Subgroups — #36554664

Order 22784.

778 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028816	C₂	1	2	yes	maximal
2	#26020393	C₂	1	4	yes	maximal, complemented
3	#26020390	C₂	1	4	yes	maximal, complemented
4	#26019979	C₂	1	2	yes	maximal
5	C₂×#19151363	C₂	1	2	yes	maximal
6	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₂²	1	4	yes	—
8	#19151625	—	2	8	no	—
9	#19151622	C₂²	1	8	yes	—
10	#19151363	C₂²	1	4	yes	—
11	#19151363	C₂²	1	8	yes	—
12	#19151363	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
16	#405×C₈₉	C₄	1	8	yes	—
17	C₂×#97×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
19	C₂×#13083558	—	2	4	no	—
20	C₂×#13083558	—	2	8	no	—
21	C₂×#13083558	—	2	8	no	—
22	C₂×#13083558	C₂³	1	8	yes	—
23	C₂²×#10623494	—	2	4	no	—
24	C₄⋊#10623494	—	2	16	no	—
25	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₈₉	—	2	4	no	—
27	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₈₉	—	2	4	no	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
32	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	#117×C₈₉	C₂×C₄	1	4	yes	—
34	#117×C₈₉	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₈₉	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₈₉	Q₈	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
38	#97×C₈₉	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
40	#13083558	—	2	8	no	—
41	#13083558	—	2	8	no	—
42	#13083558	—	4	8	no	—
43	C₂×#10623494	—	2	8	no	—
44	C₂×#10623494	—	4	8	no	—
45	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	#41	1	4	yes	—
51	C₂²×C₄×C₈₉	#41	1	4	yes	—
52	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
53	M₁₆×C₈₉	—	2	8	no	—
54	C₂×C₈×C₈₉	—	2	8	no	—
55	C₂×C₈×C₈₉	—	2	2	no	—
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	—	2	4	no	—
58	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
60	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
61	C₄⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
62	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
63	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
64	C₄²×C₈₉	C₂×D₄	1	4	yes	—
65	C₂³×C₈₉	—	4	2	no	—
66	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
67	C₂³×C₈₉	#125	1	4	yes	—
68	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	—	4	4	no	—
70	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	8	yes	—
71	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₈₉	#117	1	4	yes	—
73	C₂×C₄×C₈₉	#117	1	4	yes	—
74	C₂×C₄×C₈₉	#124	1	8	yes	—
75	C₂×C₄×C₈₉	#125	1	4	yes	—
76	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
79	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
80	C₈×C₈₉	—	2	2	no	—
81	C₈×C₈₉	—	2	2	no	—
82	C₈×C₈₉	—	2	4	no	—
83	#10623494	—	4	8	no	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	—	2	4	no	—
86	C₂²×C₈₉	—	4	4	no	—
87	C₂²×C₈₉	#382	1	4	yes	—
88	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈₉	C₂×#125	1	2	yes	—
90	C₄×C₈₉	—	2	8	no	—
91	C₄×C₈₉	#382	1	4	yes	—
92	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
93	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
94	C₄×C₈₉	C₂×#124	1	2	yes	—
95	#20768	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	—	2	2	no	—
97	C₂×C₈₉	—	2	4	no	—
98	C₂×C₈₉	C₂×#382	1	1	yes	characteristic
99	C₂×C₈₉	#2916	1	2	yes	—
100	#2905	—	89	2	no	—
101	C₂×#405	D₈₉	1	1	yes	characteristic
102	#1838	—	89	4	no	—
103	#1837	—	89	4	no	—
104	#1738	—	89	2	no	—
105	C₂×#341	—	89	2	no	—
106	C₈₉	#20768	1	1	yes	characteristic, complemented, Sylow
107	C₂²×M₁₆	—	89	4	no	—
108	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
109	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
110	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
111	#424	—	178	8	no	—
112	#423	—	89	8	no	—
113	#405	Dic₈₉	1	8	yes	—
114	C₂×#97	C₂×D₈₉	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
116	#341	—	89	4	no	—
117	#341	—	89	8	no	—
118	#341	—	89	4	no	—
119	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
120	C₂×M₁₆	—	2	4	no	—
121	C₂×M₁₆	—	89	8	no	—
122	C₂×M₁₆	—	178	8	no	—
123	C₂×M₁₆	—	178	8	no	—
124	C₂×M₁₆	—	178	4	no	—
125	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	178	4	no	—
128	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
129	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
132	#117	D₃₅₆	1	4	yes	—
133	#117	C₂×Dic₈₉	1	4	yes	—
134	C₂×C₄⋊C₄	Dic₁₇₈	1	2	yes	—
135	C₂×C₂²⋊C₄	Dic₁₇₈	1	2	yes	—
136	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
137	C₄⋊C₈	—	178	16	no	—
138	#97	C₂×Dic₈₉	1	8	yes	—
139	C₄×C₈	C₂×Dic₈₉	1	8	yes	—
140	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
142	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
143	C₂²×C₄	C₂×Dic₁₇₈	1	4	yes	—
144	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
145	C₂²×C₄	#13083574	1	4	yes	—
146	C₂²×C₄	#13083574	1	4	yes	—
147	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
148	M₁₆	—	2	8	no	—
149	M₁₆	—	356	8	no	—
150	M₁₆	—	178	8	no	—
151	M₁₆	—	178	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	356	8	no	—
157	C₂×C₈	—	178	8	no	—
158	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
159	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
160	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
161	C₄⋊C₄	C₄⋊Dic₈₉	1	8	yes	—
162	C₂²⋊C₄	C₄⋊Dic₈₉	1	8	yes	—
163	C₄²	C₂²×Dic₈₉	1	4	yes	—
164	C₄²	C₂×D₃₅₆	1	4	yes	—
165	C₂³	—	4	2	no	—
166	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
167	C₂³	#15668004	1	4	yes	—
168	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	C₂×C₄⋊Dic₈₉	1	8	yes	—
171	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
172	C₂×C₄	#15667971	1	4	yes	—
173	C₂×C₄	#15667971	1	4	yes	—
174	C₂×C₄	#15668000	1	8	yes	—
175	C₂×C₄	#15668004	1	4	yes	—
176	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
178	C₂×C₄	C₂×#13083574	1	2	yes	—
179	C₂×C₄	C₂×#13083574	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	2	4	no	—
183	C₈	—	356	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	#19151464	1	4	yes	—
188	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
189	C₂²	C₂×#15668004	1	2	yes	—
190	C₄	—	2	8	no	—
191	C₄	#19151464	1	4	yes	—
192	C₄	C₂×#15667971	1	1	yes	characteristic
193	C₄	C₂×#15667971	1	1	yes	characteristic
194	C₄	C₂×#15668000	1	2	yes	—
195	C₂	—	2	4	no	—
196	C₂	—	2	2	no	—
197	C₂	C₂×#19151464	1	1	yes	characteristic
198	C₂	#26028863	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#405×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#405	D₈₉
hypocenter	C₈₉	#20768
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #26020393	C₂	semidirect
3. #26020390	C₂	semidirect
106. C₈₉	#20768	semidirect

Maximal subgroups

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

subgroup	index
#26028816	2
#26020393	2
#26020390	2
#26019979	2
C₂×#19151363	2
C₂×#405×C₈₉	2
#20768	89