Subgroups — #36470804

Order 22784.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020424	C₂	1	1	yes	maximal, characteristic
2	#26020416	C₂	1	1	yes	maximal, characteristic
3	#26018351	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667948	—	2	1	no	—
6	C₂×#15667948	C₂²	1	2	yes	—
7	C₂×#15667946	—	2	1	no	—
8	C₂×#15667944	C₂²	1	2	yes	—
9	#19151622	—	2	1	no	—
10	#19151464	—	2	1	no	—
11	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#346×C₈₉	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₈₉	C₄	1	2	yes	complemented
15	C₂×#13083558	—	2	2	no	—
16	C₂×C₄⋊Dic₈₉	—	2	2	no	—
17	#15667948	—	2	4	no	—
18	#15667948	—	4	2	no	—
19	#15667946	—	4	2	no	—
20	#15667944	—	2	4	no	—
21	C₄⋊#10623494	—	4	2	no	—
22	#15667919	—	4	2	no	—
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
26	M₃₂×C₈₉	—	2	2	no	—
27	M₃₂×C₈₉	—	2	2	no	—
28	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
33	C₂²×Dic₈₉	—	4	2	no	—
34	#13083558	—	4	2	no	—
35	#13083558	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	D₈	1	2	yes	—
44	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
45	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₈₉	#41	1	2	yes	—
48	C₂×C₈×C₈₉	#41	1	2	yes	—
49	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₈₉	SD₁₆	1	2	yes	—
52	C₁₆×C₈₉	—	2	4	no	—
53	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	4	2	no	—
55	C₂×C₄×C₈₉	#106	1	2	yes	—
56	C₂×C₄×C₈₉	#106	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×Dic₈₉	—	8	4	no	—
64	C₂×Dic₈₉	—	8	2	no	—
65	C₈×C₈₉	#107	1	2	yes	—
66	C₈×C₈₉	#107	1	2	yes	—
67	C₈×C₈₉	#117	1	2	yes	—
68	C₈×C₈₉	#125	1	2	yes	—
69	#10623494	—	8	2	no	—
70	C₂²×C₈₉	—	2	2	no	—
71	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
72	C₂²×C₈₉	#486	1	1	yes	characteristic
73	C₂²×C₈₉	#487	1	1	yes	characteristic
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	#382	1	1	yes	characteristic
77	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
78	C₄×C₈₉	#486	1	1	yes	characteristic
79	C₄×C₈₉	#487	1	1	yes	characteristic
80	Dic₈₉	—	16	2	no	—
81	#12887	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	—	2	1	no	—
83	C₂×C₈₉	—	2	1	no	—
84	C₂×C₈₉	#1834	1	1	yes	characteristic
85	C₂×C₈₉	#2155	1	2	yes	—
86	C₄×M₃₂	D₈₉	1	1	yes	characteristic
87	#1848	—	89	1	no	—
88	#1846	—	89	1	no	—
89	#1383	—	89	4	no	—
90	C₈₉	#12887	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
92	C₂×#108	—	89	2	no	—
93	C₂×#108	—	178	1	no	—
94	C₂×#107	—	178	1	no	—
95	C₂×#106	—	89	2	no	—
96	#423	—	178	1	no	—
97	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
98	#382	—	178	1	no	—
99	#346	Dic₈₉	1	2	yes	complemented
100	C₄×C₁₆	Dic₈₉	1	2	yes	complemented
101	C₂×M₁₆	—	178	2	no	—
102	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	178	2	no	—
105	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
106	M₃₂	—	2	2	no	—
107	M₃₂	—	2	2	no	—
108	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
109	#108	—	356	2	no	—
110	#108	—	178	4	no	—
111	#107	—	356	2	no	—
112	#106	—	178	4	no	—
113	C₄⋊C₈	—	356	2	no	—
114	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
115	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
117	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
118	#95	—	356	2	no	—
119	C₂²×C₄	—	356	2	no	—
120	C₂²×C₄	#13083563	1	2	yes	—
121	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
122	M₁₆	—	356	2	no	—
123	M₁₆	—	356	2	no	—
124	C₂×C₈	—	356	2	no	—
125	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
127	C₂×C₈	D₇₁₂	1	2	yes	—
128	C₂×C₈	Dic₃₅₆	1	2	yes	—
129	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
131	C₂×C₈	#13083574	1	2	yes	—
132	C₂×C₈	#13083574	1	2	yes	—
133	C₄⋊C₄	—	356	2	no	—
134	C₄⋊C₄	—	356	2	no	—
135	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
136	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
137	C₄²	#13083563	1	2	yes	—
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×#13083563	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	712	4	no	—
142	C₂×C₄	—	712	2	no	—
143	C₂×C₄	#15667944	1	2	yes	—
144	C₂×C₄	#15667944	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
146	C₂×C₄	#15668000	1	1	yes	characteristic
147	C₂×C₄	#15668000	1	1	yes	characteristic
148	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
149	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
150	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
151	C₈	—	712	2	no	—
152	C₈	#15667946	1	2	yes	—
153	C₈	#15667946	1	2	yes	—
154	C₈	#15667971	1	2	yes	—
155	C₈	#15668004	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#15667944	1	1	yes	characteristic
158	C₂²	#19151945	1	1	yes	characteristic
159	C₂²	#19151949	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	1424	2	no	—
163	C₄	#19151464	1	1	yes	characteristic
164	C₄	C₂×#15667946	1	1	yes	characteristic
165	C₄	#19151945	1	1	yes	characteristic
166	C₄	#19151949	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#26020380	1	1	yes	characteristic
170	C₂	#26022025	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#15667944
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#106
prime radical	C₂³×C₈₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#12887
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. #346×C₈₉	C₄	semidirect
14. C₄×C₁₆×C₈₉	C₄	semidirect
90. C₈₉	#12887	semidirect
99. #346	Dic₈₉	semidirect
100. C₄×C₁₆	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020424	2
#26020416	2
#26018351	2
C₄×M₃₂×C₈₉	2
#12887	89