Subgroups — #36472808

Order 22784.

474 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667957	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151403	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151400	C₂	1	1	yes	maximal, characteristic
5	#26021118	C₂	1	1	yes	maximal, characteristic
6	#26021113	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×#15667957	—	2	2	no	—
11	C₂×#15667957	—	2	2	no	—
12	C₂×#15667957	C₄	1	4	yes	complemented
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667937	—	2	1	no	—
16	C₂×#15667934	—	2	1	no	—
17	#19151504	—	2	1	no	—
18	#19151500	—	2	1	no	—
19	#19151426	—	2	2	no	—
20	#19151403	—	2	2	no	—
21	#19151400	—	2	2	no	—
22	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₄×C₁₆×C₈₉	—	2	2	no	—
26	C₂²×Dic₁₇₈	—	2	1	no	—
27	C₂²×D₃₅₆	—	2	1	no	—
28	C₂×Dic₃₅₆	—	2	1	no	—
29	C₂×Dic₃₅₆	—	2	1	no	—
30	C₂×Dic₃₅₆	—	2	1	no	—
31	C₂×Dic₃₅₆	—	2	1	no	—
32	C₂×Dic₃₅₆	—	2	1	no	—
33	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
34	C₂×D₇₁₂	—	2	1	no	—
35	C₂×D₇₁₂	—	2	1	no	—
36	C₂×D₇₁₂	—	2	1	no	—
37	C₂×D₇₁₂	—	2	1	no	—
38	C₂×D₇₁₂	—	2	1	no	—
39	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₈₉	—	2	1	no	—
41	C₂×C₄⋊Dic₈₉	—	2	1	no	—
42	C₂×#13083551	—	4	2	no	—
43	#15667957	—	2	8	no	—
44	#15667946	—	2	1	no	—
45	#15667946	—	2	1	no	—
46	#15667946	—	2	1	no	—
47	#15667946	—	2	1	no	—
48	#15667937	—	4	2	no	—
49	#15667934	—	4	2	no	—
50	#15667919	—	4	2	no	—
51	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
54	C₂×C₁₆×C₈₉	—	2	1	no	—
55	C₂×C₁₆×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
58	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
59	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
60	C₄×C₈×C₈₉	—	2	1	no	—
61	C₄×C₈×C₈₉	—	2	1	no	—
62	C₂³×D₈₉	—	4	1	no	—
63	C₂×Dic₁₇₈	—	2	2	no	—
64	C₂×Dic₁₇₈	—	4	1	no	—
65	C₂×Dic₁₇₈	—	4	1	no	—
66	C₂×Dic₁₇₈	—	4	1	no	—
67	C₂×Dic₁₇₈	—	4	1	no	—
68	C₂×Dic₁₇₈	—	4	1	no	—
69	C₂×D₃₅₆	—	2	2	no	—
70	C₂×D₃₅₆	—	4	1	no	—
71	C₂×D₃₅₆	—	4	1	no	—
72	C₂×D₃₅₆	—	4	1	no	—
73	C₂×D₃₅₆	—	4	1	no	—
74	C₂×D₃₅₆	—	4	1	no	—
75	C₂²×Dic₈₉	—	4	1	no	—
76	C₂²×Dic₈₉	—	4	1	no	—
77	C₂²×Dic₈₉	—	4	1	no	—
78	Dic₃₅₆	—	2	2	no	—
79	Dic₃₅₆	—	2	2	no	—
80	Dic₃₅₆	—	4	1	no	—
81	Dic₃₅₆	—	4	1	no	—
82	D₇₁₂	—	2	2	no	—
83	D₇₁₂	—	2	2	no	—
84	D₇₁₂	—	4	1	no	—
85	D₇₁₂	—	4	1	no	—
86	C₄⋊Dic₈₉	—	4	1	no	—
87	C₄⋊Dic₈₉	—	4	1	no	—
88	C₄⋊Dic₈₉	—	4	1	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	#13083551	—	8	4	no	—
91	C₂²×C₄×C₈₉	D₈	1	2	yes	—
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	D₈	1	2	yes	—
96	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
101	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
102	C₄²×C₈₉	—	2	1	no	—
103	C₄²×C₈₉	—	2	1	no	—
104	C₁₆×C₈₉	—	2	2	no	—
105	C₁₆×C₈₉	C₂²⋊C₄	1	4	yes	complemented
106	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₈₉	—	4	2	no	—
108	C₂²×D₈₉	—	8	1	no	—
109	C₂²×D₈₉	—	8	1	no	—
110	C₂²×D₈₉	—	8	2	no	—
111	C₂²×D₈₉	—	8	2	no	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	2	no	—
114	C₂×C₄×C₈₉	SD₃₂	1	4	yes	complemented
115	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
121	C₂×Dic₈₉	—	4	2	no	—
122	C₂×Dic₈₉	—	8	1	no	—
123	C₂×Dic₈₉	—	8	1	no	—
124	C₂×Dic₈₉	—	8	1	no	—
125	C₂×Dic₈₉	—	8	1	no	—
126	C₂×Dic₈₉	—	8	2	no	—
127	C₂×Dic₈₉	—	8	2	no	—
128	D₃₅₆	—	4	2	no	—
129	D₃₅₆	—	4	2	no	—
130	D₃₅₆	—	8	1	no	—
131	D₃₅₆	—	8	1	no	—
132	Dic₁₇₈	—	4	2	no	—
133	Dic₁₇₈	—	4	2	no	—
134	Dic₁₇₈	—	8	1	no	—
135	Dic₁₇₈	—	8	1	no	—
136	C₈×C₈₉	—	2	2	no	—
137	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
139	C₈×C₈₉	#121	1	1	yes	characteristic
140	C₈×C₈₉	#124	1	1	yes	characteristic
141	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
142	C₂²×C₈₉	#466	1	1	yes	characteristic
143	C₂²×C₈₉	#493	1	1	yes	characteristic
144	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
145	C₂²×C₈₉	#508	1	2	yes	—
146	C₂×D₈₉	—	8	4	no	—
147	C₂×D₈₉	—	8	2	no	—
148	C₂×D₈₉	—	16	2	no	—
149	C₂×D₈₉	—	16	1	no	—
150	C₂×D₈₉	—	16	1	no	—
151	C₄×C₈₉	—	2	4	no	—
152	C₄×C₈₉	#390	1	1	yes	characteristic
153	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
154	C₄×C₈₉	#466	1	1	yes	characteristic
155	C₄×C₈₉	#495	1	1	yes	characteristic
156	Dic₈₉	—	8	2	no	—
157	Dic₈₉	—	16	1	no	—
158	Dic₈₉	—	16	1	no	—
159	#13237	—	89	1	no	maximal, Sylow
160	C₂×C₈₉	#1970	1	1	yes	characteristic
161	C₂×C₈₉	C₄×SD₃₂	1	2	yes	—
162	C₂×C₈₉	#2217	1	2	yes	—
163	C₂×C₈₉	#2248	1	1	yes	characteristic
164	C₂×C₈₉	#2249	1	1	yes	characteristic
165	D₈₉	—	16	2	no	—
166	C₂²×SD₃₂	—	89	1	no	—
167	C₂×#367	—	89	1	no	—
168	C₂×#358	—	89	1	no	—
169	C₂×#357	—	89	1	no	—
170	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
171	#1970	—	89	1	no	—
172	#1969	—	89	1	no	—
173	C₈₉	#13237	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₁₆	—	89	1	no	—
175	C₂²×D₈	—	89	1	no	—
176	C₂×SD₃₂	—	89	4	no	—
177	C₂×SD₃₂	—	178	2	no	—
178	C₂×SD₃₂	—	178	2	no	—
179	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
180	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
181	C₂×#107	—	89	1	no	—
182	C₂×#107	—	89	1	no	—
183	C₂×#103	—	178	1	no	—
184	C₂×#102	—	178	1	no	—
185	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
186	#391	—	178	1	no	—
187	#390	—	178	1	no	—
188	#367	—	178	2	no	—
189	#358	—	178	2	no	—
190	#357	—	178	2	no	—
191	C₄×C₁₆	—	2	2	no	—
192	C₂²×Q₈	—	178	1	no	—
193	C₂²×D₄	—	178	1	no	—
194	C₂×Q₁₆	—	89	2	no	—
195	C₂×Q₁₆	—	178	1	no	—
196	C₂×Q₁₆	—	178	1	no	—
197	C₂×Q₁₆	—	178	1	no	—
198	C₂×Q₁₆	—	178	1	no	—
199	C₂×Q₁₆	—	178	1	no	—
200	C₂×D₈	—	89	2	no	—
201	C₂×D₈	—	178	1	no	—
202	C₂×D₈	—	178	1	no	—
203	C₂×D₈	—	178	1	no	—
204	C₂×D₈	—	178	1	no	—
205	C₂×D₈	—	178	1	no	—
206	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	178	1	no	—
209	C₂×C₄⋊C₄	—	178	1	no	—
210	C₂×C₂²⋊C₄	—	356	2	no	—
211	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
212	SD₃₂	—	178	8	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
216	C₂×C₁₆	D₃₅₆	1	2	yes	—
217	C₂×C₁₆	D₃₅₆	1	2	yes	—
218	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
219	#107	—	178	1	no	—
220	#107	—	178	1	no	—
221	#107	—	178	1	no	—
222	#107	—	178	1	no	—
223	#103	—	356	2	no	—
224	#102	—	356	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	356	2	no	—
228	C₂⁴	—	356	1	no	—
229	C₂×Q₈	—	356	1	no	—
230	C₂×Q₈	—	356	1	no	—
231	C₂×Q₈	—	356	1	no	—
232	C₂×Q₈	—	356	1	no	—
233	C₂×Q₈	—	356	1	no	—
234	C₂×Q₈	—	178	2	no	—
235	C₂×D₄	—	356	1	no	—
236	C₂×D₄	—	356	1	no	—
237	C₂×D₄	—	356	1	no	—
238	C₂×D₄	—	356	1	no	—
239	C₂×D₄	—	356	1	no	—
240	C₂×D₄	—	178	2	no	—
241	C₂²×C₄	—	356	1	no	—
242	C₂²×C₄	—	356	1	no	—
243	C₂²×C₄	—	356	1	no	—
244	C₂²×C₄	D₇₁₂	1	2	yes	—
245	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
246	Q₁₆	—	356	1	no	—
247	Q₁₆	—	356	1	no	—
248	Q₁₆	—	178	2	no	—
249	Q₁₆	—	178	2	no	—
250	D₈	—	356	1	no	—
251	D₈	—	356	1	no	—
252	D₈	—	178	2	no	—
253	D₈	—	178	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	D₇₁₂	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
260	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
261	C₂×C₈	#13083574	1	1	yes	characteristic
262	C₂×C₈	#13083574	1	1	yes	characteristic
263	C₄⋊C₄	—	356	1	no	—
264	C₄⋊C₄	—	356	1	no	—
265	C₄⋊C₄	—	356	1	no	—
266	C₄⋊C₄	—	356	1	no	—
267	C₂²⋊C₄	—	712	4	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	#13083551	1	4	yes	complemented
272	C₂³	—	356	2	no	—
273	C₂³	—	712	1	no	—
274	C₂³	—	712	2	no	—
275	C₂³	—	712	2	no	—
276	C₂³	—	712	1	no	—
277	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
278	Q₈	—	356	2	no	—
279	Q₈	—	356	2	no	—
280	Q₈	—	712	1	no	—
281	Q₈	—	712	1	no	—
282	D₄	—	356	2	no	—
283	D₄	—	356	2	no	—
284	D₄	—	712	1	no	—
285	D₄	—	712	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	356	2	no	—
289	C₂×C₄	—	712	1	no	—
290	C₂×C₄	—	712	1	no	—
291	C₂×C₄	—	712	2	no	—
292	C₂×C₄	—	712	2	no	—
293	C₂×C₄	—	712	1	no	—
294	C₂×C₄	—	712	1	no	—
295	C₂×C₄	#15667957	1	4	yes	complemented
296	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
297	C₂×C₄	#15667986	1	1	yes	characteristic
298	C₂×C₄	#15668009	1	1	yes	characteristic
299	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
300	C₂×C₄	#15668034	1	1	yes	characteristic
301	C₂×C₄	#15668034	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	C₂×#13083551	1	1	yes	characteristic
304	C₈	C₄×D₃₅₆	1	1	yes	characteristic
305	C₈	#15667986	1	1	yes	characteristic
306	C₈	#15668000	1	1	yes	characteristic
307	C₂²	—	1424	1	no	—
308	C₂²	—	1424	1	no	—
309	C₂²	—	1424	2	no	—
310	C₂²	—	712	4	no	—
311	C₂²	—	712	2	no	—
312	C₂²	C₄×D₇₁₂	1	1	yes	characteristic
313	C₂²	#19151834	1	1	yes	characteristic
314	C₂²	#19151975	1	1	yes	characteristic
315	C₂²	C₂×#15667957	1	2	yes	—
316	C₂²	#19152030	1	2	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	1424	1	no	—
319	C₄	—	1424	1	no	—
320	C₄	—	712	2	no	—
321	C₄	#19151500	1	1	yes	characteristic
322	C₄	C₄×D₇₁₂	1	1	yes	characteristic
323	C₄	#19151834	1	1	yes	characteristic
324	C₄	#19151982	1	1	yes	characteristic
325	C₂	—	1424	2	no	—
326	C₂	#26021118	1	1	yes	characteristic
327	C₂	C₄×#15667957	1	2	yes	—
328	C₂	#26022348	1	2	yes	—
329	C₂	#26022508	1	1	yes	characteristic
330	C₂	#26022513	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#13237
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19151426	C₂	semidirect
3. C₂×#19151403	C₂	semidirect
6. #26021113	C₂	semidirect
7. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
12. C₂×#15667957	C₄	semidirect
105. C₁₆×C₈₉	C₂²⋊C₄	semidirect
114. C₂×C₄×C₈₉	SD₃₂	semidirect
170. C₂×C₄×C₁₆	D₈₉	semidirect
173. C₈₉	#13237	semidirect
271. C₁₆	#13083551	semidirect
295. C₂×C₄	#15667957	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667957	2
C₂×#19151426	2
C₂×#19151403	2
C₂×#19151400	2
#26021118	2
#26021113	2
C₂×C₄×C₁₆×C₈₉	2
#13237	89