Subgroups — #26028325

Order 11392.

456 conjugacy classes of subgroups.

The subgroup lattice has 396 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, complemented
2	C₂×#15668005	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151531	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151530	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151529	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151522	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151521	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151521	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151520	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151520	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151514	C₂	1	1	yes	maximal, characteristic
12	#19151513	C₂	1	1	yes	maximal, characteristic
13	#19151513	C₂	1	1	yes	maximal, characteristic
14	#19151467	C₂	1	1	yes	maximal, characteristic, complemented
15	#396×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
17	#15668007	—	2	4	no	—
18	#15668005	—	2	4	no	—
19	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₈₉	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×#13083553	C₂²	1	1	yes	characteristic
30	C₂×#13083553	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
34	C₂×#13083551	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×C₄×Dic₈₉	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×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
52	#95×C₈₉	C₂²	1	1	yes	characteristic
53	C₂³×D₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₈₉	—	4	1	no	—
55	C₂×C₂²⋊D₈₉	—	4	1	no	—
56	C₂×C₂²⋊D₈₉	—	4	1	no	—
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	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₈₉	—	2	2	no	—
65	C₄⋊Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	4	1	no	—
67	C₄⋊Dic₈₉	—	4	1	no	—
68	#13083553	—	2	2	no	—
69	#13083553	—	2	2	no	—
70	#13083553	—	2	2	no	—
71	#13083553	—	4	1	no	—
72	#13083553	—	4	1	no	—
73	#13083553	—	4	1	no	—
74	#13083553	—	4	1	no	—
75	#13083553	—	4	1	no	—
76	C₂²⋊Dic₈₉	—	2	2	no	—
77	C₂²⋊Dic₈₉	—	2	2	no	—
78	C₂²⋊Dic₈₉	—	4	1	no	—
79	#13083551	—	2	2	no	—
80	#13083551	—	2	2	no	—
81	#13083551	—	2	2	no	—
82	#13083551	—	4	1	no	—
83	#13083551	—	4	1	no	—
84	#13083551	—	4	1	no	—
85	C₄×Dic₈₉	D₄	1	4	yes	complemented
86	C₂⁴×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	—	4	1	no	—
95	C₄⋊C₄×C₈₉	—	4	1	no	—
96	C₂²⋊C₄×C₈₉	—	2	2	no	—
97	C₂²⋊C₄×C₈₉	—	4	1	no	—
98	C₂²⋊C₄×C₈₉	—	4	1	no	—
99	C₂³×C₈₉	—	2	1	no	—
100	C₂³×C₈₉	—	2	1	no	—
101	C₂³×C₈₉	—	2	1	no	—
102	C₂³×C₈₉	—	2	1	no	—
103	C₂³×C₈₉	—	2	1	no	—
104	C₂³×C₈₉	—	2	1	no	—
105	C₂³×C₈₉	—	2	1	no	—
106	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₂²×D₈₉	—	2	1	no	—
108	C₂²×D₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	1	no	—
111	C₂²×D₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	—	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	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
132	C₂²⋊D₈₉	—	4	4	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	—	2	1	no	—
151	C₂×Dic₈₉	—	2	1	no	—
152	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
153	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
154	C₂²×C₈₉	—	4	1	no	—
155	C₂²×C₈₉	—	4	1	no	—
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	—	4	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	4	1	no	—
170	C₂×D₈₉	—	4	1	no	—
171	C₂×D₈₉	—	4	1	no	—
172	C₂×D₈₉	—	4	1	no	—
173	C₂×D₈₉	—	4	1	no	—
174	C₂×D₈₉	—	4	1	no	—
175	C₄×C₈₉	—	2	2	no	—
176	C₄×C₈₉	—	4	1	no	—
177	C₄×C₈₉	—	4	1	no	—
178	C₄×C₈₉	—	4	1	no	—
179	C₄×C₈₉	—	4	1	no	—
180	C₄×C₈₉	—	4	1	no	—
181	Dic₈₉	—	2	2	no	—
182	Dic₈₉	—	2	2	no	—
183	Dic₈₉	—	4	1	no	—
184	Dic₈₉	—	4	1	no	—
185	Dic₈₉	—	4	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	1	no	—
188	C₂×C₈₉	—	8	1	no	—
189	C₂×C₈₉	#537	1	1	yes	characteristic
190	C₂×C₈₉	#537	1	1	yes	characteristic
191	C₂×C₈₉	#537	1	1	yes	characteristic
192	C₂×C₈₉	#560	1	1	yes	characteristic
193	C₂×C₈₉	#563	1	1	yes	characteristic
194	C₂×C₈₉	#563	1	1	yes	characteristic
195	C₂×C₈₉	#563	1	1	yes	characteristic
196	D₈₉	—	8	1	no	—
197	#2823	—	89	1	no	maximal, Sylow
198	C₈₉	#2823	1	1	yes	characteristic, complemented, Sylow
199	C₂×#126	—	89	1	no	—
200	C₂×#126	—	89	1	no	—
201	#398	—	89	1	no	—
202	#398	—	89	1	no	—
203	#398	—	89	1	no	—
204	#396	—	89	1	no	—
205	#396	—	89	1	no	—
206	#396	—	89	1	no	—
207	#396	—	89	1	no	—
208	#396	—	89	1	no	—
209	#396	D₈₉	1	1	yes	characteristic, complemented
210	#394	—	89	1	no	—
211	#394	—	89	1	no	—
212	#394	—	89	1	no	—
213	#384	—	89	1	no	—
214	C₂²×D₄	—	89	1	no	—
215	#126	—	178	4	no	—
216	#126	—	178	4	no	—
217	C₂×C₄⋊C₄	—	89	1	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×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
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×C₂²⋊C₄	—	89	1	no	—
238	C₂×C₂²⋊C₄	—	89	1	no	—
239	C₂×C₂²⋊C₄	—	89	1	no	—
240	C₂×C₂²⋊C₄	—	89	1	no	—
241	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
243	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
244	C₂×C₄²	—	89	1	no	—
245	#95	—	89	1	no	—
246	#95	—	89	1	no	—
247	#95	—	89	1	no	—
248	#95	—	89	1	no	—
249	#95	—	89	1	no	—
250	#95	C₂×D₈₉	1	1	yes	characteristic
251	C₂⁴	—	89	1	no	—
252	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×D₄	—	356	1	no	—
254	C₂×D₄	—	356	1	no	—
255	C₂×D₄	—	356	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	89	1	no	—
258	C₂²×C₄	—	89	1	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	C₂²×D₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
266	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
267	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
268	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	356	1	no	—
273	C₄⋊C₄	—	356	1	no	—
274	C₄⋊C₄	—	356	1	no	—
275	C₄⋊C₄	—	356	1	no	—
276	C₄⋊C₄	—	356	1	no	—
277	C₄⋊C₄	—	356	1	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	C₂²⋊C₄	—	2	2	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	356	1	no	—
288	C₂²⋊C₄	—	356	1	no	—
289	C₂²⋊C₄	—	356	1	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	C₂²⋊C₄	—	178	2	no	—
296	C₄²	—	89	4	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	2	1	no	—
302	C₂³	—	2	1	no	—
303	C₂³	—	2	1	no	—
304	C₂³	—	178	1	no	—
305	C₂³	—	178	1	no	—
306	C₂³	—	178	1	no	—
307	C₂³	—	178	1	no	—
308	C₂³	—	178	1	no	—
309	C₂³	—	178	1	no	—
310	C₂³	—	178	1	no	—
311	C₂³	C₂³×D₈₉	1	1	yes	characteristic
312	D₄	—	356	4	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	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	89	2	no	—
331	C₂×C₄	—	89	2	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	—	178	1	no	—
347	C₂×C₄	—	178	1	no	—
348	C₂×C₄	—	178	1	no	—
349	C₂×C₄	—	178	1	no	—
350	C₂×C₄	—	178	1	no	—
351	C₂×C₄	D₄×D₈₉	1	2	yes	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	4	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	356	1	no	—
360	C₂²	—	356	1	no	—
361	C₂²	—	356	1	no	—
362	C₂²	—	356	1	no	—
363	C₂²	—	356	1	no	—
364	C₂²	—	356	1	no	—
365	C₂²	—	356	1	no	—
366	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
367	C₂²	#15668057	1	1	yes	characteristic
368	C₂²	#15668057	1	1	yes	characteristic
369	C₂²	#15668058	1	1	yes	characteristic
370	C₂²	#15668059	1	1	yes	characteristic
371	C₂²	#15668059	1	1	yes	characteristic
372	C₂²	#15668060	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	4	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	4	1	no	—
379	C₄	—	356	1	no	—
380	C₄	—	356	1	no	—
381	C₄	—	356	1	no	—
382	C₄	—	356	1	no	—
383	C₄	—	356	1	no	—
384	C₄	—	178	2	no	—
385	C₄	—	178	2	no	—
386	C₂	—	8	1	no	—
387	C₂	—	712	1	no	—
388	C₂	#19152175	1	1	yes	characteristic
389	C₂	#19152177	1	1	yes	characteristic
390	C₂	#19152178	1	1	yes	characteristic
391	C₂	#19152336	1	1	yes	characteristic
392	C₂	#19152349	1	1	yes	characteristic
393	C₂	#19152349	1	1	yes	characteristic
394	C₂	#19152350	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#396×C₈₉	C₂
prime radical	C₂²×C₂²⋊D₈₉	C₂²
hypercenter	#396	D₈₉
hypocenter	C₈₉	#2823
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #396

Complements

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

N	H ≅ G/N	type
1. C₂×#15668007	C₂	semidirect
2. C₂×#15668005	C₂	semidirect
3. #19151531	C₂	semidirect
4. #19151530	C₂	semidirect
5. #19151529	C₂	semidirect
6. #19151522	C₂	semidirect
7. #19151521	C₂	semidirect
8. #19151521	C₂	semidirect
9. #19151520	C₂	semidirect
10. #19151520	C₂	semidirect
14. #19151467	C₂	semidirect
15. #396×C₈₉	C₂	semidirect
85. C₄×Dic₈₉	D₄	semidirect
198. C₈₉	#2823	semidirect
209. #396	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×#15668005	2
#19151531	2
#19151530	2
#19151529	2
#19151522	2
#19151521	2
#19151520	2
#19151514	2
#19151513	2
#19151467	2
#396×C₈₉	2
#2823	89