Subgroups — #36555073

Order 22784.

758 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028885	C₂	1	2	yes	maximal
2	#26020424	C₂	1	2	yes	maximal, complemented
3	#26020390	C₂	1	2	yes	maximal, complemented
4	#26020294	C₂	1	4	yes	maximal
5	#26020158	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#19151655	—	2	4	no	—
8	#19151651	—	2	4	no	—
9	C₂×#15667948	C₂²	1	8	yes	—
10	#19151622	C₂²	1	4	yes	—
11	#19151562	C₂²	1	4	yes	—
12	C₄×#13083558	C₂²	1	4	yes	—
13	#19151363	C₂²	1	8	yes	—
14	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₈₉	C₄	1	4	yes	—
17	C₄×M₁₆×C₈₉	C₄	1	4	yes	—
18	C₂×#97×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#13083558	C₂³	1	8	yes	—
22	#15667948	—	2	16	no	—
23	C₄⋊#10623494	—	2	8	no	—
24	C₂²⋊#10623494	—	2	8	no	—
25	#15667923	—	2	4	no	—
26	C₄×#10623494	—	2	4	no	—
27	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
28	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₈₉	—	2	8	no	—
30	C₂×M₁₆×C₈₉	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
32	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₈₉	D₄	1	4	yes	—
36	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	#97×C₈₉	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	#13083558	—	2	16	no	—
45	C₂×#10623494	—	2	8	no	—
46	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	—	2	4	no	—
48	C₂²×C₄×C₈₉	—	4	4	no	—
49	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₈₉	—	2	16	no	—
58	C₂×C₈×C₈₉	—	2	4	no	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₈₉	#41	1	8	yes	—
62	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
63	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
64	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
67	C₄²×C₈₉	C₂×D₄	1	2	yes	—
68	C₄²×C₈₉	C₂×D₄	1	2	yes	—
69	C₂³×C₈₉	—	2	2	no	—
70	C₂³×C₈₉	—	4	2	no	—
71	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	—	2	2	no	—
74	C₂×C₄×C₈₉	—	2	4	no	—
75	C₂×C₄×C₈₉	—	2	4	no	—
76	C₂×C₄×C₈₉	—	2	8	no	—
77	C₂×C₄×C₈₉	—	2	4	no	—
78	C₂×C₄×C₈₉	—	4	4	no	—
79	C₂×C₄×C₈₉	—	4	2	no	—
80	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
87	C₈×C₈₉	—	2	4	no	—
88	C₈×C₈₉	#117	1	8	yes	—
89	#10623494	—	4	8	no	—
90	C₂²×C₈₉	—	2	2	no	—
91	C₂²×C₈₉	—	2	4	no	—
92	C₂²×C₈₉	—	2	2	no	—
93	C₂²×C₈₉	—	4	4	no	—
94	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈₉	#533	1	2	yes	—
96	C₄×C₈₉	—	2	4	no	—
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	—	2	2	no	—
99	C₄×C₈₉	#430	1	4	yes	—
100	C₄×C₈₉	C₂×#117	1	2	yes	—
101	C₄×C₈₉	#532	1	2	yes	—
102	#20831	—	89	1	no	maximal, Sylow
103	C₂×C₈₉	—	2	2	no	—
104	C₂×C₈₉	—	4	2	no	—
105	C₂×C₈₉	#2285	1	1	yes	characteristic
106	C₂×C₈₉	C₂×#430	1	2	yes	—
107	#2920	—	89	2	no	—
108	C₂×C₄×M₁₆	D₈₉	1	1	yes	characteristic
109	#1848	—	89	2	no	—
110	#1837	—	89	2	no	—
111	#1811	—	89	4	no	—
112	#1778	—	89	4	no	—
113	C₈₉	#20831	1	1	yes	characteristic, complemented, Sylow
114	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
115	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
116	#432	—	178	4	no	—
117	#431	—	178	4	no	—
118	C₂×#108	—	89	8	no	—
119	#423	—	89	4	no	—
120	#407	—	89	4	no	—
121	C₄×M₁₆	—	89	4	no	—
122	C₄×M₁₆	Dic₈₉	1	4	yes	—
123	C₄×M₁₆	Dic₈₉	1	4	yes	—
124	C₂×#97	C₂×D₈₉	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
127	#341	—	89	8	no	—
128	C₂³×C₄	Dic₁₇₈	1	2	yes	—
129	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
130	C₂×M₁₆	—	2	8	no	—
131	C₂×M₁₆	—	89	8	no	—
132	C₂×M₁₆	C₂×Dic₈₉	1	8	yes	—
133	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
134	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
135	C₂×C₄²	Dic₁₇₈	1	2	yes	—
136	C₂×C₄²	D₃₅₆	1	4	yes	—
137	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
138	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
139	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
141	#108	—	178	16	no	—
142	C₄⋊C₈	—	178	8	no	—
143	C₂²⋊C₈	—	178	8	no	—
144	#97	—	178	4	no	—
145	#97	C₂×Dic₈₉	1	8	yes	—
146	C₄×C₈	—	178	4	no	—
147	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
148	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
149	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
150	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
151	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
155	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
156	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
157	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
158	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
160	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
161	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
162	M₁₆	—	2	16	no	—
163	M₁₆	—	178	16	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	178	8	no	—
166	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
167	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
168	C₂×C₈	#13083574	1	8	yes	—
169	C₄²	C₄⋊Dic₈₉	1	4	yes	—
170	C₄²	C₄⋊Dic₈₉	1	4	yes	—
171	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
172	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
173	C₄²	C₂²×Dic₈₉	1	2	yes	—
174	C₄²	C₂×D₃₅₆	1	2	yes	—
175	C₄²	C₂×D₃₅₆	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	4	2	no	—
178	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
179	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
189	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
190	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
191	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
193	C₂×C₄	C₂×#13083574	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	356	8	no	—
196	C₈	#15667971	1	8	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	4	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
202	C₂²	#19152155	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	#19151647	1	4	yes	—
207	C₄	C₂×#15667971	1	2	yes	—
208	C₄	#19152148	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	—	4	2	no	—
211	C₂	#26022659	1	1	yes	characteristic
212	C₂	C₂×#19151647	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×C₄×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₈₉
hypocenter	C₈₉	#20831
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #26020424	C₂	semidirect
3. #26020390	C₂	semidirect
5. #26020158	C₂	semidirect
113. C₈₉	#20831	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26020424	2
#26020390	2
#26020294	2
#26020158	2
C₂×C₄×M₁₆×C₈₉	2
#20831	89