Subgroups — #36445596

Order 22784.

390 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667922	C₂	1	1	yes	maximal, characteristic
2	#26019000	C₂	1	1	yes	maximal, characteristic
3	#26018536	C₂	1	2	yes	maximal
4	#26018084	C₂	1	2	yes	maximal
5	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₂²	1	2	yes	—
7	C₂×#15667922	C₄	1	2	yes	—
8	C₂×#15667922	C₄	1	2	yes	—
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
11	#19151351	C₄	1	2	yes	complemented
12	#19151349	C₄	1	2	yes	complemented
13	#19151320	—	2	2	no	—
14	C₈×#10623494	—	2	2	no	—
15	#431×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#335×C₈₉	C₄	1	2	yes	complemented
18	#334×C₈₉	C₄	1	2	yes	complemented
19	C₂²×#10623494	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₈₉	—	2	1	no	—
21	C₂×C₄×Dic₈₉	—	2	1	no	—
22	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
23	C₄⋊#10623494	C₂×C₄	1	4	yes	—
24	C₂²⋊#10623494	—	2	2	no	—
25	#15667922	—	2	4	no	—
26	#15667922	C₈	1	8	yes	complemented
27	C₄×#10623494	—	2	2	no	—
28	C₄×#10623494	D₄	1	2	yes	—
29	C₄×#10623494	Q₈	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₈₉	—	2	2	no	—
35	C₄×C₈×C₈₉	—	2	2	no	—
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₈₉	—	2	2	no	—
41	C₂×#10623494	—	2	4	no	—
42	C₂×#10623494	C₄⋊C₄	1	4	yes	—
43	C₂×#10623494	C₂×C₈	1	4	yes	—
44	C₂×#10623494	M₁₆	1	4	yes	—
45	C₄×Dic₈₉	—	2	2	no	—
46	C₄×Dic₈₉	—	4	2	no	—
47	C₄×Dic₈₉	—	4	2	no	—
48	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
52	C₂×C₈×C₈₉	—	2	2	no	—
53	C₂×C₈×C₈₉	—	2	4	no	—
54	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
59	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
63	C₂³×C₈₉	#117	1	1	yes	characteristic
64	C₂×C₄×C₈₉	—	2	2	no	—
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
74	C₂×Dic₈₉	—	2	4	no	—
75	C₂×Dic₈₉	—	4	2	no	—
76	C₂×Dic₈₉	—	4	2	no	—
77	C₈×C₈₉	—	2	1	no	—
78	C₈×C₈₉	—	2	1	no	—
79	C₈×C₈₉	—	4	4	no	—
80	C₈×C₈₉	C₄×C₈	1	2	yes	—
81	C₈×C₈₉	C₄×C₈	1	2	yes	—
82	#10623494	—	4	4	no	—
83	#10623494	C₄⋊C₈	1	8	yes	—
84	C₂²×C₈₉	—	2	1	no	—
85	C₂²×C₈₉	—	2	1	no	—
86	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈₉	#431	1	2	yes	—
88	C₄×C₈₉	—	2	1	no	—
89	C₄×C₈₉	—	2	1	no	—
90	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
91	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
92	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
93	C₄×C₈₉	#444	1	2	yes	—
94	C₄×C₈₉	#444	1	2	yes	—
95	Dic₈₉	—	4	4	no	—
96	#8268	—	89	1	no	maximal, Sylow
97	C₂×C₈₉	—	2	2	no	—
98	C₂×C₈₉	C₄×C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₈₉	#1773	1	1	yes	characteristic
100	C₂×C₈₉	#1773	1	1	yes	characteristic
101	C₄×#97	—	89	1	no	—
102	#1564	—	89	1	no	—
103	#1564	D₈₉	1	1	yes	characteristic
104	#1450	—	89	2	no	—
105	#1276	—	89	2	no	—
106	C₈₉	#8268	1	1	yes	characteristic, complemented, Sylow
107	#431	—	89	2	no	—
108	#431	C₂×D₈₉	1	2	yes	—
109	C₂×#97	—	89	2	no	—
110	C₂×#97	—	89	2	no	—
111	C₂×C₄×C₈	—	89	1	no	—
112	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
113	C₄³	—	89	1	no	—
114	#335	—	89	2	no	—
115	#335	Dic₈₉	1	2	yes	complemented
116	#334	—	89	2	no	—
117	#334	Dic₈₉	1	2	yes	complemented
118	#322	—	178	2	no	—
119	C₈²	—	178	2	no	—
120	C₂²×C₈	—	89	2	no	—
121	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
122	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
123	C₂×C₄²	—	89	2	no	—
124	C₂×C₄²	—	178	1	no	—
125	C₂×C₄²	—	178	1	no	—
126	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
127	C₄⋊C₈	—	89	4	no	—
128	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
129	C₂²⋊C₈	—	2	2	no	—
130	C₂²⋊C₈	—	178	2	no	—
131	#97	—	89	8	no	—
132	#97	—	178	4	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	89	2	no	—
135	C₄×C₈	—	89	2	no	—
136	C₄×C₈	—	178	2	no	—
137	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
138	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
139	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
140	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
141	C₂²×C₄	—	178	2	no	—
142	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
143	C₂²×C₄	#13083573	1	1	yes	characteristic
144	C₂²×C₄	#13083575	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	89	4	no	—
148	C₂×C₈	—	89	4	no	—
149	C₂×C₈	—	89	4	no	—
150	C₂×C₈	—	178	4	no	—
151	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
155	C₂×C₈	C₈×D₈₉	1	4	yes	—
156	C₄²	—	89	4	no	—
157	C₄²	—	356	2	no	—
158	C₄²	—	356	2	no	—
159	C₄²	—	178	2	no	—
160	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
161	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
162	C₄²	D₄×D₈₉	1	1	yes	characteristic
163	C₄²	Q₈×D₈₉	1	1	yes	characteristic
164	C₂³	#15667970	1	1	yes	characteristic
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	356	2	no	—
168	C₂×C₄	—	356	2	no	—
169	C₂×C₄	—	178	4	no	—
170	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
171	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
172	C₂×C₄	#15667974	1	1	yes	characteristic
173	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
174	C₂×C₄	#15667977	1	1	yes	characteristic
175	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
176	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
177	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	4	4	no	—
181	C₈	—	89	8	no	—
182	C₈	—	356	4	no	—
183	C₈	C₈×Dic₈₉	1	2	yes	—
184	C₈	C₈×Dic₈₉	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
188	C₂²	#19151650	1	2	yes	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	356	4	no	—
192	C₄	C₂×C₈×Dic₈₉	1	1	yes	characteristic
193	C₄	M₁₆×Dic₈₉	1	1	yes	characteristic
194	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
195	C₄	#19151719	1	2	yes	—
196	C₄	#19151720	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	C₄⋊C₈×Dic₈₉	1	1	yes	characteristic
199	C₂	#26020138	1	1	yes	characteristic
200	C₂	#26020139	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₄×C₄⋊C₄
prime radical	C₂³×C₈₉	#117
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#8268
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
11. #19151351	C₄	semidirect
12. #19151349	C₄	semidirect
17. #335×C₈₉	C₄	semidirect
18. #334×C₈₉	C₄	semidirect
26. #15667922	C₈	semidirect
106. C₈₉	#8268	semidirect
115. #335	Dic₈₉	semidirect
117. #334	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667922	2
#26019000	2
#26018536	2
#26018084	2
C₈₉×#1564	2
#8268	89