Subgroups — C₈₉×#12014 #36389862

Order 22784.

220 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×#2163	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2162	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1370	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1367	C₂	1	2	yes	maximal, complemented
6	#431×C₈₉	C₄	1	1	yes	characteristic
7	#431×C₈₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
10	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
11	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
12	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
13	#346×C₈₉	—	2	1	no	—
14	C₄×C₁₆×C₈₉	—	2	1	no	—
15	#322×C₈₉	C₄	1	1	yes	characteristic
16	#322×C₈₉	C₄	1	1	yes	characteristic
17	C₈²×C₈₉	C₂²	1	1	yes	characteristic
18	C₈²×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₈₉	—	2	2	no	—
23	C₂×C₁₆×C₈₉	—	2	2	no	—
24	C₄⋊C₈×C₈₉	—	2	1	no	—
25	C₄⋊C₈×C₈₉	C₈	1	2	yes	—
26	C₂²⋊C₈×C₈₉	—	2	1	no	—
27	C₂²⋊C₈×C₈₉	C₈	1	2	yes	—
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	—	2	1	no	—
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
52	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₈₉	—	4	2	no	—
54	C₁₆×C₈₉	—	4	2	no	—
55	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	4	1	no	—
60	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₈₉	—	2	1	no	—
67	C₈×C₈₉	—	2	1	no	—
68	C₈×C₈₉	—	2	2	no	—
69	C₈×C₈₉	—	2	2	no	—
70	C₈×C₈₉	—	4	1	no	—
71	C₈×C₈₉	—	4	1	no	—
72	C₈×C₈₉	#108	1	2	yes	—
73	C₈×C₈₉	#135	1	2	yes	—
74	C₂²×C₈₉	—	2	1	no	—
75	C₂²×C₈₉	—	2	1	no	—
76	C₂²×C₈₉	—	2	1	no	—
77	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₄×C₈₉	—	2	1	no	—
79	C₄×C₈₉	—	2	1	no	—
80	C₄×C₈₉	—	2	1	no	—
81	C₄×C₈₉	—	2	1	no	—
82	C₄×C₈₉	#427	1	1	yes	characteristic
83	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
84	C₄×C₈₉	#504	1	2	yes	—
85	#12014	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₈₉	—	4	1	no	—
87	C₂×C₈₉	#1564	1	1	yes	characteristic
88	C₂×C₈₉	#2151	1	1	yes	characteristic
89	C₂×C₈₉	#2153	1	1	yes	characteristic
90	#2163	C₂×C₈₉	1	1	yes	characteristic
91	#2162	C₂×C₈₉	1	1	yes	characteristic
92	#1450	C₂×C₈₉	1	1	yes	characteristic
93	#1370	C₂×C₈₉	1	2	yes	complemented
94	#1367	C₂×C₈₉	1	2	yes	complemented
95	C₈₉	#12014	1	1	yes	characteristic, direct product factor, Sylow
96	#431	C₄×C₈₉	1	1	yes	characteristic
97	#431	C₄×C₈₉	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
99	C₄⋊C₁₆	C₂²×C₈₉	1	2	yes	—
100	C₄⋊C₁₆	C₂²×C₈₉	1	2	yes	—
101	C₂²⋊C₁₆	—	2	1	no	—
102	C₂²⋊C₁₆	—	2	1	no	—
103	#346	—	2	1	no	—
104	C₄×C₁₆	—	2	1	no	—
105	#322	C₄×C₈₉	1	1	yes	characteristic
106	#322	C₄×C₈₉	1	1	yes	characteristic
107	C₈²	C₂²×C₈₉	1	1	yes	characteristic
108	C₈²	C₂²×C₈₉	1	1	yes	characteristic
109	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
110	C₂²×C₈	Q₈×C₈₉	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	C₈×C₈₉	1	2	yes	—
116	C₂²⋊C₈	—	2	1	no	—
117	C₂²⋊C₈	C₈×C₈₉	1	2	yes	—
118	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
120	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
121	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
122	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
123	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
124	C₄×C₈	Q₈×C₈₉	1	1	yes	characteristic
125	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
126	C₂²×C₄	C₄⋊C₄×C₈₉	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₈×C₈₉	1	1	yes	characteristic
128	C₂²×C₄	M₁₆×C₈₉	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
134	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
135	C₂×C₈	C₂×C₈×C₈₉	1	2	yes	—
136	C₂×C₈	C₂×C₈×C₈₉	1	2	yes	—
137	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
138	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
139	C₄²	C₄⋊C₄×C₈₉	1	1	yes	characteristic
140	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
141	C₄²	M₁₆×C₈₉	1	1	yes	characteristic
142	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
143	C₁₆	—	4	2	no	—
144	C₁₆	—	4	2	no	—
145	C₂³	C₄⋊C₈×C₈₉	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₄⋊C₈×C₈₉	1	1	yes	characteristic
151	C₂×C₄	C₄⋊C₈×C₈₉	1	1	yes	characteristic
152	C₂×C₄	C₄⋊C₈×C₈₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
154	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
155	C₂×C₄	C₂×M₁₆×C₈₉	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	2	no	—
159	C₈	—	2	2	no	—
160	C₈	—	4	1	no	—
161	C₈	—	4	1	no	—
162	C₈	#108×C₈₉	1	2	yes	—
163	C₈	#135×C₈₉	1	2	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	C₂×C₄⋊C₈×C₈₉	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	#427×C₈₉	1	1	yes	characteristic
173	C₄	C₂×#108×C₈₉	1	1	yes	characteristic
174	C₄	#504×C₈₉	1	2	yes	—
175	C₂	—	4	1	no	—
176	C₂	C₈₉×#1564	1	1	yes	characteristic
177	C₂	C₈₉×#2151	1	1	yes	characteristic
178	C₂	C₈₉×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₈₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₈₉
Frattini, cosocle	C₄×C₈	C₂³×C₈₉
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈₉×#1370	C₂	semidirect
5. C₈₉×#1367	C₂	semidirect
85. #12014	C₈₉	direct (N × H)
93. #1370	C₂×C₈₉	semidirect
94. #1367	C₂×C₈₉	semidirect
95. C₈₉	#12014	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2163	2
C₈₉×#2162	2
C₈₉×#1450	2
C₈₉×#1370	2
C₈₉×#1367	2
#12014	89

Subgroups — C89×#12014 #36389862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#12014 #36389862