Subgroups — #36453354

Order 22784.

510 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018642	C₂	1	2	yes	maximal, complemented
4	#26018572	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₁₂	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₅₆	C₂²	1	2	yes	—
8	C₄×D₇₁₂	—	2	1	no	—
9	C₄×D₇₁₂	—	2	1	no	—
10	C₄×D₇₁₂	—	2	1	no	—
11	C₄×D₇₁₂	—	2	1	no	—
12	#19151651	C₂²	1	2	yes	—
13	C₂×#15667946	C₄	1	1	yes	characteristic
14	C₂×#15667934	C₄	1	2	yes	—
15	#19151351	—	2	1	no	—
16	#19151349	—	2	1	no	—
17	#19151325	—	2	4	no	—
18	#19151321	—	2	4	no	—
19	#431×C₈₉	C₂²	1	1	yes	characteristic
20	#431×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#322×C₈₉	—	2	1	no	—
23	C₈²×C₈₉	—	2	1	no	—
24	C₂²×D₃₅₆	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₈₉	—	2	2	no	—
26	C₂×D₇₁₂	—	2	1	no	—
27	C₂×D₇₁₂	—	2	1	no	—
28	C₂×D₇₁₂	—	2	2	no	—
29	C₂×D₇₁₂	C₈	1	4	yes	complemented
30	C₄×D₃₅₆	—	2	2	no	—
31	C₄×D₃₅₆	—	2	2	no	—
32	C₄×D₃₅₆	—	2	2	no	—
33	C₄×D₃₅₆	—	2	2	no	—
34	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
35	C₂×#13083551	—	2	2	no	—
36	#15667946	—	2	1	no	—
37	#15667946	—	2	1	no	—
38	C₄⋊#10623494	—	2	2	no	—
39	#15667934	—	2	2	no	—
40	#15667934	—	2	2	no	—
41	C₂²⋊#10623494	—	2	2	no	—
42	C₄×#10623494	—	2	2	no	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₈₉	—	2	1	no	—
47	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
48	C₂²⋊C₈×C₈₉	—	2	1	no	—
49	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
50	C₄×C₈×C₈₉	—	2	1	no	—
51	C₄×C₈×C₈₉	—	2	1	no	—
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₂³×D₈₉	—	2	2	no	—
57	C₂×D₃₅₆	—	2	2	no	—
58	C₂×D₃₅₆	—	2	2	no	—
59	C₂×D₃₅₆	—	2	2	no	—
60	C₂×D₃₅₆	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×D₃₅₆	C₂×C₈	1	4	yes	—
63	C₂²×Dic₈₉	—	2	2	no	—
64	C₂×C₄×D₈₉	—	4	2	no	—
65	C₂×C₄×D₈₉	—	4	2	no	—
66	C₂×C₄×D₈₉	—	8	4	no	—
67	D₇₁₂	—	2	4	no	—
68	D₇₁₂	—	2	4	no	—
69	C₂×#10623494	—	4	2	no	—
70	C₄⋊Dic₈₉	—	2	2	no	—
71	C₄⋊Dic₈₉	—	2	2	no	—
72	#13083551	—	4	2	no	—
73	#13083551	—	4	2	no	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
77	C₂×C₈×C₈₉	—	2	1	no	—
78	C₂×C₈×C₈₉	—	2	1	no	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
87	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₈₉	#41	1	1	yes	characteristic
91	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
92	C₂²×D₈₉	—	2	4	no	—
93	C₂²×D₈₉	—	4	2	no	—
94	C₂²×D₈₉	—	4	2	no	—
95	C₂²×D₈₉	—	8	2	no	—
96	C₂²×D₈₉	—	8	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	4	1	no	—
99	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
109	C₂×Dic₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	8	2	no	—
111	D₃₅₆	—	2	4	no	—
112	D₃₅₆	—	2	4	no	—
113	D₃₅₆	—	4	2	no	—
114	D₃₅₆	—	4	2	no	—
115	C₄×D₈₉	—	8	4	no	—
116	C₄×D₈₉	—	8	4	no	—
117	C₈×C₈₉	—	2	1	no	—
118	C₈×C₈₉	—	2	1	no	—
119	C₈×C₈₉	—	4	2	no	—
120	C₈×C₈₉	—	4	2	no	—
121	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
122	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
123	#10623494	—	8	2	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	—	2	1	no	—
126	C₂²×C₈₉	#390	1	1	yes	characteristic
127	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
128	C₂²×C₈₉	#436	1	1	yes	characteristic
129	C₂×D₈₉	—	4	4	no	—
130	C₂×D₈₉	—	8	4	no	—
131	C₂×D₈₉	—	8	2	no	—
132	C₂×D₈₉	—	8	2	no	—
133	C₂×D₈₉	—	8	4	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₄×C₈₉	#408	1	1	yes	characteristic
139	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
140	C₄×C₈₉	#436	1	1	yes	characteristic
141	C₄×C₈₉	#472	1	2	yes	—
142	Dic₈₉	—	8	2	no	—
143	#9668	—	89	1	no	maximal, Sylow
144	C₂×C₈₉	—	2	1	no	—
145	C₂×C₈₉	—	2	1	no	—
146	C₂×C₈₉	#1954	1	1	yes	characteristic
147	C₂×C₈₉	#1976	1	1	yes	characteristic
148	C₂×C₈₉	#1978	1	1	yes	characteristic
149	D₈₉	—	8	4	no	—
150	C₂×C₄×D₈	—	89	1	no	—
151	#1564	—	89	1	no	—
152	#1478	—	89	2	no	—
153	#1460	—	89	2	no	—
154	#1450	D₈₉	1	1	yes	characteristic, complemented
155	C₈₉	#9668	1	1	yes	characteristic, complemented, Sylow
156	C₂²×D₈	—	89	1	no	—
157	C₂×C₄×D₄	—	89	2	no	—
158	C₄×D₈	—	178	1	no	—
159	C₄×D₈	—	178	1	no	—
160	C₄×D₈	—	178	1	no	—
161	C₄×D₈	—	178	1	no	—
162	#431	—	89	2	no	—
163	#431	C₂×D₈₉	1	1	yes	characteristic
164	#431	C₂×D₈₉	1	1	yes	characteristic
165	C₂×#107	—	89	1	no	—
166	C₂×#102	—	89	2	no	—
167	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
168	#335	—	178	1	no	—
169	#334	—	178	1	no	—
170	D₄⋊C₈	—	178	4	no	—
171	#323	—	178	4	no	—
172	#322	—	2	1	no	—
173	C₈²	—	2	1	no	—
174	C₂²×D₄	—	89	2	no	—
175	C₂³×C₄	—	178	2	no	—
176	C₂×D₈	—	89	4	no	—
177	C₂×D₈	—	178	1	no	—
178	C₂×D₈	—	178	1	no	—
179	C₂×D₈	—	178	2	no	—
180	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
182	C₄×D₄	—	178	2	no	—
183	C₄×D₄	—	178	2	no	—
184	C₄×D₄	—	178	2	no	—
185	C₄×D₄	—	178	2	no	—
186	C₂×C₄⋊C₄	—	89	2	no	—
187	C₂×C₂²⋊C₄	—	178	2	no	—
188	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
189	#107	—	178	1	no	—
190	#107	—	178	1	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	178	2	no	—
193	C₄⋊C₈	D₃₅₆	1	2	yes	—
194	#102	—	178	2	no	—
195	#102	—	178	2	no	—
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	178	2	no	—
198	C₂²⋊C₈	D₃₅₆	1	2	yes	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	178	2	no	—
202	C₄×C₈	D₃₅₆	1	1	yes	characteristic
203	C₄×C₈	D₃₅₆	1	1	yes	characteristic
204	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
206	C₂⁴	—	178	2	no	—
207	C₂×D₄	—	89	4	no	—
208	C₂×D₄	—	178	2	no	—
209	C₂×D₄	—	178	2	no	—
210	C₂×D₄	—	178	2	no	—
211	C₂×D₄	—	178	2	no	—
212	C₂×D₄	—	178	2	no	—
213	C₂²×C₄	—	356	2	no	—
214	C₂²×C₄	—	356	2	no	—
215	C₂²×C₄	—	178	2	no	—
216	C₂²×C₄	—	712	4	no	—
217	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
219	C₂²×C₄	#13083574	1	1	yes	characteristic
220	D₈	—	178	4	no	—
221	D₈	—	178	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	356	2	no	—
227	C₂×C₈	#13083551	1	1	yes	characteristic
228	C₂×C₈	#13083551	1	1	yes	characteristic
229	C₂×C₈	#13083551	1	1	yes	characteristic
230	C₂×C₈	#13083551	1	1	yes	characteristic
231	C₂×C₈	C₈×D₈₉	1	2	yes	—
232	C₂×C₈	#13083557	1	2	yes	—
233	C₄⋊C₄	—	178	2	no	—
234	C₄⋊C₄	—	178	2	no	—
235	C₂²⋊C₄	—	356	2	no	—
236	C₂²⋊C₄	—	356	2	no	—
237	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
238	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
239	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
240	C₄²	#13083574	1	1	yes	characteristic
241	C₂³	—	356	2	no	—
242	C₂³	—	356	2	no	—
243	C₂³	—	178	4	no	—
244	C₂³	—	712	2	no	—
245	C₂³	—	712	2	no	—
246	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
247	D₄	—	356	2	no	—
248	D₄	—	356	2	no	—
249	D₄	—	178	4	no	—
250	D₄	—	178	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	356	2	no	—
254	C₂×C₄	—	712	4	no	—
255	C₂×C₄	—	712	4	no	—
256	C₂×C₄	—	712	2	no	—
257	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
258	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
259	C₂×C₄	#15667986	1	1	yes	characteristic
260	C₂×C₄	#15667986	1	1	yes	characteristic
261	C₂×C₄	#15668000	1	1	yes	characteristic
262	C₂×C₄	#15668009	1	1	yes	characteristic
263	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
265	C₂×C₄	#15668018	1	1	yes	characteristic
266	C₂×C₄	#15668018	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	712	2	no	—
272	C₈	#15667924	1	2	yes	—
273	C₈	#15667924	1	2	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	356	4	no	—
277	C₂²	—	712	4	no	—
278	C₂²	—	712	4	no	—
279	C₂²	—	712	2	no	—
280	C₂²	—	712	2	no	—
281	C₂²	#19151500	1	1	yes	characteristic
282	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
283	C₂²	#19151677	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	712	2	no	—
288	C₄	C₂×#15667924	1	1	yes	characteristic
289	C₄	#19151564	1	1	yes	characteristic
290	C₄	C₈×D₃₅₆	1	1	yes	characteristic
291	C₄	#19151677	1	1	yes	characteristic
292	C₄	#19151868	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	712	4	no	—
296	C₂	#26021046	1	1	yes	characteristic
297	C₂	#26021150	1	1	yes	characteristic
298	C₂	#26021162	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667986
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1450	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂²×D₇₁₂	C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1450	D₈₉
hypocenter	C₈₉	#9668
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #26019000	C₂	semidirect
3. #26018642	C₂	semidirect
4. #26018572	C₂	semidirect
5. C₈₉×#1450	C₂	semidirect
29. C₂×D₇₁₂	C₈	semidirect
154. #1450	D₈₉	semidirect
155. C₈₉	#9668	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
#26019000	2
#26018642	2
#26018572	2
C₈₉×#1450	2
#9668	89