Subgroups — #36707842

Order 22784.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033388	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151716	C₂	1	2	yes	maximal, complemented
3	#26022139	C₂	1	4	yes	maximal, complemented
4	#26022134	C₂	1	4	yes	maximal, complemented
5	#26022047	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021981	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	2	yes	—
9	C₂×#15668018	—	2	2	no	—
10	#19152041	C₄	1	4	yes	complemented
11	#19152036	C₄	1	4	yes	complemented
12	#19152030	C₂²	1	4	yes	—
13	#19151716	—	2	2	no	—
14	#19151716	—	2	2	no	—
15	#19151716	C₂²	1	4	yes	complemented
16	#19151716	C₂²	1	4	yes	complemented
17	C₂×#15667948	C₂²	1	2	yes	—
18	C₂×#15667940	—	2	2	no	—
19	#19151428	C₂²	1	4	yes	complemented
20	#19151406	C₂²	1	8	yes	complemented
21	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#346×C₈₉	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₈₉	C₂²	1	2	yes	complemented
26	C₂×#13083574	—	2	2	no	—
27	#15668034	—	2	2	no	—
28	#15668034	—	2	2	no	—
29	#15668034	C₂×C₄	1	4	yes	—
30	#15668034	C₂³	1	4	yes	—
31	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
32	C₂×#13083563	—	2	2	no	—
33	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
34	#15668018	—	2	4	no	—
35	#15668018	—	2	4	no	—
36	#15668018	—	4	2	no	—
37	#15668018	—	4	2	no	—
38	C₂×#13083558	—	2	2	no	—
39	C₂×#13083557	—	4	2	no	—
40	C₂×C₈×D₈₉	—	4	2	no	—
41	Dic₇₁₂	C₂×C₄	1	4	yes	complemented
42	#15667957	C₂×C₄	1	8	yes	complemented
43	D₁₄₂₄	C₂×C₄	1	4	yes	complemented
44	#15667948	—	2	2	no	—
45	#15667948	C₂³	1	4	yes	—
46	#15667940	—	2	4	no	—
47	#15667940	—	2	4	no	—
48	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	M₃₂×C₈₉	—	2	2	no	—
52	M₃₂×C₈₉	C₂×C₄	1	4	yes	—
53	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
54	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
55	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	#13083574	—	2	4	no	—
60	#13083574	—	2	4	no	—
61	#13083574	—	4	2	no	—
62	#13083574	—	4	2	no	—
63	C₂×Dic₁₇₈	—	2	2	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×C₂²⋊D₈₉	—	4	2	no	—
66	C₂×C₄×D₈₉	—	4	2	no	—
67	Dic₃₅₆	—	2	2	no	—
68	Dic₃₅₆	C₂²×C₄	1	4	yes	—
69	#13083563	—	2	4	no	—
70	#13083563	—	2	4	no	—
71	D₇₁₂	—	2	2	no	—
72	D₇₁₂	C₂²×C₄	1	4	yes	—
73	#13083558	—	2	4	no	—
74	#13083558	—	4	2	no	—
75	#13083557	—	4	4	no	—
76	#13083557	—	4	4	no	—
77	C₂×#10623494	—	4	2	no	—
78	C₈×D₈₉	—	4	4	no	—
79	C₈×D₈₉	—	4	4	no	—
80	C₂²×C₄×C₈₉	—	2	1	no	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	D₈	1	2	yes	—
83	C₂×C₈×C₈₉	D₈	1	2	yes	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	#41	1	2	yes	—
89	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₄²×C₈₉	—	2	1	no	—
91	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₈₉	—	2	2	no	—
94	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
95	C₂³×C₈₉	#135	1	1	yes	characteristic
96	C₂²×D₈₉	—	4	2	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₈₉	—	4	4	no	—
105	C₂²⋊D₈₉	—	4	4	no	—
106	C₂×Dic₈₉	—	4	2	no	—
107	D₃₅₆	—	2	4	no	—
108	D₃₅₆	—	4	2	no	—
109	C₄×D₈₉	—	4	4	no	—
110	C₄×D₈₉	—	4	4	no	—
111	Dic₁₇₈	—	2	4	no	—
112	Dic₁₇₈	—	4	2	no	—
113	C₈×C₈₉	C₄×D₄	1	2	yes	—
114	C₈×C₈₉	C₂×D₈	1	2	yes	—
115	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
116	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
117	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
119	#10623494	—	4	4	no	—
120	C₂²×C₈₉	—	4	1	no	—
121	C₂²×C₈₉	C₄×D₈	1	2	yes	—
122	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
123	C₂×D₈₉	—	4	4	no	—
124	C₂×D₈₉	—	8	4	no	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	4	1	no	—
127	C₄×C₈₉	C₄×D₈	1	2	yes	—
128	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
130	Dic₈₉	—	4	4	no	—
131	#33463	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	—	2	1	no	—
133	C₂×C₈₉	—	2	2	no	—
134	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
135	D₈₉	—	8	4	no	—
136	#3414	—	89	1	no	—
137	C₂×#443	—	89	2	no	—
138	#2179	—	89	4	no	—
139	#2178	—	89	4	no	—
140	#2161	—	89	1	no	—
141	#2143	—	89	2	no	—
142	C₄×M₃₂	D₈₉	1	1	yes	characteristic, complemented
143	C₈₉	#33463	1	1	yes	characteristic, complemented, Sylow
144	C₂×#135	—	89	2	no	—
145	C₂×#131	—	178	2	no	—
146	#510	—	89	4	no	—
147	#509	—	89	4	no	—
148	#508	—	89	4	no	—
149	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
150	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
151	#443	—	89	4	no	—
152	#443	—	89	4	no	—
153	#443	—	178	2	no	—
154	#443	—	178	2	no	—
155	C₂×#108	—	89	2	no	—
156	C₂×#104	—	178	2	no	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	#368	—	89	4	no	—
159	#359	—	89	8	no	—
160	#346	C₂×D₈₉	1	2	yes	complemented
161	C₄×C₁₆	C₂×D₈₉	1	2	yes	complemented
162	C₂×#41	—	178	2	no	—
163	#135	—	89	4	no	—
164	#135	—	89	4	no	—
165	#135	—	178	2	no	—
166	#135	—	178	2	no	—
167	C₂×Q₁₆	—	89	2	no	—
168	C₂×SD₁₆	—	178	2	no	—
169	C₂×D₈	—	89	2	no	—
170	#131	—	356	2	no	—
171	#131	—	356	2	no	—
172	#131	—	178	4	no	—
173	#131	—	178	4	no	—
174	C₂×M₁₆	—	356	2	no	—
175	C₂×M₁₆	—	178	2	no	—
176	C₂²×C₈	—	356	2	no	—
177	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
179	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
180	Q₃₂	—	89	4	no	—
181	SD₃₂	—	89	8	no	—
182	D₁₆	—	89	4	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	C₄×D₈₉	1	4	yes	—
185	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
186	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
187	#108	—	89	4	no	—
188	#108	—	178	2	no	—
189	#104	—	178	4	no	—
190	#104	—	178	4	no	—
191	C₄×C₈	D₃₅₆	1	1	yes	characteristic
192	C₄×C₈	D₃₅₆	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
195	#41	—	356	2	no	—
196	#41	—	356	2	no	—
197	#41	—	178	4	no	—
198	#41	—	178	4	no	—
199	C₂×Q₈	—	178	2	no	—
200	C₂×D₄	—	356	2	no	—
201	C₂×D₄	—	178	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	356	2	no	—
204	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
205	Q₁₆	—	89	4	no	—
206	Q₁₆	—	178	2	no	—
207	SD₁₆	—	178	4	no	—
208	SD₁₆	—	178	4	no	—
209	D₈	—	89	4	no	—
210	D₈	—	178	2	no	—
211	M₁₆	—	356	2	no	—
212	M₁₆	—	356	4	no	—
213	M₁₆	—	356	4	no	—
214	M₁₆	—	178	4	no	—
215	C₂×C₈	—	356	2	no	—
216	C₂×C₈	—	356	4	no	—
217	C₂×C₈	—	356	4	no	—
218	C₂×C₈	D₇₁₂	1	2	yes	—
219	C₂×C₈	D₇₁₂	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
221	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
222	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
223	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
224	C₂×C₈	#13083574	1	2	yes	—
225	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
226	C₄²	—	2	1	no	—
227	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
228	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
229	C₁₆	—	2	2	no	—
230	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
231	C₂³	—	356	2	no	—
232	C₂³	#15668034	1	1	yes	characteristic
233	Q₈	—	356	2	no	—
234	Q₈	—	178	4	no	—
235	D₄	—	356	4	no	—
236	D₄	—	356	4	no	—
237	D₄	—	356	2	no	—
238	D₄	—	178	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	356	2	no	—
243	C₂×C₄	—	356	4	no	—
244	C₂×C₄	—	356	4	no	—
245	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
246	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
247	C₂×C₄	#15668034	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
249	C₈	—	356	4	no	—
250	C₈	C₄×D₃₅₆	1	2	yes	—
251	C₈	C₂×D₇₁₂	1	2	yes	—
252	C₈	C₂×D₇₁₂	1	1	yes	characteristic
253	C₈	C₂×D₇₁₂	1	1	yes	characteristic
254	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
255	C₈	C₂×#13083574	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	356	4	no	—
258	C₂²	—	712	4	no	—
259	C₂²	C₄×D₇₁₂	1	2	yes	—
260	C₂²	C₂×#15668034	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	356	4	no	—
264	C₄	C₄×D₇₁₂	1	2	yes	—
265	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
266	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	712	4	no	—
270	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂×C₈₉	C₂×C₄×D₈
prime radical	#26033388	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#33463
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#19151716	C₂	semidirect
3. #26022139	C₂	semidirect
4. #26022134	C₂	semidirect
5. #26022047	C₂	semidirect
6. #26021981	C₂	semidirect
7. C₄×M₃₂×C₈₉	C₂	semidirect
10. #19152041	C₄	semidirect
11. #19152036	C₄	semidirect
15. #19151716	C₂²	semidirect
16. #19151716	C₂²	semidirect
19. #19151428	C₂²	semidirect
20. #19151406	C₂²	semidirect
24. #346×C₈₉	C₂²	semidirect
25. C₄×C₁₆×C₈₉	C₂²	semidirect
41. Dic₇₁₂	C₂×C₄	semidirect
42. #15667957	C₂×C₄	semidirect
43. D₁₄₂₄	C₂×C₄	semidirect
142. C₄×M₃₂	D₈₉	semidirect
143. C₈₉	#33463	semidirect
160. #346	C₂×D₈₉	semidirect
161. C₄×C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033388	2
C₂×#19151716	2
#26022139	2
#26022134	2
#26022047	2
#26021981	2
C₄×M₃₂×C₈₉	2
#33463	89