Subgroups — #36707849

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 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	#26029170	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	C₂×#19151406	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	2	yes	—
9	#19152364	—	2	2	no	—
10	#19152041	C₄	1	4	yes	complemented
11	#19152036	C₄	1	4	yes	complemented
12	#19152030	C₂²	1	4	yes	—
13	#19151721	—	2	4	no	—
14	#19151716	C₂²	1	8	yes	complemented
15	C₂×#15667948	C₂²	1	2	yes	—
16	#19151616	—	2	2	no	—
17	#19151428	C₂²	1	4	yes	complemented
18	#19151406	C₂²	1	4	yes	complemented
19	#19151406	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	#405×C₈₉	C₂²	1	1	yes	characteristic
23	#346×C₈₉	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₈₉	C₂²	1	2	yes	complemented
25	C₂×#13083574	—	2	2	no	—
26	#15668034	—	2	2	no	—
27	#15668034	—	2	2	no	—
28	#15668034	C₂×C₄	1	4	yes	—
29	#15668034	C₂³	1	4	yes	—
30	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
31	C₂×#13083563	—	2	2	no	—
32	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
33	#15668020	—	4	4	no	—
34	#15668018	—	2	8	no	—
35	C₂×#13083558	—	2	2	no	—
36	M₁₆×D₈₉	—	4	4	no	—
37	Dic₇₁₂	C₂×C₄	1	4	yes	—
38	#15667957	C₂×C₄	1	8	yes	—
39	D₁₄₂₄	C₂×C₄	1	4	yes	—
40	#15667948	—	2	2	no	—
41	#15667948	C₂³	1	4	yes	—
42	#15667940	—	2	8	no	—
43	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	#117×C₈₉	D₄	1	1	yes	characteristic
46	M₃₂×C₈₉	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₈₉	—	2	2	no	—
48	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
51	#97×C₈₉	D₄	1	2	yes	—
52	C₄×C₈×C₈₉	C₂³	1	2	yes	—
53	#13083574	—	2	4	no	—
54	#13083574	—	2	4	no	—
55	#13083574	—	4	2	no	—
56	#13083574	—	4	2	no	—
57	C₂×Dic₁₇₈	—	2	2	no	—
58	C₂×D₃₅₆	—	2	2	no	—
59	C₂×C₂²⋊D₈₉	—	4	2	no	—
60	C₂×C₄×D₈₉	—	4	2	no	—
61	Dic₃₅₆	—	2	2	no	—
62	Dic₃₅₆	C₂²×C₄	1	4	yes	—
63	#13083563	—	2	4	no	—
64	#13083563	—	2	4	no	—
65	D₇₁₂	—	2	2	no	—
66	D₇₁₂	C₂²×C₄	1	4	yes	—
67	#13083558	—	2	4	no	—
68	#13083558	—	4	2	no	—
69	#13083557	—	4	8	no	—
70	C₂×#10623494	—	4	2	no	—
71	C₈×D₈₉	—	4	8	no	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	M₁₆×C₈₉	D₈	1	4	yes	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₈₉	#41	1	2	yes	—
78	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂²⋊C₄×C₈₉	—	2	1	no	—
81	C₄²×C₈₉	C₂×D₄	1	2	yes	—
82	C₁₆×C₈₉	—	2	1	no	—
83	C₁₆×C₈₉	—	2	1	no	—
84	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
85	C₂³×C₈₉	#135	1	1	yes	characteristic
86	C₂²×D₈₉	—	4	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
92	C₂²⋊D₈₉	—	4	4	no	—
93	C₂²⋊D₈₉	—	4	4	no	—
94	C₂×Dic₈₉	—	4	2	no	—
95	D₃₅₆	—	2	4	no	—
96	D₃₅₆	—	4	2	no	—
97	C₄×D₈₉	—	4	4	no	—
98	C₄×D₈₉	—	4	4	no	—
99	Dic₁₇₈	—	2	4	no	—
100	Dic₁₇₈	—	4	2	no	—
101	C₈×C₈₉	C₄×D₄	1	2	yes	—
102	C₈×C₈₉	C₂×D₈	1	4	yes	—
103	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
105	#10623494	—	4	4	no	—
106	C₂²×C₈₉	—	4	1	no	—
107	C₂²×C₈₉	C₄×D₈	1	2	yes	—
108	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
109	C₂×D₈₉	—	4	4	no	—
110	C₂×D₈₉	—	16	2	no	—
111	C₄×C₈₉	—	4	2	no	—
112	C₄×C₈₉	C₄×D₈	1	2	yes	—
113	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
115	Dic₈₉	—	4	4	no	—
116	#33464	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	—	2	1	no	—
118	C₂×C₈₉	—	2	2	no	—
119	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
120	D₈₉	—	8	4	no	—
121	#3414	—	89	1	no	—
122	#2957	—	89	2	no	—
123	#2179	—	89	4	no	—
124	#2178	—	89	4	no	—
125	#2161	—	89	1	no	—
126	C₂×#359	—	89	2	no	—
127	#2108	D₈₉	1	1	yes	characteristic, complemented
128	C₈₉	#33464	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	89	2	no	—
130	#568	—	178	2	no	—
131	#510	—	89	4	no	—
132	#509	—	89	4	no	—
133	#508	—	89	4	no	—
134	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
135	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
136	#444	—	178	4	no	—
137	#443	—	89	8	no	—
138	C₂×#108	—	89	2	no	—
139	#421	—	178	2	no	—
140	#405	C₂×D₈₉	1	1	yes	characteristic
141	#368	—	89	4	no	—
142	#359	—	89	4	no	—
143	#359	—	89	4	no	—
144	#346	C₂×D₈₉	1	2	yes	complemented
145	C₄×C₁₆	C₂×D₈₉	1	2	yes	complemented
146	C₂×#41	—	178	2	no	—
147	#135	—	89	4	no	—
148	#135	—	89	4	no	—
149	#135	—	178	2	no	—
150	#135	—	178	2	no	—
151	C₂×Q₁₆	—	89	2	no	—
152	C₂×SD₁₆	—	178	2	no	—
153	C₂×D₈	—	89	2	no	—
154	#131	—	356	4	no	—
155	#131	—	178	8	no	—
156	C₂×M₁₆	—	356	4	no	—
157	C₂×M₁₆	—	178	2	no	—
158	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
160	#117	D₃₅₆	1	1	yes	characteristic
161	Q₃₂	—	89	4	no	—
162	SD₃₂	—	89	8	no	—
163	D₁₆	—	89	4	no	—
164	M₃₂	C₄×D₈₉	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
167	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
168	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
169	#108	—	89	4	no	—
170	#108	—	178	2	no	—
171	#104	—	178	8	no	—
172	#97	D₃₅₆	1	2	yes	—
173	C₄×C₈	C₂²×D₈₉	1	2	yes	—
174	#41	—	356	2	no	—
175	#41	—	356	2	no	—
176	#41	—	178	4	no	—
177	#41	—	178	4	no	—
178	C₂×Q₈	—	178	2	no	—
179	C₂×D₄	—	356	2	no	—
180	C₂×D₄	—	178	2	no	—
181	C₂²×C₄	—	356	2	no	—
182	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
183	Q₁₆	—	89	4	no	—
184	Q₁₆	—	178	2	no	—
185	SD₁₆	—	178	4	no	—
186	SD₁₆	—	178	4	no	—
187	D₈	—	89	4	no	—
188	D₈	—	178	2	no	—
189	M₁₆	—	356	2	no	—
190	M₁₆	—	356	8	no	—
191	M₁₆	—	178	4	no	—
192	M₁₆	D₇₁₂	1	4	yes	—
193	C₂×C₈	—	356	2	no	—
194	C₂×C₈	—	356	8	no	—
195	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
196	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
197	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
198	C₂×C₈	#13083574	1	2	yes	—
199	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₃₅₆	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
206	C₂³	—	356	2	no	—
207	C₂³	#15668034	1	1	yes	characteristic
208	Q₈	—	356	2	no	—
209	Q₈	—	178	4	no	—
210	D₄	—	356	4	no	—
211	D₄	—	356	4	no	—
212	D₄	—	356	2	no	—
213	D₄	—	178	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	356	2	no	—
216	C₂×C₄	—	356	4	no	—
217	C₂×C₄	—	356	4	no	—
218	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
219	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
220	C₂×C₄	#15668034	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
222	C₈	—	356	4	no	—
223	C₈	C₄×D₃₅₆	1	2	yes	—
224	C₈	C₂×D₇₁₂	1	4	yes	—
225	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
226	C₈	C₂×#13083574	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	356	4	no	—
229	C₂²	—	1424	2	no	—
230	C₂²	C₄×D₇₁₂	1	2	yes	—
231	C₂²	C₂×#15668034	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	356	4	no	—
234	C₄	C₄×D₇₁₂	1	2	yes	—
235	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
236	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	712	4	no	—
240	C₂	C₂×C₄×D₇₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂×C₈₉	C₂×C₄×D₈
prime radical	#26033388	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#33464
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #26029170	C₂	semidirect
3. #26022139	C₂	semidirect
4. #26022134	C₂	semidirect
5. #26022047	C₂	semidirect
6. C₂×#19151406	C₂	semidirect
7. C₈₉×#2108	C₂	semidirect
10. #19152041	C₄	semidirect
11. #19152036	C₄	semidirect
14. #19151716	C₂²	semidirect
17. #19151428	C₂²	semidirect
18. #19151406	C₂²	semidirect
19. #19151406	C₂²	semidirect
23. #346×C₈₉	C₂²	semidirect
24. C₄×C₁₆×C₈₉	C₂²	semidirect
127. #2108	D₈₉	semidirect
128. C₈₉	#33464	semidirect
144. #346	C₂×D₈₉	semidirect
145. 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
#26029170	2
#26022139	2
#26022134	2
#26022047	2
C₂×#19151406	2
C₈₉×#2108	2
#33464	89