Subgroups — C₈₉×#7860 #36385708

Order 22784.

666 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₈₉	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1275	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₈₉	C₄	1	2	yes	—
5	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
6	#431×C₈₉	—	2	2	no	—
7	#431×C₈₉	—	2	2	no	—
8	#431×C₈₉	C₄	1	4	yes	—
9	#431×C₈₉	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₈₉	—	2	2	no	—
11	C₂×C₂²⋊C₈×C₈₉	—	2	2	no	—
12	C₄×M₁₆×C₈₉	—	2	2	no	—
13	C₄×M₁₆×C₈₉	—	2	2	no	—
14	C₂×#97×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₄	1	4	yes	—
17	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₈₉	—	2	4	no	—
21	C₂×M₁₆×C₈₉	—	2	4	no	—
22	C₂×M₁₆×C₈₉	C₈	1	8	yes	complemented
23	C₂²×C₈×C₈₉	—	2	4	no	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	D₄	1	2	yes	—
30	C₂×C₄²×C₈₉	D₄	1	2	yes	—
31	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₉	—	2	4	no	—
34	C₄⋊C₈×C₈₉	—	2	4	no	—
35	C₂²⋊C₈×C₈₉	—	2	4	no	—
36	C₂²⋊C₈×C₈₉	—	2	4	no	—
37	#97×C₈₉	—	2	4	no	—
38	C₄×C₈×C₈₉	—	2	2	no	—
39	C₄×C₈×C₈₉	—	2	2	no	—
40	C₄×C₈×C₈₉	—	2	4	no	—
41	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
42	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	—	4	2	no	—
44	C₂²×C₄×C₈₉	—	4	2	no	—
45	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
46	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	M₁₆×C₈₉	—	2	16	no	—
60	C₂×C₈×C₈₉	—	2	4	no	—
61	C₂×C₈×C₈₉	—	2	4	no	—
62	C₂×C₈×C₈₉	—	2	8	no	—
63	C₂×C₈×C₈₉	—	4	8	no	—
64	C₂×C₈×C₈₉	C₂×C₈	1	8	yes	complemented
65	C₄²×C₈₉	C₄²	1	1	yes	characteristic
66	C₄²×C₈₉	C₄²	1	1	yes	characteristic
67	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
73	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
74	C₄²×C₈₉	C₂×D₄	1	2	yes	—
75	C₂³×C₈₉	—	4	1	no	—
76	C₂³×C₈₉	—	4	1	no	—
77	C₂³×C₈₉	#95	1	2	yes	—
78	C₂³×C₈₉	C₄⋊C₈	1	4	yes	—
79	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	—	2	4	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	4	no	—
84	C₂×C₄×C₈₉	—	4	2	no	—
85	C₂×C₄×C₈₉	—	4	2	no	—
86	C₂×C₄×C₈₉	—	4	1	no	—
87	C₂×C₄×C₈₉	—	4	1	no	—
88	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#95	1	2	yes	—
91	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₄×C₈	1	4	yes	—
94	C₂×C₄×C₈₉	#97	1	4	yes	—
95	C₂×C₄×C₈₉	C₂²⋊C₈	1	8	yes	—
96	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
97	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
103	C₈×C₈₉	—	2	8	no	—
104	C₈×C₈₉	—	4	8	no	—
105	C₂²×C₈₉	—	2	2	no	—
106	C₂²×C₈₉	—	2	1	no	—
107	C₂²×C₈₉	—	2	2	no	—
108	C₂²×C₈₉	—	2	1	no	—
109	C₂²×C₈₉	—	4	4	no	—
110	C₂²×C₈₉	#336	1	4	yes	—
111	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
112	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
113	C₄×C₈₉	—	2	2	no	—
114	C₄×C₈₉	—	2	1	no	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	2	2	no	—
117	C₄×C₈₉	#336	1	4	yes	—
118	C₄×C₈₉	H(3,ℤ/4)	1	2	yes	—
119	C₄×C₈₉	#338	1	2	yes	—
120	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
121	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
122	C₄×C₈₉	C₂×C₂²⋊C₈	1	2	yes	—
123	#7860	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₈₉	—	2	4	no	—
125	C₂×C₈₉	—	2	2	no	—
126	C₂×C₈₉	C₂×#336	1	1	yes	characteristic
127	C₂×C₈₉	C₂×H(3,ℤ/4)	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#338	1	1	yes	characteristic
129	C₂×#431	C₂×C₈₉	1	2	yes	—
130	C₂×C₄×M₁₆	C₂×C₈₉	1	1	yes	characteristic
131	#1275	C₂×C₈₉	1	4	yes	complemented
132	C₈₉	#7860	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²×M₁₆	C₄×C₈₉	1	2	yes	—
134	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
135	#431	—	2	2	no	—
136	#431	—	2	2	no	—
137	#431	C₄×C₈₉	1	4	yes	—
138	#431	C₂²×C₈₉	1	4	yes	—
139	C₂×C₄⋊C₈	—	2	2	no	—
140	C₂×C₂²⋊C₈	—	2	2	no	—
141	C₄×M₁₆	—	2	2	no	—
142	C₄×M₁₆	—	2	2	no	—
143	C₂×#97	C₂²×C₈₉	1	2	yes	—
144	C₂×C₄×C₈	C₄×C₈₉	1	2	yes	—
145	C₂×C₄×C₈	C₄×C₈₉	1	4	yes	—
146	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
147	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
148	C₂³×C₄	Q₈×C₈₉	1	1	yes	characteristic
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	C₈×C₈₉	1	8	yes	complemented
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₂×C₄×C₈₉	1	4	yes	—
154	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
155	C₂×C₄²	C₂×C₄×C₈₉	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₈₉	1	2	yes	—
157	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
158	C₂×C₄²	D₄×C₈₉	1	2	yes	—
159	C₂×C₄²	D₄×C₈₉	1	2	yes	—
160	C₂×C₄²	Q₈×C₈₉	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	4	no	—
164	C₂²⋊C₈	—	2	4	no	—
165	C₂²⋊C₈	—	2	4	no	—
166	#97	—	2	4	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	2	4	no	—
170	C₄×C₈	C₂×C₄×C₈₉	1	8	yes	—
171	C₂⁴	C₄⋊C₄×C₈₉	1	1	yes	characteristic
172	C₂²×C₄	—	4	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₄²×C₈₉	1	2	yes	—
175	C₂²×C₄	C₂²⋊C₄×C₈₉	1	2	yes	—
176	C₂²×C₄	C₂²⋊C₄×C₈₉	1	2	yes	—
177	C₂²×C₄	C₂²⋊C₄×C₈₉	1	2	yes	—
178	C₂²×C₄	C₄⋊C₄×C₈₉	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊C₄×C₈₉	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₈₉	1	2	yes	—
181	C₂²×C₄	C₄⋊C₄×C₈₉	1	2	yes	—
182	C₂²×C₄	C₄⋊C₄×C₈₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₈×C₈₉	1	4	yes	—
184	C₂²×C₄	M₁₆×C₈₉	1	4	yes	—
185	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₈₉	1	1	yes	characteristic
188	M₁₆	—	2	16	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	4	8	no	—
193	C₂×C₈	C₂×C₈×C₈₉	1	8	yes	complemented
194	C₄²	C₄²×C₈₉	1	1	yes	characteristic
195	C₄²	C₄²×C₈₉	1	1	yes	characteristic
196	C₄²	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
197	C₄²	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
198	C₄²	C₂²⋊C₄×C₈₉	1	2	yes	—
199	C₄²	C₂²⋊C₄×C₈₉	1	2	yes	—
200	C₄²	C₄⋊C₄×C₈₉	1	2	yes	—
201	C₄²	C₄⋊C₄×C₈₉	1	2	yes	—
202	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
203	C₄²	C₂×D₄×C₈₉	1	2	yes	—
204	C₂³	—	4	1	no	—
205	C₂³	—	4	1	no	—
206	C₂³	#95×C₈₉	1	2	yes	—
207	C₂³	C₄⋊C₈×C₈₉	1	4	yes	—
208	C₂³	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	#95×C₈₉	1	1	yes	characteristic
218	C₂×C₄	#95×C₈₉	1	1	yes	characteristic
219	C₂×C₄	#95×C₈₉	1	2	yes	—
220	C₂×C₄	#95×C₈₉	1	1	yes	characteristic
221	C₂×C₄	#95×C₈₉	1	1	yes	characteristic
222	C₂×C₄	C₄×C₈×C₈₉	1	4	yes	—
223	C₂×C₄	#97×C₈₉	1	4	yes	—
224	C₂×C₄	C₂²⋊C₈×C₈₉	1	8	yes	—
225	C₂×C₄	C₄⋊C₈×C₈₉	1	4	yes	—
226	C₂×C₄	C₂×C₄²×C₈₉	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	2	yes	—
230	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
231	C₂×C₄	C₂×M₁₆×C₈₉	1	2	yes	—
232	C₈	—	2	8	no	—
233	C₈	—	4	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	4	no	—
239	C₂²	#336×C₈₉	1	4	yes	—
240	C₂²	C₂×#95×C₈₉	1	1	yes	characteristic
241	C₂²	C₂×C₄⋊C₈×C₈₉	1	2	yes	—
242	C₄	—	2	2	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	#336×C₈₉	1	4	yes	—
247	C₄	H(3,ℤ/4)×C₈₉	1	2	yes	—
248	C₄	#338×C₈₉	1	2	yes	—
249	C₄	C₂×C₄×C₈×C₈₉	1	1	yes	characteristic
250	C₄	C₂×#97×C₈₉	1	1	yes	characteristic
251	C₄	C₂×C₂²⋊C₈×C₈₉	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	2	4	no	—
254	C₂	C₂×#336×C₈₉	1	1	yes	characteristic
255	C₂	C₂×H(3,ℤ/4)×C₈₉	1	1	yes	characteristic
256	C₂	C₂×#338×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₈₉	C₄²
commutator	C₄	C₂×C₄×C₈×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
socle	C₂²×C₈₉	C₂×#95
prime radical	C₂⁴×C₈₉	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#1275	C₂	semidirect
22. C₂×M₁₆×C₈₉	C₈	semidirect
64. C₂×C₈×C₈₉	C₂×C₈	semidirect
123. #7860	C₈₉	direct (N × H)
131. #1275	C₂×C₈₉	semidirect
132. C₈₉	#7860	direct (N × H)
151. C₂×M₁₆	C₈×C₈₉	semidirect
193. C₂×C₈	C₂×C₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₈₉	2
C₂×C₄×M₁₆×C₈₉	2
C₈₉×#1275	2
#7860	89

Subgroups — C89×#7860 #36385708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7860 #36385708