Subgroups — C₈₉×#35285 #36413133

Order 22784.

3206 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#127×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2858	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2641	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2548	C₂	1	8	yes	maximal, complemented
6	C₈₉×#2435	C₂	1	8	yes	maximal, complemented
7	C₈₉×#2309	C₂	1	4	yes	maximal, complemented
8	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₈₉	C₂²	1	2	yes	—
10	C₂³×D₄×C₈₉	C₂²	1	4	yes	—
11	#533×C₈₉	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
13	C₂×#127×C₈₉	—	2	8	no	—
14	C₂×#127×C₈₉	C₂²	1	2	yes	—
15	C₂×#127×C₈₉	C₂²	1	2	yes	—
16	C₂×#127×C₈₉	C₂²	1	4	yes	—
17	C₂×#125×C₈₉	C₂²	1	2	yes	complemented
18	C₂×#124×C₈₉	C₂²	1	16	yes	complemented
19	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	complemented
21	C₂×#117×C₈₉	C₂²	1	2	yes	—
22	C₂×#117×C₈₉	C₂²	1	2	yes	—
23	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
24	#394×C₈₉	C₂²	1	32	yes	complemented
25	#390×C₈₉	C₂²	1	16	yes	complemented
26	#390×C₈₉	C₂²	1	16	yes	complemented
27	#390×C₈₉	C₂²	1	16	yes	complemented
28	#379×C₈₉	C₂²	1	16	yes	—
29	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
30	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
31	#376×C₈₉	C₂²	1	2	yes	complemented
32	C₄³×C₈₉	C₂²	1	2	yes	complemented
33	C₂⁵×C₈₉	C₂³	1	4	yes	—
34	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
35	C₂²×D₄×C₈₉	—	2	16	no	—
36	C₂²×D₄×C₈₉	—	2	8	no	—
37	C₂²×D₄×C₈₉	—	2	16	no	—
38	C₂²×D₄×C₈₉	—	4	16	no	—
39	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
40	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
41	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
42	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
43	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
44	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₈₉	—	2	8	no	—
46	#127×C₈₉	—	2	16	no	—
47	#127×C₈₉	—	2	8	no	—
48	#127×C₈₉	—	2	8	no	—
49	#125×C₈₉	—	2	8	no	—
50	#124×C₈₉	—	4	32	no	—
51	C₂²⋊Q₈×C₈₉	—	2	16	no	—
52	C₄×Q₈×C₈₉	—	2	16	no	—
53	#117×C₈₉	—	2	8	no	—
54	#117×C₈₉	—	2	8	no	—
55	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
57	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
59	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₈₉	C₂³	1	64	yes	—
62	C₂×C₄²×C₈₉	D₄	1	8	yes	—
63	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
64	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
65	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
69	#95×C₈₉	C₂³	1	8	yes	—
70	C₂⁴×C₈₉	—	2	8	no	—
71	C₂⁴×C₈₉	—	2	16	no	—
72	C₂⁴×C₈₉	—	4	16	no	—
73	C₂⁴×C₈₉	C₂⁴	1	8	yes	—
74	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
75	C₂×Q₈×C₈₉	—	2	8	no	—
76	C₂×Q₈×C₈₉	—	2	16	no	—
77	C₂×D₄×C₈₉	—	2	16	no	—
78	C₂×D₄×C₈₉	—	2	8	no	—
79	C₂×D₄×C₈₉	—	2	8	no	—
80	C₂×D₄×C₈₉	—	2	16	no	—
81	C₂×D₄×C₈₉	—	2	16	no	—
82	C₂×D₄×C₈₉	—	2	16	no	—
83	C₂×D₄×C₈₉	—	2	32	no	—
84	C₂×D₄×C₈₉	—	4	16	no	—
85	C₂×D₄×C₈₉	—	4	64	no	—
86	C₂×D₄×C₈₉	—	4	32	no	—
87	C₂²×C₄×C₈₉	—	2	8	no	—
88	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
89	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
90	C₂²×C₄×C₈₉	#41	1	8	yes	—
91	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
92	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
93	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
94	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
97	C₄⋊C₄×C₈₉	—	2	8	no	—
98	C₄⋊C₄×C₈₉	—	2	16	no	—
99	C₄⋊C₄×C₈₉	—	2	8	no	—
100	C₄⋊C₄×C₈₉	—	2	8	no	—
101	C₂²⋊C₄×C₈₉	—	2	8	no	—
102	C₂²⋊C₄×C₈₉	—	2	8	no	—
103	C₂²⋊C₄×C₈₉	—	4	64	no	—
104	C₄²×C₈₉	—	2	8	no	—
105	C₄²×C₈₉	—	2	8	no	—
106	C₄²×C₈₉	—	2	16	no	—
107	C₄²×C₈₉	C₂×D₄	1	8	yes	—
108	C₄²×C₈₉	C₂×D₄	1	4	yes	—
109	C₄²×C₈₉	C₂×D₄	1	4	yes	—
110	C₂³×C₈₉	—	2	4	no	—
111	C₂³×C₈₉	—	2	8	no	—
112	C₂³×C₈₉	—	2	16	no	—
113	C₂³×C₈₉	—	2	32	no	—
114	C₂³×C₈₉	—	4	32	no	—
115	C₂³×C₈₉	—	4	16	no	—
116	C₂³×C₈₉	—	8	32	no	—
117	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
118	C₂³×C₈₉	C₂×#41	1	4	yes	—
119	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
120	Q₈×C₈₉	—	4	16	no	—
121	D₄×C₈₉	—	4	32	no	—
122	D₄×C₈₉	—	4	16	no	—
123	D₄×C₈₉	—	4	16	no	—
124	D₄×C₈₉	—	4	32	no	—
125	C₂×C₄×C₈₉	—	2	4	no	—
126	C₂×C₄×C₈₉	—	2	16	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	#124	1	16	yes	complemented
132	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
133	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
135	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
136	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
137	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
138	C₂×C₄×C₈₉	H(5,𝔽₂)	1	8	yes	—
139	C₂²×C₈₉	—	2	4	no	—
140	C₂²×C₈₉	—	2	8	no	—
141	C₂²×C₈₉	—	4	8	no	—
142	C₂²×C₈₉	—	4	16	no	—
143	C₂²×C₈₉	—	4	32	no	—
144	C₂²×C₈₉	—	8	32	no	—
145	C₂²×C₈₉	C₂×#124	1	4	yes	—
146	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
147	C₂²×C₈₉	C₂²×#41	1	2	yes	—
148	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	4	yes	—
149	C₄×C₈₉	—	2	4	no	—
150	C₄×C₈₉	—	4	8	no	—
151	C₄×C₈₉	—	4	8	no	—
152	C₄×C₈₉	C₂×#124	1	4	yes	—
153	C₄×C₈₉	C₂×#124	1	4	yes	—
154	C₄×C₈₉	#535	1	8	yes	—
155	#35285	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₈₉	—	2	4	no	—
157	C₂×C₈₉	—	8	8	no	—
158	C₂×C₈₉	C₂²×#124	1	1	yes	characteristic
159	C₂×C₈₉	C₂×#535	1	2	yes	—
160	C₂×C₈₉	#3508	1	4	yes	—
161	C₂×#533	C₂×C₈₉	1	1	yes	characteristic, complemented
162	C₂²×#127	C₂×C₈₉	1	1	yes	characteristic
163	#2858	C₂×C₈₉	1	4	yes	complemented
164	#2641	C₂×C₈₉	1	4	yes	complemented
165	#2548	C₂×C₈₉	1	8	yes	complemented
166	#2435	C₂×C₈₉	1	8	yes	complemented
167	#2309	C₂×C₈₉	1	4	yes	complemented
168	C₄×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
169	C₈₉	#35285	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×D₄	C₂²×C₈₉	1	2	yes	—
171	C₂³×D₄	C₂²×C₈₉	1	4	yes	—
172	#533	—	2	8	no	—
173	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	complemented
174	C₂×#127	—	2	8	no	—
175	C₂×#127	C₂²×C₈₉	1	2	yes	—
176	C₂×#127	C₂²×C₈₉	1	2	yes	—
177	C₂×#127	C₂²×C₈₉	1	4	yes	—
178	C₂×#125	C₂²×C₈₉	1	2	yes	complemented
179	C₂×#124	C₂²×C₈₉	1	16	yes	complemented
180	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
181	C₂×C₄×Q₈	C₂²×C₈₉	1	4	yes	complemented
182	C₂×#117	C₂²×C₈₉	1	2	yes	—
183	C₂×#117	C₂²×C₈₉	1	2	yes	—
184	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
185	#394	C₂²×C₈₉	1	32	yes	complemented
186	#390	C₂²×C₈₉	1	16	yes	complemented
187	#390	C₂²×C₈₉	1	16	yes	complemented
188	#390	C₂²×C₈₉	1	16	yes	complemented
189	#379	C₂²×C₈₉	1	16	yes	—
190	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
191	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
192	#376	C₂²×C₈₉	1	2	yes	complemented
193	C₄³	C₂²×C₈₉	1	2	yes	complemented
194	C₂⁵	C₂³×C₈₉	1	4	yes	—
195	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
196	C₂²×D₄	—	2	16	no	—
197	C₂²×D₄	—	2	8	no	—
198	C₂²×D₄	—	2	16	no	—
199	C₂²×D₄	—	4	16	no	—
200	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
201	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
202	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
203	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
204	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
205	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
206	C₄⋊Q₈	—	2	8	no	—
207	#127	—	2	16	no	—
208	#127	—	2	8	no	—
209	#127	—	2	8	no	—
210	#125	—	2	8	no	—
211	#124	—	4	32	no	—
212	C₂²⋊Q₈	—	2	16	no	—
213	C₄×Q₈	—	2	16	no	—
214	#117	—	2	8	no	—
215	#117	—	2	8	no	—
216	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
217	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
218	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
219	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
220	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
221	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
222	C₂×C₂²⋊C₄	C₂³×C₈₉	1	64	yes	—
223	C₂×C₄²	D₄×C₈₉	1	8	yes	—
224	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
225	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
226	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
227	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
228	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
229	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
230	#95	C₂³×C₈₉	1	8	yes	—
231	C₂⁴	—	2	8	no	—
232	C₂⁴	—	2	16	no	—
233	C₂⁴	—	4	16	no	—
234	C₂⁴	C₂⁴×C₈₉	1	8	yes	—
235	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
236	C₂×Q₈	—	2	8	no	—
237	C₂×Q₈	—	2	16	no	—
238	C₂×D₄	—	2	16	no	—
239	C₂×D₄	—	2	8	no	—
240	C₂×D₄	—	2	8	no	—
241	C₂×D₄	—	2	16	no	—
242	C₂×D₄	—	2	16	no	—
243	C₂×D₄	—	2	16	no	—
244	C₂×D₄	—	2	32	no	—
245	C₂×D₄	—	4	16	no	—
246	C₂×D₄	—	4	64	no	—
247	C₂×D₄	—	4	32	no	—
248	C₂²×C₄	—	2	8	no	—
249	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
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	2	yes	—
255	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
257	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	16	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₂²⋊C₄	—	2	8	no	—
263	C₂²⋊C₄	—	2	8	no	—
264	C₂²⋊C₄	—	4	64	no	—
265	C₄²	—	2	8	no	—
266	C₄²	—	2	8	no	—
267	C₄²	—	2	16	no	—
268	C₄²	C₂×D₄×C₈₉	1	8	yes	—
269	C₄²	C₂×D₄×C₈₉	1	4	yes	—
270	C₄²	C₂×D₄×C₈₉	1	4	yes	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	8	no	—
273	C₂³	—	2	16	no	—
274	C₂³	—	2	32	no	—
275	C₂³	—	4	32	no	—
276	C₂³	—	4	16	no	—
277	C₂³	—	8	32	no	—
278	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
279	C₂³	C₂×#41×C₈₉	1	4	yes	—
280	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
281	Q₈	—	4	16	no	—
282	D₄	—	4	32	no	—
283	D₄	—	4	16	no	—
284	D₄	—	4	16	no	—
285	D₄	—	4	32	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	16	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	16	no	—
291	C₂×C₄	—	2	16	no	—
292	C₂×C₄	#124×C₈₉	1	16	yes	complemented
293	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
294	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
295	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
296	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
297	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
298	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
299	C₂×C₄	H(5,𝔽₂)×C₈₉	1	8	yes	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	8	no	—
302	C₂²	—	4	8	no	—
303	C₂²	—	4	16	no	—
304	C₂²	—	4	32	no	—
305	C₂²	—	8	32	no	—
306	C₂²	C₂×#124×C₈₉	1	4	yes	—
307	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
308	C₂²	C₂²×#41×C₈₉	1	2	yes	—
309	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	4	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	8	no	—
312	C₄	—	4	8	no	—
313	C₄	C₂×#124×C₈₉	1	4	yes	—
314	C₄	C₂×#124×C₈₉	1	4	yes	—
315	C₄	#535×C₈₉	1	8	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	8	8	no	—
318	C₂	C₂²×#124×C₈₉	1	1	yes	characteristic
319	C₂	C₂×#535×C₈₉	1	2	yes	—
320	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
1. C₂×#533×C₈₉	C₂	semidirect
3. C₈₉×#2858	C₂	semidirect
4. C₈₉×#2641	C₂	semidirect
5. C₈₉×#2548	C₂	semidirect
6. C₈₉×#2435	C₂	semidirect
7. C₈₉×#2309	C₂	semidirect
8. C₄×#117×C₈₉	C₂	semidirect
12. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
17. C₂×#125×C₈₉	C₂²	semidirect
18. C₂×#124×C₈₉	C₂²	semidirect
20. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
24. #394×C₈₉	C₂²	semidirect
25. #390×C₈₉	C₂²	semidirect
26. #390×C₈₉	C₂²	semidirect
27. #390×C₈₉	C₂²	semidirect
29. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
31. #376×C₈₉	C₂²	semidirect
32. C₄³×C₈₉	C₂²	semidirect
131. C₂×C₄×C₈₉	#124	semidirect
155. #35285	C₈₉	direct (N × H)
161. C₂×#533	C₂×C₈₉	semidirect
163. #2858	C₂×C₈₉	semidirect
164. #2641	C₂×C₈₉	semidirect
165. #2548	C₂×C₈₉	semidirect
166. #2435	C₂×C₈₉	semidirect
167. #2309	C₂×C₈₉	semidirect
168. C₄×#117	C₂×C₈₉	semidirect
169. C₈₉	#35285	direct (N × H)
173. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
178. C₂×#125	C₂²×C₈₉	semidirect
179. C₂×#124	C₂²×C₈₉	semidirect
181. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
185. #394	C₂²×C₈₉	semidirect
186. #390	C₂²×C₈₉	semidirect
187. #390	C₂²×C₈₉	semidirect
188. #390	C₂²×C₈₉	semidirect
190. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
192. #376	C₂²×C₈₉	semidirect
193. C₄³	C₂²×C₈₉	semidirect
292. C₂×C₄	#124×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₈₉	2
C₂²×#127×C₈₉	2
C₈₉×#2858	2
C₈₉×#2641	2
C₈₉×#2548	2
C₈₉×#2435	2
C₈₉×#2309	2
C₄×#117×C₈₉	2
#35285	89

Subgroups — C89×#35285 #36413133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35285 #36413133