Subgroups — #36524258

Order 22784.

710 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029292	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
3	#26019167	C₂	1	4	yes	maximal, complemented
4	#26019138	C₂	1	4	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018644	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₁₂	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃₅₆	C₂²	1	2	yes	—
10	C₄×D₇₁₂	C₄	1	2	yes	—
11	C₄×D₇₁₂	C₄	1	2	yes	—
12	C₄×D₇₁₂	C₂²	1	2	yes	—
13	C₄×D₇₁₂	C₂²	1	2	yes	—
14	#19151677	C₂²	1	8	yes	complemented
15	#19151675	—	2	4	no	—
16	#19151651	C₂²	1	2	yes	—
17	C₂×#15667946	C₄	1	1	yes	characteristic
18	C₂×#15667934	C₄	1	2	yes	—
19	#19151560	—	2	4	no	—
20	#19151351	C₂²	1	2	yes	complemented
21	#19151349	C₂²	1	2	yes	complemented
22	#19151327	C₂²	1	8	yes	complemented
23	#431×C₈₉	C₂²	1	1	yes	characteristic
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#322×C₈₉	C₂²	1	2	yes	complemented
27	C₈²×C₈₉	C₂²	1	2	yes	complemented
28	C₂²×D₃₅₆	C₂×C₄	1	2	yes	—
29	C₂²×C₄×D₈₉	—	2	2	no	—
30	C₂×D₇₁₂	—	2	4	no	—
31	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
32	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
33	C₂×#13083557	—	2	8	no	—
34	C₄×D₃₅₆	—	2	2	no	—
35	C₄×D₃₅₆	—	2	2	no	—
36	C₄×D₃₅₆	C₂×C₄	1	4	yes	—
37	C₄×D₃₅₆	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
39	C₂×#13083551	—	2	2	no	—
40	#15667946	C₂×C₄	1	2	yes	—
41	#15667946	C₂×C₄	1	2	yes	—
42	C₄⋊#10623494	C₂³	1	4	yes	—
43	#15667934	C₂×C₄	1	4	yes	—
44	#15667934	C₂×C₄	1	4	yes	—
45	C₂²⋊#10623494	—	2	2	no	—
46	#15667924	—	2	8	no	—
47	C₄×#10623494	—	2	2	no	—
48	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₈₉	—	2	1	no	—
52	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
53	C₂²⋊C₈×C₈₉	—	2	1	no	—
54	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
55	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂³	1	2	yes	—
61	C₂³×D₈₉	—	2	2	no	—
62	C₂×D₃₅₆	—	2	4	no	—
63	C₂×D₃₅₆	—	2	4	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×D₃₅₆	C₂²×C₄	1	4	yes	—
66	C₂²×Dic₈₉	—	2	2	no	—
67	C₂×C₄×D₈₉	—	2	4	no	—
68	C₂×C₄×D₈₉	—	2	4	no	—
69	C₂×C₄×D₈₉	—	8	4	no	—
70	D₇₁₂	—	2	4	no	—
71	D₇₁₂	—	2	4	no	—
72	#13083557	—	8	8	no	—
73	C₂×#10623494	—	2	4	no	—
74	C₄⋊Dic₈₉	—	2	2	no	—
75	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
76	#13083551	—	2	4	no	—
77	#13083551	—	2	4	no	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	M₁₆	1	4	yes	complemented
84	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
89	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₈₉	#41	1	1	yes	characteristic
92	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
93	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
94	C₂²×D₈₉	—	2	4	no	—
95	C₂²×D₈₉	—	4	4	no	—
96	C₂²×D₈₉	—	8	2	no	—
97	C₂²×D₈₉	—	8	2	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	4	1	no	—
100	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₈₉	#131	1	2	yes	—
105	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂×Dic₈₉	—	2	4	no	—
109	C₂×Dic₈₉	—	8	2	no	—
110	D₃₅₆	—	2	4	no	—
111	D₃₅₆	—	2	4	no	—
112	D₃₅₆	—	4	4	no	—
113	C₄×D₈₉	—	4	8	no	—
114	C₄×D₈₉	—	8	4	no	—
115	C₈×C₈₉	—	2	1	no	—
116	C₈×C₈₉	—	2	1	no	—
117	C₈×C₈₉	—	2	4	no	—
118	C₈×C₈₉	—	4	2	no	—
119	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
120	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
121	#10623494	—	4	4	no	—
122	C₂²×C₈₉	—	2	1	no	—
123	C₂²×C₈₉	—	2	1	no	—
124	C₂²×C₈₉	#436	1	2	yes	—
125	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂×D₈₉	—	4	4	no	—
127	C₂×D₈₉	—	4	4	no	—
128	C₂×D₈₉	—	4	4	no	—
129	C₂×D₈₉	—	8	8	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	#436	1	2	yes	—
134	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
135	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
136	C₄×C₈₉	#576	1	2	yes	—
137	Dic₈₉	—	4	4	no	—
138	#17895	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	—	2	2	no	—
140	C₂×C₈₉	C₂×#436	1	1	yes	characteristic
141	C₂×C₈₉	#2948	1	1	yes	characteristic
142	C₂×C₈₉	#2957	1	1	yes	characteristic
143	D₈₉	—	8	4	no	—
144	#2975	—	89	2	no	—
145	C₂×C₄×D₈	—	89	1	no	—
146	#1594	—	89	4	no	—
147	#1589	—	89	4	no	—
148	#1564	—	89	1	no	—
149	#1478	—	89	2	no	—
150	#1450	D₈₉	1	1	yes	characteristic, complemented
151	C₈₉	#17895	1	1	yes	characteristic, complemented, Sylow
152	C₂²×D₈	—	89	1	no	—
153	C₂×C₄×D₄	—	89	2	no	—
154	C₄×D₈	—	89	2	no	—
155	C₄×D₈	—	89	2	no	—
156	C₄×D₈	—	89	2	no	—
157	C₄×D₈	—	89	2	no	—
158	#436	—	89	8	no	—
159	#436	—	178	4	no	—
160	#431	—	89	2	no	—
161	#431	C₂×D₈₉	1	1	yes	characteristic
162	#431	C₂×D₈₉	1	1	yes	characteristic
163	C₂×#107	—	89	1	no	—
164	C₂×#102	—	89	2	no	—
165	#407	—	178	4	no	—
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	#335	—	89	2	no	—
168	#334	—	89	2	no	—
169	D₄⋊C₈	—	89	8	no	—
170	#322	C₂×D₈₉	1	2	yes	complemented
171	C₈²	C₂×D₈₉	1	2	yes	complemented
172	C₂²×D₄	—	89	2	no	—
173	C₂³×C₄	—	178	2	no	—
174	C₂×D₈	—	89	2	no	—
175	C₂×D₈	—	89	2	no	—
176	C₂×D₈	—	178	4	no	—
177	C₂×M₁₆	—	178	8	no	—
178	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
180	C₄×D₄	—	89	4	no	—
181	C₄×D₄	—	89	4	no	—
182	C₄×D₄	—	178	2	no	—
183	C₄×D₄	—	178	2	no	—
184	C₂×C₄⋊C₄	—	89	2	no	—
185	C₂×C₂²⋊C₄	—	178	2	no	—
186	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
187	#107	—	89	2	no	—
188	#107	—	89	2	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	—	89	4	no	—
191	C₄⋊C₈	D₃₅₆	1	2	yes	—
192	#102	—	89	4	no	—
193	#102	—	89	4	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₂²⋊C₈	—	178	2	no	—
196	C₂²⋊C₈	—	178	8	no	—
197	C₂²⋊C₈	D₃₅₆	1	2	yes	—
198	C₄×C₈	—	178	2	no	—
199	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
200	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₈₉	1	2	yes	—
202	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₈₉	1	2	yes	—
205	C₂⁴	—	178	2	no	—
206	C₂×D₄	—	89	4	no	—
207	C₂×D₄	—	178	2	no	—
208	C₂×D₄	—	178	4	no	—
209	C₂×D₄	—	178	4	no	—
210	C₂²×C₄	—	178	4	no	—
211	C₂²×C₄	—	178	2	no	—
212	C₂²×C₄	—	178	4	no	—
213	C₂²×C₄	—	712	4	no	—
214	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂²×C₄	#13083574	1	1	yes	characteristic
217	D₈	—	178	4	no	—
218	D₈	—	178	4	no	—
219	M₁₆	—	712	8	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	178	4	no	—
223	C₂×C₈	#13083557	1	4	yes	complemented
224	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
228	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
229	C₄⋊C₄	—	89	4	no	—
230	C₄⋊C₄	—	178	2	no	—
231	C₂²⋊C₄	—	178	4	no	—
232	C₂²⋊C₄	—	178	4	no	—
233	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
234	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
235	C₄²	#13083574	1	1	yes	characteristic
236	C₄²	C₂³×D₈₉	1	1	yes	characteristic
237	C₂³	—	356	4	no	—
238	C₂³	—	178	4	no	—
239	C₂³	—	712	2	no	—
240	C₂³	—	712	2	no	—
241	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
242	D₄	—	356	4	no	—
243	D₄	—	178	4	no	—
244	D₄	—	178	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	356	8	no	—
248	C₂×C₄	—	178	4	no	—
249	C₂×C₄	—	712	4	no	—
250	C₂×C₄	—	712	2	no	—
251	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
252	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
253	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
254	C₂×C₄	C₂×#13083557	1	2	yes	—
255	C₂×C₄	#15668018	1	2	yes	—
256	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
258	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
259	C₈	—	2	4	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	4	2	no	—
263	C₈	—	356	4	no	—
264	C₈	C₂×#13083557	1	2	yes	—
265	C₈	C₂×#13083557	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	356	4	no	—
269	C₂²	—	356	4	no	—
270	C₂²	—	356	4	no	—
271	C₂²	—	712	8	no	—
272	C₂²	#19151677	1	2	yes	—
273	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	356	4	no	—
278	C₄	#19151677	1	2	yes	—
279	C₄	C₂²×#13083557	1	1	yes	characteristic
280	C₄	C₂×#15668018	1	1	yes	characteristic
281	C₄	#19152403	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	712	4	no	—
284	C₂	C₂×#19151677	1	1	yes	characteristic
285	C₂	#26029105	1	1	yes	characteristic
286	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1450	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₇₁₂	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1450	D₈₉
hypocenter	C₈₉	#17895
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #26029292	C₂	semidirect
3. #26019167	C₂	semidirect
4. #26019138	C₂	semidirect
5. #26019000	C₂	semidirect
6. #26018644	C₂	semidirect
7. C₈₉×#1450	C₂	semidirect
14. #19151677	C₂²	semidirect
20. #19151351	C₂²	semidirect
21. #19151349	C₂²	semidirect
22. #19151327	C₂²	semidirect
26. #322×C₈₉	C₂²	semidirect
27. C₈²×C₈₉	C₂²	semidirect
83. C₂×C₈×C₈₉	M₁₆	semidirect
150. #1450	D₈₉	semidirect
151. C₈₉	#17895	semidirect
170. #322	C₂×D₈₉	semidirect
171. C₈²	C₂×D₈₉	semidirect
223. C₂×C₈	#13083557	semidirect

Maximal subgroups

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

subgroup	index
#26029292	2
C₂×C₄×D₇₁₂	2
#26019167	2
#26019138	2
#26019000	2
#26018644	2
C₈₉×#1450	2
#17895	89