Subgroups — C₈₉×#29221 #36407069

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2100	C₂	1	2	yes	maximal
4	C₈₉×#2099	C₂	1	2	yes	maximal
5	C₈₉×#1991	C₂	1	2	yes	maximal
6	C₈₉×#1989	C₂	1	4	yes	maximal
7	C₈₉×#1835	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#125×C₈₉	C₂²	1	2	yes	—
10	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
13	#389×C₈₉	—	2	2	no	—
14	#389×C₈₉	—	2	2	no	—
15	#389×C₈₉	C₂²	1	2	yes	—
16	#387×C₈₉	—	2	2	no	—
17	#384×C₈₉	C₂²	1	1	yes	characteristic
18	#384×C₈₉	C₂²	1	1	yes	characteristic
19	#382×C₈₉	C₂²	1	1	yes	characteristic
20	#382×C₈₉	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
22	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
23	C₄³×C₈₉	C₂²	1	1	yes	characteristic
24	#340×C₈₉	C₂²	1	4	yes	—
25	#340×C₈₉	C₂²	1	4	yes	—
26	#340×C₈₉	C₂²	1	4	yes	—
27	#340×C₈₉	C₂²	1	4	yes	—
28	#336×C₈₉	C₂²	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
30	#125×C₈₉	—	2	2	no	—
31	#125×C₈₉	—	2	8	no	—
32	#125×C₈₉	Q₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
43	C₂×C₄²×C₈₉	—	2	2	no	—
44	C₂×C₄²×C₈₉	D₄	1	2	yes	—
45	C₂×C₄²×C₈₉	D₄	1	2	yes	—
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₈₉	—	2	4	no	—
50	C₄×C₈×C₈₉	—	2	4	no	—
51	#95×C₈₉	—	2	2	no	—
52	#95×C₈₉	—	2	2	no	—
53	#95×C₈₉	—	2	2	no	—
54	#95×C₈₉	—	2	4	no	—
55	C₂²×C₄×C₈₉	—	2	2	no	—
56	C₂²×C₄×C₈₉	—	2	2	no	—
57	C₂²×C₄×C₈₉	—	2	2	no	—
58	C₂²×C₄×C₈₉	—	2	2	no	—
59	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₈₉	—	2	4	no	—
65	C₂×C₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	—	2	4	no	—
67	C₄⋊C₄×C₈₉	—	2	8	no	—
68	C₄⋊C₄×C₈₉	—	2	2	no	—
69	C₄⋊C₄×C₈₉	—	2	4	no	—
70	C₄⋊C₄×C₈₉	—	2	4	no	—
71	C₄⋊C₄×C₈₉	—	2	4	no	—
72	C₄⋊C₄×C₈₉	—	4	4	no	—
73	C₄⋊C₄×C₈₉	—	4	4	no	—
74	C₄⋊C₄×C₈₉	—	4	4	no	—
75	C₄⋊C₄×C₈₉	—	4	4	no	—
76	C₄⋊C₄×C₈₉	—	4	4	no	—
77	C₄⋊C₄×C₈₉	C₂×Q₈	1	4	yes	—
78	C₄²×C₈₉	—	2	4	no	—
79	C₄²×C₈₉	—	2	2	no	—
80	C₄²×C₈₉	—	4	2	no	—
81	C₄²×C₈₉	—	4	4	no	—
82	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
83	C₄²×C₈₉	#41	1	2	yes	—
84	C₄²×C₈₉	#41	1	4	yes	—
85	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	4	no	—
93	C₂×C₄×C₈₉	—	4	2	no	—
94	C₂×C₄×C₈₉	—	4	2	no	—
95	C₂×C₄×C₈₉	—	4	2	no	—
96	C₂×C₄×C₈₉	—	4	2	no	—
97	C₂×C₄×C₈₉	—	4	2	no	—
98	C₂×C₄×C₈₉	—	4	4	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	—	4	4	no	—
102	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₈₉	#124	1	2	yes	—
104	C₂×C₄×C₈₉	#124	1	2	yes	—
105	C₂×C₄×C₈₉	#135	1	4	yes	—
106	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
109	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
110	C₈×C₈₉	—	4	4	no	—
111	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
112	C₂²×C₈₉	C₂×#124	1	1	yes	characteristic
113	C₂²×C₈₉	#536	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×#135	1	2	yes	—
115	C₂²×C₈₉	#575	1	2	yes	—
116	C₄×C₈₉	—	2	4	no	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	4	no	—
120	C₄×C₈₉	—	4	4	no	—
121	C₄×C₈₉	—	8	2	no	—
122	C₄×C₈₉	—	8	2	no	—
123	C₄×C₈₉	—	8	2	no	—
124	C₄×C₈₉	#556	1	2	yes	—
125	C₄×C₈₉	#557	1	2	yes	—
126	#29221	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₈₉	#2568	1	1	yes	characteristic
128	C₂×C₈₉	#3076	1	2	yes	—
129	C₂×C₈₉	#3131	1	1	yes	characteristic
130	C₂×C₈₉	#3142	1	1	yes	characteristic
131	C₂×C₈₉	#3232	1	2	yes	—
132	#2859	C₂×C₈₉	1	1	yes	characteristic
133	C₄×#125	C₂×C₈₉	1	1	yes	characteristic
134	#2100	C₂×C₈₉	1	2	yes	—
135	#2099	C₂×C₈₉	1	2	yes	—
136	#1991	C₂×C₈₉	1	2	yes	—
137	#1989	C₂×C₈₉	1	4	yes	—
138	#1835	C₂×C₈₉	1	2	yes	—
139	C₄×C₄⋊C₈	C₂×C₈₉	1	1	yes	characteristic
140	C₈₉	#29221	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×#125	C₂²×C₈₉	1	2	yes	—
142	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
143	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
145	#389	—	2	2	no	—
146	#389	—	2	2	no	—
147	#389	C₂²×C₈₉	1	2	yes	—
148	#387	—	2	2	no	—
149	#384	C₂²×C₈₉	1	1	yes	characteristic
150	#384	C₂²×C₈₉	1	1	yes	characteristic
151	#382	C₂²×C₈₉	1	1	yes	characteristic
152	#382	C₂²×C₈₉	1	2	yes	—
153	C₄×C₄⋊C₄	—	2	2	no	—
154	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
155	C₄³	C₂²×C₈₉	1	1	yes	characteristic
156	#340	C₂²×C₈₉	1	4	yes	—
157	#340	C₂²×C₈₉	1	4	yes	—
158	#340	C₂²×C₈₉	1	4	yes	—
159	#340	C₂²×C₈₉	1	4	yes	—
160	#336	C₂²×C₈₉	1	2	yes	—
161	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
162	#125	—	2	2	no	—
163	#125	—	2	8	no	—
164	#125	Q₈×C₈₉	1	4	yes	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
172	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
173	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
175	C₂×C₄²	—	2	2	no	—
176	C₂×C₄²	D₄×C₈₉	1	2	yes	—
177	C₂×C₄²	D₄×C₈₉	1	2	yes	—
178	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
180	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
181	C₄⋊C₈	—	2	4	no	—
182	C₄×C₈	—	2	4	no	—
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	#95	—	2	2	no	—
186	#95	—	2	4	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₄⋊C₄	—	2	8	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₄⋊C₄	C₂×Q₈×C₈₉	1	4	yes	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
215	C₄²	#41×C₈₉	1	2	yes	—
216	C₄²	#41×C₈₉	1	4	yes	—
217	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
235	C₂×C₄	#124×C₈₉	1	2	yes	—
236	C₂×C₄	#124×C₈₉	1	2	yes	—
237	C₂×C₄	#135×C₈₉	1	4	yes	—
238	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
240	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
241	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
242	C₈	—	4	4	no	—
243	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
244	C₂²	C₂×#124×C₈₉	1	1	yes	characteristic
245	C₂²	#536×C₈₉	1	1	yes	characteristic
246	C₂²	C₂×#135×C₈₉	1	2	yes	—
247	C₂²	#575×C₈₉	1	2	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	4	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	8	2	no	—
256	C₄	#556×C₈₉	1	2	yes	—
257	C₄	#557×C₈₉	1	2	yes	—
258	C₂	C₈₉×#2568	1	1	yes	characteristic
259	C₂	C₈₉×#3076	1	2	yes	—
260	C₂	C₈₉×#3131	1	1	yes	characteristic
261	C₂	C₈₉×#3142	1	1	yes	characteristic
262	C₂	C₈₉×#3232	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
126. #29221	C₈₉	direct (N × H)
140. C₈₉	#29221	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×#2859	2
C₄×#125×C₈₉	2
C₈₉×#2100	2
C₈₉×#2099	2
C₈₉×#1991	2
C₈₉×#1989	2
C₈₉×#1835	2
C₄×C₄⋊C₈×C₈₉	2
#29221	89

Subgroups — C89×#29221 #36407069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29221 #36407069