Subgroups — C₈₉×#33740 #36411588

Order 22784.

532 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3176	C₂	1	2	yes	maximal, complemented
2	C₈₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2244	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2218	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2215	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2177	C₂	1	2	yes	maximal, complemented
8	C₄×D₁₆×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	#540×C₈₉	—	2	2	no	—
11	C₂×D₁₆×C₈₉	C₂²	1	2	yes	—
12	C₂×D₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	#499×C₈₉	C₂²	1	2	yes	—
16	#493×C₈₉	C₂²	1	2	yes	—
17	#490×C₈₉	—	2	2	no	—
18	#469×C₈₉	—	2	2	no	—
19	#466×C₈₉	C₂²	1	4	yes	—
20	#442×C₈₉	—	2	2	no	—
21	C₄×D₈×C₈₉	C₂²	1	4	yes	—
22	#405×C₈₉	C₂²	1	1	yes	characteristic
23	#367×C₈₉	C₂²	1	2	yes	—
24	#366×C₈₉	C₂²	1	2	yes	—
25	#357×C₈₉	C₂²	1	4	yes	—
26	#357×C₈₉	C₂²	1	4	yes	—
27	#346×C₈₉	C₂²	1	1	yes	characteristic
28	#346×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×D₈×C₈₉	—	2	2	no	—
32	C₂×D₈×C₈₉	C₂³	1	4	yes	—
33	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	#127×C₈₉	—	2	2	no	—
36	#125×C₈₉	—	2	2	no	—
37	#123×C₈₉	—	4	4	no	—
38	#121×C₈₉	—	2	4	no	—
39	#121×C₈₉	—	4	4	no	—
40	C₄×D₄×C₈₉	—	2	4	no	—
41	#117×C₈₉	D₄	1	1	yes	characteristic
42	D₁₆×C₈₉	—	2	4	no	—
43	D₁₆×C₈₉	—	2	4	no	—
44	M₃₂×C₈₉	—	2	2	no	—
45	C₂×C₁₆×C₈₉	—	2	2	no	—
46	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
50	#107×C₈₉	C₂³	1	4	yes	—
51	#106×C₈₉	—	2	2	no	—
52	#102×C₈₉	—	2	4	no	—
53	#102×C₈₉	—	2	4	no	—
54	#97×C₈₉	D₄	1	1	yes	characteristic
55	#97×C₈₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₈₉	—	2	4	no	—
59	C₂×D₄×C₈₉	—	4	4	no	—
60	C₂×D₄×C₈₉	—	4	4	no	—
61	C₂²×C₄×C₈₉	—	4	4	no	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	D₈×C₈₉	—	2	4	no	—
64	D₈×C₈₉	—	2	4	no	—
65	D₈×C₈₉	—	4	4	no	—
66	M₁₆×C₈₉	—	2	2	no	—
67	C₂×C₈×C₈₉	—	2	1	no	—
68	C₂×C₈×C₈₉	—	2	1	no	—
69	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₈₉	—	2	4	no	—
74	C₄⋊C₄×C₈₉	—	4	4	no	—
75	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
76	C₂²⋊C₄×C₈₉	—	4	4	no	—
77	C₂²⋊C₄×C₈₉	—	4	4	no	—
78	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
79	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₈₉	—	2	1	no	—
82	C₁₆×C₈₉	—	2	1	no	—
83	C₁₆×C₈₉	#41	1	2	yes	—
84	C₁₆×C₈₉	#41	1	2	yes	—
85	C₂³×C₈₉	—	4	4	no	—
86	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
87	D₄×C₈₉	—	4	4	no	—
88	D₄×C₈₉	—	4	4	no	—
89	D₄×C₈₉	—	16	4	no	—
90	D₄×C₈₉	—	16	4	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	4	4	no	—
94	C₂×C₄×C₈₉	—	16	4	no	—
95	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₈₉	—	2	1	no	—
100	C₈×C₈₉	—	2	1	no	—
101	C₈×C₈₉	#135	1	2	yes	—
102	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
103	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
104	C₂²×C₈₉	—	2	2	no	—
105	C₂²×C₈₉	—	2	1	no	—
106	C₂²×C₈₉	—	8	4	no	—
107	C₂²×C₈₉	—	8	4	no	—
108	C₂²×C₈₉	—	8	4	no	—
109	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₈₉	—	2	2	no	—
111	C₄×C₈₉	—	2	2	no	—
112	C₄×C₈₉	—	2	1	no	—
113	C₄×C₈₉	—	8	4	no	—
114	C₄×C₈₉	#532	1	1	yes	characteristic
115	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
116	#33740	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₈₉	—	4	1	no	—
118	C₂×C₈₉	—	16	4	no	—
119	C₂×C₈₉	#3169	1	1	yes	characteristic
120	C₂×C₈₉	#3415	1	2	yes	—
121	#3176	C₂×C₈₉	1	2	yes	complemented
122	#2253	C₂×C₈₉	1	1	yes	characteristic, complemented
123	#2246	C₂×C₈₉	1	1	yes	characteristic, complemented
124	#2244	C₂×C₈₉	1	2	yes	complemented
125	#2218	C₂×C₈₉	1	2	yes	complemented
126	#2215	C₂×C₈₉	1	2	yes	complemented
127	#2177	C₂×C₈₉	1	2	yes	complemented
128	C₄×D₁₆	C₂×C₈₉	1	2	yes	complemented
129	#2108	C₂×C₈₉	1	1	yes	characteristic, complemented
130	C₈₉	#33740	1	1	yes	characteristic, direct product factor, Sylow
131	#540	—	2	2	no	—
132	C₂×D₁₆	C₂²×C₈₉	1	2	yes	—
133	C₂×D₁₆	C₂²×C₈₉	1	2	yes	—
134	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
135	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
136	#499	C₂²×C₈₉	1	2	yes	—
137	#493	C₂²×C₈₉	1	2	yes	—
138	#490	—	2	2	no	—
139	#469	—	2	2	no	—
140	#466	C₂²×C₈₉	1	4	yes	—
141	#442	—	2	2	no	—
142	C₄×D₈	C₂²×C₈₉	1	4	yes	—
143	#405	C₂²×C₈₉	1	1	yes	characteristic
144	#367	C₂²×C₈₉	1	2	yes	—
145	#366	C₂²×C₈₉	1	2	yes	—
146	#357	C₂²×C₈₉	1	4	yes	—
147	#357	C₂²×C₈₉	1	4	yes	—
148	#346	C₂²×C₈₉	1	1	yes	characteristic
149	#346	C₂²×C₈₉	1	1	yes	characteristic
150	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
151	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	C₂³×C₈₉	1	4	yes	—
154	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
156	#127	—	2	2	no	—
157	#125	—	2	2	no	—
158	#123	—	4	4	no	—
159	#121	—	2	4	no	—
160	#121	—	4	4	no	—
161	C₄×D₄	—	2	4	no	—
162	#117	D₄×C₈₉	1	1	yes	characteristic
163	D₁₆	—	2	4	no	—
164	D₁₆	—	2	4	no	—
165	M₃₂	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
168	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
169	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
170	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
171	#107	C₂³×C₈₉	1	4	yes	—
172	#106	—	2	2	no	—
173	#102	—	2	4	no	—
174	#102	—	2	4	no	—
175	#97	D₄×C₈₉	1	1	yes	characteristic
176	#97	D₄×C₈₉	1	1	yes	characteristic
177	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
184	D₈	—	2	4	no	—
185	D₈	—	2	4	no	—
186	D₈	—	4	4	no	—
187	M₁₆	—	2	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
191	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
193	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	D₈×C₈₉	1	2	yes	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₂²⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	D₈×C₈₉	1	2	yes	—
200	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
202	C₁₆	—	2	1	no	—
203	C₁₆	—	2	1	no	—
204	C₁₆	#41×C₈₉	1	2	yes	—
205	C₁₆	#41×C₈₉	1	2	yes	—
206	C₂³	—	4	4	no	—
207	C₂³	C₂×D₈×C₈₉	1	1	yes	characteristic
208	D₄	—	4	4	no	—
209	D₄	—	4	4	no	—
210	D₄	—	16	4	no	—
211	D₄	—	16	4	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	16	4	no	—
216	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
217	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
218	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	#135×C₈₉	1	2	yes	—
223	C₈	C₂×#41×C₈₉	1	1	yes	characteristic
224	C₈	C₂×#41×C₈₉	1	1	yes	characteristic
225	C₂²	—	2	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	4	no	—
229	C₂²	—	8	4	no	—
230	C₂²	C₂²×D₈×C₈₉	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	8	4	no	—
235	C₄	#532×C₈₉	1	1	yes	characteristic
236	C₄	C₂×#135×C₈₉	1	1	yes	characteristic
237	C₂	—	4	1	no	—
238	C₂	—	16	4	no	—
239	C₂	C₈₉×#3169	1	1	yes	characteristic
240	C₂	C₈₉×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ #117×C₈₉ ◁ #405×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3176	C₂	semidirect
2. C₈₉×#2253	C₂	semidirect
3. C₈₉×#2246	C₂	semidirect
4. C₈₉×#2244	C₂	semidirect
5. C₈₉×#2218	C₂	semidirect
6. C₈₉×#2215	C₂	semidirect
7. C₈₉×#2177	C₂	semidirect
8. C₄×D₁₆×C₈₉	C₂	semidirect
9. C₈₉×#2108	C₂	semidirect
116. #33740	C₈₉	direct (N × H)
121. #3176	C₂×C₈₉	semidirect
122. #2253	C₂×C₈₉	semidirect
123. #2246	C₂×C₈₉	semidirect
124. #2244	C₂×C₈₉	semidirect
125. #2218	C₂×C₈₉	semidirect
126. #2215	C₂×C₈₉	semidirect
127. #2177	C₂×C₈₉	semidirect
128. C₄×D₁₆	C₂×C₈₉	semidirect
129. #2108	C₂×C₈₉	semidirect
130. C₈₉	#33740	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×#3176	2
C₈₉×#2253	2
C₈₉×#2246	2
C₈₉×#2244	2
C₈₉×#2218	2
C₈₉×#2215	2
C₈₉×#2177	2
C₄×D₁₆×C₈₉	2
C₈₉×#2108	2
#33740	89

Subgroups — C89×#33740 #36411588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33740 #36411588