Subgroups — #36707799

Order 22784.

682 conjugacy classes of subgroups.

The subgroup lattice has 302 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	#26029121	C₂	1	1	yes	maximal, characteristic, complemented
3	#26029105	C₂	1	1	yes	maximal, characteristic
4	#26022118	C₂	1	4	yes	maximal, complemented
5	C₄×#15667957	C₂	1	4	yes	maximal, complemented
6	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021973	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021969	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
11	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
12	#19152075	—	2	1	no	—
13	#19152072	—	2	1	no	—
14	C₂×#15667957	C₄	1	8	yes	complemented
15	C₂×#15667957	C₂²	1	2	yes	—
16	C₂×#15667957	C₂²	1	2	yes	—
17	#19151711	—	2	2	no	—
18	#19151706	—	2	2	no	—
19	C₄×Dic₃₅₆	C₂²	1	4	yes	complemented
20	C₄×D₇₁₂	C₂²	1	4	yes	—
21	#19151636	C₂²	1	1	yes	characteristic
22	#19151636	C₂²	1	1	yes	characteristic
23	#19151608	—	2	1	no	—
24	#19151604	—	2	1	no	—
25	#19151426	C₂²	1	2	yes	complemented
26	#19151424	C₂²	1	2	yes	complemented
27	#19151403	C₂²	1	4	yes	complemented
28	#19151400	C₂²	1	4	yes	—
29	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	#405×C₈₉	C₂²	1	1	yes	characteristic
32	#346×C₈₉	C₂²	1	2	yes	complemented
33	C₄×C₁₆×C₈₉	C₂²	1	2	yes	complemented
34	C₂²×Dic₁₇₈	—	2	1	no	—
35	C₂²×D₃₅₆	—	2	1	no	—
36	C₂×Dic₃₅₆	—	2	1	no	—
37	C₂×Dic₃₅₆	—	2	2	no	—
38	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	complemented
39	C₂×Dic₃₅₆	C₂³	1	2	yes	—
40	C₂×D₇₁₂	—	2	1	no	—
41	C₂×D₇₁₂	—	2	2	no	—
42	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
43	C₂×D₇₁₂	C₂³	1	2	yes	—
44	C₄×Dic₁₇₈	—	2	4	no	—
45	#15667977	—	4	2	no	—
46	C₄×D₃₅₆	—	2	4	no	—
47	#15667974	—	4	2	no	—
48	#15667971	—	2	1	no	—
49	#15667971	—	2	1	no	—
50	#15667968	—	4	2	no	—
51	C₂²⋊C₄×D₈₉	—	4	2	no	—
52	#15667957	C₂×C₄	1	8	yes	complemented
53	#15667957	C₂×C₄	1	8	yes	complemented
54	#15667946	C₂³	1	2	yes	—
55	#15667946	C₂³	1	2	yes	—
56	#15667944	—	2	1	no	—
57	#15667944	—	2	1	no	—
58	#15667937	—	2	4	no	—
59	#15667934	—	2	4	no	—
60	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	#117×C₈₉	D₄	1	1	yes	characteristic
63	M₃₂×C₈₉	—	2	2	no	—
64	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
65	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
66	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
68	#97×C₈₉	D₄	1	2	yes	—
69	C₄×C₈×C₈₉	C₂³	1	2	yes	—
70	C₂³×D₈₉	—	4	1	no	—
71	C₂×Dic₁₇₈	—	2	4	no	—
72	C₂×Dic₁₇₈	—	2	2	no	—
73	C₂×Dic₁₇₈	—	4	1	no	—
74	C₂×Dic₁₇₈	—	4	2	no	—
75	C₂×D₃₅₆	—	2	4	no	—
76	C₂×D₃₅₆	—	2	2	no	—
77	C₂×D₃₅₆	—	4	1	no	—
78	C₂×D₃₅₆	—	4	2	no	—
79	C₂²×Dic₈₉	—	4	1	no	—
80	C₂×C₄×D₈₉	—	4	4	no	—
81	Dic₃₅₆	—	2	4	no	—
82	Dic₃₅₆	—	2	2	no	—
83	Dic₃₅₆	C₂²×C₄	1	4	yes	—
84	D₇₁₂	—	2	4	no	—
85	D₇₁₂	—	2	2	no	—
86	D₇₁₂	C₂²×C₄	1	4	yes	—
87	C₄⋊Dic₈₉	—	2	2	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	#13083553	—	4	4	no	—
90	C₂²⋊Dic₈₉	—	4	1	no	—
91	C₂²⋊Dic₈₉	—	4	1	no	—
92	#13083551	—	4	4	no	—
93	C₄×Dic₈₉	—	4	4	no	—
94	C₄×Dic₈₉	—	4	1	no	—
95	C₄×Dic₈₉	—	4	1	no	—
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	M₁₆×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	#41	1	2	yes	—
102	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
104	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
105	C₄²×C₈₉	C₂×D₄	1	2	yes	—
106	C₁₆×C₈₉	—	2	2	no	—
107	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
108	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
109	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
110	C₂²×D₈₉	—	4	2	no	—
111	C₂²×D₈₉	—	4	4	no	—
112	C₂²×D₈₉	—	16	2	no	—
113	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
115	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#135	1	2	yes	—
117	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×Dic₈₉	—	4	2	no	—
119	C₂×Dic₈₉	—	4	4	no	—
120	C₂×Dic₈₉	—	4	2	no	—
121	C₂×Dic₈₉	—	4	2	no	—
122	D₃₅₆	—	2	4	no	—
123	D₃₅₆	—	4	2	no	—
124	D₃₅₆	—	4	4	no	—
125	C₄×D₈₉	—	16	4	no	—
126	Dic₁₇₈	—	2	4	no	—
127	Dic₁₇₈	—	4	2	no	—
128	Dic₁₇₈	—	4	4	no	—
129	C₈×C₈₉	—	2	2	no	—
130	C₈×C₈₉	C₄×D₄	1	2	yes	—
131	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
132	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	C₄×D₈	1	2	yes	—
135	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
136	C₂×D₈₉	—	4	4	no	—
137	C₂×D₈₉	—	8	4	no	—
138	C₂×D₈₉	—	16	4	no	—
139	C₄×C₈₉	—	2	4	no	—
140	C₄×C₈₉	C₄×D₈	1	2	yes	—
141	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
142	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
143	Dic₈₉	—	4	4	no	—
144	Dic₈₉	—	8	2	no	—
145	Dic₈₉	—	8	2	no	—
146	#33457	—	89	1	no	maximal, Sylow
147	C₂×C₈₉	—	2	2	no	—
148	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
149	C₂×C₈₉	#3416	1	2	yes	—
150	D₈₉	—	8	4	no	—
151	C₂²×SD₃₂	—	89	1	no	—
152	#2950	—	89	1	no	—
153	#2948	—	89	1	no	—
154	#2175	—	89	4	no	—
155	C₄×SD₃₂	—	89	4	no	—
156	#2158	—	89	1	no	—
157	#2141	—	89	1	no	—
158	#2140	—	89	1	no	—
159	#2108	D₈₉	1	1	yes	characteristic, complemented
160	C₈₉	#33457	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₁₆	—	89	1	no	—
162	C₂²×D₈	—	89	1	no	—
163	#519	—	178	1	no	—
164	#518	—	178	1	no	—
165	C₂×SD₃₂	—	89	8	no	—
166	C₂×SD₃₂	—	89	2	no	—
167	C₂×SD₃₂	—	89	2	no	—
168	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
169	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
170	#442	—	178	2	no	—
171	#441	—	178	2	no	—
172	C₄×Q₁₆	—	89	4	no	—
173	C₄×D₈	—	89	4	no	—
174	#427	—	89	1	no	—
175	#427	—	89	1	no	—
176	#419	—	178	1	no	—
177	#418	—	178	1	no	—
178	#405	C₂×D₈₉	1	1	yes	characteristic
179	#367	—	89	2	no	—
180	#366	—	89	2	no	—
181	#358	—	89	4	no	—
182	#357	—	89	4	no	—
183	#346	C₂×D₈₉	1	2	yes	complemented
184	C₄×C₁₆	C₂×D₈₉	1	2	yes	complemented
185	C₂²×Q₈	—	178	1	no	—
186	C₂²×D₄	—	178	1	no	—
187	C₂×Q₁₆	—	89	4	no	—
188	C₂×Q₁₆	—	89	2	no	—
189	C₂×Q₁₆	—	178	1	no	—
190	C₂×Q₁₆	—	178	2	no	—
191	C₂×D₈	—	89	4	no	—
192	C₂×D₈	—	89	2	no	—
193	C₂×D₈	—	178	1	no	—
194	C₂×D₈	—	178	2	no	—
195	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
197	C₄×Q₈	—	356	2	no	—
198	C₄×Q₈	—	178	4	no	—
199	C₄×D₄	—	356	2	no	—
200	C₄×D₄	—	178	4	no	—
201	#117	—	356	2	no	—
202	#117	—	178	1	no	—
203	#117	—	178	1	no	—
204	#117	D₃₅₆	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	356	2	no	—
206	SD₃₂	—	89	8	no	—
207	SD₃₂	—	89	8	no	—
208	M₃₂	—	2	2	no	—
209	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
210	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
211	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
212	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
213	#107	—	89	2	no	—
214	#107	—	89	2	no	—
215	#106	—	178	1	no	—
216	#106	—	178	1	no	—
217	#103	—	178	4	no	—
218	#102	—	178	4	no	—
219	#97	D₃₅₆	1	2	yes	—
220	C₄×C₈	C₂²×D₈₉	1	2	yes	—
221	C₂⁴	—	356	1	no	—
222	C₂×Q₈	—	356	1	no	—
223	C₂×Q₈	—	356	2	no	—
224	C₂×Q₈	—	178	4	no	—
225	C₂×Q₈	—	178	2	no	—
226	C₂×D₄	—	356	1	no	—
227	C₂×D₄	—	356	2	no	—
228	C₂×D₄	—	178	4	no	—
229	C₂×D₄	—	178	2	no	—
230	C₂²×C₄	—	356	1	no	—
231	C₂²×C₄	—	356	4	no	—
232	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
233	Q₁₆	—	89	4	no	—
234	Q₁₆	—	178	4	no	—
235	Q₁₆	—	178	2	no	—
236	D₈	—	89	4	no	—
237	D₈	—	178	4	no	—
238	D₈	—	178	2	no	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
241	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
242	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
243	C₂×C₈	#13083574	1	2	yes	—
244	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	—	356	4	no	—
246	C₄⋊C₄	—	178	2	no	—
247	C₄⋊C₄	—	178	2	no	—
248	C₄⋊C₄	D₇₁₂	1	2	yes	—
249	C₂²⋊C₄	—	356	1	no	—
250	C₂²⋊C₄	—	356	1	no	—
251	C₂²⋊C₄	—	356	4	no	—
252	C₂²⋊C₄	D₇₁₂	1	2	yes	—
253	C₄²	—	356	4	no	—
254	C₄²	—	356	1	no	—
255	C₄²	—	356	1	no	—
256	C₄²	C₂×D₃₅₆	1	2	yes	—
257	C₁₆	—	2	2	no	—
258	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
259	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
260	C₂³	—	356	4	no	—
261	C₂³	—	356	2	no	—
262	C₂³	—	1424	2	no	—
263	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
264	Q₈	—	356	4	no	—
265	Q₈	—	356	2	no	—
266	Q₈	—	178	4	no	—
267	D₄	—	356	4	no	—
268	D₄	—	356	2	no	—
269	D₄	—	178	4	no	—
270	C₂×C₄	—	356	4	no	—
271	C₂×C₄	—	356	2	no	—
272	C₂×C₄	—	356	2	no	—
273	C₂×C₄	—	356	2	no	—
274	C₂×C₄	—	1424	4	no	—
275	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
276	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
277	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
278	C₂×C₄	#15668034	1	2	yes	—
279	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	C₄×D₃₅₆	1	2	yes	—
282	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
283	C₈	C₂×#13083574	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	356	4	no	—
286	C₂²	—	1424	4	no	—
287	C₂²	—	712	4	no	—
288	C₂²	C₄×D₇₁₂	1	2	yes	—
289	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
290	C₄	—	2	4	no	—
291	C₄	—	356	4	no	—
292	C₄	—	712	2	no	—
293	C₄	—	712	2	no	—
294	C₄	C₄×D₇₁₂	1	2	yes	—
295	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
296	C₄	C₂×#15668034	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	712	4	no	—
299	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic
300	C₂	#26033400	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#33457
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #26029121	C₂	semidirect
4. #26022118	C₂	semidirect
5. C₄×#15667957	C₂	semidirect
6. #26022036	C₂	semidirect
7. #26021973	C₂	semidirect
9. C₈₉×#2108	C₂	semidirect
14. C₂×#15667957	C₄	semidirect
19. C₄×Dic₃₅₆	C₂²	semidirect
25. #19151426	C₂²	semidirect
26. #19151424	C₂²	semidirect
27. #19151403	C₂²	semidirect
32. #346×C₈₉	C₂²	semidirect
33. C₄×C₁₆×C₈₉	C₂²	semidirect
38. C₂×Dic₃₅₆	C₂×C₄	semidirect
52. #15667957	C₂×C₄	semidirect
53. #15667957	C₂×C₄	semidirect
64. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
159. #2108	D₈₉	semidirect
160. C₈₉	#33457	semidirect
183. #346	C₂×D₈₉	semidirect
184. C₄×C₁₆	C₂×D₈₉	semidirect
209. C₂×C₁₆	C₄×D₈₉	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
#26029121	2
#26029105	2
#26022118	2
C₄×#15667957	2
#26022036	2
#26021973	2
#26021969	2
C₈₉×#2108	2
#33457	89