Subgroups — #36557850

Order 22784.

882 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667948	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151636	C₂	1	1	yes	maximal, characteristic
3	#26028815	C₂	1	2	yes	maximal
4	#26021005	C₂	1	4	yes	maximal, complemented
5	#26021000	C₂	1	4	yes	maximal, complemented
6	#26019962	C₂	1	2	yes	maximal
7	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13083558	C₂²	1	2	yes	—
9	C₂×#15667971	C₂²	1	2	yes	—
10	C₂×#15667948	—	2	4	no	—
11	C₂×#15667948	C₂²	1	2	yes	—
12	C₂×#15667948	C₂²	1	2	yes	—
13	#19151636	C₄	1	2	yes	complemented
14	#19151636	C₄	1	2	yes	complemented
15	#19151636	C₂²	1	2	yes	—
16	#19151636	C₂²	1	2	yes	—
17	C₂×#15667946	C₄	1	2	yes	complemented
18	C₂×#15667944	C₄	1	2	yes	complemented
19	#19151626	—	2	4	no	—
20	#19151624	C₂²	1	8	yes	—
21	#19151621	—	2	4	no	—
22	#19151358	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
26	#405×C₈₉	—	2	4	no	—
27	C₂×#97×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
29	C₂×#13083558	—	2	2	no	—
30	C₂×#13083558	—	2	4	no	—
31	C₂×#13083558	—	2	4	no	—
32	C₂×#13083558	C₂³	1	4	yes	—
33	C₂²×#10623494	—	2	2	no	—
34	#15667971	—	2	2	no	—
35	#15667971	—	2	2	no	—
36	#15667971	C₂×C₄	1	4	yes	—
37	#15667971	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
40	C₂×C₄×Dic₈₉	—	2	2	no	—
41	#15667948	—	2	4	no	—
42	#15667948	—	2	4	no	—
43	#15667946	C₂×C₄	1	4	yes	—
44	#15667946	C₂×C₄	1	4	yes	—
45	#15667944	C₂×C₄	1	4	yes	—
46	#15667944	C₂×C₄	1	4	yes	—
47	#15667941	—	2	16	no	—
48	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₈₉	—	2	2	no	—
50	C₂×M₁₆×C₈₉	—	2	2	no	—
51	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
52	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
53	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	D₄	1	4	yes	—
58	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
59	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	#117×C₈₉	—	2	2	no	—
63	#117×C₈₉	Q₈	1	4	yes	—
64	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
66	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
67	#97×C₈₉	—	2	4	no	—
68	C₄×C₈×C₈₉	—	2	4	no	—
69	C₂²×Dic₈₉	—	2	4	no	—
70	#13083558	—	2	4	no	—
71	#13083558	—	2	4	no	—
72	#13083558	—	4	4	no	—
73	C₂×#10623494	—	2	4	no	—
74	C₂×#10623494	—	4	4	no	—
75	C₄⋊Dic₈₉	—	2	4	no	—
76	C₄⋊Dic₈₉	C₂²×C₄	1	8	yes	—
77	C₂²⋊Dic₈₉	—	2	4	no	—
78	C₂²⋊Dic₈₉	—	2	4	no	—
79	C₄×Dic₈₉	—	2	4	no	—
80	C₄×Dic₈₉	—	2	4	no	—
81	C₂⁴×C₈₉	#41	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	—	2	2	no	—
83	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	#41	1	2	yes	—
92	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₈₉	—	2	4	no	—
94	M₁₆×C₈₉	—	2	4	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
98	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
99	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
100	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
106	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
110	C₄⋊C₄×C₈₉	—	2	4	no	—
111	C₂²⋊C₄×C₈₉	—	2	4	no	—
112	C₄²×C₈₉	—	2	2	no	—
113	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
114	C₂³×C₈₉	—	4	2	no	—
115	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
116	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
117	C₂³×C₈₉	C₂²⋊Q₈	1	2	yes	—
118	C₂³×C₈₉	#125	1	2	yes	—
119	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₈₉	—	2	4	no	—
121	C₂×C₄×C₈₉	—	4	4	no	—
122	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₈₉	#121	1	2	yes	—
129	C₂×C₄×C₈₉	#121	1	2	yes	—
130	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₈₉	#125	1	2	yes	—
132	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₈₉	—	2	8	no	—
141	C₂×Dic₈₉	—	8	4	no	—
142	C₈×C₈₉	—	2	4	no	—
143	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
144	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
145	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
146	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
147	#10623494	—	4	4	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	—	4	2	no	—
152	C₂²×C₈₉	—	4	2	no	—
153	C₂²×C₈₉	#389	1	2	yes	—
154	C₂²×C₈₉	#389	1	2	yes	—
155	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₈₉	C₂×#125	1	1	yes	characteristic
158	C₄×C₈₉	—	4	4	no	—
159	C₄×C₈₉	#389	1	2	yes	—
160	C₄×C₈₉	#389	1	2	yes	—
161	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
163	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
164	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	Dic₈₉	—	4	8	no	—
166	#21163	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	1	no	—
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	2	2	no	—
170	C₂×C₈₉	—	2	2	no	—
171	C₂×C₈₉	C₂×#389	1	1	yes	characteristic
172	C₂×C₈₉	#2957	1	2	yes	—
173	C₂²×#108	—	89	1	no	—
174	C₂×#427	—	89	1	no	—
175	#2905	—	89	2	no	—
176	C₂×#405	D₈₉	1	1	yes	characteristic
177	#1946	—	89	4	no	—
178	#1945	—	89	4	no	—
179	#1733	—	89	2	no	—
180	C₈₉	#21163	1	1	yes	characteristic, complemented, Sylow
181	C₂²×M₁₆	—	89	2	no	—
182	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
183	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
184	C₂×#117	—	89	2	no	—
185	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
186	C₂×#108	—	89	2	no	—
187	C₂×#108	—	89	2	no	—
188	C₂×#108	—	178	4	no	—
189	#427	—	89	2	no	—
190	#427	—	89	2	no	—
191	#427	—	89	2	no	—
192	#427	—	89	2	no	—
193	C₂×#107	—	89	2	no	—
194	C₂×#106	—	89	2	no	—
195	#424	—	89	8	no	—
196	#424	—	178	4	no	—
197	#423	—	178	4	no	—
198	#405	—	2	4	no	—
199	C₂×#97	C₂×D₈₉	1	2	yes	—
200	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
201	#339	—	89	8	no	—
202	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	89	4	no	—
206	C₂×M₁₆	—	178	4	no	—
207	C₂×M₁₆	—	178	4	no	—
208	C₂×M₁₆	—	178	2	no	—
209	C₂×M₁₆	D₃₅₆	1	2	yes	—
210	C₂×M₁₆	D₃₅₆	1	2	yes	—
211	C₂²×C₈	—	178	2	no	—
212	C₂²×C₈	Dic₁₇₈	1	4	yes	—
213	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
216	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
218	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
219	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
221	#117	—	2	2	no	—
222	#117	—	89	4	no	—
223	#117	—	89	4	no	—
224	#117	—	178	2	no	—
225	#117	—	178	2	no	—
226	#117	Dic₁₇₈	1	4	yes	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	89	4	no	—
229	C₂×C₂²⋊C₄	—	2	1	no	—
230	C₂×C₂²⋊C₄	—	178	2	no	—
231	C₂×C₄²	—	178	2	no	—
232	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
233	#108	—	178	4	no	—
234	#108	—	178	4	no	—
235	#107	—	89	4	no	—
236	#107	—	89	4	no	—
237	#106	—	89	4	no	—
238	#106	—	89	4	no	—
239	C₄⋊C₈	—	178	16	no	—
240	#97	—	2	4	no	—
241	C₄×C₈	—	2	4	no	—
242	C₂⁴	#13083574	1	1	yes	characteristic
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	178	4	no	—
245	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
248	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
250	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
251	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
252	C₂²×C₄	#13083574	1	1	yes	characteristic
253	C₂²×C₄	#13083574	1	2	yes	—
254	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	356	4	no	—
258	M₁₆	—	178	4	no	—
259	M₁₆	—	178	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	356	4	no	—
263	C₂×C₈	—	178	4	no	—
264	C₂×C₈	#13083553	1	4	yes	—
265	C₂×C₈	#13083553	1	4	yes	—
266	C₂×C₈	#13083553	1	4	yes	—
267	C₂×C₈	#13083553	1	4	yes	—
268	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
274	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
275	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
276	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	89	8	no	—
279	C₄⋊C₄	—	178	4	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	178	4	no	—
282	C₂²⋊C₄	—	178	4	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	178	4	no	—
285	C₄²	—	178	4	no	—
286	C₄²	C₂×Dic₁₇₈	1	4	yes	—
287	C₂³	—	4	2	no	—
288	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
289	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
290	C₂³	#15667991	1	2	yes	—
291	C₂³	#15668004	1	2	yes	—
292	C₂³	C₂×#13083574	1	1	yes	characteristic
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	178	8	no	—
296	C₂×C₄	—	712	4	no	—
297	C₂×C₄	C₂×#13083553	1	2	yes	—
298	C₂×C₄	C₂×#13083553	1	2	yes	—
299	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
300	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
301	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
302	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
303	C₂×C₄	#15667986	1	2	yes	—
304	C₂×C₄	#15667986	1	2	yes	—
305	C₂×C₄	#15667991	1	2	yes	—
306	C₂×C₄	#15668004	1	2	yes	—
307	C₂×C₄	#15668012	1	2	yes	—
308	C₂×C₄	#15668012	1	2	yes	—
309	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
312	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
313	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
314	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
315	C₈	—	2	4	no	—
316	C₈	—	356	4	no	—
317	C₈	C₂×#13083553	1	2	yes	—
318	C₈	C₂×#13083553	1	2	yes	—
319	C₈	C₂×#13083553	1	2	yes	—
320	C₈	C₂×#13083553	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	#19151496	1	2	yes	—
327	C₂²	#19151496	1	2	yes	—
328	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
329	C₂²	C₂×#15667991	1	1	yes	characteristic
330	C₂²	C₂×#15668004	1	1	yes	characteristic
331	C₄	—	4	4	no	—
332	C₄	—	356	8	no	—
333	C₄	#19151496	1	2	yes	—
334	C₄	#19151496	1	2	yes	—
335	C₄	C₂²×#13083553	1	1	yes	characteristic
336	C₄	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
337	C₄	C₂×#15667986	1	1	yes	characteristic
338	C₄	C₂×#15668012	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	C₂×#19151496	1	1	yes	characteristic
344	C₂	#26029170	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#405×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂⁴×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#405	D₈₉
hypocenter	C₈₉	#21163
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #26021005	C₂	semidirect
5. #26021000	C₂	semidirect
13. #19151636	C₄	semidirect
14. #19151636	C₄	semidirect
17. C₂×#15667946	C₄	semidirect
18. C₂×#15667944	C₄	semidirect
180. C₈₉	#21163	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667948	2
C₂×#19151636	2
#26028815	2
#26021005	2
#26021000	2
#26019962	2
C₂×#405×C₈₉	2
#21163	89