Subgroups — C₈₉×#35554 #36413402

Order 22784.

3310 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2445	C₂	1	16	yes	maximal, complemented
4	C₈₉×#2405	C₂	1	8	yes	maximal, complemented
5	C₈₉×#2309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2309	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2296	C₂	1	2	yes	maximal, complemented
8	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₈₉	C₂²	1	4	yes	—
11	C₂×#127×C₈₉	—	2	8	no	—
12	C₂×#127×C₈₉	C₂²	1	8	yes	—
13	C₂×#123×C₈₉	C₂²	1	16	yes	complemented
14	#518×C₈₉	—	2	8	no	—
15	C₂×C₄×D₄×C₈₉	C₂²	1	8	yes	complemented
16	C₂×#117×C₈₉	C₂²	1	2	yes	—
17	C₂×#117×C₈₉	C₂²	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
19	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#394×C₈₉	C₂²	1	32	yes	complemented
21	#390×C₈₉	C₂²	1	8	yes	complemented
22	#390×C₈₉	C₂²	1	8	yes	complemented
23	#390×C₈₉	C₂²	1	8	yes	complemented
24	#389×C₈₉	C₂²	1	8	yes	complemented
25	#386×C₈₉	C₂²	1	16	yes	complemented
26	#385×C₈₉	C₂²	1	8	yes	—
27	#379×C₈₉	C₂²	1	4	yes	—
28	#379×C₈₉	C₂²	1	8	yes	—
29	C₄×C₄⋊C₄×C₈₉	C₂²	1	8	yes	complemented
30	C₂⁵×C₈₉	C₂³	1	2	yes	—
31	C₂⁵×C₈₉	C₂³	1	2	yes	—
32	C₂²×D₄×C₈₉	—	2	16	no	—
33	C₂²×D₄×C₈₉	—	2	4	no	—
34	C₂²×D₄×C₈₉	—	2	8	no	—
35	C₂²×D₄×C₈₉	—	2	8	no	—
36	C₂²×D₄×C₈₉	—	4	16	no	—
37	C₂²×D₄×C₈₉	D₄	1	8	yes	—
38	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
39	C₂²×D₄×C₈₉	C₂³	1	16	yes	—
40	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
41	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
42	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
43	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	#127×C₈₉	—	2	32	no	—
45	#123×C₈₉	—	4	32	no	—
46	C₄×D₄×C₈₉	—	2	32	no	—
47	#117×C₈₉	—	2	8	no	—
48	#117×C₈₉	—	2	8	no	—
49	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
50	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₈₉	—	2	16	no	—
53	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
54	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
55	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
56	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
57	C₂×C₂²⋊C₄×C₈₉	C₂³	1	32	yes	—
58	C₂×C₄²×C₈₉	—	2	4	no	—
59	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
60	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
61	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
62	#95×C₈₉	C₂³	1	16	yes	—
63	C₂⁴×C₈₉	—	2	8	no	—
64	C₂⁴×C₈₉	—	2	4	no	—
65	C₂⁴×C₈₉	—	2	8	no	—
66	C₂⁴×C₈₉	—	4	8	no	—
67	C₂⁴×C₈₉	—	4	8	no	—
68	C₂⁴×C₈₉	C₂×D₄	1	8	yes	—
69	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
70	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
71	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₂×D₄×C₈₉	—	2	32	no	—
73	C₂×D₄×C₈₉	—	2	32	no	—
74	C₂×D₄×C₈₉	—	2	8	no	—
75	C₂×D₄×C₈₉	—	2	8	no	—
76	C₂×D₄×C₈₉	—	2	8	no	—
77	C₂×D₄×C₈₉	—	2	8	no	—
78	C₂×D₄×C₈₉	—	2	8	no	—
79	C₂×D₄×C₈₉	—	2	32	no	—
80	C₂×D₄×C₈₉	—	4	16	no	—
81	C₂×D₄×C₈₉	—	4	64	no	—
82	C₂×D₄×C₈₉	—	4	16	no	—
83	C₂×D₄×C₈₉	C₂×D₄	1	16	yes	—
84	C₂²×C₄×C₈₉	—	2	4	no	—
85	C₂²×C₄×C₈₉	—	2	8	no	—
86	C₂²×C₄×C₈₉	—	2	16	no	—
87	C₂²×C₄×C₈₉	—	4	16	no	—
88	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₈₉	#41	1	8	yes	—
90	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
91	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
92	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
93	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
94	C₄⋊C₄×C₈₉	—	2	8	no	—
95	C₄⋊C₄×C₈₉	—	2	8	no	—
96	C₄⋊C₄×C₈₉	—	2	32	no	—
97	C₂²⋊C₄×C₈₉	—	2	8	no	—
98	C₂²⋊C₄×C₈₉	—	2	16	no	—
99	C₂²⋊C₄×C₈₉	—	2	8	no	—
100	C₂²⋊C₄×C₈₉	—	4	16	no	—
101	C₂²⋊C₄×C₈₉	—	4	32	no	—
102	C₄²×C₈₉	—	2	8	no	—
103	C₄²×C₈₉	—	2	8	no	—
104	C₄²×C₈₉	—	2	8	no	—
105	C₂³×C₈₉	—	2	4	no	—
106	C₂³×C₈₉	—	2	8	no	—
107	C₂³×C₈₉	—	2	16	no	—
108	C₂³×C₈₉	—	2	16	no	—
109	C₂³×C₈₉	—	2	4	no	—
110	C₂³×C₈₉	—	2	8	no	—
111	C₂³×C₈₉	—	2	16	no	—
112	C₂³×C₈₉	—	4	16	no	—
113	C₂³×C₈₉	—	4	16	no	—
114	C₂³×C₈₉	—	4	8	no	—
115	C₂³×C₈₉	—	8	16	no	—
116	C₂³×C₈₉	—	8	16	no	—
117	C₂³×C₈₉	C₂²×D₄	1	8	yes	—
118	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
119	C₂³×C₈₉	C₂×#41	1	4	yes	—
120	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
121	D₄×C₈₉	—	2	32	no	—
122	D₄×C₈₉	—	4	64	no	—
123	D₄×C₈₉	—	4	16	no	—
124	C₂×C₄×C₈₉	—	2	16	no	—
125	C₂×C₄×C₈₉	—	2	4	no	—
126	C₂×C₄×C₈₉	—	2	4	no	—
127	C₂×C₄×C₈₉	—	2	8	no	—
128	C₂×C₄×C₈₉	—	2	8	no	—
129	C₂×C₄×C₈₉	—	2	16	no	—
130	C₂×C₄×C₈₉	—	2	16	no	—
131	C₂×C₄×C₈₉	—	8	16	no	—
132	C₂×C₄×C₈₉	#123	1	16	yes	complemented
133	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
134	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
135	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
136	C₂×C₄×C₈₉	H(5,𝔽₂)	1	8	yes	—
137	C₂²×C₈₉	—	2	4	no	—
138	C₂²×C₈₉	—	2	8	no	—
139	C₂²×C₈₉	—	2	8	no	—
140	C₂²×C₈₉	—	2	16	no	—
141	C₂²×C₈₉	—	4	16	no	—
142	C₂²×C₈₉	—	4	4	no	—
143	C₂²×C₈₉	—	4	8	no	—
144	C₂²×C₈₉	—	4	16	no	—
145	C₂²×C₈₉	—	8	16	no	—
146	C₂²×C₈₉	—	8	16	no	—
147	C₂²×C₈₉	C₂×#123	1	4	yes	—
148	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
149	C₂²×C₈₉	C₂²×#41	1	2	yes	—
150	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	4	yes	—
151	C₄×C₈₉	—	2	8	no	—
152	C₄×C₈₉	—	4	8	no	—
153	C₄×C₈₉	—	4	8	no	—
154	C₄×C₈₉	C₂×#123	1	8	yes	—
155	#35554	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₈₉	—	2	4	no	—
157	C₂×C₈₉	—	4	8	no	—
158	C₂×C₈₉	—	8	4	no	—
159	C₂×C₈₉	C₂²×#123	1	1	yes	characteristic
160	C₂×C₈₉	#3484	1	2	yes	—
161	C₂×C₈₉	#3508	1	4	yes	—
162	C₂²×#127	C₂×C₈₉	1	1	yes	characteristic
163	C₂×#518	C₂×C₈₉	1	1	yes	characteristic, complemented
164	#2445	C₂×C₈₉	1	16	yes	complemented
165	#2405	C₂×C₈₉	1	8	yes	complemented
166	#2309	C₂×C₈₉	1	1	yes	characteristic, complemented
167	#2309	C₂×C₈₉	1	2	yes	complemented
168	#2296	C₂×C₈₉	1	2	yes	complemented
169	C₈₉	#35554	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
171	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
172	C₂³×D₄	C₂²×C₈₉	1	4	yes	—
173	C₂×#127	—	2	8	no	—
174	C₂×#127	C₂²×C₈₉	1	8	yes	—
175	C₂×#123	C₂²×C₈₉	1	16	yes	complemented
176	#518	—	2	8	no	—
177	C₂×C₄×D₄	C₂²×C₈₉	1	8	yes	complemented
178	C₂×#117	C₂²×C₈₉	1	2	yes	—
179	C₂×#117	C₂²×C₈₉	1	2	yes	—
180	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
181	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
182	#394	C₂²×C₈₉	1	32	yes	complemented
183	#390	C₂²×C₈₉	1	8	yes	complemented
184	#390	C₂²×C₈₉	1	8	yes	complemented
185	#390	C₂²×C₈₉	1	8	yes	complemented
186	#389	C₂²×C₈₉	1	8	yes	complemented
187	#386	C₂²×C₈₉	1	16	yes	complemented
188	#385	C₂²×C₈₉	1	8	yes	—
189	#379	C₂²×C₈₉	1	8	yes	—
190	#379	C₂²×C₈₉	1	4	yes	—
191	C₄×C₄⋊C₄	C₂²×C₈₉	1	8	yes	complemented
192	C₂⁵	C₂³×C₈₉	1	2	yes	—
193	C₂⁵	C₂³×C₈₉	1	2	yes	—
194	C₂²×D₄	—	2	16	no	—
195	C₂²×D₄	—	2	4	no	—
196	C₂²×D₄	—	2	8	no	—
197	C₂²×D₄	—	2	8	no	—
198	C₂²×D₄	—	4	16	no	—
199	C₂²×D₄	D₄×C₈₉	1	8	yes	—
200	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
201	C₂²×D₄	C₂³×C₈₉	1	16	yes	—
202	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
203	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
204	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
205	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
206	#127	—	2	32	no	—
207	#123	—	4	32	no	—
208	C₄×D₄	—	2	32	no	—
209	#117	—	2	8	no	—
210	#117	—	2	8	no	—
211	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
212	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
213	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
214	C₂×C₂²⋊C₄	—	2	16	no	—
215	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
216	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
217	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
218	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
219	C₂×C₂²⋊C₄	C₂³×C₈₉	1	32	yes	—
220	C₂×C₄²	—	2	4	no	—
221	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
222	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
223	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
224	#95	C₂³×C₈₉	1	16	yes	—
225	C₂⁴	—	2	8	no	—
226	C₂⁴	—	2	4	no	—
227	C₂⁴	—	2	8	no	—
228	C₂⁴	—	4	8	no	—
229	C₂⁴	—	4	8	no	—
230	C₂⁴	C₂×D₄×C₈₉	1	8	yes	—
231	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
232	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
233	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
234	C₂×D₄	—	2	32	no	—
235	C₂×D₄	—	2	32	no	—
236	C₂×D₄	—	2	8	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	2	8	no	—
239	C₂×D₄	—	2	8	no	—
240	C₂×D₄	—	2	8	no	—
241	C₂×D₄	—	2	32	no	—
242	C₂×D₄	—	4	16	no	—
243	C₂×D₄	—	4	64	no	—
244	C₂×D₄	—	4	16	no	—
245	C₂×D₄	C₂×D₄×C₈₉	1	16	yes	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	2	16	no	—
249	C₂²×C₄	—	4	16	no	—
250	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
251	C₂²×C₄	#41×C₈₉	1	8	yes	—
252	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
253	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
254	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
255	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	32	no	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	2	16	no	—
261	C₂²⋊C₄	—	2	8	no	—
262	C₂²⋊C₄	—	4	16	no	—
263	C₂²⋊C₄	—	4	32	no	—
264	C₄²	—	2	8	no	—
265	C₄²	—	2	8	no	—
266	C₄²	—	2	8	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	2	8	no	—
269	C₂³	—	2	16	no	—
270	C₂³	—	2	16	no	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	8	no	—
273	C₂³	—	2	16	no	—
274	C₂³	—	4	16	no	—
275	C₂³	—	4	16	no	—
276	C₂³	—	4	8	no	—
277	C₂³	—	8	16	no	—
278	C₂³	—	8	16	no	—
279	C₂³	C₂²×D₄×C₈₉	1	8	yes	—
280	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
281	C₂³	C₂×#41×C₈₉	1	4	yes	—
282	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
283	D₄	—	2	32	no	—
284	D₄	—	4	64	no	—
285	D₄	—	4	16	no	—
286	C₂×C₄	—	2	16	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	2	16	no	—
292	C₂×C₄	—	2	16	no	—
293	C₂×C₄	—	8	16	no	—
294	C₂×C₄	#123×C₈₉	1	16	yes	complemented
295	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
296	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
297	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
298	C₂×C₄	H(5,𝔽₂)×C₈₉	1	8	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	2	8	no	—
302	C₂²	—	2	16	no	—
303	C₂²	—	4	16	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	8	no	—
306	C₂²	—	4	16	no	—
307	C₂²	—	8	16	no	—
308	C₂²	—	8	16	no	—
309	C₂²	C₂×#123×C₈₉	1	4	yes	—
310	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
311	C₂²	C₂²×#41×C₈₉	1	2	yes	—
312	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	4	yes	—
313	C₄	—	2	8	no	—
314	C₄	—	4	8	no	—
315	C₄	—	4	8	no	—
316	C₄	C₂×#123×C₈₉	1	8	yes	—
317	C₂	—	2	4	no	—
318	C₂	—	4	8	no	—
319	C₂	—	8	4	no	—
320	C₂	C₂²×#123×C₈₉	1	1	yes	characteristic
321	C₂	C₈₉×#3484	1	2	yes	—
322	C₂	C₈₉×#3508	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂²×#127×C₈₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#518×C₈₉	C₂	semidirect
3. C₈₉×#2445	C₂	semidirect
4. C₈₉×#2405	C₂	semidirect
5. C₈₉×#2309	C₂	semidirect
6. C₈₉×#2309	C₂	semidirect
7. C₈₉×#2296	C₂	semidirect
13. C₂×#123×C₈₉	C₂²	semidirect
15. C₂×C₄×D₄×C₈₉	C₂²	semidirect
20. #394×C₈₉	C₂²	semidirect
21. #390×C₈₉	C₂²	semidirect
22. #390×C₈₉	C₂²	semidirect
23. #390×C₈₉	C₂²	semidirect
24. #389×C₈₉	C₂²	semidirect
25. #386×C₈₉	C₂²	semidirect
29. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
132. C₂×C₄×C₈₉	#123	semidirect
155. #35554	C₈₉	direct (N × H)
163. C₂×#518	C₂×C₈₉	semidirect
164. #2445	C₂×C₈₉	semidirect
165. #2405	C₂×C₈₉	semidirect
166. #2309	C₂×C₈₉	semidirect
167. #2309	C₂×C₈₉	semidirect
168. #2296	C₂×C₈₉	semidirect
169. C₈₉	#35554	direct (N × H)
175. C₂×#123	C₂²×C₈₉	semidirect
177. C₂×C₄×D₄	C₂²×C₈₉	semidirect
182. #394	C₂²×C₈₉	semidirect
183. #390	C₂²×C₈₉	semidirect
184. #390	C₂²×C₈₉	semidirect
185. #390	C₂²×C₈₉	semidirect
186. #389	C₂²×C₈₉	semidirect
187. #386	C₂²×C₈₉	semidirect
191. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
294. C₂×C₄	#123×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₈₉	2
C₂×#518×C₈₉	2
C₈₉×#2445	2
C₈₉×#2405	2
C₈₉×#2309	2
C₈₉×#2296	2
#35554	89

Subgroups — C89×#35554 #36413402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35554 #36413402