Subgroups — #36470470

Order 22784.

510 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151400	C₂	1	2	yes	maximal
3	#26020412	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018344	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₃₅₆	—	2	1	no	—
8	C₄×D₇₁₂	—	2	2	no	—
9	C₄×D₇₁₂	C₄	1	4	yes	complemented
10	C₂×#15667946	—	2	1	no	—
11	C₂×#15667946	C₂²	1	2	yes	—
12	C₂×#15667946	C₂²	1	1	yes	characteristic
13	C₂×#15667934	—	2	1	no	—
14	#19151468	—	2	1	no	—
15	#19151400	—	2	4	no	—
16	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₈₉	—	2	2	no	—
19	C₂²×D₃₅₆	—	2	1	no	—
20	C₂²×C₄×D₈₉	—	4	1	no	—
21	C₂×D₇₁₂	—	2	1	no	—
22	C₂×D₇₁₂	—	2	1	no	—
23	C₂×D₇₁₂	—	2	1	no	—
24	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
25	C₂×D₇₁₂	D₄	1	2	yes	—
26	C₂×D₇₁₂	Q₈	1	2	yes	—
27	C₄×D₃₅₆	—	2	4	no	—
28	C₄×D₃₅₆	—	4	2	no	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	—	2	2	no	—
31	C₂×C₄⋊Dic₈₉	—	4	1	no	—
32	C₂×C₄⋊Dic₈₉	—	4	1	no	—
33	C₂×#13083551	—	4	1	no	—
34	#15667946	—	2	1	no	—
35	#15667946	—	2	2	no	—
36	#15667946	—	2	2	no	—
37	#15667946	—	4	2	no	—
38	#15667946	C₂×C₄	1	2	yes	—
39	#15667934	—	2	4	no	—
40	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₈₉	—	2	2	no	—
44	C₂×C₁₆×C₈₉	—	2	2	no	—
45	C₂×C₁₆×C₈₉	—	2	2	no	—
46	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₈₉	D₄	1	2	yes	—
48	C₂³×D₈₉	—	4	1	no	—
49	C₂×D₃₅₆	—	2	2	no	—
50	C₂×D₃₅₆	—	2	2	no	—
51	C₂×D₃₅₆	—	2	2	no	—
52	C₂×D₃₅₆	—	4	1	no	—
53	C₂×D₃₅₆	—	4	1	no	—
54	C₂×D₃₅₆	—	4	1	no	—
55	C₂²×Dic₈₉	—	4	1	no	—
56	C₂²×Dic₈₉	—	4	2	no	—
57	C₂×C₄×D₈₉	—	4	4	no	—
58	C₂×C₄×D₈₉	—	8	2	no	—
59	C₂×C₄×D₈₉	—	8	2	no	—
60	D₇₁₂	—	2	2	no	—
61	D₇₁₂	—	2	2	no	—
62	D₇₁₂	—	2	2	no	—
63	D₇₁₂	C₄⋊C₄	1	4	yes	complemented
64	C₄⋊Dic₈₉	—	2	2	no	—
65	C₄⋊Dic₈₉	—	4	1	no	—
66	C₄⋊Dic₈₉	—	4	2	no	—
67	C₄⋊Dic₈₉	—	4	2	no	—
68	C₄⋊Dic₈₉	—	8	2	no	—
69	C₄⋊Dic₈₉	—	8	2	no	—
70	#13083551	—	4	4	no	—
71	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	—	2	1	no	—
75	C₂×C₈×C₈₉	—	2	1	no	—
76	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
83	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₈₉	SD₁₆	1	2	yes	—
85	C₁₆×C₈₉	—	4	2	no	—
86	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
87	C₂²×D₈₉	—	4	2	no	—
88	C₂²×D₈₉	—	4	2	no	—
89	C₂²×D₈₉	—	4	2	no	—
90	C₂²×D₈₉	—	8	1	no	—
91	C₂²×D₈₉	—	8	1	no	—
92	C₂²×D₈₉	—	8	1	no	—
93	C₂²×D₈₉	—	8	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	#102	1	2	yes	—
97	C₂×C₄×C₈₉	#102	1	2	yes	—
98	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
99	C₂×C₄×C₈₉	SD₃₂	1	2	yes	—
100	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
106	C₂×Dic₈₉	—	4	2	no	—
107	C₂×Dic₈₉	—	8	1	no	—
108	C₂×Dic₈₉	—	8	1	no	—
109	C₂×Dic₈₉	—	8	2	no	—
110	C₂×Dic₈₉	—	8	2	no	—
111	C₂×Dic₈₉	—	8	2	no	—
112	D₃₅₆	—	2	4	no	—
113	D₃₅₆	—	4	2	no	—
114	D₃₅₆	—	4	2	no	—
115	D₃₅₆	—	4	2	no	—
116	C₄×D₈₉	—	8	8	no	—
117	C₈×C₈₉	—	2	2	no	—
118	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₈₉	#117	1	1	yes	characteristic
120	C₈×C₈₉	#121	1	1	yes	characteristic
121	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
123	C₂²×C₈₉	#459	1	1	yes	characteristic
124	C₂²×C₈₉	#474	1	1	yes	characteristic
125	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
126	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
127	C₂²×C₈₉	#509	1	1	yes	characteristic
128	C₂²×C₈₉	#510	1	1	yes	characteristic
129	C₂×D₈₉	—	4	4	no	—
130	C₂×D₈₉	—	8	2	no	—
131	C₂×D₈₉	—	8	2	no	—
132	C₂×D₈₉	—	8	2	no	—
133	C₂×D₈₉	—	8	2	no	—
134	C₂×D₈₉	—	8	2	no	—
135	C₂×D₈₉	—	8	2	no	—
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	#357	1	4	yes	complemented
138	C₄×C₈₉	#380	1	1	yes	characteristic
139	C₄×C₈₉	#417	1	1	yes	characteristic
140	C₄×C₈₉	#461	1	1	yes	characteristic
141	C₄×C₈₉	#476	1	1	yes	characteristic
142	Dic₈₉	—	8	2	no	—
143	Dic₈₉	—	16	2	no	—
144	#12811	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	#1802	1	1	yes	characteristic
146	C₂×C₈₉	C₂×#357	1	1	yes	characteristic
147	C₂×C₈₉	#2139	1	1	yes	characteristic
148	C₂×C₈₉	#2206	1	1	yes	characteristic
149	C₂×C₈₉	#2208	1	1	yes	characteristic
150	C₂×C₈₉	#2224	1	1	yes	characteristic
151	C₂×C₈₉	#2226	1	1	yes	characteristic
152	D₈₉	—	8	4	no	—
153	C₂×C₄×D₈	—	89	1	no	—
154	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic, complemented
155	C₂×#357	—	89	2	no	—
156	#1845	—	89	1	no	—
157	#1380	—	89	2	no	—
158	C₈₉	#12811	1	1	yes	characteristic, complemented, Sylow
159	C₂²×D₈	—	89	1	no	—
160	C₂×C₄×D₄	—	178	1	no	—
161	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
162	C₄×D₈	—	89	4	no	—
163	C₄×D₈	—	178	2	no	—
164	C₂×#107	—	89	1	no	—
165	C₂×#107	—	89	2	no	—
166	C₂×#107	—	178	1	no	—
167	C₂×#102	—	178	1	no	—
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	#384	—	178	1	no	—
170	C₄⋊C₁₆	—	2	2	no	—
171	#357	—	178	4	no	—
172	C₂²×D₄	—	178	1	no	—
173	C₂³×C₄	—	356	1	no	—
174	C₂×D₈	—	89	2	no	—
175	C₂×D₈	—	89	2	no	—
176	C₂×D₈	—	89	2	no	—
177	C₂×D₈	—	178	1	no	—
178	C₂×D₈	—	178	1	no	—
179	C₂×D₈	—	178	1	no	—
180	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
182	C₄×D₄	—	356	2	no	—
183	C₄×D₄	—	178	4	no	—
184	C₂×C₄⋊C₄	—	356	1	no	—
185	C₂×C₄⋊C₄	—	356	1	no	—
186	C₂×C₄⋊C₄	—	178	2	no	—
187	C₂×C₄⋊C₄	—	178	1	no	—
188	C₂×C₂²⋊C₄	—	356	1	no	—
189	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	—	2	2	no	—
193	#107	—	89	2	no	—
194	#107	—	356	2	no	—
195	#107	—	178	2	no	—
196	#107	—	178	2	no	—
197	#107	—	178	1	no	—
198	#102	—	178	4	no	—
199	C₄×C₈	C₄×D₈₉	1	2	yes	—
200	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
201	C₂⁴	—	356	1	no	—
202	C₂×D₄	—	356	1	no	—
203	C₂×D₄	—	356	1	no	—
204	C₂×D₄	—	356	1	no	—
205	C₂×D₄	—	178	2	no	—
206	C₂×D₄	—	178	2	no	—
207	C₂×D₄	—	178	2	no	—
208	C₂²×C₄	—	356	1	no	—
209	C₂²×C₄	—	356	4	no	—
210	C₂²×C₄	—	356	2	no	—
211	C₂²×C₄	—	712	2	no	—
212	C₂²×C₄	—	712	2	no	—
213	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
214	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
216	D₈	—	89	4	no	—
217	D₈	—	178	2	no	—
218	D₈	—	178	2	no	—
219	D₈	—	178	2	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	#13083551	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
224	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
227	C₂×C₈	#13083573	1	1	yes	characteristic
228	C₂×C₈	#13083575	1	1	yes	characteristic
229	C₄⋊C₄	—	356	1	no	—
230	C₄⋊C₄	—	356	2	no	—
231	C₄⋊C₄	—	356	2	no	—
232	C₄⋊C₄	—	178	2	no	—
233	C₄⋊C₄	—	712	2	no	—
234	C₄⋊C₄	—	712	2	no	—
235	C₂²⋊C₄	—	356	4	no	—
236	C₄²	#13083551	1	2	yes	—
237	C₄²	#13083563	1	2	yes	—
238	C₁₆	—	4	2	no	—
239	C₂³	—	356	2	no	—
240	C₂³	—	356	2	no	—
241	C₂³	—	356	2	no	—
242	C₂³	—	712	1	no	—
243	C₂³	—	712	1	no	—
244	C₂³	—	712	1	no	—
245	C₂³	—	712	1	no	—
246	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
247	D₄	—	356	2	no	—
248	D₄	—	356	2	no	—
249	D₄	—	356	2	no	—
250	D₄	—	178	4	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	356	2	no	—
254	C₂×C₄	—	712	2	no	—
255	C₂×C₄	—	712	2	no	—
256	C₂×C₄	—	712	2	no	—
257	C₂×C₄	—	712	1	no	—
258	C₂×C₄	—	712	1	no	—
259	C₂×C₄	—	712	8	no	—
260	C₂×C₄	#15667934	1	2	yes	—
261	C₂×C₄	#15667934	1	2	yes	—
262	C₂×C₄	D₁₄₂₄	1	2	yes	—
263	C₂×C₄	#15667957	1	2	yes	—
264	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
265	C₂×C₄	#15667987	1	1	yes	characteristic
266	C₂×C₄	#15667992	1	1	yes	characteristic
267	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
268	C₂×C₄	#15668040	1	1	yes	characteristic
269	C₂×C₄	#15668045	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
272	C₈	#15667970	1	1	yes	characteristic
273	C₈	#15667983	1	1	yes	characteristic
274	C₈	#15667988	1	1	yes	characteristic
275	C₂²	—	356	4	no	—
276	C₂²	—	712	2	no	—
277	C₂²	—	712	2	no	—
278	C₂²	—	712	2	no	—
279	C₂²	—	712	2	no	—
280	C₂²	—	712	2	no	—
281	C₂²	—	712	2	no	—
282	C₂²	C₂×#15667934	1	1	yes	characteristic
283	C₂²	#19151794	1	1	yes	characteristic
284	C₂²	#19151879	1	1	yes	characteristic
285	C₂²	C₂×D₁₄₂₄	1	1	yes	characteristic
286	C₂²	C₂×#15667957	1	1	yes	characteristic
287	C₂²	#19152036	1	1	yes	characteristic
288	C₂²	#19152041	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	1424	2	no	—
291	C₄	—	712	2	no	—
292	C₄	#19151400	1	4	yes	complemented
293	C₄	#19151455	1	1	yes	characteristic
294	C₄	#19151600	1	1	yes	characteristic
295	C₄	#19151804	1	1	yes	characteristic
296	C₄	#19151889	1	1	yes	characteristic
297	C₂	—	712	4	no	—
298	C₂	#26020253	1	1	yes	characteristic
299	C₂	C₂×#19151400	1	1	yes	characteristic
300	C₂	#26021965	1	1	yes	characteristic
301	C₂	#26022288	1	1	yes	characteristic
302	C₂	#26022298	1	1	yes	characteristic
303	C₂	#26022390	1	1	yes	characteristic
304	C₂	#26022400	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	#12811
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
3. #26020412	C₂	semidirect
4. #26018344	C₂	semidirect
5. C₂×C₄⋊C₁₆×C₈₉	C₂	semidirect
9. C₄×D₇₁₂	C₄	semidirect
63. D₇₁₂	C₄⋊C₄	semidirect
137. C₄×C₈₉	#357	semidirect
154. C₂×C₄⋊C₁₆	D₈₉	semidirect
158. C₈₉	#12811	semidirect
292. C₄	#19151400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₁₂	2
C₂×#19151400	2
#26020412	2
#26018344	2
C₂×C₄⋊C₁₆×C₈₉	2
#12811	89