Subgroups — C₈₉×#60308 #36438156

Order 22784.

1964 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3483	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2991	C₂	1	6	yes	maximal, complemented
3	C₈₉×#2979	C₂	1	9	yes	maximal, complemented
4	C₈×#41×C₈₉	C₂	1	6	yes	maximal, complemented
5	C₈₉×#2931	C₂	1	9	yes	maximal, complemented
6	C₂×#131×C₈₉	C₂²	1	6	yes	—
7	Q₈²×C₈₉	C₄	1	1	yes	characteristic
8	#555×C₈₉	C₄	1	9	yes	—
9	#550×C₈₉	C₄	1	6	yes	—
10	#532×C₈₉	C₂²	1	9	yes	—
11	C₄×#41×C₈₉	C₂²	1	6	yes	—
12	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
13	C₈×Q₈×C₈₉	C₂²	1	6	yes	—
14	#436×C₈₉	C₂²	1	36	yes	—
15	C₈×D₄×C₈₉	C₂²	1	18	yes	—
16	C₈×D₄×C₈₉	C₂²	1	18	yes	—
17	#431×C₈₉	C₂²	1	18	yes	—
18	#423×C₈₉	C₂²	1	9	yes	—
19	#408×C₈₉	C₂²	1	18	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	9	yes	—
21	C₂×#41×C₈₉	C₂×C₄	1	6	yes	—
22	C₂×#41×C₈₉	C₂³	1	6	yes	—
23	#131×C₈₉	—	4	12	no	—
24	C₂×M₁₆×C₈₉	C₂³	1	18	yes	—
25	C₂²×C₈×C₈₉	C₂³	1	9	yes	—
26	C₂²×C₈×C₈₉	C₂³	1	9	yes	—
27	C₄⋊Q₈×C₈₉	C₂×C₄	1	9	yes	—
28	#127×C₈₉	C₂×C₄	1	9	yes	—
29	#124×C₈₉	C₂×C₄	1	18	yes	—
30	#121×C₈₉	C₂×C₄	1	36	yes	—
31	C₄×Q₈×C₈₉	C₂×C₄	1	6	yes	—
32	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
33	C₄×D₄×C₈₉	C₂×C₄	1	18	yes	—
34	C₄×D₄×C₈₉	C₂³	1	18	yes	—
35	#117×C₈₉	C₂×C₄	1	18	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	9	yes	—
37	C₄⋊C₈×C₈₉	C₂³	1	18	yes	—
38	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
39	C₂²⋊C₈×C₈₉	C₂³	1	36	yes	—
40	C₄×C₈×C₈₉	C₂³	1	18	yes	—
41	C₄×C₈×C₈₉	C₂³	1	6	yes	—
42	#41×C₈₉	—	2	24	no	—
43	#41×C₈₉	—	4	6	no	—
44	#41×C₈₉	—	4	6	no	—
45	C₂×Q₈×C₈₉	C₂²×C₄	1	6	yes	—
46	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
47	C₂×D₄×C₈₉	C₂²×C₄	1	18	yes	—
48	C₂×D₄×C₈₉	C₂²×C₄	1	18	yes	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	18	yes	—
50	C₂²×C₄×C₈₉	C₂⁴	1	9	yes	—
51	M₁₆×C₈₉	—	4	18	no	—
52	C₂×C₈×C₈₉	—	2	18	no	—
53	C₂×C₈×C₈₉	—	4	9	no	—
54	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
55	C₂×C₈×C₈₉	C₂⁴	1	9	yes	—
56	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
58	C₄⋊C₄×C₈₉	C₂²×C₄	1	18	yes	—
59	C₂²⋊C₄×C₈₉	C₂²×C₄	1	36	yes	—
60	C₄²×C₈₉	C₂²×C₄	1	9	yes	—
61	C₄²×C₈₉	C₂²×C₄	1	9	yes	—
62	C₄²×C₈₉	C₂⁴	1	6	yes	—
63	C₂³×C₈₉	C₂³×C₄	1	9	yes	—
64	Q₈×C₈₉	—	4	6	no	—
65	Q₈×C₈₉	#131	1	8	yes	complemented
66	D₄×C₈₉	—	2	36	no	—
67	D₄×C₈₉	—	4	18	no	—
68	C₂×C₄×C₈₉	—	2	36	no	—
69	C₂×C₄×C₈₉	—	2	9	no	—
70	C₂×C₄×C₈₉	—	2	9	no	—
71	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
72	C₂×C₄×C₈₉	C₂³×C₄	1	9	yes	—
73	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
74	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
75	C₈×C₈₉	—	2	9	no	—
76	C₈×C₈₉	—	2	6	no	—
77	C₈×C₈₉	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₈₉	—	2	9	no	—
79	C₂²×C₈₉	—	2	18	no	—
80	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
81	C₄×C₈₉	—	2	9	no	—
82	C₄×C₈₉	—	2	6	no	—
83	C₄×C₈₉	C₂×#131	1	12	yes	—
84	C₄×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
85	C₄×C₈₉	#585	1	1	yes	characteristic
86	#60308	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂×C₈₉	—	4	9	no	—
88	C₂×C₈₉	C₄×H(5,𝔽₂)	1	1	yes	characteristic
89	C₂×C₈₉	C₂²×#131	1	2	yes	—
90	#3483	C₂×C₈₉	1	1	yes	characteristic
91	#2991	C₂×C₈₉	1	6	yes	complemented
92	#2979	C₂×C₈₉	1	9	yes	complemented
93	C₈×#41	C₂×C₈₉	1	6	yes	complemented
94	#2931	C₂×C₈₉	1	9	yes	complemented
95	C₈₉	#60308	1	1	yes	characteristic, direct product factor, Sylow
96	C₂×#131	C₂²×C₈₉	1	6	yes	—
97	Q₈²	C₄×C₈₉	1	1	yes	characteristic
98	#555	C₄×C₈₉	1	9	yes	—
99	#550	C₄×C₈₉	1	6	yes	—
100	#532	C₂²×C₈₉	1	9	yes	—
101	C₄×#41	C₂²×C₈₉	1	6	yes	—
102	C₈×Q₈	C₂²×C₈₉	1	2	yes	—
103	C₈×Q₈	C₂²×C₈₉	1	6	yes	—
104	#436	C₂²×C₈₉	1	36	yes	—
105	C₈×D₄	C₂²×C₈₉	1	18	yes	—
106	C₈×D₄	C₂²×C₈₉	1	18	yes	—
107	#431	C₂²×C₈₉	1	18	yes	—
108	#423	C₂²×C₈₉	1	9	yes	—
109	#408	C₂²×C₈₉	1	18	yes	—
110	C₂×C₄×C₈	C₂²×C₈₉	1	9	yes	—
111	C₂×#41	C₂×C₄×C₈₉	1	6	yes	—
112	C₂×#41	C₂³×C₈₉	1	6	yes	—
113	#131	—	4	12	no	—
114	C₂×M₁₆	C₂³×C₈₉	1	18	yes	—
115	C₂²×C₈	C₂³×C₈₉	1	9	yes	—
116	C₂²×C₈	C₂³×C₈₉	1	9	yes	—
117	C₄⋊Q₈	C₂×C₄×C₈₉	1	9	yes	—
118	#127	C₂×C₄×C₈₉	1	9	yes	—
119	#124	C₂×C₄×C₈₉	1	18	yes	—
120	#121	C₂×C₄×C₈₉	1	36	yes	—
121	C₄×Q₈	C₂×C₄×C₈₉	1	6	yes	—
122	C₄×Q₈	C₂³×C₈₉	1	2	yes	—
123	C₄×D₄	C₂×C₄×C₈₉	1	18	yes	—
124	C₄×D₄	C₂³×C₈₉	1	18	yes	—
125	#117	C₂×C₄×C₈₉	1	18	yes	—
126	C₂×C₄²	C₂³×C₈₉	1	9	yes	—
127	C₄⋊C₈	C₂³×C₈₉	1	18	yes	—
128	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
129	C₂²⋊C₈	C₂³×C₈₉	1	36	yes	—
130	C₄×C₈	C₂³×C₈₉	1	6	yes	—
131	C₄×C₈	C₂³×C₈₉	1	18	yes	—
132	#41	—	2	24	no	—
133	#41	—	4	6	no	—
134	#41	—	4	6	no	—
135	C₂×Q₈	C₂²×C₄×C₈₉	1	6	yes	—
136	C₂×Q₈	C₂²×C₄×C₈₉	1	2	yes	—
137	C₂×D₄	C₂²×C₄×C₈₉	1	18	yes	—
138	C₂×D₄	C₂²×C₄×C₈₉	1	18	yes	—
139	C₂²×C₄	C₂²×C₄×C₈₉	1	18	yes	—
140	C₂²×C₄	C₂⁴×C₈₉	1	9	yes	—
141	M₁₆	—	4	18	no	—
142	C₂×C₈	—	2	18	no	—
143	C₂×C₈	—	4	9	no	—
144	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
145	C₂×C₈	C₂⁴×C₈₉	1	9	yes	—
146	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
147	C₄⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
148	C₄⋊C₄	C₂²×C₄×C₈₉	1	18	yes	—
149	C₂²⋊C₄	C₂²×C₄×C₈₉	1	36	yes	—
150	C₄²	C₂²×C₄×C₈₉	1	9	yes	—
151	C₄²	C₂²×C₄×C₈₉	1	9	yes	—
152	C₄²	C₂⁴×C₈₉	1	6	yes	—
153	C₂³	C₂³×C₄×C₈₉	1	9	yes	—
154	Q₈	—	4	6	no	—
155	Q₈	#131×C₈₉	1	8	yes	complemented
156	D₄	—	2	36	no	—
157	D₄	—	4	18	no	—
158	C₂×C₄	—	2	36	no	—
159	C₂×C₄	—	2	9	no	—
160	C₂×C₄	—	2	9	no	—
161	C₂×C₄	C₂³×C₄×C₈₉	1	6	yes	—
162	C₂×C₄	C₂³×C₄×C₈₉	1	9	yes	—
163	C₂×C₄	C₂³×C₄×C₈₉	1	6	yes	—
164	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
165	C₈	—	2	9	no	—
166	C₈	—	2	6	no	—
167	C₈	H(5,𝔽₂)×C₈₉	1	2	yes	—
168	C₂²	—	2	9	no	—
169	C₂²	—	2	18	no	—
170	C₂²	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
171	C₄	—	2	9	no	—
172	C₄	—	2	6	no	—
173	C₄	C₂×#131×C₈₉	1	12	yes	—
174	C₄	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
175	C₄	#585×C₈₉	1	1	yes	characteristic
176	C₂	—	4	9	no	—
177	C₂	C₄×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
178	C₂	C₂²×#131×C₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₈₉	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂⁴×C₄
prime radical	C₈₉×#3483	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2991	C₂	semidirect
3. C₈₉×#2979	C₂	semidirect
4. C₈×#41×C₈₉	C₂	semidirect
5. C₈₉×#2931	C₂	semidirect
65. Q₈×C₈₉	#131	semidirect
86. #60308	C₈₉	direct (N × H)
91. #2991	C₂×C₈₉	semidirect
92. #2979	C₂×C₈₉	semidirect
93. C₈×#41	C₂×C₈₉	semidirect
94. #2931	C₂×C₈₉	semidirect
95. C₈₉	#60308	direct (N × H)
155. Q₈	#131×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3483	2
C₈₉×#2991	2
C₈₉×#2979	2
C₈×#41×C₈₉	2
C₈₉×#2931	2
#60308	89

Subgroups — C89×#60308 #36438156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60308 #36438156