Subgroups — #36470814

Order 22784.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020416	C₂	1	1	yes	maximal, characteristic
2	#26020412	C₂	1	1	yes	maximal, characteristic
3	#26018344	C₂	1	4	yes	maximal
4	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667946	—	2	1	no	—
6	C₂×#15667946	C₂²	1	2	yes	—
7	C₂×#15667946	C₂²	1	2	yes	—
8	C₂×#15667944	—	2	1	no	—
9	#19151468	—	2	1	no	—
10	#19151464	—	2	1	no	—
11	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#346×C₈₉	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₈₉	—	2	2	no	—
15	C₂×C₄⋊Dic₈₉	—	2	2	no	—
16	C₂×C₄⋊Dic₈₉	—	4	2	no	—
17	#15667946	—	2	4	no	—
18	#15667946	—	2	4	no	—
19	#15667946	—	4	2	no	—
20	#15667944	—	4	2	no	—
21	#15667919	—	4	2	no	—
22	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	—	2	2	no	—
26	C₂×C₁₆×C₈₉	—	2	2	no	—
27	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₈₉	Q₈	1	2	yes	—
30	C₂²×Dic₈₉	—	4	2	no	—
31	C₂²×Dic₈₉	—	4	2	no	—
32	C₄⋊Dic₈₉	—	4	2	no	—
33	C₄⋊Dic₈₉	—	4	2	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₄⋊Dic₈₉	—	8	4	no	—
37	C₂²×C₄×C₈₉	D₈	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
41	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
42	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₈₉	#41	1	2	yes	—
45	C₂×C₈×C₈₉	#41	1	2	yes	—
46	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₄²×C₈₉	Q₁₆	1	2	yes	—
48	C₁₆×C₈₉	—	2	4	no	—
49	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	#107	1	4	yes	—
53	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
55	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₈₉	—	8	2	no	—
60	C₂×Dic₈₉	—	8	2	no	—
61	C₂×Dic₈₉	—	8	4	no	—
62	C₂×Dic₈₉	—	8	2	no	—
63	C₂×Dic₈₉	—	8	2	no	—
64	C₈×C₈₉	#106	1	4	yes	—
65	C₈×C₈₉	#117	1	2	yes	—
66	C₈×C₈₉	#125	1	2	yes	—
67	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
68	C₂²×C₈₉	#486	1	1	yes	characteristic
69	C₂²×C₈₉	#486	1	1	yes	characteristic
70	C₂²×C₈₉	#509	1	2	yes	—
71	C₂²×C₈₉	#510	1	2	yes	—
72	C₄×C₈₉	—	2	4	no	—
73	C₄×C₈₉	#382	1	1	yes	characteristic
74	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
75	C₄×C₈₉	#487	1	1	yes	characteristic
76	C₄×C₈₉	#487	1	1	yes	characteristic
77	Dic₈₉	—	16	2	no	—
78	Dic₈₉	—	16	2	no	—
79	#12890	—	89	1	no	maximal, Sylow
80	C₂×C₈₉	#1834	1	1	yes	characteristic
81	C₂×C₈₉	#2159	1	2	yes	—
82	C₂×C₈₉	#2243	1	2	yes	—
83	C₂×C₈₉	#2244	1	1	yes	characteristic
84	C₂×C₈₉	#2245	1	1	yes	characteristic
85	C₂×#346	D₈₉	1	1	yes	characteristic
86	#1846	—	89	1	no	—
87	#1845	—	89	1	no	—
88	#1380	—	89	4	no	—
89	C₈₉	#12890	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
91	C₂×#107	—	89	2	no	—
92	C₂×#107	—	89	2	no	—
93	C₂×#107	—	178	1	no	—
94	C₂×#106	—	178	1	no	—
95	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
96	#384	—	178	1	no	—
97	#382	—	178	1	no	—
98	#346	Dic₈₉	1	4	yes	complemented
99	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	356	2	no	—
102	C₂×C₄⋊C₄	—	178	2	no	—
103	C₂×C₄⋊C₄	—	178	2	no	—
104	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
108	#107	—	356	2	no	—
109	#107	—	178	4	no	—
110	#107	—	178	4	no	—
111	#106	—	356	2	no	—
112	C₄×C₈	Dic₁₇₈	1	2	yes	—
113	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
114	#95	—	356	2	no	—
115	C₂²×C₄	—	356	2	no	—
116	C₂²×C₄	—	356	2	no	—
117	C₂²×C₄	D₇₁₂	1	2	yes	—
118	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
120	C₂×C₈	#13083563	1	2	yes	—
121	C₂×C₈	#13083563	1	2	yes	—
122	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
124	C₂×C₈	#13083574	1	2	yes	—
125	C₂×C₈	#13083574	1	2	yes	—
126	C₄⋊C₄	—	356	2	no	—
127	C₄⋊C₄	—	356	2	no	—
128	C₄⋊C₄	—	356	2	no	—
129	C₄⋊C₄	—	356	2	no	—
130	C₄⋊C₄	—	712	4	no	—
131	C₄²	C₄⋊Dic₈₉	1	2	yes	—
132	C₄²	Dic₃₅₆	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	712	2	no	—
138	C₂×C₄	—	712	2	no	—
139	C₂×C₄	—	712	4	no	—
140	C₂×C₄	—	712	2	no	—
141	C₂×C₄	—	712	2	no	—
142	C₂×C₄	#15667946	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
144	C₂×C₄	#15668000	1	1	yes	characteristic
145	C₂×C₄	#15668000	1	1	yes	characteristic
146	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
147	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
149	C₈	#15667944	1	4	yes	—
150	C₈	#15667971	1	2	yes	—
151	C₈	#15668004	1	2	yes	—
152	C₂²	C₂×#15667946	1	1	yes	characteristic
153	C₂²	#19151945	1	1	yes	characteristic
154	C₂²	#19151945	1	1	yes	characteristic
155	C₂²	#19152036	1	2	yes	—
156	C₂²	#19152041	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	1424	2	no	—
159	C₄	—	1424	2	no	—
160	C₄	#19151464	1	1	yes	characteristic
161	C₄	C₂×#15667944	1	1	yes	characteristic
162	C₄	#19151949	1	1	yes	characteristic
163	C₄	#19151949	1	1	yes	characteristic
164	C₂	#26020380	1	1	yes	characteristic
165	C₂	#26022040	1	2	yes	—
166	C₂	#26022489	1	2	yes	—
167	C₂	#26022494	1	1	yes	characteristic
168	C₂	#26022498	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#12890
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. #346×C₈₉	C₄	semidirect
89. C₈₉	#12890	semidirect
98. #346	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020416	2
#26020412	2
#26018344	2
C₂×#346×C₈₉	2
#12890	89