	no	—
103	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₂²×D₈₉	—	2	1	no	—
105	C₂²×D₈₉	—	2	1	no	—
106	C₂²×D₈₉	—	2	1	no	—
107	C₂²×D₈₉	—	2	1	no	—
108	C₂²×D₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	C₂×Q₈	1	2	yes	—
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	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
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	—	2	1	no	—
127	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
131	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
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	—	2	1	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	2	1	no	—
145	C₂×Dic₈₉	—	2	1	no	—
146	C₂×Dic₈₉	—	2	1	no	—
147	C₂×Dic₈₉	—	2	1	no	—
148	C₂×Dic₈₉	—	2	1	no	—
149	C₂×Dic₈₉	—	2	1	no	—
150	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
151	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
152	C₂×Dic₈₉	C₂×Q₈	1	2	yes	—
153	C₄×D₈₉	—	4	4	no	—
154	Dic₁₇₈	—	4	4	no	—
155	Dic₁₇₈	—	4	4	no	—
156	Dic₁₇₈	—	4	4	no	—
157	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
159	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
160	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
164	C₂×D₈₉	—	2	2	no	—
165	C₂×D₈₉	—	2	2	no	—
166	C₂×D₈₉	—	2	2	no	—
167	C₂×D₈₉	—	4	1	no	—
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	4	1	no	—
170	C₂×D₈₉	—	4	1	no	—
171	C₄×C₈₉	—	2	2	no	—
172	C₄×C₈₉	—	2	2	no	—
173	C₄×C₈₉	—	2	2	no	—
174	C₄×C₈₉	—	2	2	no	—
175	C₄×C₈₉	—	2	2	no	—
176	C₄×C₈₉	—	4	1	no	—
177	C₄×C₈₉	—	4	1	no	—
178	Dic₈₉	—	2	2	no	—
179	Dic₈₉	—	2	2	no	—
180	Dic₈₉	—	2	2	no	—
181	Dic₈₉	—	4	1	no	—
182	Dic₈₉	—	4	1	no	—
183	Dic₈₉	—	4	1	no	—
184	Dic₈₉	—	4	1	no	—
185	C₂×C₈₉	#536	1	1	yes	characteristic
186	C₂×C₈₉	#536	1	1	yes	characteristic
187	C₂×C₈₉	#544	1	1	yes	characteristic
188	C₂×C₈₉	D₄²	1	1	yes	characteristic
189	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
190	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
191	C₂×C₈₉	#563	1	1	yes	characteristic
192	D₈₉	—	4	2	no	—
193	#2677	—	89	1	no	maximal, Sylow
194	C₈₉	#2677	1	1	yes	characteristic, complemented, Sylow
195	C₂×C₄⋊Q₈	—	89	1	no	—
196	C₂×C₄⋊Q₈	—	89	1	no	—
197	C₂×C₂²⋊Q₈	—	89	1	no	—
198	C₂×C₂²⋊Q₈	—	89	1	no	—
199	#399	—	89	1	no	—
200	#398	—	89	1	no	—
201	#398	—	89	1	no	—
202	#395	—	89	1	no	—
203	#395	—	89	1	no	—
204	#393	—	89	1	no	—
205	#388	—	89	1	no	—
206	#388	—	89	1	no	—
207	#387	—	89	1	no	—
208	#387	D₈₉	1	1	yes	characteristic, complemented
209	#385	—	89	1	no	—
210	C₂²×Q₈	—	89	1	no	—
211	C₂²×Q₈	—	89	1	no	—
212	C₂²×Q₈	—	89	1	no	—
213	C₂³×C₄	—	89	1	no	—
214	C₄⋊Q₈	—	178	4	no	—
215	C₄⋊Q₈	—	178	4	no	—
216	C₂²⋊Q₈	—	178	4	no	—
217	C₂²⋊Q₈	—	178	4	no	—
218	C₂×C₄⋊C₄	—	89	1	no	—
219	C₂×C₄⋊C₄	—	89	1	no	—
220	C₂×C₄⋊C₄	—	89	1	no	—
221	C₂×C₄⋊C₄	—	89	1	no	—
222	C₂×C₄⋊C₄	—	89	1	no	—
223	C₂×C₄⋊C₄	—	89	1	no	—
224	C₂×C₄⋊C₄	—	89	1	no	—
225	C₂×C₄⋊C₄	—	89	1	no	—
226	C₂×C₄⋊C₄	—	89	1	no	—
227	C₂×C₄⋊C₄	—	89	1	no	—
228	C₂×C₄⋊C₄	—	89	1	no	—
229	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
230	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	—	89	1	no	—
232	C₂×C₂²⋊C₄	—	89	1	no	—
233	C₂×C₂²⋊C₄	—	89	1	no	—
234	C₂×C₂²⋊C₄	—	89	1	no	—
235	C₂×C₂²⋊C₄	—	89	1	no	—
236	C₂×C₂²⋊C₄	—	89	1	no	—
237	C₂×C₄²	—	89	1	no	—
238	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
239	#95	—	89	1	no	—
240	#95	—	89	1	no	—
241	#95	—	89	1	no	—
242	#95	—	89	1	no	—
243	#95	—	89	1	no	—
244	#95	—	89	1	no	—
245	#95	C₂×D₈₉	1	1	yes	characteristic
246	#95	C₂×D₈₉	1	1	yes	characteristic
247	#95	C₂×D₈₉	1	1	yes	characteristic
248	#95	C₂×D₈₉	1	1	yes	characteristic
249	C₂⁴	—	89	1	no	—
250	C₂×Q₈	—	356	1	no	—
251	C₂×Q₈	—	356	1	no	—
252	C₂×Q₈	—	356	1	no	—
253	C₂×Q₈	—	178	2	no	—
254	C₂×Q₈	—	178	2	no	—
255	C₂×Q₈	—	178	2	no	—
256	C₂×Q₈	—	178	2	no	—
257	C₂×Q₈	—	178	2	no	—
258	C₂×Q₈	—	178	2	no	—
259	C₂²×C₄	—	89	1	no	—
260	C₂²×C₄	—	89	1	no	—
261	C₂²×C₄	—	89	1	no	—
262	C₂²×C₄	—	89	1	no	—
263	C₂²×C₄	—	89	1	no	—
264	C₂²×C₄	—	89	1	no	—
265	C₂²×C₄	—	89	1	no	—
266	C₂²×C₄	—	356	1	no	—
267	C₂²×C₄	—	178	2	no	—
268	C₂²×C₄	—	178	2	no	—
269	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
270	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
271	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
272	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
273	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
274	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
275	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	89	4	no	—
278	C₄⋊C₄	—	356	1	no	—
279	C₄⋊C₄	—	178	2	no	—
280	C₄⋊C₄	—	178	2	no	—
281	C₄⋊C₄	—	178	2	no	—
282	C₄⋊C₄	—	178	2	no	—
283	C₄⋊C₄	—	178	2	no	—
284	C₄⋊C₄	—	178	2	no	—
285	C₄⋊C₄	—	178	2	no	—
286	C₄⋊C₄	—	178	2	no	—
287	C₄⋊C₄	—	178	2	no	—
288	C₄⋊C₄	D₃₅₆	1	4	yes	—
289	C₂²⋊C₄	—	89	4	no	—
290	C₂²⋊C₄	—	356	1	no	—
291	C₂²⋊C₄	—	178	2	no	—
292	C₂²⋊C₄	—	178	2	no	—
293	C₂²⋊C₄	—	178	2	no	—
294	C₂²⋊C₄	—	178	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	178	2	no	—
297	C₂³	—	89	2	no	—
298	C₂³	—	178	1	no	—
299	C₂³	—	178	1	no	—
300	C₂³	—	178	1	no	—
301	C₂³	—	178	1	no	—
302	C₂³	—	178	1	no	—
303	C₂³	—	178	1	no	—
304	C₂³	C₂³×D₈₉	1	1	yes	characteristic
305	Q₈	—	356	4	no	—
306	Q₈	—	356	4	no	—
307	Q₈	—	356	4	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	89	2	no	—
325	C₂×C₄	—	89	2	no	—
326	C₂×C₄	—	89	2	no	—
327	C₂×C₄	—	356	4	no	—
328	C₂×C₄	—	178	1	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	—	178	1	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	—	178	1	no	—
339	C₂×C₄	—	178	1	no	—
340	C₂×C₄	—	178	1	no	—
341	C₂×C₄	—	178	1	no	—
342	C₂×C₄	—	178	1	no	—
343	C₂×C₄	—	178	1	no	—
344	C₂×C₄	—	178	1	no	—
345	C₂×C₄	—	178	1	no	—
346	C₂×C₄	D₄×D₈₉	1	2	yes	—
347	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
348	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
349	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
350	C₂×C₄	Q₈×D₈₉	1	2	yes	—
351	C₂²	—	356	1	no	—
352	C₂²	—	356	1	no	—
353	C₂²	—	356	1	no	—
354	C₂²	—	356	1	no	—
355	C₂²	—	178	2	no	—
356	C₂²	—	178	2	no	—
357	C₂²	—	178	2	no	—
358	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
359	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
360	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
361	C₂²	#15668058	1	1	yes	characteristic
362	C₂²	#15668059	1	1	yes	characteristic
363	C₂²	#15668059	1	1	yes	characteristic
364	C₂²	#15668059	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	356	1	no	—
373	C₄	—	356	1	no	—
374	C₄	—	356	1	no	—
375	C₄	—	356	1	no	—
376	C₄	—	178	2	no	—
377	C₄	—	178	2	no	—
378	C₄	—	178	2	no	—
379	C₂	—	356	2	no	—
380	C₂	#19152170	1	1	yes	characteristic
381	C₂	#19152174	1	1	yes	characteristic
382	C₂	#19152228	1	1	yes	characteristic
383	C₂	D₄×D₃₅₆	1	1	yes	characteristic
384	C₂	#19152261	1	1	yes	characteristic
385	C₂	Q₈×D₃₅₆	1	1	yes	characteristic
386	C₂	#19152350	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#387×C₈₉	C₂
prime radical	C₂³×D₈₉	C₂³
hypercenter	#387	D₈₉
hypocenter	C₈₉	#2677
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #387

Complements

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

N	H ≅ G/N	type
1. C₂×#15668012	C₂	semidirect
2. C₂×#15668011	C₂	semidirect
6. #19151530	C₂	semidirect
7. #19151528	C₂	semidirect
8. #19151519	C₂	semidirect
9. #19151519	C₂	semidirect
13. #19151483	C₂	semidirect
15. #387×C₈₉	C₂	semidirect
91. #13083551	Q₈	semidirect
194. C₈₉	#2677	semidirect
208. #387	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668012	2
C₂×#15668011	2
C₂×#15667992	2
C₂×#15667990	2
#19151535	2
#19151530	2
#19151528	2
#19151519	2
#19151511	2
#19151490	2
#19151489	2
#19151483	2
#19151472	2
#387×C₈₉	2
#2677	89