Subgroups — C₈₉×#35293 #36413141

Order 22784.

2986 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₈₉×#2867	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2858	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2619	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2544	C₂	1	8	yes	maximal, complemented
7	C₈₉×#2431	C₂	1	8	yes	maximal, complemented
8	C₈₉×#2328	C₂	1	4	yes	maximal, complemented
9	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₈₉	C₂²	1	4	yes	—
11	#532×C₈₉	—	2	16	no	—
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
13	C₂×#127×C₈₉	C₂²	1	4	yes	—
14	C₂×#127×C₈₉	C₂²	1	2	yes	—
15	C₂×#127×C₈₉	C₂²	1	8	yes	—
16	C₂×#124×C₈₉	C₂²	1	4	yes	—
17	C₂×#124×C₈₉	C₂²	1	8	yes	—
18	C₂×#121×C₈₉	C₂²	1	4	yes	—
19	C₂×#121×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
22	C₂×#117×C₈₉	C₂²	1	2	yes	—
23	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
24	#394×C₈₉	C₂²	1	16	yes	—
25	#392×C₈₉	C₂²	1	16	yes	—
26	#391×C₈₉	C₂²	1	8	yes	—
27	#390×C₈₉	C₂²	1	8	yes	—
28	#390×C₈₉	C₂²	1	8	yes	—
29	#386×C₈₉	C₂²	1	32	yes	—
30	#382×C₈₉	C₂²	1	4	yes	—
31	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
32	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
33	#376×C₈₉	C₂²	1	2	yes	—
34	C₄³×C₈₉	C₂²	1	1	yes	characteristic
35	C₄³×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#41×C₈₉	—	2	16	no	—
37	C₂×#41×C₈₉	—	4	16	no	—
38	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
39	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
40	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
41	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
42	C₂²×D₄×C₈₉	C₂³	1	16	yes	—
43	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
44	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
45	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
47	C₄⋊Q₈×C₈₉	—	2	8	no	—
48	#127×C₈₉	—	2	8	no	—
49	#127×C₈₉	—	2	16	no	—
50	#127×C₈₉	—	4	16	no	—
51	#124×C₈₉	—	2	16	no	—
52	#124×C₈₉	—	4	16	no	—
53	#121×C₈₉	—	2	16	no	—
54	#121×C₈₉	—	2	16	no	—
55	C₄×D₄×C₈₉	—	2	16	no	—
56	C₄×D₄×C₈₉	—	2	16	no	—
57	#117×C₈₉	—	2	8	no	—
58	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
62	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
63	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
64	C₂×C₂²⋊C₄×C₈₉	C₂³	1	32	yes	—
65	C₂×C₄²×C₈₉	D₄	1	8	yes	complemented
66	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
67	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
68	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
69	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
72	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
73	#95×C₈₉	C₂³	1	8	yes	—
74	#95×C₈₉	C₂³	1	16	yes	—
75	C₂⁴×C₈₉	C₂⁴	1	8	yes	—
76	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
77	#41×C₈₉	—	4	64	no	—
78	C₂×Q₈×C₈₉	—	2	8	no	—
79	C₂×Q₈×C₈₉	—	2	16	no	—
80	C₂×D₄×C₈₉	—	2	16	no	—
81	C₂×D₄×C₈₉	—	2	8	no	—
82	C₂×D₄×C₈₉	—	2	8	no	—
83	C₂×D₄×C₈₉	—	2	8	no	—
84	C₂×D₄×C₈₉	—	2	16	no	—
85	C₂×D₄×C₈₉	—	2	32	no	—
86	C₂×D₄×C₈₉	—	2	16	no	—
87	C₂×D₄×C₈₉	—	2	16	no	—
88	C₂×D₄×C₈₉	—	4	16	no	—
89	C₂×D₄×C₈₉	—	4	32	no	—
90	C₂²×C₄×C₈₉	—	2	4	no	—
91	C₂²×C₄×C₈₉	—	2	8	no	—
92	C₂²×C₄×C₈₉	—	2	16	no	—
93	C₂²×C₄×C₈₉	—	4	16	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
95	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
96	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
97	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
98	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
99	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
101	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
103	C₄⋊C₄×C₈₉	—	2	8	no	—
104	C₄⋊C₄×C₈₉	—	2	8	no	—
105	C₄⋊C₄×C₈₉	—	2	8	no	—
106	C₂²⋊C₄×C₈₉	—	2	16	no	—
107	C₂²⋊C₄×C₈₉	—	2	16	no	—
108	C₂²⋊C₄×C₈₉	—	2	8	no	—
109	C₂²⋊C₄×C₈₉	—	4	32	no	—
110	C₄²×C₈₉	—	2	4	no	—
111	C₄²×C₈₉	—	2	8	no	—
112	C₄²×C₈₉	—	2	8	no	—
113	C₄²×C₈₉	C₂×D₄	1	4	yes	—
114	C₄²×C₈₉	C₂×D₄	1	8	yes	—
115	C₄²×C₈₉	C₂×D₄	1	4	yes	—
116	C₄²×C₈₉	#41	1	8	yes	complemented
117	C₂³×C₈₉	—	2	2	no	—
118	C₂³×C₈₉	—	2	4	no	—
119	C₂³×C₈₉	—	2	8	no	—
120	C₂³×C₈₉	—	2	16	no	—
121	C₂³×C₈₉	—	2	16	no	—
122	C₂³×C₈₉	—	2	16	no	—
123	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
124	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
125	Q₈×C₈₉	—	4	16	no	—
126	D₄×C₈₉	—	4	32	no	—
127	D₄×C₈₉	—	4	16	no	—
128	D₄×C₈₉	—	4	16	no	—
129	D₄×C₈₉	—	4	16	no	—
130	D₄×C₈₉	—	8	32	no	—
131	C₂×C₄×C₈₉	—	2	2	no	—
132	C₂×C₄×C₈₉	—	2	8	no	—
133	C₂×C₄×C₈₉	—	2	4	no	—
134	C₂×C₄×C₈₉	—	2	8	no	—
135	C₂×C₄×C₈₉	—	2	16	no	—
136	C₂×C₄×C₈₉	—	2	8	no	—
137	C₂×C₄×C₈₉	—	2	4	no	—
138	C₂×C₄×C₈₉	—	2	8	no	—
139	C₂×C₄×C₈₉	—	2	16	no	—
140	C₂×C₄×C₈₉	—	4	32	no	—
141	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
143	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
145	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
146	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
147	C₂×C₄×C₈₉	H(5,𝔽₂)	1	8	yes	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	2	4	no	—
150	C₂²×C₈₉	—	4	4	no	—
151	C₂²×C₈₉	—	4	8	no	—
152	C₂²×C₈₉	—	4	8	no	—
153	C₂²×C₈₉	—	4	8	no	—
154	C₂²×C₈₉	—	4	32	no	—
155	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
156	C₂²×C₈₉	C₂²×#41	1	2	yes	—
157	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	4	yes	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	4	no	—
160	C₄×C₈₉	—	2	8	no	—
161	C₄×C₈₉	—	4	8	no	—
162	C₄×C₈₉	—	4	4	no	—
163	C₄×C₈₉	#532	1	4	yes	—
164	C₄×C₈₉	#535	1	8	yes	—
165	#35293	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂×C₈₉	—	4	2	no	—
167	C₂×C₈₉	—	8	8	no	—
168	C₂×C₈₉	C₂×#532	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#535	1	2	yes	—
170	C₂×C₈₉	#3508	1	4	yes	—
171	C₂×#532	C₂×C₈₉	1	2	yes	complemented
172	#2867	C₂×C₈₉	1	1	yes	characteristic, complemented
173	#2862	C₂×C₈₉	1	1	yes	characteristic, complemented
174	#2858	C₂×C₈₉	1	2	yes	complemented
175	#2619	C₂×C₈₉	1	4	yes	complemented
176	#2544	C₂×C₈₉	1	8	yes	complemented
177	#2431	C₂×C₈₉	1	8	yes	complemented
178	#2328	C₂×C₈₉	1	4	yes	complemented
179	C₄×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
180	C₈₉	#35293	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²×#41	C₂²×C₈₉	1	4	yes	—
182	#532	—	2	16	no	—
183	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
184	C₂×#127	C₂²×C₈₉	1	4	yes	—
185	C₂×#127	C₂²×C₈₉	1	2	yes	—
186	C₂×#127	C₂²×C₈₉	1	8	yes	—
187	C₂×#124	C₂²×C₈₉	1	4	yes	—
188	C₂×#124	C₂²×C₈₉	1	8	yes	—
189	C₂×#121	C₂²×C₈₉	1	4	yes	—
190	C₂×#121	C₂²×C₈₉	1	4	yes	—
191	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
192	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
193	C₂×#117	C₂²×C₈₉	1	2	yes	—
194	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
195	#394	C₂²×C₈₉	1	16	yes	—
196	#392	C₂²×C₈₉	1	16	yes	—
197	#391	C₂²×C₈₉	1	8	yes	—
198	#390	C₂²×C₈₉	1	8	yes	—
199	#390	C₂²×C₈₉	1	8	yes	—
200	#386	C₂²×C₈₉	1	32	yes	—
201	#382	C₂²×C₈₉	1	4	yes	—
202	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
203	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
204	#376	C₂²×C₈₉	1	2	yes	—
205	C₄³	C₂²×C₈₉	1	1	yes	characteristic
206	C₄³	C₂²×C₈₉	1	1	yes	characteristic
207	C₂×#41	—	2	16	no	—
208	C₂×#41	—	4	16	no	—
209	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
210	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
211	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
212	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
213	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
214	C₂²×D₄	C₂³×C₈₉	1	16	yes	—
215	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
216	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
217	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
218	C₄⋊Q₈	—	2	8	no	—
219	#127	—	2	8	no	—
220	#127	—	2	16	no	—
221	#127	—	4	16	no	—
222	#124	—	2	16	no	—
223	#124	—	4	16	no	—
224	#121	—	2	16	no	—
225	#121	—	2	16	no	—
226	C₄×D₄	—	2	16	no	—
227	C₄×D₄	—	2	16	no	—
228	#117	—	2	8	no	—
229	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
230	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
231	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
232	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
233	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
234	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
235	C₂×C₂²⋊C₄	C₂³×C₈₉	1	32	yes	—
236	C₂×C₄²	D₄×C₈₉	1	8	yes	complemented
237	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
238	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
239	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
240	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
242	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
244	#95	C₂³×C₈₉	1	8	yes	—
245	#95	C₂³×C₈₉	1	16	yes	—
246	C₂⁴	C₂⁴×C₈₉	1	8	yes	—
247	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
248	#41	—	4	64	no	—
249	C₂×Q₈	—	2	8	no	—
250	C₂×Q₈	—	2	16	no	—
251	C₂×D₄	—	2	16	no	—
252	C₂×D₄	—	2	8	no	—
253	C₂×D₄	—	2	8	no	—
254	C₂×D₄	—	2	8	no	—
255	C₂×D₄	—	2	16	no	—
256	C₂×D₄	—	2	32	no	—
257	C₂×D₄	—	2	16	no	—
258	C₂×D₄	—	2	16	no	—
259	C₂×D₄	—	4	16	no	—
260	C₂×D₄	—	4	32	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	8	no	—
263	C₂²×C₄	—	2	16	no	—
264	C₂²×C₄	—	4	16	no	—
265	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
266	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
267	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
268	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
269	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
270	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
272	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
273	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
274	C₄⋊C₄	—	2	8	no	—
275	C₄⋊C₄	—	2	8	no	—
276	C₄⋊C₄	—	2	8	no	—
277	C₂²⋊C₄	—	2	16	no	—
278	C₂²⋊C₄	—	2	16	no	—
279	C₂²⋊C₄	—	2	8	no	—
280	C₂²⋊C₄	—	4	32	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	2	8	no	—
283	C₄²	—	2	8	no	—
284	C₄²	C₂×D₄×C₈₉	1	4	yes	—
285	C₄²	C₂×D₄×C₈₉	1	8	yes	—
286	C₄²	C₂×D₄×C₈₉	1	4	yes	—
287	C₄²	#41×C₈₉	1	8	yes	complemented
288	C₂³	—	2	2	no	—
289	C₂³	—	2	4	no	—
290	C₂³	—	2	8	no	—
291	C₂³	—	2	16	no	—
292	C₂³	—	2	16	no	—
293	C₂³	—	2	16	no	—
294	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
295	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
296	Q₈	—	4	16	no	—
297	D₄	—	4	32	no	—
298	D₄	—	4	16	no	—
299	D₄	—	4	16	no	—
300	D₄	—	4	16	no	—
301	D₄	—	8	32	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	8	no	—
306	C₂×C₄	—	2	16	no	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	8	no	—
310	C₂×C₄	—	2	16	no	—
311	C₂×C₄	—	4	32	no	—
312	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
313	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
314	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
315	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
316	C₂×C₄	C₂×#41×C₈₉	1	8	yes	—
317	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
318	C₂×C₄	H(5,𝔽₂)×C₈₉	1	8	yes	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	4	4	no	—
322	C₂²	—	4	8	no	—
323	C₂²	—	4	8	no	—
324	C₂²	—	4	8	no	—
325	C₂²	—	4	32	no	—
326	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
327	C₂²	C₂²×#41×C₈₉	1	2	yes	—
328	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	4	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	2	4	no	—
331	C₄	—	2	8	no	—
332	C₄	—	4	8	no	—
333	C₄	—	4	4	no	—
334	C₄	#532×C₈₉	1	4	yes	—
335	C₄	#535×C₈₉	1	8	yes	—
336	C₂	—	4	2	no	—
337	C₂	—	8	8	no	—
338	C₂	C₂×#532×C₈₉	1	1	yes	characteristic
339	C₂	C₂×#535×C₈₉	1	2	yes	—
340	C₂	C₈₉×#3508	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
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
1. C₂×#532×C₈₉	C₂	semidirect
2. C₈₉×#2867	C₂	semidirect
3. C₈₉×#2862	C₂	semidirect
4. C₈₉×#2858	C₂	semidirect
5. C₈₉×#2619	C₂	semidirect
6. C₈₉×#2544	C₂	semidirect
7. C₈₉×#2431	C₂	semidirect
8. C₈₉×#2328	C₂	semidirect
9. C₄×#117×C₈₉	C₂	semidirect
65. C₂×C₄²×C₈₉	D₄	semidirect
116. C₄²×C₈₉	#41	semidirect
165. #35293	C₈₉	direct (N × H)
171. C₂×#532	C₂×C₈₉	semidirect
172. #2867	C₂×C₈₉	semidirect
173. #2862	C₂×C₈₉	semidirect
174. #2858	C₂×C₈₉	semidirect
175. #2619	C₂×C₈₉	semidirect
176. #2544	C₂×C₈₉	semidirect
177. #2431	C₂×C₈₉	semidirect
178. #2328	C₂×C₈₉	semidirect
179. C₄×#117	C₂×C₈₉	semidirect
180. C₈₉	#35293	direct (N × H)
236. C₂×C₄²	D₄×C₈₉	semidirect
287. C₄²	#41×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₈₉	2
C₈₉×#2867	2
C₈₉×#2862	2
C₈₉×#2858	2
C₈₉×#2619	2
C₈₉×#2544	2
C₈₉×#2431	2
C₈₉×#2328	2
C₄×#117×C₈₉	2
#35293	89

Subgroups — C89×#35293 #36413141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35293 #36413141