Subgroups — #36460132

Order 22784.

626 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019991	C₂	1	2	yes	maximal
2	C₂×#19151363	C₂	1	2	yes	maximal
3	C₂×#19151356	C₂	1	2	yes	maximal
4	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	4	yes	—
6	#19151367	—	2	4	no	—
7	#19151363	—	2	4	no	—
8	#19151356	C₄	1	8	yes	complemented
9	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
12	#405×C₈₉	—	2	4	no	—
13	C₂×#97×C₈₉	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
15	C₂×#13083558	—	2	4	no	—
16	C₂×#13083558	—	2	4	no	—
17	C₂×#13083558	—	2	4	no	—
18	C₂×#13083558	—	2	4	no	—
19	C₂×#13083558	—	2	4	no	—
20	C₂×#13083558	C₂×C₄	1	8	yes	—
21	C₂²×#10623494	—	2	4	no	—
22	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₈₉	—	2	2	no	—
24	C₂×M₁₆×C₈₉	—	2	2	no	—
25	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	D₄	1	4	yes	—
32	#117×C₈₉	—	2	2	no	—
33	#117×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₈₉	Q₈	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
37	#97×C₈₉	—	2	4	no	—
38	C₄×C₈×C₈₉	—	2	4	no	—
39	#13083558	—	2	8	no	—
40	#13083558	—	2	8	no	—
41	#13083558	—	4	4	no	—
42	#13083558	—	4	4	no	—
43	C₂×#10623494	—	2	8	no	—
44	C₂×#10623494	—	4	4	no	—
45	C₂×#10623494	—	4	4	no	—
46	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	#41	1	2	yes	—
53	C₂²×C₄×C₈₉	#41	1	2	yes	—
54	C₂²×C₄×C₈₉	#41	1	2	yes	—
55	C₂²×C₄×C₈₉	#41	1	2	yes	—
56	M₁₆×C₈₉	—	2	8	no	—
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	2	1	no	—
59	C₂×C₈×C₈₉	—	2	1	no	—
60	C₂×C₈×C₈₉	—	2	1	no	—
61	C₂×C₈×C₈₉	—	2	1	no	—
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	—	2	8	no	—
64	C₂×C₈×C₈₉	C₄²	1	4	yes	—
65	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₈₉	—	2	4	no	—
68	C₂²⋊C₄×C₈₉	—	2	4	no	—
69	C₄²×C₈₉	—	2	2	no	—
70	C₄²×C₈₉	—	2	2	no	—
71	C₂³×C₈₉	—	4	2	no	—
72	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
74	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
75	C₂³×C₈₉	#125	1	2	yes	—
76	C₂×C₄×C₈₉	—	2	4	no	—
77	C₂×C₄×C₈₉	—	4	4	no	—
78	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#117	1	4	yes	—
81	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
86	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₈₉	#121	1	2	yes	—
88	C₂×C₄×C₈₉	#123	1	4	yes	—
89	C₂×C₄×C₈₉	#124	1	2	yes	—
90	C₂×C₄×C₈₉	#126	1	4	yes	—
91	C₈×C₈₉	—	2	4	no	—
92	C₈×C₈₉	—	2	2	no	—
93	C₈×C₈₉	—	2	2	no	—
94	#10623494	—	4	8	no	—
95	C₂²×C₈₉	—	2	2	no	—
96	C₂²×C₈₉	—	2	4	no	—
97	C₂²×C₈₉	—	4	2	no	—
98	C₂²×C₈₉	—	4	2	no	—
99	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₈₉	#380	1	1	yes	characteristic
101	C₂²×C₈₉	#382	1	1	yes	characteristic
102	C₂²×C₈₉	#387	1	4	yes	—
103	C₄×C₈₉	—	4	4	no	—
104	C₄×C₈₉	#376	1	1	yes	characteristic
105	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₈₉	#381	1	1	yes	characteristic
107	C₄×C₈₉	#383	1	1	yes	characteristic
108	C₄×C₈₉	#388	1	4	yes	—
109	#10815	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	—	2	1	no	—
111	C₂×C₈₉	—	2	1	no	—
112	C₂×C₈₉	—	2	4	no	—
113	C₂×C₈₉	#1434	1	1	yes	characteristic
114	C₂×C₈₉	#1813	1	2	yes	—
115	C₂×#405	D₈₉	1	1	yes	characteristic
116	#1742	—	89	2	no	—
117	C₂×#341	—	89	2	no	—
118	C₂×#338	—	89	2	no	—
119	C₈₉	#10815	1	1	yes	characteristic, complemented, Sylow
120	C₂²×M₁₆	—	89	4	no	—
121	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
122	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
123	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
124	#405	—	2	4	no	—
125	C₂×#97	Dic₈₉	1	2	yes	—
126	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
127	#343	—	178	4	no	—
128	#341	—	178	4	no	—
129	#338	—	89	8	no	—
130	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	89	8	no	—
134	C₂×M₁₆	—	178	4	no	—
135	C₂×M₁₆	—	178	4	no	—
136	C₂×M₁₆	—	178	4	no	—
137	C₂×M₁₆	—	178	4	no	—
138	C₂×M₁₆	—	178	4	no	—
139	C₂×M₁₆	C₄×D₈₉	1	4	yes	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	178	4	no	—
143	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
144	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
147	#117	—	2	2	no	—
148	#117	—	2	2	no	—
149	C₂×C₄⋊C₄	D₃₅₆	1	2	yes	—
150	C₂×C₂²⋊C₄	Dic₁₇₈	1	2	yes	—
151	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	4	no	—
154	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
155	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
156	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
159	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
160	C₂²×C₄	#13083574	1	2	yes	—
161	C₂²×C₄	#13083574	1	2	yes	—
162	C₂²×C₄	#13083574	1	2	yes	—
163	C₂²×C₄	#13083574	1	2	yes	—
164	M₁₆	—	2	8	no	—
165	M₁₆	—	356	4	no	—
166	M₁₆	—	356	4	no	—
167	M₁₆	—	178	8	no	—
168	M₁₆	—	178	8	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	356	4	no	—
177	C₂×C₈	—	356	4	no	—
178	C₂×C₈	—	178	8	no	—
179	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
180	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
181	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₂³	—	4	2	no	—
187	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
188	C₂³	C₄×Dic₁₇₈	1	2	yes	—
189	C₂³	#15667991	1	2	yes	—
190	C₂³	#15668004	1	2	yes	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
195	C₂×C₄	#15667969	1	4	yes	—
196	C₂×C₄	#15667971	1	1	yes	characteristic
197	C₂×C₄	#15667971	1	1	yes	characteristic
198	C₂×C₄	#15667971	1	1	yes	characteristic
199	C₂×C₄	#15667971	1	1	yes	characteristic
200	C₂×C₄	C₄×C₂²⋊D₈₉	1	4	yes	—
201	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
202	C₂×C₄	#15667986	1	2	yes	—
203	C₂×C₄	#15667995	1	4	yes	—
204	C₂×C₄	#15668000	1	2	yes	—
205	C₂×C₄	#15668006	1	4	yes	—
206	C₈	—	2	4	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	356	8	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
215	C₂²	#19151458	1	1	yes	characteristic
216	C₂²	#19151464	1	1	yes	characteristic
217	C₂²	#19151482	1	4	yes	—
218	C₄	—	4	4	no	—
219	C₄	#19151444	1	1	yes	characteristic
220	C₄	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
221	C₄	#19151461	1	1	yes	characteristic
222	C₄	#19151466	1	1	yes	characteristic
223	C₄	#19151489	1	4	yes	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	—	2	4	no	—
227	C₂	#26018479	1	1	yes	characteristic
228	C₂	#26020302	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667971
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#405×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#405	D₈₉
hypocenter	C₈₉	#10815
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
8. #19151356	C₄	semidirect
119. C₈₉	#10815	semidirect

Maximal subgroups

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

subgroup	index
#26019991	2
C₂×#19151363	2
C₂×#19151356	2
C₂×#405×C₈₉	2
#10815	89