Subgroups — #36460164

Order 22784.

718 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151363	C₂	1	2	yes	maximal
2	C₂×#19151363	C₂	1	2	yes	maximal
3	#26019946	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	4	yes	—
6	#19151363	—	2	4	no	—
7	#19151363	C₄	1	8	yes	complemented
8	#19151353	—	2	4	no	—
9	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₈₉	—	2	4	no	—
13	C₂×C₄×C₈×C₈₉	C₄	1	4	yes	—
14	C₂×#13083558	—	2	4	no	—
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	C₂×C₄	1	8	yes	—
20	C₂²×#10623494	—	2	4	no	—
21	C₂³×C₄×C₈₉	Q₈	1	2	yes	—
22	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	—	2	2	no	—
24	C₂²×C₈×C₈₉	—	2	2	no	—
25	C₂²×C₈×C₈₉	—	2	2	no	—
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₉	D₄	1	4	yes	—
31	C₂×C₄²×C₈₉	—	2	2	no	—
32	C₂×C₄²×C₈₉	—	2	2	no	—
33	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₈₉	D₄	1	2	yes	—
35	C₄×C₈×C₈₉	—	2	8	no	—
36	#13083558	—	2	8	no	—
37	#13083558	—	2	8	no	—
38	#13083558	—	4	4	no	—
39	#13083558	—	4	4	no	—
40	C₂×#10623494	—	2	8	no	—
41	C₂×#10623494	—	4	4	no	—
42	C₂×#10623494	—	4	4	no	—
43	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	—	2	4	no	—
45	C₂²×C₄×C₈₉	—	2	8	no	—
46	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	#41	1	2	yes	—
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	C₂×C₈×C₈₉	—	2	2	no	—
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	—	2	2	no	—
59	C₂×C₈×C₈₉	—	2	8	no	—
60	C₂×C₈×C₈₉	—	2	8	no	—
61	C₂×C₈×C₈₉	C₄²	1	4	yes	—
62	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
63	C₄²×C₈₉	—	2	2	no	—
64	C₄²×C₈₉	—	2	2	no	—
65	C₄²×C₈₉	—	2	4	no	—
66	C₂³×C₈₉	—	2	4	no	—
67	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
69	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
70	C₂³×C₈₉	#125	1	2	yes	—
71	C₂×C₄×C₈₉	—	2	16	no	—
72	C₂×C₄×C₈₉	—	2	4	no	—
73	C₂×C₄×C₈₉	—	2	4	no	—
74	C₂×C₄×C₈₉	#108	1	8	yes	—
75	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#117	1	4	yes	—
78	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₈₉	#121	1	2	yes	—
85	C₂×C₄×C₈₉	#123	1	4	yes	—
86	C₂×C₄×C₈₉	#124	1	2	yes	—
87	C₂×C₄×C₈₉	#126	1	4	yes	—
88	C₈×C₈₉	—	2	4	no	—
89	C₈×C₈₉	—	2	4	no	—
90	#10623494	—	4	8	no	—
91	C₂²×C₈₉	—	2	4	no	—
92	C₂²×C₈₉	—	2	4	no	—
93	C₂²×C₈₉	—	2	4	no	—
94	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₈₉	#380	1	1	yes	characteristic
96	C₂²×C₈₉	#382	1	1	yes	characteristic
97	C₂²×C₈₉	#387	1	4	yes	—
98	C₂²×C₈₉	C₂×#108	1	4	yes	—
99	C₄×C₈₉	—	2	8	no	—
100	C₄×C₈₉	#376	1	1	yes	characteristic
101	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₈₉	#381	1	1	yes	characteristic
103	C₄×C₈₉	#383	1	1	yes	characteristic
104	C₄×C₈₉	#388	1	4	yes	—
105	#10822	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	—	2	4	no	—
107	C₂×C₈₉	#1434	1	1	yes	characteristic
108	C₂×C₈₉	C₄×#108	1	2	yes	—
109	C₂×C₈₉	#1810	1	2	yes	—
110	C₂×C₈₉	#1837	1	2	yes	—
111	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
112	C₂×#341	—	89	2	no	—
113	C₂×#341	—	89	2	no	—
114	#1728	—	89	2	no	—
115	C₈₉	#10822	1	1	yes	characteristic, complemented, Sylow
116	C₂²×M₁₆	—	89	4	no	—
117	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
118	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
119	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
120	C₂×C₄×C₈	—	2	4	no	—
121	C₂×C₄×C₈	Dic₈₉	1	4	yes	—
122	#341	—	89	8	no	—
123	#341	—	178	4	no	—
124	#336	—	178	4	no	—
125	C₂³×C₄	Dic₁₇₈	1	2	yes	—
126	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
127	C₂×M₁₆	—	89	8	no	—
128	C₂×M₁₆	—	178	4	no	—
129	C₂×M₁₆	—	178	4	no	—
130	C₂×M₁₆	—	178	4	no	—
131	C₂×M₁₆	—	178	4	no	—
132	C₂×M₁₆	—	178	4	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	178	4	no	—
138	C₂²×C₈	C₄×D₈₉	1	4	yes	—
139	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
140	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
141	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	D₃₅₆	1	2	yes	—
145	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
146	C₄×C₈	—	2	8	no	—
147	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
151	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
152	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
154	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
155	C₂²×C₄	#13083574	1	2	yes	—
156	C₂²×C₄	#13083574	1	2	yes	—
157	C₂²×C₄	#13083574	1	2	yes	—
158	C₂²×C₄	#13083574	1	2	yes	—
159	M₁₆	—	356	4	no	—
160	M₁₆	—	356	4	no	—
161	M₁₆	—	178	8	no	—
162	M₁₆	—	178	8	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	356	4	no	—
170	C₂×C₈	—	356	4	no	—
171	C₂×C₈	—	178	8	no	—
172	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
173	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₂³	—	2	4	no	—
178	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
179	C₂³	C₄×Dic₁₇₈	1	2	yes	—
180	C₂³	#15667991	1	2	yes	—
181	C₂³	#15668004	1	2	yes	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	#15667948	1	8	yes	—
186	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
188	C₂×C₄	#15667969	1	4	yes	—
189	C₂×C₄	#15667971	1	1	yes	characteristic
190	C₂×C₄	#15667971	1	1	yes	characteristic
191	C₂×C₄	#15667971	1	1	yes	characteristic
192	C₂×C₄	#15667971	1	1	yes	characteristic
193	C₂×C₄	C₄×C₂²⋊D₈₉	1	4	yes	—
194	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
195	C₂×C₄	#15667986	1	2	yes	—
196	C₂×C₄	#15667995	1	4	yes	—
197	C₂×C₄	#15668000	1	2	yes	—
198	C₂×C₄	#15668006	1	4	yes	—
199	C₈	—	2	4	no	—
200	C₈	—	2	4	no	—
201	C₈	—	356	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
206	C₂²	#19151458	1	1	yes	characteristic
207	C₂²	#19151464	1	1	yes	characteristic
208	C₂²	#19151482	1	4	yes	—
209	C₂²	C₂×#15667948	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	#19151444	1	1	yes	characteristic
212	C₄	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
213	C₄	#19151461	1	1	yes	characteristic
214	C₄	#19151466	1	1	yes	characteristic
215	C₄	#19151489	1	4	yes	—
216	C₂	—	2	4	no	—
217	C₂	#26018479	1	1	yes	characteristic
218	C₂	C₄×#15667948	1	2	yes	—
219	C₂	#26020290	1	2	yes	—
220	C₂	#26020390	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	#10822
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #19151363	C₄	semidirect
115. C₈₉	#10822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151363	2
#26019946	2
C₂²×C₄×C₈×C₈₉	2
#10822	89