Subgroups — C₈₉×#9853 #36387701

Order 22784.

422 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1994	C₂	1	2	yes	maximal
3	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1493	C₂	1	2	yes	maximal
6	#533×C₈₉	C₂²	1	2	yes	—
7	#500×C₈₉	C₄	1	1	yes	characteristic, complemented
8	#494×C₈₉	C₄	1	1	yes	characteristic, complemented
9	#487×C₈₉	C₄	1	2	yes	complemented
10	#467×C₈₉	—	2	2	no	—
11	C₄×Q₁₆×C₈₉	—	2	2	no	—
12	#423×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	#339×C₈₉	—	2	4	no	—
17	#336×C₈₉	—	2	2	no	—
18	#334×C₈₉	—	2	2	no	—
19	Q₈⋊C₈×C₈₉	—	2	4	no	—
20	C₂×Q₁₆×C₈₉	—	2	2	no	—
21	C₂×M₁₆×C₈₉	—	2	2	no	—
22	C₂²×C₈×C₈₉	—	2	1	no	—
23	C₂²×C₈×C₈₉	D₄	1	2	yes	—
24	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
26	#125×C₈₉	—	2	2	no	—
27	C₂²⋊Q₈×C₈₉	—	2	2	no	—
28	C₂²⋊Q₈×C₈₉	—	2	2	no	—
29	C₄×Q₈×C₈₉	—	2	2	no	—
30	C₄×Q₈×C₈₉	—	2	2	no	—
31	#117×C₈₉	—	2	2	no	—
32	C₂×C₄²×C₈₉	—	2	1	no	—
33	C₂×C₄²×C₈₉	—	2	1	no	—
34	C₂×C₄²×C₈₉	D₄	1	2	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	#107×C₈₉	—	2	2	no	—
37	C₄⋊C₈×C₈₉	—	2	2	no	—
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	#103×C₈₉	—	2	4	no	—
40	C₄×C₈×C₈₉	—	2	1	no	—
41	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	C₄×C₈×C₈₉	D₄	1	2	yes	—
45	C₂×Q₈×C₈₉	—	2	2	no	—
46	C₂×Q₈×C₈₉	—	2	2	no	—
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
51	Q₁₆×C₈₉	—	4	4	no	—
52	M₁₆×C₈₉	—	4	4	no	—
53	C₂×C₈×C₈₉	—	2	2	no	—
54	C₂×C₈×C₈₉	—	4	2	no	—
55	C₂×C₈×C₈₉	—	4	2	no	—
56	C₂×C₈×C₈₉	—	4	2	no	—
57	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₈₉	—	2	2	no	—
60	C₄⋊C₄×C₈₉	—	2	2	no	—
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₄⋊C₄×C₈₉	—	4	2	no	—
64	C₄⋊C₄×C₈₉	—	4	2	no	—
65	C₂²⋊C₄×C₈₉	—	4	2	no	—
66	C₄²×C₈₉	—	2	4	no	—
67	C₄²×C₈₉	—	2	4	no	—
68	C₄²×C₈₉	—	4	2	no	—
69	C₄²×C₈₉	—	4	2	no	—
70	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₈₉	#41	1	1	yes	characteristic
74	C₂³×C₈₉	#121	1	1	yes	characteristic
75	Q₈×C₈₉	—	4	4	no	—
76	Q₈×C₈₉	—	4	4	no	—
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	4	no	—
80	C₂×C₄×C₈₉	—	4	2	no	—
81	C₂×C₄×C₈₉	—	4	2	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	—	4	2	no	—
84	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#137	1	4	yes	—
91	C₈×C₈₉	—	2	2	no	—
92	C₈×C₈₉	—	2	2	no	—
93	C₈×C₈₉	—	4	2	no	—
94	C₈×C₈₉	—	8	2	no	—
95	C₂²×C₈₉	—	2	1	no	—
96	C₂²×C₈₉	—	2	1	no	—
97	C₂²×C₈₉	#390	1	1	yes	characteristic
98	C₂²×C₈₉	#470	1	2	yes	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	4	4	no	—
102	C₄×C₈₉	—	8	2	no	—
103	C₄×C₈₉	—	8	2	no	—
104	C₄×C₈₉	#421	1	2	yes	—
105	C₄×C₈₉	#441	1	2	yes	—
106	C₄×C₈₉	#443	1	2	yes	—
107	C₄×C₈₉	#497	1	2	yes	—
108	#9853	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₈₉	—	2	2	no	—
110	C₂×C₈₉	#1963	1	1	yes	characteristic
111	C₂×C₈₉	#1971	1	1	yes	characteristic
112	C₂×C₈₉	#1974	1	1	yes	characteristic
113	#3170	C₂×C₈₉	1	1	yes	characteristic
114	#1994	C₂×C₈₉	1	2	yes	—
115	#1744	C₂×C₈₉	1	1	yes	characteristic
116	#1566	C₂×C₈₉	1	1	yes	characteristic
117	#1493	C₂×C₈₉	1	2	yes	—
118	C₈₉	#9853	1	1	yes	characteristic, direct product factor, Sylow
119	#533	C₂²×C₈₉	1	2	yes	—
120	#500	C₄×C₈₉	1	1	yes	characteristic, complemented
121	#494	C₄×C₈₉	1	1	yes	characteristic, complemented
122	#487	C₄×C₈₉	1	2	yes	complemented
123	#467	—	2	2	no	—
124	C₄×Q₁₆	—	2	2	no	—
125	#423	C₂²×C₈₉	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
128	C₄³	C₂²×C₈₉	1	1	yes	characteristic
129	#339	—	2	4	no	—
130	#336	—	2	2	no	—
131	#334	—	2	2	no	—
132	Q₈⋊C₈	—	2	4	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	D₄×C₈₉	1	2	yes	—
137	C₄⋊Q₈	C₂×C₄×C₈₉	1	2	yes	—
138	C₄⋊Q₈	C₂×C₄×C₈₉	1	2	yes	—
139	#125	—	2	2	no	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₂²⋊Q₈	—	2	2	no	—
142	C₄×Q₈	—	2	2	no	—
143	C₄×Q₈	—	2	2	no	—
144	#117	—	2	2	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	D₄×C₈₉	1	2	yes	—
148	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
149	#107	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	#103	—	2	4	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
155	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
156	C₄×C₈	D₄×C₈₉	1	2	yes	—
157	C₄×C₈	D₄×C₈₉	1	2	yes	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
163	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
164	Q₁₆	—	4	4	no	—
165	M₁₆	—	4	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
171	C₂×C₈	C₂²⋊C₄×C₈₉	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	2	4	no	—
180	C₄²	—	2	4	no	—
181	C₄²	—	4	2	no	—
182	C₄²	—	4	2	no	—
183	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
184	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
185	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
186	C₄²	#41×C₈₉	1	1	yes	characteristic
187	C₂³	#121×C₈₉	1	1	yes	characteristic
188	Q₈	—	4	4	no	—
189	Q₈	—	4	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
198	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
199	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
200	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
201	C₂×C₄	#124×C₈₉	1	1	yes	characteristic
202	C₂×C₄	#127×C₈₉	1	1	yes	characteristic
203	C₂×C₄	#137×C₈₉	1	4	yes	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	4	2	no	—
207	C₈	—	8	2	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#390×C₈₉	1	1	yes	characteristic
211	C₂²	#470×C₈₉	1	2	yes	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	8	2	no	—
216	C₄	—	8	2	no	—
217	C₄	#421×C₈₉	1	2	yes	—
218	C₄	#441×C₈₉	1	2	yes	—
219	C₄	#443×C₈₉	1	2	yes	—
220	C₄	#497×C₈₉	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	C₈₉×#1963	1	1	yes	characteristic
223	C₂	C₈₉×#1971	1	1	yes	characteristic
224	C₂	C₈₉×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈₉	#127
commutator	C₄²	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
7. #500×C₈₉	C₄	semidirect
8. #494×C₈₉	C₄	semidirect
9. #487×C₈₉	C₄	semidirect
108. #9853	C₈₉	direct (N × H)
118. C₈₉	#9853	direct (N × H)
120. #500	C₄×C₈₉	semidirect
121. #494	C₄×C₈₉	semidirect
122. #487	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3170	2
C₈₉×#1994	2
C₈₉×#1744	2
C₈₉×#1566	2
C₈₉×#1493	2
#9853	89

Subgroups — C89×#9853 #36387701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#9853 #36387701