Subgroups — C₈₉×#13278 #36391126

Order 22784.

358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3160	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×#106×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1386	C₂	1	4	yes	maximal
5	#533×C₈₉	—	2	1	no	—
6	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
7	#500×C₈₉	C₄	1	1	yes	characteristic, complemented
8	#500×C₈₉	C₄	1	1	yes	characteristic, complemented
9	#499×C₈₉	—	2	1	no	—
10	#499×C₈₉	C₄	1	2	yes	complemented
11	#498×C₈₉	—	2	1	no	—
12	#427×C₈₉	—	2	1	no	—
13	#427×C₈₉	C₂²	1	2	yes	—
14	C₂×#106×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
17	C₄⋊C₁₆×C₈₉	—	2	1	no	—
18	C₄⋊C₁₆×C₈₉	—	2	1	no	—
19	#340×C₈₉	—	2	1	no	—
20	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈×C₈₉	—	2	1	no	—
23	C₄⋊Q₈×C₈₉	—	2	1	no	—
24	#125×C₈₉	—	2	2	no	—
25	C₂²⋊Q₈×C₈₉	—	4	1	no	—
26	C₂²⋊Q₈×C₈₉	—	4	1	no	—
27	C₄×Q₈×C₈₉	—	4	1	no	—
28	C₄×Q₈×C₈₉	—	4	1	no	—
29	#117×C₈₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₄²×C₈₉	—	4	1	no	—
32	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
33	M₃₂×C₈₉	—	2	4	no	—
34	C₂×C₁₆×C₈₉	—	2	2	no	—
35	#107×C₈₉	—	2	1	no	—
36	#107×C₈₉	—	2	1	no	—
37	#107×C₈₉	C₂×C₄	1	2	yes	—
38	#107×C₈₉	C₂×C₄	1	2	yes	—
39	#106×C₈₉	—	2	2	no	—
40	#106×C₈₉	—	2	2	no	—
41	#106×C₈₉	—	2	2	no	—
42	#106×C₈₉	Q₈	1	4	yes	complemented
43	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
47	#95×C₈₉	—	4	1	no	—
48	C₂×Q₈×C₈₉	—	4	1	no	—
49	C₂×Q₈×C₈₉	—	4	1	no	—
50	C₂²×C₄×C₈₉	—	4	2	no	—
51	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₈₉	#41	1	2	yes	—
61	C₄⋊C₄×C₈₉	—	2	2	no	—
62	C₄⋊C₄×C₈₉	—	2	2	no	—
63	C₄⋊C₄×C₈₉	—	2	4	no	—
64	C₄⋊C₄×C₈₉	—	4	2	no	—
65	C₄⋊C₄×C₈₉	—	4	2	no	—
66	C₄⋊C₄×C₈₉	—	4	1	no	—
67	C₄⋊C₄×C₈₉	—	4	1	no	—
68	C₄⋊C₄×C₈₉	—	4	2	no	—
69	C₂²⋊C₄×C₈₉	—	4	2	no	—
70	C₄²×C₈₉	—	4	2	no	—
71	C₄²×C₈₉	—	8	2	no	—
72	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
75	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
76	C₁₆×C₈₉	—	4	2	no	—
77	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₈₉	—	8	2	no	—
79	Q₈×C₈₉	—	8	2	no	—
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	4	1	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	—	4	2	no	—
84	C₂×C₄×C₈₉	—	4	4	no	—
85	C₂×C₄×C₈₉	—	8	2	no	—
86	C₂×C₄×C₈₉	—	8	2	no	—
87	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₈₉	#137	1	2	yes	—
95	C₈×C₈₉	—	2	1	no	—
96	C₈×C₈₉	—	2	1	no	—
97	C₈×C₈₉	C₄×Q₈	1	2	yes	—
98	C₈×C₈₉	#124	1	1	yes	characteristic
99	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
100	C₂²×C₈₉	—	2	1	no	—
101	C₂²×C₈₉	—	2	1	no	—
102	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
103	C₂²×C₈₉	#474	1	2	yes	—
104	C₄×C₈₉	—	2	1	no	—
105	C₄×C₈₉	—	2	1	no	—
106	C₄×C₈₉	—	2	1	no	—
107	C₄×C₈₉	—	8	4	no	—
108	C₄×C₈₉	—	8	2	no	—
109	C₄×C₈₉	—	8	2	no	—
110	C₄×C₈₉	#360	1	2	yes	—
111	C₄×C₈₉	#391	1	1	yes	characteristic
112	C₄×C₈₉	#419	1	1	yes	characteristic
113	C₄×C₈₉	#475	1	2	yes	—
114	#13278	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₈₉	—	2	2	no	—
116	C₂×C₈₉	#1988	1	1	yes	characteristic
117	C₂×C₈₉	C₂×#360	1	1	yes	characteristic
118	C₂×C₈₉	#2145	1	1	yes	characteristic
119	#3160	C₂×C₈₉	1	1	yes	characteristic
120	#2150	C₂×C₈₉	1	1	yes	characteristic
121	C₄×#106	C₂×C₈₉	1	1	yes	characteristic
122	#1386	C₂×C₈₉	1	4	yes	—
123	C₈₉	#13278	1	1	yes	characteristic, direct product factor, Sylow
124	#533	—	2	1	no	—
125	C₂×M₃₂	C₂²×C₈₉	1	2	yes	—
126	#500	C₄×C₈₉	1	1	yes	characteristic, complemented
127	#500	C₄×C₈₉	1	1	yes	characteristic, complemented
128	#499	—	2	1	no	—
129	#499	C₄×C₈₉	1	2	yes	complemented
130	#498	—	2	1	no	—
131	#427	—	2	1	no	—
132	#427	C₂²×C₈₉	1	2	yes	—
133	C₂×#106	C₂²×C₈₉	1	2	yes	—
134	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₁₆	—	2	1	no	—
137	C₄⋊C₁₆	—	2	1	no	—
138	#340	—	2	1	no	—
139	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	1	no	—
142	C₄⋊Q₈	—	2	1	no	—
143	#125	—	2	2	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₂²⋊Q₈	—	4	1	no	—
146	C₄×Q₈	—	4	1	no	—
147	C₄×Q₈	—	4	1	no	—
148	#117	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄²	—	4	1	no	—
151	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
152	M₃₂	—	2	4	no	—
153	C₂×C₁₆	—	2	2	no	—
154	#107	—	2	1	no	—
155	#107	—	2	1	no	—
156	#107	C₂×C₄×C₈₉	1	2	yes	—
157	#107	C₂×C₄×C₈₉	1	2	yes	—
158	#106	—	2	2	no	—
159	#106	—	2	2	no	—
160	#106	—	2	2	no	—
161	#106	Q₈×C₈₉	1	4	yes	complemented
162	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
163	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
164	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
165	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
166	#95	—	4	1	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂×Q₈	—	4	1	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	D₈×C₈₉	1	1	yes	characteristic
171	C₂²×C₄	D₈×C₈₉	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
178	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
179	C₂×C₈	#41×C₈₉	1	2	yes	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	—	8	2	no	—
191	C₄²	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
192	C₄²	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
193	C₄²	SD₁₆×C₈₉	1	1	yes	characteristic
194	C₄²	SD₁₆×C₈₉	1	1	yes	characteristic
195	C₁₆	—	4	2	no	—
196	C₂³	C₂×D₈×C₈₉	1	1	yes	characteristic
197	Q₈	—	8	2	no	—
198	Q₈	—	8	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	#102×C₈₉	1	1	yes	characteristic
207	C₂×C₄	#102×C₈₉	1	1	yes	characteristic
208	C₂×C₄	#102×C₈₉	1	1	yes	characteristic
209	C₂×C₄	#102×C₈₉	1	1	yes	characteristic
210	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
211	C₂×C₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
212	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
213	C₂×C₄	#137×C₈₉	1	2	yes	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	C₄×Q₈×C₈₉	1	2	yes	—
217	C₈	#124×C₈₉	1	1	yes	characteristic
218	C₈	C₄⋊Q₈×C₈₉	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×#102×C₈₉	1	1	yes	characteristic
222	C₂²	#474×C₈₉	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	4	no	—
227	C₄	—	8	2	no	—
228	C₄	—	8	2	no	—
229	C₄	#360×C₈₉	1	2	yes	—
230	C₄	#391×C₈₉	1	1	yes	characteristic
231	C₄	#419×C₈₉	1	1	yes	characteristic
232	C₄	#475×C₈₉	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	C₈₉×#1988	1	1	yes	characteristic
235	C₂	C₂×#360×C₈₉	1	1	yes	characteristic
236	C₂	C₈₉×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
prime radical	C₂³×C₈₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
7. #500×C₈₉	C₄	semidirect
8. #500×C₈₉	C₄	semidirect
10. #499×C₈₉	C₄	semidirect
42. #106×C₈₉	Q₈	semidirect
114. #13278	C₈₉	direct (N × H)
123. C₈₉	#13278	direct (N × H)
126. #500	C₄×C₈₉	semidirect
127. #500	C₄×C₈₉	semidirect
129. #499	C₄×C₈₉	semidirect
161. #106	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3160	2
C₈₉×#2150	2
C₄×#106×C₈₉	2
C₈₉×#1386	2
#13278	89

Subgroups — C89×#13278 #36391126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#13278 #36391126