Subgroups — #36448645

Order 22784.

602 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26023152	C₂	1	1	yes	maximal, characteristic
2	#26021011	C₂	1	1	yes	maximal, characteristic, complemented
3	#26020379	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020253	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151325	C₂	1	1	yes	maximal, characteristic
6	#26018091	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
9	C₂×#15667974	—	2	1	no	—
10	C₂×C₄⋊C₄×D₈₉	—	2	1	no	—
11	C₂×#15667946	—	2	1	no	—
12	C₂×#15667934	—	2	1	no	—
13	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
14	#19151494	—	2	1	no	—
15	#19151469	—	2	2	no	—
16	#19151468	C₂²	1	1	yes	characteristic
17	#19151455	—	2	1	no	—
18	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	#19151361	—	2	1	no	—
20	#19151361	—	2	1	no	—
21	#19151360	—	2	2	no	—
22	#19151325	C₄	1	4	yes	complemented
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#384×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₈₉	—	2	2	no	—
28	C₂²×C₄×D₈₉	—	2	1	no	—
29	C₂²×#10623494	—	2	1	no	—
30	C₄×D₃₅₆	—	2	1	no	—
31	C₄×D₃₅₆	—	2	1	no	—
32	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
33	C₄×D₃₅₆	Q₈	1	2	yes	—
34	#15667974	—	4	2	no	—
35	#15667974	—	4	2	no	—
36	C₂×C₄⋊Dic₈₉	—	2	1	no	—
37	C₂×C₄⋊Dic₈₉	—	2	1	no	—
38	C₂×C₄⋊Dic₈₉	—	2	1	no	—
39	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
42	C₂×#13083553	—	2	2	no	—
43	C₄⋊C₄×D₈₉	—	4	2	no	—
44	C₄⋊C₄×D₈₉	—	4	2	no	—
45	C₂×#13083551	—	2	2	no	—
46	C₂×#13083551	—	2	1	no	—
47	C₂×C₄×Dic₈₉	—	2	1	no	—
48	#15667946	—	4	2	no	—
49	#15667934	—	4	2	no	—
50	C₄×#10623494	—	2	1	no	—
51	C₄×#10623494	C₂×C₄	1	2	yes	complemented
52	#15667919	—	2	2	no	—
53	#15667919	—	2	1	no	—
54	C₂²×C₈×C₈₉	—	2	1	no	—
55	C₂²×C₈×C₈₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₄⋊C₈×C₈₉	—	2	1	no	—
62	C₄⋊C₈×C₈₉	—	2	1	no	—
63	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	complemented
64	C₂³×D₈₉	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
69	C₂×D₃₅₆	D₈	1	2	yes	—
70	C₂×D₃₅₆	Q₁₆	1	2	yes	—
71	C₂²×Dic₈₉	—	2	1	no	—
72	C₂²×Dic₈₉	—	2	1	no	—
73	C₂²×Dic₈₉	—	2	1	no	—
74	C₂²×Dic₈₉	—	4	1	no	—
75	C₂×C₄×D₈₉	—	2	2	no	—
76	C₂×C₄×D₈₉	—	2	2	no	—
77	C₂×C₄×D₈₉	—	4	1	no	—
78	C₂×C₄×D₈₉	—	4	1	no	—
79	C₂×C₄×D₈₉	—	4	1	no	—
80	C₂×C₄×D₈₉	—	4	1	no	—
81	C₂×C₄×D₈₉	—	4	4	no	—
82	C₂×C₄×D₈₉	—	4	4	no	—
83	C₂×C₄×D₈₉	—	8	2	no	—
84	C₂×#10623494	—	2	2	no	—
85	C₂×#10623494	—	4	1	no	—
86	C₂×#10623494	—	4	1	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	1	no	—
89	C₄⋊Dic₈₉	—	2	1	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	—	4	1	no	—
93	C₄⋊Dic₈₉	—	4	1	no	—
94	C₄⋊Dic₈₉	—	4	1	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
97	C₄⋊Dic₈₉	SD₁₆	1	2	yes	—
98	#13083553	—	4	4	no	—
99	#13083551	—	2	2	no	—
100	#13083551	—	2	2	no	—
101	#13083551	—	4	4	no	—
102	C₄×Dic₈₉	—	4	1	no	—
103	C₄×Dic₈₉	—	4	1	no	—
104	C₂²×C₄×C₈₉	—	2	2	no	—
105	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	4	1	no	—
110	C₂×C₈×C₈₉	—	4	1	no	—
111	C₂×C₈×C₈₉	—	4	1	no	—
112	C₂×C₈×C₈₉	—	4	1	no	—
113	C₂×C₈×C₈₉	—	4	1	no	—
114	C₄⋊C₄×C₈₉	—	4	2	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₄⋊C₄×C₈₉	—	4	1	no	—
118	C₄⋊C₄×C₈₉	—	4	1	no	—
119	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
121	C₄²×C₈₉	#41	1	1	yes	characteristic
122	C₄²×C₈₉	#41	1	1	yes	characteristic
123	C₂³×C₈₉	#120	1	1	yes	characteristic
124	C₂²×D₈₉	—	2	2	no	—
125	C₂²×D₈₉	—	2	2	no	—
126	C₂²×D₈₉	—	2	2	no	—
127	C₂²×D₈₉	—	4	1	no	—
128	C₂²×D₈₉	—	4	1	no	—
129	C₂²×D₈₉	—	4	1	no	—
130	C₂²×D₈₉	—	4	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
146	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
150	C₂×Dic₈₉	—	2	2	no	—
151	C₂×Dic₈₉	—	2	2	no	—
152	C₂×Dic₈₉	—	4	1	no	—
153	C₂×Dic₈₉	—	4	1	no	—
154	C₂×Dic₈₉	—	4	1	no	—
155	C₂×Dic₈₉	—	4	1	no	—
156	C₂×Dic₈₉	—	4	1	no	—
157	C₂×Dic₈₉	—	4	1	no	—
158	C₂×Dic₈₉	—	4	1	no	—
159	C₂×Dic₈₉	—	8	1	no	—
160	C₂×Dic₈₉	—	8	1	no	—
161	C₂×Dic₈₉	—	8	1	no	—
162	D₃₅₆	—	2	2	no	—
163	D₃₅₆	—	2	2	no	—
164	D₃₅₆	—	2	2	no	—
165	D₃₅₆	#107	1	4	yes	complemented
166	C₄×D₈₉	—	4	2	no	—
167	C₄×D₈₉	—	4	2	no	—
168	C₄×D₈₉	—	4	2	no	—
169	C₄×D₈₉	—	4	2	no	—
170	C₄×D₈₉	—	8	8	no	—
171	C₈×C₈₉	—	4	2	no	—
172	C₈×C₈₉	—	8	1	no	—
173	#10623494	—	4	2	no	—
174	C₂²×C₈₉	#385	1	1	yes	characteristic
175	C₂²×C₈₉	#447	1	1	yes	characteristic
176	C₂²×C₈₉	#449	1	1	yes	characteristic
177	C₂²×C₈₉	#450	1	1	yes	characteristic
178	C₂²×C₈₉	#451	1	1	yes	characteristic
179	C₂²×C₈₉	#454	1	1	yes	characteristic
180	C₂²×C₈₉	#455	1	1	yes	characteristic
181	C₂×D₈₉	—	2	4	no	—
182	C₂×D₈₉	—	4	2	no	—
183	C₂×D₈₉	—	4	2	no	—
184	C₂×D₈₉	—	4	2	no	—
185	C₂×D₈₉	—	4	2	no	—
186	C₂×D₈₉	—	4	2	no	—
187	C₂×D₈₉	—	4	2	no	—
188	C₄×C₈₉	—	2	1	no	—
189	C₄×C₈₉	—	2	1	no	—
190	C₄×C₈₉	—	8	2	no	—
191	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
192	C₄×C₈₉	#428	1	1	yes	characteristic
193	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
194	C₄×C₈₉	#442	1	1	yes	characteristic
195	C₄×C₈₉	#476	1	1	yes	characteristic
196	C₄×C₈₉	#478	1	1	yes	characteristic
197	C₄×C₈₉	#480	1	1	yes	characteristic
198	C₄×C₈₉	#484	1	1	yes	characteristic
199	Dic₈₉	—	4	2	no	—
200	Dic₈₉	—	4	2	no	—
201	Dic₈₉	—	8	1	no	—
202	Dic₈₉	—	16	1	no	—
203	#8890	—	89	1	no	maximal, Sylow
204	C₂×C₈₉	#1625	1	1	yes	characteristic
205	C₂×C₈₉	#1637	1	1	yes	characteristic
206	C₂×C₈₉	#1639	1	1	yes	characteristic
207	C₂×C₈₉	#1644	1	1	yes	characteristic
208	C₂×C₈₉	#1852	1	1	yes	characteristic
209	C₂×C₈₉	#1860	1	1	yes	characteristic
210	C₂×C₈₉	#1864	1	1	yes	characteristic
211	D₈₉	—	4	4	no	—
212	D₄×C₄⋊C₄	—	89	1	no	—
213	#1947	—	89	1	no	—
214	#1834	—	89	1	no	—
215	#1802	—	89	1	no	—
216	C₂×D₄⋊C₈	—	89	1	no	—
217	#1296	D₈₉	1	1	yes	characteristic, complemented
218	#1279	—	89	1	no	—
219	C₈₉	#8890	1	1	yes	characteristic, complemented, Sylow
220	C₂×C₄×D₄	—	89	1	no	—
221	C₂×C₄×D₄	—	178	1	no	—
222	C₂²×C₄⋊C₄	—	178	1	no	—
223	C₂×#107	—	178	1	no	—
224	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
225	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
226	C₂×#102	—	178	1	no	—
227	C₂×C₄×C₈	—	89	1	no	—
228	#389	—	178	1	no	—
229	#385	—	178	2	no	—
230	#384	—	89	1	no	—
231	#384	C₂×D₈₉	1	1	yes	characteristic
232	#380	—	178	1	no	—
233	C₄×C₄⋊C₄	—	89	1	no	—
234	#340	—	178	1	no	—
235	#340	—	178	1	no	—
236	#340	—	178	2	no	—
237	D₄⋊C₈	—	89	4	no	—
238	C₂²×D₄	—	89	1	no	—
239	C₂³×C₄	—	178	2	no	—
240	C₂³×C₄	—	178	1	no	—
241	C₂²×C₈	—	2	1	no	—
242	C₂²×C₈	—	2	1	no	—
243	C₂²×C₈	—	178	1	no	—
244	C₄×D₄	—	89	2	no	—
245	C₄×D₄	—	89	2	no	—
246	C₄×D₄	—	356	2	no	—
247	C₄×D₄	—	356	2	no	—
248	C₄×D₄	—	178	1	no	—
249	C₄×D₄	—	178	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	89	1	no	—
253	C₂×C₄⋊C₄	—	89	1	no	—
254	C₂×C₄⋊C₄	—	89	1	no	—
255	C₂×C₄⋊C₄	—	356	2	no	—
256	C₂×C₄⋊C₄	—	356	2	no	—
257	C₂×C₄⋊C₄	—	178	2	no	—
258	C₂×C₄⋊C₄	—	178	1	no	—
259	C₂×C₄⋊C₄	—	178	1	no	—
260	C₂×C₄⋊C₄	—	178	1	no	—
261	C₂×C₄⋊C₄	D₃₅₆	1	2	yes	—
262	C₂×C₂²⋊C₄	—	178	2	no	—
263	C₂×C₂²⋊C₄	—	178	1	no	—
264	C₂×C₄²	—	178	1	no	—
265	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
266	#107	—	356	2	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	C₄×D₈₉	1	2	yes	complemented
271	#102	—	356	2	no	—
272	C₄×C₈	—	89	2	no	—
273	C₄×C₈	—	178	1	no	—
274	#95	—	178	2	no	—
275	#95	—	178	1	no	—
276	C₂⁴	—	178	1	no	—
277	C₂×D₄	—	89	2	no	—
278	C₂×D₄	—	89	2	no	—
279	C₂×D₄	—	89	2	no	—
280	C₂×D₄	—	178	1	no	—
281	C₂×D₄	—	178	1	no	—
282	C₂×D₄	—	178	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	356	4	no	—
285	C₂²×C₄	—	356	4	no	—
286	C₂²×C₄	—	356	1	no	—
287	C₂²×C₄	—	356	1	no	—
288	C₂²×C₄	—	356	1	no	—
289	C₂²×C₄	—	356	1	no	—
290	C₂²×C₄	—	356	1	no	—
291	C₂²×C₄	—	178	1	no	—
292	C₂²×C₄	—	178	1	no	—
293	C₂²×C₄	—	178	1	no	—
294	C₂²×C₄	—	178	2	no	—
295	C₂²×C₄	—	178	2	no	—
296	C₂²×C₄	—	712	2	no	—
297	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
298	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
299	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	356	1	no	—
307	C₂×C₈	—	356	1	no	—
308	C₂×C₈	—	178	2	no	—
309	C₄⋊C₄	—	4	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	89	2	no	—
315	C₄⋊C₄	—	89	2	no	—
316	C₄⋊C₄	—	356	2	no	—
317	C₄⋊C₄	—	356	4	no	—
318	C₄⋊C₄	—	356	1	no	—
319	C₄⋊C₄	—	356	1	no	—
320	C₄⋊C₄	—	356	1	no	—
321	C₄⋊C₄	—	356	1	no	—
322	C₄⋊C₄	—	178	1	no	—
323	C₄⋊C₄	—	178	1	no	—
324	C₄⋊C₄	—	178	1	no	—
325	C₄⋊C₄	—	178	1	no	—
326	C₂²⋊C₄	—	356	4	no	—
327	C₂²⋊C₄	—	178	2	no	—
328	C₂²⋊C₄	—	178	2	no	—
329	C₄²	—	356	1	no	—
330	C₄²	—	356	1	no	—
331	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
332	C₄²	Q₈×D₈₉	1	1	yes	characteristic
333	C₄²	#13083573	1	1	yes	characteristic
334	C₄²	#13083574	1	1	yes	characteristic
335	C₂³	—	356	1	no	—
336	C₂³	—	356	1	no	—
337	C₂³	—	356	1	no	—
338	C₂³	—	356	1	no	—
339	C₂³	—	178	2	no	—
340	C₂³	—	178	2	no	—
341	C₂³	—	178	2	no	—
342	C₂³	#15667982	1	1	yes	characteristic
343	D₄	—	89	4	no	—
344	D₄	—	178	2	no	—
345	D₄	—	178	2	no	—
346	D₄	—	178	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	356	1	no	—
353	C₂×C₄	—	356	1	no	—
354	C₂×C₄	—	356	1	no	—
355	C₂×C₄	—	356	1	no	—
356	C₂×C₄	—	356	2	no	—
357	C₂×C₄	—	356	2	no	—
358	C₂×C₄	—	356	2	no	—
359	C₂×C₄	—	356	2	no	—
360	C₂×C₄	—	356	1	no	—
361	C₂×C₄	—	356	1	no	—
362	C₂×C₄	—	356	1	no	—
363	C₂×C₄	—	178	2	no	—
364	C₂×C₄	—	178	2	no	—
365	C₂×C₄	—	712	8	no	—
366	C₂×C₄	—	712	1	no	—
367	C₂×C₄	—	712	1	no	—
368	C₂×C₄	—	712	1	no	—
369	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
370	C₂×C₄	#15667972	1	1	yes	characteristic
371	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
372	C₂×C₄	#15667990	1	1	yes	characteristic
373	C₂×C₄	#15667992	1	1	yes	characteristic
374	C₂×C₄	#15667994	1	1	yes	characteristic
375	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
376	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
377	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
378	C₂×C₄	#15668035	1	1	yes	characteristic
379	C₂×C₄	#15668036	1	1	yes	characteristic
380	C₂×C₄	#15668039	1	1	yes	characteristic
381	C₂×C₄	#15668039	1	1	yes	characteristic
382	C₂×C₄	#15668041	1	1	yes	characteristic
383	C₈	—	4	2	no	—
384	C₈	—	8	1	no	—
385	C₈	—	356	2	no	—
386	C₂²	—	356	2	no	—
387	C₂²	—	356	2	no	—
388	C₂²	—	356	2	no	—
389	C₂²	—	356	2	no	—
390	C₂²	—	356	2	no	—
391	C₂²	—	356	2	no	—
392	C₂²	—	178	4	no	—
393	C₂²	#19151472	1	1	yes	characteristic
394	C₂²	#19151733	1	1	yes	characteristic
395	C₂²	#19151746	1	1	yes	characteristic
396	C₂²	#19151750	1	1	yes	characteristic
397	C₂²	#19151755	1	1	yes	characteristic
398	C₂²	#19151773	1	1	yes	characteristic
399	C₂²	#19151778	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	8	2	no	—
403	C₄	—	356	2	no	—
404	C₄	—	356	2	no	—
405	C₄	—	1424	1	no	—
406	C₄	—	712	1	no	—
407	C₄	#107×D₈₉	1	1	yes	characteristic
408	C₄	#19151637	1	1	yes	characteristic
409	C₄	#19151683	1	1	yes	characteristic
410	C₄	#19151707	1	1	yes	characteristic
411	C₄	#19151887	1	1	yes	characteristic
412	C₄	#19151901	1	1	yes	characteristic
413	C₄	#19151910	1	1	yes	characteristic
414	C₄	#19151932	1	1	yes	characteristic
415	C₂	—	356	4	no	—
416	C₂	#26019342	1	1	yes	characteristic
417	C₂	#26019418	1	1	yes	characteristic
418	C₂	#26019432	1	1	yes	characteristic
419	C₂	#26019467	1	1	yes	characteristic
420	C₂	#26020441	1	1	yes	characteristic
421	C₂	#26020488	1	1	yes	characteristic
422	C₂	#26020514	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667982
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1296	C₂
socle	C₂³×C₈₉	#120
prime radical	C₂²×D₃₅₆	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1296	D₈₉
hypocenter	C₈₉	#8890
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #26021011	C₂	semidirect
3. #26020379	C₂	semidirect
6. #26018091	C₂	semidirect
7. C₈₉×#1296	C₂	semidirect
22. #19151325	C₄	semidirect
51. C₄×#10623494	C₂×C₄	semidirect
63. C₄⋊C₈×C₈₉	C₂×C₄	semidirect
165. D₃₅₆	#107	semidirect
217. #1296	D₈₉	semidirect
219. C₈₉	#8890	semidirect
270. C₄⋊C₈	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26023152	2
#26021011	2
#26020379	2
#26020253	2
C₂×#19151325	2
#26018091	2
C₈₉×#1296	2
#8890	89