Subgroups — C₈₉×#23659 #36401507

Order 22784.

902 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₈₉	C₂	1	2	yes	maximal
2	C₈₉×#2296	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2085	C₂	1	8	yes	maximal, complemented
5	C₈₉×#1736	C₂	1	2	yes	maximal
6	C₈₉×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
8	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	#428×C₈₉	—	2	8	no	—
13	C₂×#107×C₈₉	C₂²	1	4	yes	—
14	C₂×#106×C₈₉	C₂²	1	4	yes	—
15	#389×C₈₉	C₂²	1	4	yes	—
16	#385×C₈₉	—	2	2	no	—
17	#382×C₈₉	—	2	2	no	—
18	#382×C₈₉	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
20	C₂×#95×C₈₉	—	2	2	no	—
21	#340×C₈₉	C₂²	1	8	yes	—
22	#336×C₈₉	C₂²	1	4	yes	—
23	C₂³×C₄×C₈₉	—	2	2	no	—
24	C₂³×C₄×C₈₉	D₄	1	2	yes	—
25	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₈₉	—	2	4	no	—
27	C₂×M₁₆×C₈₉	—	2	4	no	—
28	C₂×M₁₆×C₈₉	Q₈	1	8	yes	—
29	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
30	#117×C₈₉	—	2	2	no	—
31	#117×C₈₉	—	2	4	no	—
32	#117×C₈₉	D₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
34	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
40	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
42	C₂×C₄²×C₈₉	—	2	1	no	—
43	C₂×C₄²×C₈₉	—	2	4	no	—
44	C₂×C₄²×C₈₉	—	2	1	no	—
45	C₂×C₄²×C₈₉	D₄	1	2	yes	—
46	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
47	#107×C₈₉	—	2	8	no	—
48	#106×C₈₉	—	2	8	no	—
49	#95×C₈₉	—	2	8	no	—
50	#95×C₈₉	—	2	4	no	—
51	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	—	2	4	no	—
53	C₂²×C₄×C₈₉	—	2	2	no	—
54	C₂²×C₄×C₈₉	—	2	4	no	—
55	C₂²×C₄×C₈₉	—	2	2	no	—
56	C₂²×C₄×C₈₉	—	4	4	no	—
57	C₂²×C₄×C₈₉	—	4	8	no	—
58	C₂²×C₄×C₈₉	—	4	4	no	—
59	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₈₉	#41	1	2	yes	—
64	C₂²×C₄×C₈₉	#41	1	4	yes	—
65	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₈₉	—	2	16	no	—
67	C₂×C₈×C₈₉	—	2	4	no	—
68	C₂×C₈×C₈₉	—	2	4	no	—
69	C₂×C₈×C₈₉	C₂×Q₈	1	8	yes	—
70	C₄⋊C₄×C₈₉	—	2	2	no	—
71	C₄⋊C₄×C₈₉	—	2	4	no	—
72	C₄⋊C₄×C₈₉	—	2	4	no	—
73	C₄⋊C₄×C₈₉	—	2	4	no	—
74	C₄⋊C₄×C₈₉	—	4	8	no	—
75	C₄⋊C₄×C₈₉	—	4	4	no	—
76	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
77	C₂²⋊C₄×C₈₉	—	2	4	no	—
78	C₂²⋊C₄×C₈₉	—	4	2	no	—
79	C₄²×C₈₉	—	2	4	no	—
80	C₄²×C₈₉	—	2	4	no	—
81	C₄²×C₈₉	—	2	4	no	—
82	C₄²×C₈₉	—	4	2	no	—
83	C₂³×C₈₉	—	2	4	no	—
84	C₂³×C₈₉	#121	1	2	yes	—
85	C₂³×C₈₉	#123	1	2	yes	—
86	C₂³×C₈₉	C₄⋊Q₈	1	2	yes	—
87	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₈₉	—	2	4	no	—
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	—	2	4	no	—
91	C₂×C₄×C₈₉	—	2	4	no	—
92	C₂×C₄×C₈₉	—	4	8	no	—
93	C₂×C₄×C₈₉	—	4	4	no	—
94	C₂×C₄×C₈₉	—	4	4	no	—
95	C₂×C₄×C₈₉	—	4	2	no	—
96	C₂×C₄×C₈₉	—	4	4	no	—
97	C₂×C₄×C₈₉	—	4	8	no	—
98	C₂×C₄×C₈₉	—	8	8	no	—
99	C₂×C₄×C₈₉	#121	1	2	yes	—
100	C₂×C₄×C₈₉	C₂²⋊Q₈	1	8	yes	—
101	C₂×C₄×C₈₉	#123	1	2	yes	—
102	C₂×C₄×C₈₉	#125	1	8	yes	—
103	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
104	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₈×C₈₉	—	2	8	no	—
109	C₂²×C₈₉	—	2	4	no	—
110	C₂²×C₈₉	—	2	4	no	—
111	C₂²×C₈₉	—	2	4	no	—
112	C₂²×C₈₉	#398	1	4	yes	—
113	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
114	C₂²×C₈₉	C₂×#123	1	1	yes	characteristic
115	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₈₉	#575	1	4	yes	—
117	C₄×C₈₉	—	4	4	no	—
118	C₄×C₈₉	—	4	4	no	—
119	C₄×C₈₉	—	8	2	no	—
120	C₄×C₈₉	—	8	4	no	—
121	C₄×C₈₉	#398	1	4	yes	—
122	C₄×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
123	C₄×C₈₉	C₂×#125	1	2	yes	—
124	#23659	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₈₉	—	2	4	no	—
126	C₂×C₈₉	C₂×#398	1	1	yes	characteristic
127	C₂×C₈₉	#3040	1	2	yes	—
128	C₂×C₈₉	#3093	1	2	yes	—
129	C₂×C₈₉	#3166	1	2	yes	—
130	C₂×#428	C₂×C₈₉	1	2	yes	—
131	#2296	C₂×C₈₉	1	1	yes	characteristic
132	C₂×#382	C₂×C₈₉	1	1	yes	characteristic
133	#2085	C₂×C₈₉	1	8	yes	complemented
134	#1736	C₂×C₈₉	1	2	yes	—
135	#1728	C₂×C₈₉	1	1	yes	characteristic
136	C₈₉	#23659	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
138	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
139	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
140	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
141	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
142	#428	—	2	8	no	—
143	C₂×#107	C₂²×C₈₉	1	4	yes	—
144	C₂×#106	C₂²×C₈₉	1	4	yes	—
145	#389	C₂²×C₈₉	1	4	yes	—
146	#385	—	2	2	no	—
147	#382	—	2	2	no	—
148	#382	C₂²×C₈₉	1	4	yes	—
149	C₄×C₄⋊C₄	—	2	2	no	—
150	C₂×#95	—	2	2	no	—
151	#340	C₂²×C₈₉	1	8	yes	—
152	#336	C₂²×C₈₉	1	4	yes	—
153	C₂³×C₄	—	2	2	no	—
154	C₂³×C₄	D₄×C₈₉	1	2	yes	—
155	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	Q₈×C₈₉	1	8	yes	—
159	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
160	#117	—	2	2	no	—
161	#117	—	2	4	no	—
162	#117	D₄×C₈₉	1	4	yes	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
168	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
169	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₂²⋊C₄	—	2	1	no	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	—	2	4	no	—
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	D₄×C₈₉	1	2	yes	—
176	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
177	#107	—	2	8	no	—
178	#106	—	2	8	no	—
179	#95	—	2	8	no	—
180	#95	—	2	4	no	—
181	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
190	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
192	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
193	C₂²×C₄	#41×C₈₉	1	2	yes	—
194	C₂²×C₄	#41×C₈₉	1	4	yes	—
195	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
196	M₁₆	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	C₂×Q₈×C₈₉	1	8	yes	—
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	8	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	4	2	no	—
213	C₂³	—	2	4	no	—
214	C₂³	#121×C₈₉	1	2	yes	—
215	C₂³	#123×C₈₉	1	2	yes	—
216	C₂³	C₄⋊Q₈×C₈₉	1	2	yes	—
217	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	8	8	no	—
229	C₂×C₄	#121×C₈₉	1	2	yes	—
230	C₂×C₄	C₂²⋊Q₈×C₈₉	1	8	yes	—
231	C₂×C₄	#123×C₈₉	1	2	yes	—
232	C₂×C₄	#125×C₈₉	1	8	yes	—
233	C₂×C₄	C₄⋊Q₈×C₈₉	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
235	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
236	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
237	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
238	C₈	—	2	8	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	#398×C₈₉	1	4	yes	—
243	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
244	C₂²	C₂×#123×C₈₉	1	1	yes	characteristic
245	C₂²	C₂×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
246	C₂²	#575×C₈₉	1	4	yes	—
247	C₄	—	4	4	no	—
248	C₄	—	4	4	no	—
249	C₄	—	8	2	no	—
250	C₄	—	8	4	no	—
251	C₄	#398×C₈₉	1	4	yes	—
252	C₄	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
253	C₄	C₂×#125×C₈₉	1	2	yes	—
254	C₂	—	2	4	no	—
255	C₂	C₂×#398×C₈₉	1	1	yes	characteristic
256	C₂	C₈₉×#3040	1	2	yes	—
257	C₂	C₈₉×#3093	1	2	yes	—
258	C₂	C₈₉×#3166	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈₉×#2085	C₂	semidirect
124. #23659	C₈₉	direct (N × H)
133. #2085	C₂×C₈₉	semidirect
136. C₈₉	#23659	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#428×C₈₉	2
C₈₉×#2296	2
C₂×#382×C₈₉	2
C₈₉×#2085	2
C₈₉×#1736	2
C₈₉×#1728	2
#23659	89

Subgroups — C89×#23659 #36401507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23659 #36401507