Subgroups — C₈₉×#22816 #36400664

Order 22784.

850 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2357	C₂	1	2	yes	maximal, complemented
3	C₈₉×#1972	C₂	1	2	yes	maximal
4	C₈₉×#1925	C₂	1	4	yes	maximal, complemented
5	C₈₉×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1804	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1776	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
11	C₄×D₈×C₈₉	C₄	1	8	yes	complemented
12	C₂×#107×C₈₉	C₂²	1	2	yes	—
13	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#106×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#102×C₈₉	C₂²	1	4	yes	—
17	C₂×#102×C₈₉	C₂²	1	2	yes	—
18	C₂×#97×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#390×C₈₉	C₂²	1	4	yes	—
21	#388×C₈₉	—	2	4	no	—
22	#382×C₈₉	C₂²	1	2	yes	—
23	#381×C₈₉	—	2	2	no	—
24	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
25	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊C₄×C₈₉	—	2	2	no	—
27	#340×C₈₉	C₂²	1	4	yes	—
28	#336×C₈₉	C₂²	1	2	yes	—
29	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
30	C₂³×C₄×C₈₉	—	2	2	no	—
31	C₂×D₈×C₈₉	—	2	2	no	—
32	C₂×D₈×C₈₉	—	2	2	no	—
33	C₂×D₈×C₈₉	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₄×D₄×C₈₉	—	2	4	no	—
38	C₄×D₄×C₈₉	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
45	C₂×C₄²×C₈₉	—	2	4	no	—
46	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	#107×C₈₉	—	4	2	no	—
49	#107×C₈₉	C₂×C₄	1	4	yes	—
50	#106×C₈₉	—	2	4	no	—
51	C₄⋊C₈×C₈₉	—	2	2	no	—
52	#102×C₈₉	—	4	4	no	—
53	#102×C₈₉	C₂×C₄	1	8	yes	—
54	#97×C₈₉	—	2	4	no	—
55	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
56	#95×C₈₉	—	2	2	no	—
57	#95×C₈₉	—	2	4	no	—
58	#95×C₈₉	—	2	2	no	—
59	#95×C₈₉	D₄	1	2	yes	—
60	C₂⁴×C₈₉	—	2	2	no	—
61	C₂×D₄×C₈₉	—	2	2	no	—
62	C₂×D₄×C₈₉	—	2	2	no	—
63	C₂×D₄×C₈₉	—	2	2	no	—
64	C₂×D₄×C₈₉	—	2	2	no	—
65	C₂×D₄×C₈₉	—	2	2	no	—
66	C₂×D₄×C₈₉	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	—	2	8	no	—
69	C₂²×C₄×C₈₉	—	2	4	no	—
70	C₂²×C₄×C₈₉	—	8	4	no	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
75	D₈×C₈₉	—	2	8	no	—
76	C₂×C₈×C₈₉	—	2	1	no	—
77	C₂×C₈×C₈₉	—	2	1	no	—
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₈₉	#41	1	2	yes	—
84	C₂×C₈×C₈₉	#41	1	2	yes	—
85	C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₄⋊C₄×C₈₉	—	2	4	no	—
87	C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₄⋊C₄×C₈₉	—	4	2	no	—
89	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
90	C₂²⋊C₄×C₈₉	—	2	8	no	—
91	C₂²⋊C₄×C₈₉	—	4	8	no	—
92	C₂²⋊C₄×C₈₉	—	8	4	no	—
93	C₄²×C₈₉	—	2	2	no	—
94	C₄²×C₈₉	—	2	2	no	—
95	C₄²×C₈₉	—	8	4	no	—
96	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
97	C₄²×C₈₉	#41	1	2	yes	—
98	C₂³×C₈₉	—	2	4	no	—
99	C₂³×C₈₉	—	4	2	no	—
100	C₂³×C₈₉	—	4	2	no	—
101	C₂³×C₈₉	—	4	4	no	—
102	C₂³×C₈₉	—	4	4	no	—
103	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₈₉	—	2	4	no	—
105	D₄×C₈₉	—	2	4	no	—
106	D₄×C₈₉	—	4	2	no	—
107	D₄×C₈₉	—	4	2	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	4	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	2	4	no	—
114	C₂×C₄×C₈₉	—	4	4	no	—
115	C₂×C₄×C₈₉	—	4	4	no	—
116	C₂×C₄×C₈₉	—	4	4	no	—
117	C₂×C₄×C₈₉	—	4	4	no	—
118	C₂×C₄×C₈₉	—	8	8	no	—
119	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
127	C₈×C₈₉	—	2	2	no	—
128	C₈×C₈₉	—	4	2	no	—
129	C₈×C₈₉	#117	1	4	yes	—
130	C₂²×C₈₉	—	4	4	no	—
131	C₂²×C₈₉	—	4	4	no	—
132	C₂²×C₈₉	—	4	4	no	—
133	C₂²×C₈₉	—	8	4	no	—
134	C₂²×C₈₉	—	8	4	no	—
135	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₈₉	#547	1	1	yes	characteristic
137	C₂²×C₈₉	#550	1	1	yes	characteristic
138	C₂²×C₈₉	#575	1	2	yes	—
139	C₂²×C₈₉	#576	1	2	yes	—
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	—	2	2	no	—
142	C₄×C₈₉	—	4	4	no	—
143	C₄×C₈₉	—	4	4	no	—
144	C₄×C₈₉	—	8	4	no	—
145	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
146	C₄×C₈₉	#520	1	1	yes	characteristic
147	C₄×C₈₉	#555	1	1	yes	characteristic
148	C₄×C₈₉	#556	1	1	yes	characteristic
149	#22816	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂×C₈₉	—	8	4	no	—
151	C₂×C₈₉	#2354	1	1	yes	characteristic
152	C₂×C₈₉	#2943	1	1	yes	characteristic
153	C₂×C₈₉	#2948	1	1	yes	characteristic
154	C₂×C₈₉	#3355	1	2	yes	—
155	C₂×C₈₉	#3403	1	2	yes	—
156	C₂×C₄×D₈	C₂×C₈₉	1	1	yes	characteristic
157	#2357	C₂×C₈₉	1	2	yes	complemented
158	#1972	C₂×C₈₉	1	2	yes	—
159	#1925	C₂×C₈₉	1	4	yes	complemented
160	#1846	C₂×C₈₉	1	1	yes	characteristic, complemented
161	#1804	C₂×C₈₉	1	2	yes	complemented
162	#1776	C₂×C₈₉	1	2	yes	complemented
163	#1770	C₂×C₈₉	1	1	yes	characteristic, complemented
164	C₈₉	#22816	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
166	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
167	C₄×D₈	C₄×C₈₉	1	8	yes	complemented
168	C₂×#107	C₂²×C₈₉	1	2	yes	—
169	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
170	C₂×#106	C₂²×C₈₉	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
172	C₂×#102	C₂²×C₈₉	1	4	yes	—
173	C₂×#102	C₂²×C₈₉	1	2	yes	—
174	C₂×#97	C₂²×C₈₉	1	2	yes	—
175	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
176	#390	C₂²×C₈₉	1	4	yes	—
177	#388	—	2	4	no	—
178	#382	C₂²×C₈₉	1	2	yes	—
179	#381	—	2	2	no	—
180	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
181	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
182	C₄×C₂²⋊C₄	—	2	2	no	—
183	#340	C₂²×C₈₉	1	4	yes	—
184	#336	C₂²×C₈₉	1	2	yes	—
185	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
186	C₂³×C₄	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	C₂×C₄×C₈₉	1	4	yes	—
190	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
191	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
193	C₄×D₄	—	2	4	no	—
194	C₄×D₄	C₂×C₄×C₈₉	1	8	yes	—
195	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
197	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	2	4	no	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
203	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
204	#107	—	4	2	no	—
205	#107	C₂×C₄×C₈₉	1	4	yes	—
206	#106	—	2	4	no	—
207	C₄⋊C₈	—	2	2	no	—
208	#102	—	4	4	no	—
209	#102	C₂×C₄×C₈₉	1	8	yes	—
210	#97	—	2	4	no	—
211	C₄×C₈	C₂×C₄×C₈₉	1	4	yes	—
212	#95	—	2	2	no	—
213	#95	—	2	4	no	—
214	#95	—	2	2	no	—
215	#95	D₄×C₈₉	1	2	yes	—
216	C₂⁴	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	C₂²×C₄×C₈₉	1	4	yes	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	8	4	no	—
227	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
228	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
231	D₈	—	2	8	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂²×C₄×C₈₉	1	2	yes	—
238	C₂×C₈	C₂²×C₄×C₈₉	1	2	yes	—
239	C₂×C₈	#41×C₈₉	1	2	yes	—
240	C₂×C₈	#41×C₈₉	1	2	yes	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	C₂²×C₄×C₈₉	1	4	yes	—
246	C₂²⋊C₄	—	2	8	no	—
247	C₂²⋊C₄	—	4	8	no	—
248	C₂²⋊C₄	—	8	4	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	8	4	no	—
252	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
253	C₄²	#41×C₈₉	1	2	yes	—
254	C₂³	—	2	4	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	4	no	—
258	C₂³	—	4	4	no	—
259	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
260	D₄	—	2	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	8	no	—
275	C₂×C₄	C₄×D₄×C₈₉	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₈₉	1	2	yes	—
277	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
281	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
282	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	#117×C₈₉	1	4	yes	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	4	no	—
291	C₂²	C₂×C₄×D₄×C₈₉	1	1	yes	characteristic
292	C₂²	#547×C₈₉	1	1	yes	characteristic
293	C₂²	#550×C₈₉	1	1	yes	characteristic
294	C₂²	#575×C₈₉	1	2	yes	—
295	C₂²	#576×C₈₉	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	4	4	no	—
300	C₄	—	8	4	no	—
301	C₄	C₂×#117×C₈₉	1	1	yes	characteristic
302	C₄	#520×C₈₉	1	1	yes	characteristic
303	C₄	#555×C₈₉	1	1	yes	characteristic
304	C₄	#556×C₈₉	1	1	yes	characteristic
305	C₂	—	8	4	no	—
306	C₂	C₈₉×#2354	1	1	yes	characteristic
307	C₂	C₈₉×#2943	1	1	yes	characteristic
308	C₂	C₈₉×#2948	1	1	yes	characteristic
309	C₂	C₈₉×#3355	1	2	yes	—
310	C₂	C₈₉×#3403	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂²×D₈×C₈₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄×C₄⋊C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2357	C₂	semidirect
4. C₈₉×#1925	C₂	semidirect
5. C₈₉×#1846	C₂	semidirect
6. C₈₉×#1804	C₂	semidirect
7. C₈₉×#1776	C₂	semidirect
8. C₈₉×#1770	C₂	semidirect
11. C₄×D₈×C₈₉	C₄	semidirect
149. #22816	C₈₉	direct (N × H)
157. #2357	C₂×C₈₉	semidirect
159. #1925	C₂×C₈₉	semidirect
160. #1846	C₂×C₈₉	semidirect
161. #1804	C₂×C₈₉	semidirect
162. #1776	C₂×C₈₉	semidirect
163. #1770	C₂×C₈₉	semidirect
164. C₈₉	#22816	direct (N × H)
167. C₄×D₈	C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₈₉	2
C₈₉×#2357	2
C₈₉×#1972	2
C₈₉×#1925	2
C₈₉×#1846	2
C₈₉×#1804	2
C₈₉×#1776	2
C₈₉×#1770	2
#22816	89

Subgroups — C89×#22816 #36400664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#22816 #36400664