Subgroups — C₈₉×#33726 #36411574

Order 22784.

490 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	C₈₉×#3160	C₂	1	2	yes	maximal
2	C₈₉×#2254	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2253	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2252	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2158	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#533×C₈₉	—	2	2	no	—
8	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
9	#500×C₈₉	C₂²	1	4	yes	—
10	#499×C₈₉	—	2	2	no	—
11	#499×C₈₉	C₂²	1	4	yes	—
12	#498×C₈₉	—	2	2	no	—
13	#427×C₈₉	—	2	2	no	—
14	#427×C₈₉	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#367×C₈₉	C₂²	1	8	yes	—
17	#366×C₈₉	C₂²	1	8	yes	—
18	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₈₉	—	2	4	no	—
24	#125×C₈₉	—	2	4	no	—
25	C₂²⋊Q₈×C₈₉	—	4	4	no	—
26	C₄×Q₈×C₈₉	—	4	4	no	—
27	#117×C₈₉	—	2	4	no	—
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₈₉	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
31	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
32	#107×C₈₉	—	2	4	no	—
33	#107×C₈₉	C₂³	1	8	yes	—
34	#106×C₈₉	—	2	4	no	—
35	#106×C₈₉	—	2	4	no	—
36	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×Q₈×C₈₉	—	4	4	no	—
41	C₂²×C₄×C₈₉	D₈	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
44	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₈₉	—	2	8	no	—
50	C₄⋊C₄×C₈₉	—	4	8	no	—
51	C₄⋊C₄×C₈₉	—	4	4	no	—
52	C₂²⋊C₄×C₈₉	—	4	4	no	—
53	C₄²×C₈₉	—	4	4	no	—
54	C₄²×C₈₉	D₈	1	2	yes	—
55	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
58	C₁₆×C₈₉	C₂×Q₈	1	4	yes	—
59	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₈₉	—	8	8	no	—
61	C₂×C₄×C₈₉	—	2	4	no	—
62	C₂×C₄×C₈₉	—	4	8	no	—
63	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
66	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
69	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
70	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
71	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
72	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
73	C₂²×C₈₉	—	2	2	no	—
74	C₂²×C₈₉	#500	1	2	yes	—
75	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
76	C₄×C₈₉	—	2	2	no	—
77	C₄×C₈₉	—	8	8	no	—
78	C₄×C₈₉	#500	1	2	yes	—
79	C₄×C₈₉	#508	1	4	yes	—
80	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
82	#33726	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₈₉	—	2	2	no	—
84	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
85	C₂×C₈₉	C₂×#508	1	2	yes	—
86	#3160	C₂×C₈₉	1	2	yes	—
87	#2254	C₂×C₈₉	1	2	yes	complemented
88	#2253	C₂×C₈₉	1	4	yes	complemented
89	#2252	C₂×C₈₉	1	2	yes	complemented
90	#2158	C₂×C₈₉	1	4	yes	—
91	C₂×C₄×C₁₆	C₂×C₈₉	1	1	yes	characteristic
92	C₈₉	#33726	1	1	yes	characteristic, direct product factor, Sylow
93	#533	—	2	2	no	—
94	C₂²×C₁₆	C₂²×C₈₉	1	2	yes	—
95	#500	C₂²×C₈₉	1	4	yes	—
96	#499	—	2	2	no	—
97	#499	C₂²×C₈₉	1	4	yes	—
98	#498	—	2	2	no	—
99	#427	—	2	2	no	—
100	#427	C₂²×C₈₉	1	4	yes	—
101	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
102	#367	C₂²×C₈₉	1	8	yes	—
103	#366	C₂²×C₈₉	1	8	yes	—
104	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
105	C₄×C₁₆	C₂²×C₈₉	1	2	yes	—
106	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
107	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
108	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
109	C₄⋊Q₈	—	2	4	no	—
110	#125	—	2	4	no	—
111	C₂²⋊Q₈	—	4	4	no	—
112	C₄×Q₈	—	4	4	no	—
113	#117	—	2	4	no	—
114	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
115	C₂×C₁₆	Q₈×C₈₉	1	8	yes	complemented
116	C₂×C₁₆	C₂³×C₈₉	1	2	yes	—
117	C₂×C₁₆	C₂³×C₈₉	1	2	yes	—
118	#107	—	2	4	no	—
119	#107	C₂³×C₈₉	1	8	yes	—
120	#106	—	2	4	no	—
121	#106	—	2	4	no	—
122	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
123	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
124	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
125	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
126	C₂×Q₈	—	4	4	no	—
127	C₂²×C₄	D₈×C₈₉	1	2	yes	—
128	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
129	C₂×C₈	Q₁₆×C₈₉	1	4	yes	—
130	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
131	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
132	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
133	C₂×C₈	C₂×Q₈×C₈₉	1	4	yes	—
134	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	4	8	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₂²⋊C₄	—	4	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	D₈×C₈₉	1	2	yes	—
141	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
142	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
143	C₁₆	C₂×Q₈×C₈₉	1	4	yes	—
144	C₁₆	C₂×Q₈×C₈₉	1	4	yes	—
145	C₂³	C₂×D₈×C₈₉	1	1	yes	characteristic
146	Q₈	—	8	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	C₄⋊Q₈×C₈₉	1	2	yes	—
150	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
151	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
153	C₂×C₄	C₂×Q₁₆×C₈₉	1	2	yes	—
154	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
155	C₈	C₄⋊Q₈×C₈₉	1	2	yes	—
156	C₈	C₂×Q₁₆×C₈₉	1	2	yes	—
157	C₈	C₂×Q₁₆×C₈₉	1	2	yes	—
158	C₈	C₂²×Q₈×C₈₉	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#500×C₈₉	1	2	yes	—
161	C₂²	C₂²×D₈×C₈₉	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	8	8	no	—
164	C₄	#500×C₈₉	1	2	yes	—
165	C₄	#508×C₈₉	1	4	yes	—
166	C₄	C₂×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
167	C₄	C₂²×Q₁₆×C₈₉	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#500×C₈₉	1	1	yes	characteristic
170	C₂	C₂×#508×C₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2254	C₂	semidirect
3. C₈₉×#2253	C₂	semidirect
4. C₈₉×#2252	C₂	semidirect
29. C₂×C₁₆×C₈₉	Q₈	semidirect
82. #33726	C₈₉	direct (N × H)
87. #2254	C₂×C₈₉	semidirect
88. #2253	C₂×C₈₉	semidirect
89. #2252	C₂×C₈₉	semidirect
92. C₈₉	#33726	direct (N × H)
115. C₂×C₁₆	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3160	2
C₈₉×#2254	2
C₈₉×#2253	2
C₈₉×#2252	2
C₈₉×#2158	2
C₂×C₄×C₁₆×C₈₉	2
#33726	89

Subgroups — C89×#33726 #36411574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33726 #36411574