Subgroups — #26027191

Order 11392.

434 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668007	C₂	1	1	yes	maximal, characteristic
2	C₂×#15668002	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151539	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151527	C₂	1	1	yes	maximal, characteristic
5	#19151522	C₂	1	1	yes	maximal, characteristic
6	#19151522	C₂	1	1	yes	maximal, characteristic
7	#19151520	C₂	1	1	yes	maximal, characteristic
8	#19151495	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151494	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151493	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151493	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151488	C₂	1	1	yes	maximal, characteristic
13	#19151488	C₂	1	1	yes	maximal, characteristic
14	#19151481	C₂	1	1	yes	maximal, characteristic, complemented
15	#398×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	#15668007	—	2	4	no	—
17	#15668002	—	2	4	no	—
18	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
20	C₂×#13083553	C₂²	1	1	yes	characteristic
21	C₂×#13083553	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×#13083553	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×#13083551	C₂²	1	1	yes	characteristic
35	C₂×#13083551	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	C₂×C₄⋊C₄×C₈₉	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×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
50	#95×C₈₉	C₂²	1	1	yes	characteristic
51	#95×C₈₉	C₂²	1	1	yes	characteristic
52	#95×C₈₉	C₂²	1	1	yes	characteristic
53	C₂³×D₈₉	C₂³	1	1	yes	characteristic
54	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
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	C₄⋊Dic₈₉	—	2	2	no	—
62	C₄⋊Dic₈₉	D₄	1	4	yes	—
63	#13083553	—	2	2	no	—
64	#13083553	—	2	2	no	—
65	#13083553	—	2	2	no	—
66	#13083553	—	2	2	no	—
67	#13083553	—	2	2	no	—
68	#13083553	—	2	2	no	—
69	#13083553	—	2	2	no	—
70	#13083553	—	2	2	no	—
71	#13083553	—	4	1	no	—
72	#13083551	—	2	2	no	—
73	#13083551	—	2	2	no	—
74	#13083551	—	2	2	no	—
75	#13083551	—	2	2	no	—
76	#13083551	—	2	2	no	—
77	#13083551	—	4	1	no	—
78	#13083551	—	4	1	no	—
79	C₄×Dic₈₉	—	2	2	no	—
80	C₄×Dic₈₉	—	2	2	no	—
81	C₄×Dic₈₉	—	4	1	no	—
82	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₈₉	—	2	2	no	—
90	C₄⋊C₄×C₈₉	—	2	2	no	—
91	C₄⋊C₄×C₈₉	—	4	1	no	—
92	C₄⋊C₄×C₈₉	—	4	1	no	—
93	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
94	C₂²×D₈₉	—	2	1	no	—
95	C₂²×D₈₉	—	2	1	no	—
96	C₂²×D₈₉	—	2	1	no	—
97	C₂²×D₈₉	—	2	1	no	—
98	C₂²×D₈₉	—	2	1	no	—
99	C₂²×D₈₉	—	2	1	no	—
100	C₂²×D₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
140	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
141	C₂×Dic₈₉	#41	1	2	yes	—
142	C₂×Dic₈₉	#41	1	2	yes	—
143	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
146	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
149	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂×D₈₉	—	4	1	no	—
151	C₂×D₈₉	—	4	1	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₂×D₈₉	—	4	1	no	—
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	Dic₈₉	—	2	2	no	—
165	Dic₈₉	—	2	2	no	—
166	Dic₈₉	—	2	2	no	—
167	Dic₈₉	—	2	2	no	—
168	Dic₈₉	—	4	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	C₂×C₈₉	#537	1	1	yes	characteristic
172	C₂×C₈₉	#541	1	1	yes	characteristic
173	C₂×C₈₉	#541	1	1	yes	characteristic
174	C₂×C₈₉	#547	1	1	yes	characteristic
175	C₂×C₈₉	#548	1	1	yes	characteristic
176	C₂×C₈₉	#553	1	1	yes	characteristic
177	C₂×C₈₉	#563	1	1	yes	characteristic
178	D₈₉	—	8	1	no	—
179	#2725	—	89	1	no	maximal, Sylow
180	C₈₉	#2725	1	1	yes	characteristic, complemented, Sylow
181	C₂×#126	—	89	1	no	—
182	C₂×#125	—	89	1	no	—
183	#400	—	89	1	no	—
184	#398	D₈₉	1	1	yes	characteristic, complemented
185	#397	—	89	1	no	—
186	#396	—	89	1	no	—
187	#396	—	89	1	no	—
188	#396	—	89	1	no	—
189	#389	—	89	1	no	—
190	#389	—	89	1	no	—
191	#389	—	89	1	no	—
192	#389	—	89	1	no	—
193	#388	—	89	1	no	—
194	#388	—	89	1	no	—
195	#387	—	89	1	no	—
196	#126	—	178	4	no	—
197	#125	—	178	4	no	—
198	C₂×C₄⋊C₄	—	89	1	no	—
199	C₂×C₄⋊C₄	—	89	1	no	—
200	C₂×C₄⋊C₄	—	89	1	no	—
201	C₂×C₄⋊C₄	—	89	1	no	—
202	C₂×C₄⋊C₄	—	89	1	no	—
203	C₂×C₄⋊C₄	—	89	1	no	—
204	C₂×C₄⋊C₄	—	89	1	no	—
205	C₂×C₄⋊C₄	—	89	1	no	—
206	C₂×C₄⋊C₄	—	89	1	no	—
207	C₂×C₄⋊C₄	—	89	1	no	—
208	C₂×C₄⋊C₄	—	89	1	no	—
209	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
210	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	89	1	no	—
214	C₂×C₂²⋊C₄	—	89	1	no	—
215	C₂×C₂²⋊C₄	—	89	1	no	—
216	C₂×C₂²⋊C₄	—	89	1	no	—
217	C₂×C₂²⋊C₄	—	89	1	no	—
218	C₂×C₂²⋊C₄	—	89	1	no	—
219	C₂×C₂²⋊C₄	—	89	1	no	—
220	C₂×C₄²	—	89	1	no	—
221	C₂×C₄²	—	89	1	no	—
222	C₂×C₄²	—	89	1	no	—
223	#95	—	89	1	no	—
224	#95	—	89	1	no	—
225	#95	—	89	1	no	—
226	#95	—	89	1	no	—
227	#95	—	89	1	no	—
228	#95	—	89	1	no	—
229	#95	—	89	1	no	—
230	#95	C₂×D₈₉	1	1	yes	characteristic
231	#95	C₂×D₈₉	1	1	yes	characteristic
232	#95	C₂×D₈₉	1	1	yes	characteristic
233	C₂⁴	—	89	1	no	—
234	C₂²×C₄	—	89	1	no	—
235	C₂²×C₄	—	89	1	no	—
236	C₂²×C₄	—	89	1	no	—
237	C₂²×C₄	—	89	1	no	—
238	C₂²×C₄	—	89	1	no	—
239	C₂²×C₄	—	89	1	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	89	4	no	—
253	C₄⋊C₄	—	356	1	no	—
254	C₄⋊C₄	—	178	2	no	—
255	C₄⋊C₄	—	178	2	no	—
256	C₄⋊C₄	—	178	2	no	—
257	C₄⋊C₄	—	178	2	no	—
258	C₄⋊C₄	—	178	2	no	—
259	C₄⋊C₄	—	178	2	no	—
260	C₄⋊C₄	—	178	2	no	—
261	C₄⋊C₄	—	178	2	no	—
262	C₄⋊C₄	—	178	2	no	—
263	C₂²⋊C₄	—	356	1	no	—
264	C₂²⋊C₄	—	356	1	no	—
265	C₂²⋊C₄	—	178	2	no	—
266	C₂²⋊C₄	—	178	2	no	—
267	C₂²⋊C₄	—	178	2	no	—
268	C₂²⋊C₄	—	178	2	no	—
269	C₂²⋊C₄	—	178	2	no	—
270	C₄²	—	356	1	no	—
271	C₄²	—	178	2	no	—
272	C₄²	—	178	2	no	—
273	C₂³	—	178	1	no	—
274	C₂³	—	178	1	no	—
275	C₂³	—	178	1	no	—
276	C₂³	—	178	1	no	—
277	C₂³	—	178	1	no	—
278	C₂³	—	178	1	no	—
279	C₂³	—	178	1	no	—
280	C₂³	C₂³×D₈₉	1	1	yes	characteristic
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	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	—	89	2	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	89	2	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	—	178	1	no	—
314	C₂×C₄	—	178	1	no	—
315	C₂×C₄	—	178	1	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	D₄×D₈₉	1	2	yes	—
323	C₂²	—	356	1	no	—
324	C₂²	—	356	1	no	—
325	C₂²	—	356	1	no	—
326	C₂²	—	356	1	no	—
327	C₂²	—	356	1	no	—
328	C₂²	—	356	1	no	—
329	C₂²	—	356	1	no	—
330	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
331	C₂²	#41×D₈₉	1	1	yes	characteristic
332	C₂²	#41×D₈₉	1	1	yes	characteristic
333	C₂²	#15668057	1	1	yes	characteristic
334	C₂²	#15668059	1	1	yes	characteristic
335	C₂²	#15668060	1	1	yes	characteristic
336	C₂²	#15668060	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	356	1	no	—
345	C₄	—	356	1	no	—
346	C₄	—	356	1	no	—
347	C₄	—	178	2	no	—
348	C₄	—	178	2	no	—
349	C₄	—	178	2	no	—
350	C₄	—	178	2	no	—
351	C₂	—	712	1	no	—
352	C₂	#19152175	1	1	yes	characteristic
353	C₂	#19152207	1	1	yes	characteristic
354	C₂	#19152207	1	1	yes	characteristic
355	C₂	#19152250	1	1	yes	characteristic
356	C₂	#19152258	1	1	yes	characteristic
357	C₂	#19152293	1	1	yes	characteristic
358	C₂	#19152347	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#398×C₈₉	C₂
prime radical	C₂³×D₈₉	C₂³
hypercenter	#398	D₈₉
hypocenter	C₈₉	#2725
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #398

Complements

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

N	H ≅ G/N	type
2. C₂×#15668002	C₂	semidirect
3. #19151539	C₂	semidirect
8. #19151495	C₂	semidirect
9. #19151494	C₂	semidirect
10. #19151493	C₂	semidirect
11. #19151493	C₂	semidirect
14. #19151481	C₂	semidirect
15. #398×C₈₉	C₂	semidirect
180. C₈₉	#2725	semidirect
184. #398	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668007	2
C₂×#15668002	2
#19151539	2
#19151527	2
#19151522	2
#19151520	2
#19151495	2
#19151494	2
#19151493	2
#19151488	2
#19151481	2
#398×C₈₉	2
#2725	89