Subgroups — #36708925

Order 22784.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029900	C₂	1	2	yes	maximal, complemented
3	#26022355	C₂	1	4	yes	maximal, complemented
4	#26022338	C₂	1	4	yes	maximal, complemented
5	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021969	C₂	1	2	yes	maximal, complemented
7	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₁₂	C₂²	1	2	yes	—
9	#19152162	—	2	2	no	—
10	C₂×D₁₄₂₄	—	2	4	no	—
11	C₂×D₁₄₂₄	C₂²	1	2	yes	—
12	C₂×D₁₄₂₄	C₂²	1	2	yes	—
13	#19151851	—	2	4	no	—
14	#19151834	C₂²	1	8	yes	complemented
15	#19151636	C₂²	1	2	yes	—
16	#19151604	—	2	2	no	—
17	#19151426	C₂²	1	2	yes	complemented
18	#19151424	C₂²	1	2	yes	complemented
19	#19151400	C₂²	1	8	yes	complemented
20	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
21	#504×C₈₉	—	2	4	no	—
22	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₈₉	C₂²	1	2	yes	complemented
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	complemented
26	C₂²×D₃₅₆	—	2	2	no	—
27	C₂×D₇₁₂	—	2	4	no	—
28	C₂×D₇₁₂	—	2	2	no	—
29	C₂×D₇₁₂	—	2	4	no	—
30	C₂×D₇₁₂	C₂³	1	4	yes	—
31	#15668008	—	4	2	no	—
32	#15668001	—	4	2	no	—
33	#15667986	—	2	8	no	—
34	#15667980	—	4	4	no	—
35	#15667971	—	2	2	no	—
36	D₁₄₂₄	—	2	4	no	—
37	D₁₄₂₄	—	2	4	no	—
38	#15667946	C₂³	1	4	yes	—
39	#15667944	—	2	2	no	—
40	#15667934	—	2	8	no	—
41	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
42	#131×C₈₉	—	2	2	no	—
43	#131×C₈₉	—	2	2	no	—
44	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
45	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	M₃₂×C₈₉	—	2	2	no	—
49	M₃₂×C₈₉	—	2	4	no	—
50	C₂×C₁₆×C₈₉	—	2	4	no	—
51	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
52	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
53	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
55	C₂³×D₈₉	—	4	2	no	—
56	C₂×D₃₅₆	—	2	4	no	—
57	C₂×D₃₅₆	—	4	4	no	—
58	C₂×D₃₅₆	—	4	4	no	—
59	C₂×D₃₅₆	—	4	2	no	—
60	C₂×C₂²⋊D₈₉	—	4	8	no	—
61	D₇₁₂	—	2	4	no	—
62	D₇₁₂	—	2	4	no	—
63	D₇₁₂	—	4	4	no	—
64	C₄⋊Dic₈₉	—	2	4	no	—
65	C₂²⋊Dic₈₉	—	4	2	no	—
66	#13083551	—	4	8	no	—
67	C₄×Dic₈₉	—	4	2	no	—
68	#41×C₈₉	—	2	2	no	—
69	#41×C₈₉	—	2	2	no	—
70	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
71	C₂×D₄×C₈₉	D₈	1	2	yes	—
72	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	M₁₆×C₈₉	—	2	2	no	—
76	M₁₆×C₈₉	—	2	4	no	—
77	C₂×C₈×C₈₉	—	2	2	no	—
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₈₉	#41	1	2	yes	—
83	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₁₆×C₈₉	—	2	2	no	—
85	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
86	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
87	C₂³×C₈₉	C₂×D₈	1	2	yes	—
88	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
89	C₂²×D₈₉	—	4	4	no	—
90	C₂²×D₈₉	—	8	4	no	—
91	C₂²×D₈₉	—	16	4	no	—
92	Q₈×C₈₉	—	2	2	no	—
93	D₄×C₈₉	—	2	4	no	—
94	D₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	#121	1	2	yes	—
98	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#135	1	2	yes	—
101	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
102	C₂²⋊D₈₉	—	16	8	no	—
103	C₂×Dic₈₉	—	4	4	no	—
104	D₃₅₆	—	4	4	no	—
105	D₃₅₆	—	4	4	no	—
106	D₃₅₆	—	8	4	no	—
107	C₈×C₈₉	—	2	2	no	—
108	C₈×C₈₉	#121	1	2	yes	—
109	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
110	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
111	C₂²×C₈₉	—	2	2	no	—
112	C₂²×C₈₉	—	2	4	no	—
113	C₂²×C₈₉	—	2	2	no	—
114	C₂²×C₈₉	#466	1	2	yes	—
115	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
116	C₂×D₈₉	—	8	8	no	—
117	C₂×D₈₉	—	8	4	no	—
118	C₂×D₈₉	—	16	8	no	—
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	#466	1	2	yes	—
121	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
122	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
123	Dic₈₉	—	8	4	no	—
124	#33607	—	89	1	no	maximal, Sylow
125	C₂×C₈₉	—	2	2	no	—
126	C₂×C₈₉	—	4	2	no	—
127	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
128	C₂×C₈₉	#3415	1	2	yes	—
129	D₈₉	—	16	4	no	—
130	C₂²×D₁₆	—	89	1	no	—
131	C₂×#504	D₈₉	1	1	yes	characteristic, complemented
132	#3058	—	89	2	no	—
133	#2218	—	89	4	no	—
134	#2215	—	89	4	no	—
135	#2158	—	89	1	no	—
136	#2140	—	89	2	no	—
137	C₈₉	#33607	1	1	yes	characteristic, complemented, Sylow
138	C₂²×D₈	—	89	2	no	—
139	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
140	#535	—	178	2	no	—
141	C₂×D₁₆	—	89	2	no	—
142	C₂×D₁₆	—	89	2	no	—
143	C₂×D₁₆	—	178	4	no	—
144	#504	—	2	4	no	—
145	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
146	C₂×M₃₂	C₂×D₈₉	1	2	yes	complemented
147	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
148	C₂²×C₁₆	C₂×D₈₉	1	2	yes	complemented
149	#469	—	178	4	no	—
150	#466	—	89	8	no	—
151	#427	—	89	2	no	—
152	#418	—	178	2	no	—
153	#367	—	89	2	no	—
154	#366	—	89	2	no	—
155	#357	—	89	8	no	—
156	C₂×#41	D₃₅₆	1	1	yes	characteristic
157	C₂²×D₄	—	178	2	no	—
158	C₂×D₈	—	89	4	no	—
159	C₂×D₈	—	178	4	no	—
160	C₂×D₈	—	178	2	no	—
161	C₂×D₈	—	178	4	no	—
162	#131	—	2	2	no	—
163	#131	—	2	2	no	—
164	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
165	C₂×M₁₆	D₃₅₆	1	2	yes	—
166	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
167	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
168	#127	—	356	2	no	—
169	#124	—	356	2	no	—
170	#121	—	178	8	no	—
171	#120	—	356	4	no	—
172	#117	—	178	2	no	—
173	D₁₆	—	178	4	no	—
174	D₁₆	—	178	4	no	—
175	M₃₂	—	2	2	no	—
176	M₃₂	—	2	4	no	—
177	C₂×C₁₆	—	2	4	no	—
178	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	complemented
179	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
180	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
181	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
182	#107	—	89	4	no	—
183	#106	—	178	2	no	—
184	#102	—	178	8	no	—
185	C₂⁴	—	356	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
189	C₂×D₄	—	356	8	no	—
190	C₂×D₄	—	356	4	no	—
191	C₂×D₄	—	356	2	no	—
192	C₂×D₄	—	356	4	no	—
193	C₂×D₄	—	178	4	no	—
194	C₂×D₄	D₇₁₂	1	2	yes	—
195	C₂×D₄	D₇₁₂	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
197	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
198	D₈	—	356	4	no	—
199	D₈	—	178	4	no	—
200	D₈	—	178	4	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
206	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
207	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
208	C₂×C₈	#13083574	1	2	yes	—
209	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
210	C₄⋊C₄	—	178	4	no	—
211	C₂²⋊C₄	—	356	2	no	—
212	C₂²⋊C₄	—	356	8	no	—
213	C₄²	—	356	2	no	—
214	C₁₆	—	2	2	no	—
215	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
216	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
217	C₂³	—	356	4	no	—
218	C₂³	—	1424	4	no	—
219	C₂³	—	712	4	no	—
220	C₂³	C₂×D₇₁₂	1	2	yes	—
221	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	356	4	no	—
226	D₄	—	356	4	no	—
227	D₄	—	1424	8	no	—
228	D₄	—	712	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	356	4	no	—
232	C₂×C₄	#15667986	1	2	yes	—
233	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
234	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
235	C₂×C₄	#15668034	1	2	yes	—
236	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	#15667986	1	2	yes	—
239	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
240	C₈	C₂×#13083574	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	1424	8	no	—
245	C₂²	—	712	8	no	—
246	C₂²	—	712	4	no	—
247	C₂²	#19151834	1	2	yes	—
248	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	712	4	no	—
251	C₄	#19151834	1	2	yes	—
252	C₄	C₂×#15667986	1	1	yes	characteristic
253	C₄	C₂×#15668034	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	1424	4	no	—
257	C₂	C₂×#19151834	1	1	yes	characteristic
258	C₂	#26033393	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#504×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	#33607
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₄₂₄	C₂	semidirect
2. #26029900	C₂	semidirect
3. #26022355	C₂	semidirect
4. #26022338	C₂	semidirect
5. #26022036	C₂	semidirect
6. #26021969	C₂	semidirect
7. C₂×#504×C₈₉	C₂	semidirect
14. #19151834	C₂²	semidirect
17. #19151426	C₂²	semidirect
18. #19151424	C₂²	semidirect
19. #19151400	C₂²	semidirect
23. C₂×M₃₂×C₈₉	C₂²	semidirect
25. C₂²×C₁₆×C₈₉	C₂²	semidirect
51. C₂×C₁₆×C₈₉	D₄	semidirect
131. C₂×#504	D₈₉	semidirect
137. C₈₉	#33607	semidirect
146. C₂×M₃₂	C₂×D₈₉	semidirect
148. C₂²×C₁₆	C₂×D₈₉	semidirect
178. C₂×C₁₆	C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₄₂₄	2
#26029900	2
#26022355	2
#26022338	2
#26022036	2
#26021969	2
C₂×#504×C₈₉	2
#33607	89