Subgroups — #36708932

Order 22784.

650 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029907	C₂	1	2	yes	maximal
3	#26022360	C₂	1	4	yes	maximal, complemented
4	#26022343	C₂	1	4	yes	maximal, complemented
5	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021973	C₂	1	2	yes	maximal, complemented
7	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
9	#19152169	—	2	2	no	—
10	C₂×Dic₇₁₂	—	2	4	no	—
11	C₂×Dic₇₁₂	C₂²	1	2	yes	—
12	C₂×Dic₇₁₂	C₂²	1	2	yes	—
13	#19151856	—	2	4	no	—
14	#19151839	C₂²	1	8	yes	—
15	#19151636	C₂²	1	2	yes	—
16	#19151608	—	2	2	no	—
17	#19151426	C₂²	1	2	yes	—
18	#19151424	C₂²	1	2	yes	—
19	#19151403	C₂²	1	8	yes	—
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	—
24	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₇₈	—	2	2	no	—
27	C₂×Dic₃₅₆	—	2	4	no	—
28	C₂×Dic₃₅₆	—	2	2	no	—
29	C₂×Dic₃₅₆	—	2	4	no	—
30	C₂×Dic₃₅₆	C₂³	1	4	yes	—
31	#15668010	—	4	2	no	—
32	#15667999	—	4	2	no	—
33	#15667993	—	4	4	no	—
34	#15667991	—	2	8	no	—
35	#15667971	—	2	2	no	—
36	Dic₇₁₂	—	2	4	no	—
37	Dic₇₁₂	—	2	4	no	—
38	#15667946	C₂³	1	4	yes	—
39	#15667944	—	2	2	no	—
40	#15667937	—	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	—
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	C₂×Dic₁₇₈	—	2	4	no	—
56	C₂×Dic₁₇₈	—	4	4	no	—
57	C₂×Dic₁₇₈	—	4	4	no	—
58	C₂×Dic₁₇₈	—	4	2	no	—
59	C₂²×Dic₈₉	—	4	2	no	—
60	Dic₃₅₆	—	2	4	no	—
61	Dic₃₅₆	—	2	4	no	—
62	Dic₃₅₆	—	4	4	no	—
63	C₄⋊Dic₈₉	—	2	4	no	—
64	#13083553	—	4	8	no	—
65	C₂²⋊Dic₈₉	—	4	2	no	—
66	C₂²⋊Dic₈₉	—	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	Q₈×C₈₉	—	2	2	no	—
90	D₄×C₈₉	—	2	4	no	—
91	D₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	#121	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#135	1	2	yes	—
98	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₈₉	—	4	4	no	—
100	C₂×Dic₈₉	—	4	4	no	—
101	C₂×Dic₈₉	—	8	4	no	—
102	Dic₁₇₈	—	4	4	no	—
103	Dic₁₇₈	—	4	4	no	—
104	Dic₁₇₈	—	8	4	no	—
105	C₈×C₈₉	—	2	2	no	—
106	C₈×C₈₉	#121	1	2	yes	—
107	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂²×C₈₉	—	2	2	no	—
110	C₂²×C₈₉	—	2	4	no	—
111	C₂²×C₈₉	—	2	2	no	—
112	C₂²×C₈₉	#466	1	2	yes	—
113	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
114	C₄×C₈₉	—	2	2	no	—
115	C₄×C₈₉	#466	1	2	yes	—
116	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
117	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
118	Dic₈₉	—	8	4	no	—
119	Dic₈₉	—	8	4	no	—
120	#33608	—	89	1	no	maximal, Sylow
121	C₂×C₈₉	—	2	2	no	—
122	C₂×C₈₉	—	4	2	no	—
123	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
124	C₂×C₈₉	#3417	1	2	yes	—
125	C₂²×Q₃₂	—	89	1	no	—
126	C₂×#504	D₈₉	1	1	yes	characteristic
127	#3059	—	89	2	no	—
128	#2219	—	89	4	no	—
129	#2216	—	89	4	no	—
130	#2158	—	89	1	no	—
131	#2141	—	89	2	no	—
132	C₈₉	#33608	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₁₆	—	89	2	no	—
134	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
135	#536	—	178	2	no	—
136	C₂×Q₃₂	—	89	2	no	—
137	C₂×Q₃₂	—	89	2	no	—
138	C₂×Q₃₂	—	178	4	no	—
139	#504	—	2	4	no	—
140	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
141	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
142	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
143	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
144	#470	—	178	4	no	—
145	#467	—	89	8	no	—
146	#427	—	89	2	no	—
147	#419	—	178	2	no	—
148	#367	—	89	2	no	—
149	#366	—	89	2	no	—
150	#358	—	89	8	no	—
151	C₂×#41	D₃₅₆	1	1	yes	characteristic
152	C₂²×Q₈	—	178	2	no	—
153	C₂×Q₁₆	—	89	4	no	—
154	C₂×Q₁₆	—	178	4	no	—
155	C₂×Q₁₆	—	178	2	no	—
156	C₂×Q₁₆	—	178	4	no	—
157	#131	—	2	2	no	—
158	#131	—	2	2	no	—
159	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
160	C₂×M₁₆	D₃₅₆	1	2	yes	—
161	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
162	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
163	C₄⋊Q₈	—	356	2	no	—
164	#124	—	356	2	no	—
165	#123	—	356	4	no	—
166	C₂²⋊Q₈	—	178	8	no	—
167	#117	—	178	2	no	—
168	Q₃₂	—	178	4	no	—
169	Q₃₂	—	178	4	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	4	no	—
172	C₂×C₁₆	—	2	4	no	—
173	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	—
174	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
175	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
176	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
177	#107	—	89	4	no	—
178	#106	—	178	2	no	—
179	#103	—	178	8	no	—
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	356	2	no	—
183	C₂×Q₈	—	356	4	no	—
184	C₂×Q₈	—	356	4	no	—
185	C₂×Q₈	—	178	4	no	—
186	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
187	C₂×D₄	D₇₁₂	1	2	yes	—
188	C₂×D₄	D₇₁₂	1	1	yes	characteristic
189	C₂²×C₄	—	356	2	no	—
190	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
191	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
192	Q₁₆	—	356	4	no	—
193	Q₁₆	—	178	4	no	—
194	Q₁₆	—	178	4	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
200	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
201	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
202	C₂×C₈	#13083574	1	2	yes	—
203	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
204	C₄⋊C₄	—	356	8	no	—
205	C₄⋊C₄	—	178	4	no	—
206	C₂²⋊C₄	—	356	2	no	—
207	C₂²⋊C₄	—	356	8	no	—
208	C₄²	—	356	2	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
211	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
212	C₂³	C₂×D₇₁₂	1	2	yes	—
213	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	356	4	no	—
216	Q₈	—	356	4	no	—
217	Q₈	—	712	4	no	—
218	D₄	—	2	4	no	—
219	D₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	356	4	no	—
223	C₂×C₄	—	356	4	no	—
224	C₂×C₄	—	712	4	no	—
225	C₂×C₄	#15667986	1	2	yes	—
226	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
227	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
228	C₂×C₄	#15668034	1	2	yes	—
229	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	#15667986	1	2	yes	—
232	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
233	C₈	C₂×#13083574	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	#19151834	1	2	yes	—
238	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	712	4	no	—
241	C₄	—	712	4	no	—
242	C₄	#19151834	1	2	yes	—
243	C₄	C₂×#15667986	1	1	yes	characteristic
244	C₄	C₂×#15668034	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	4	2	no	—
247	C₂	C₂×#19151834	1	1	yes	characteristic
248	C₂	#26033405	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
prime radical	C₂×#41×C₈₉	D₄
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	#33608
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×Dic₇₁₂	C₂	semidirect
3. #26022360	C₂	semidirect
4. #26022343	C₂	semidirect
5. #26022036	C₂	semidirect
6. #26021973	C₂	semidirect
132. C₈₉	#33608	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₇₁₂	2
#26029907	2
#26022360	2
#26022343	2
#26022036	2
#26021973	2
C₂×#504×C₈₉	2
#33608	89