Subgroups — #36448619

Order 22784.

618 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#26021015	C₂	1	1	yes	maximal, characteristic, complemented
3	#26020409	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020253	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151327	C₂	1	1	yes	maximal, characteristic
6	#26018136	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₅₆	—	2	1	no	—
9	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₄×D₈₉	—	2	1	no	—
11	C₂×#15667944	—	2	2	no	—
12	C₂×#15667943	—	2	1	no	—
13	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	C₂×#15667934	—	2	1	no	—
15	#19151496	—	2	1	no	—
16	#19151472	—	2	2	no	—
17	#19151468	C₂²	1	1	yes	characteristic
18	#19151468	C₂²	1	1	yes	characteristic
19	#19151455	—	2	1	no	—
20	#19151361	—	2	1	no	—
21	#19151360	—	2	1	no	—
22	#19151327	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	C₄×C₄⋊C₄×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	—	4	2	no	—
33	C₄×D₃₅₆	—	4	2	no	—
34	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
35	C₄×D₃₅₆	Q₈	1	2	yes	—
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	—	2	1	no	—
40	C₂×C₄⋊Dic₈₉	—	2	1	no	—
41	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
43	C₂×#13083553	—	2	2	no	—
44	C₄⋊C₄×D₈₉	—	4	2	no	—
45	C₄⋊C₄×D₈₉	—	4	2	no	—
46	C₂×#13083551	—	2	2	no	—
47	C₂×#13083551	—	2	1	no	—
48	#15667944	—	4	4	no	—
49	#15667943	—	4	2	no	—
50	C₄⋊#10623494	—	2	1	no	—
51	C₄⋊#10623494	C₂×C₄	1	2	yes	complemented
52	#15667934	—	4	2	no	—
53	#15667919	—	2	2	no	—
54	C₂²×C₈×C₈₉	—	2	1	no	—
55	C₂²×C₈×C₈₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	—	2	1	no	—
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	#95×C₈₉	—	2	1	no	—
65	C₂³×D₈₉	—	2	1	no	—
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	—	2	1	no	—
69	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
70	C₂×D₃₅₆	SD₁₆	1	2	yes	—
71	C₂×D₃₅₆	SD₁₆	1	2	yes	—
72	C₂²×Dic₈₉	—	2	1	no	—
73	C₂²×Dic₈₉	—	2	2	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	—	4	1	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	—	4	1	no	—
91	C₄⋊Dic₈₉	—	4	1	no	—
92	C₄⋊Dic₈₉	—	4	2	no	—
93	C₄⋊Dic₈₉	—	4	2	no	—
94	C₄⋊Dic₈₉	—	4	1	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
99	#13083553	—	4	4	no	—
100	#13083551	—	2	2	no	—
101	#13083551	—	2	2	no	—
102	#13083551	—	4	4	no	—
103	C₂²×C₄×C₈₉	—	2	1	no	—
104	C₂²×C₄×C₈₉	—	2	1	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	—	2	2	no	—
110	C₂×C₈×C₈₉	—	2	2	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	—	2	1	no	—
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	—	2	1	no	—
117	C₄⋊C₄×C₈₉	SD₁₆	1	2	yes	—
118	C₄²×C₈₉	—	4	1	no	—
119	C₄²×C₈₉	—	4	1	no	—
120	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
122	C₄²×C₈₉	#41	1	1	yes	characteristic
123	C₄²×C₈₉	#41	1	1	yes	characteristic
124	C₂³×C₈₉	#120	1	1	yes	characteristic
125	C₂²×D₈₉	—	2	2	no	—
126	C₂²×D₈₉	—	2	2	no	—
127	C₂²×D₈₉	—	2	2	no	—
128	C₂²×D₈₉	—	4	1	no	—
129	C₂²×D₈₉	—	4	1	no	—
130	C₂²×D₈₉	—	4	1	no	—
131	C₂²×D₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	4	1	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
150	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
154	C₂×Dic₈₉	—	2	2	no	—
155	C₂×Dic₈₉	—	4	1	no	—
156	C₂×Dic₈₉	—	4	1	no	—
157	C₂×Dic₈₉	—	4	2	no	—
158	C₂×Dic₈₉	—	4	2	no	—
159	C₂×Dic₈₉	—	4	2	no	—
160	C₂×Dic₈₉	—	8	1	no	—
161	C₂×Dic₈₉	—	8	1	no	—
162	C₂×Dic₈₉	—	8	1	no	—
163	D₃₅₆	—	2	2	no	—
164	D₃₅₆	—	2	2	no	—
165	D₃₅₆	—	2	2	no	—
166	D₃₅₆	#106	1	4	yes	complemented
167	C₄×D₈₉	—	4	2	no	—
168	C₄×D₈₉	—	4	2	no	—
169	C₄×D₈₉	—	4	2	no	—
170	C₄×D₈₉	—	4	2	no	—
171	C₄×D₈₉	—	8	8	no	—
172	C₈×C₈₉	—	2	4	no	—
173	C₈×C₈₉	—	8	1	no	—
174	#10623494	—	4	2	no	—
175	C₂²×C₈₉	#385	1	1	yes	characteristic
176	C₂²×C₈₉	#448	1	1	yes	characteristic
177	C₂²×C₈₉	#449	1	1	yes	characteristic
178	C₂²×C₈₉	#450	1	1	yes	characteristic
179	C₂²×C₈₉	#450	1	1	yes	characteristic
180	C₂²×C₈₉	#453	1	1	yes	characteristic
181	C₂²×C₈₉	#456	1	1	yes	characteristic
182	C₂×D₈₉	—	2	4	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	C₂×D₈₉	—	4	2	no	—
189	C₄×C₈₉	—	2	1	no	—
190	C₄×C₈₉	—	2	1	no	—
191	C₄×C₈₉	—	4	2	no	—
192	C₄×C₈₉	—	8	1	no	—
193	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
194	C₄×C₈₉	#428	1	1	yes	characteristic
195	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
196	C₄×C₈₉	#442	1	1	yes	characteristic
197	C₄×C₈₉	#476	1	1	yes	characteristic
198	C₄×C₈₉	#478	1	1	yes	characteristic
199	C₄×C₈₉	#481	1	1	yes	characteristic
200	C₄×C₈₉	#483	1	1	yes	characteristic
201	Dic₈₉	—	4	2	no	—
202	Dic₈₉	—	8	2	no	—
203	Dic₈₉	—	16	1	no	—
204	#8886	—	89	1	no	maximal, Sylow
205	C₂×C₈₉	#1622	1	1	yes	characteristic
206	C₂×C₈₉	#1629	1	1	yes	characteristic
207	C₂×C₈₉	#1651	1	1	yes	characteristic
208	C₂×C₈₉	#1654	1	1	yes	characteristic
209	C₂×C₈₉	#1853	1	1	yes	characteristic
210	C₂×C₈₉	#1859	1	1	yes	characteristic
211	C₂×C₈₉	#1864	1	1	yes	characteristic
212	D₈₉	—	4	4	no	—
213	D₄×C₄⋊C₄	—	89	1	no	—
214	#1948	—	89	1	no	—
215	#1844	—	89	1	no	—
216	#1802	—	89	1	no	—
217	C₂×D₄⋊C₈	—	89	1	no	—
218	#1296	—	89	1	no	—
219	#1279	D₈₉	1	1	yes	characteristic, complemented
220	C₈₉	#8886	1	1	yes	characteristic, complemented, Sylow
221	C₂×C₄×D₄	—	89	1	no	—
222	C₂×C₄×D₄	—	178	1	no	—
223	C₂²×C₄⋊C₄	—	178	1	no	—
224	C₂×#106	—	178	2	no	—
225	C₂×#106	—	178	1	no	—
226	C₂×C₄⋊C₈	—	89	1	no	—
227	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
228	C₂×#102	—	178	1	no	—
229	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
230	#389	—	178	1	no	—
231	#385	—	178	2	no	—
232	#384	—	89	1	no	—
233	#384	—	89	1	no	—
234	#380	—	178	1	no	—
235	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
236	#340	—	178	1	no	—
237	#340	—	178	1	no	—
238	D₄⋊C₈	—	89	4	no	—
239	C₂²×D₄	—	89	1	no	—
240	C₂³×C₄	—	178	2	no	—
241	C₂³×C₄	—	178	1	no	—
242	C₂²×C₈	—	2	1	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	178	1	no	—
245	C₄×D₄	—	89	2	no	—
246	C₄×D₄	—	89	2	no	—
247	C₄×D₄	—	356	2	no	—
248	C₄×D₄	—	356	2	no	—
249	C₄×D₄	—	178	1	no	—
250	C₄×D₄	—	178	1	no	—
251	C₂×C₄⋊C₄	—	89	1	no	—
252	C₂×C₄⋊C₄	—	89	2	no	—
253	C₂×C₄⋊C₄	—	356	2	no	—
254	C₂×C₄⋊C₄	—	356	2	no	—
255	C₂×C₄⋊C₄	—	178	2	no	—
256	C₂×C₄⋊C₄	—	178	1	no	—
257	C₂×C₄⋊C₄	—	178	1	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	1	yes	characteristic
262	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
263	C₂×C₂²⋊C₄	—	178	2	no	—
264	C₂×C₂²⋊C₄	—	178	1	no	—
265	C₂×C₄²	—	2	1	no	—
266	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
267	#106	—	356	2	no	—
268	#106	—	356	4	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	89	2	no	—
272	C₄⋊C₈	—	178	1	no	—
273	#102	—	356	2	no	—
274	C₄×C₈	—	2	1	no	—
275	C₄×C₈	C₄×D₈₉	1	2	yes	complemented
276	#95	—	2	1	no	—
277	#95	—	178	2	no	—
278	C₂⁴	—	178	1	no	—
279	C₂×D₄	—	89	2	no	—
280	C₂×D₄	—	89	2	no	—
281	C₂×D₄	—	89	2	no	—
282	C₂×D₄	—	178	1	no	—
283	C₂×D₄	—	178	1	no	—
284	C₂×D₄	—	178	1	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	356	4	no	—
288	C₂²×C₄	—	356	4	no	—
289	C₂²×C₄	—	356	1	no	—
290	C₂²×C₄	—	356	1	no	—
291	C₂²×C₄	—	356	1	no	—
292	C₂²×C₄	—	356	1	no	—
293	C₂²×C₄	—	356	1	no	—
294	C₂²×C₄	—	178	2	no	—
295	C₂²×C₄	—	178	1	no	—
296	C₂²×C₄	—	178	2	no	—
297	C₂²×C₄	—	178	2	no	—
298	C₂²×C₄	—	712	2	no	—
299	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
300	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
301	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	356	1	no	—
309	C₂×C₈	—	356	1	no	—
310	C₂×C₈	—	178	2	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	89	2	no	—
315	C₄⋊C₄	—	356	2	no	—
316	C₄⋊C₄	—	356	4	no	—
317	C₄⋊C₄	—	356	2	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	—	356	1	no	—
323	C₄⋊C₄	—	356	1	no	—
324	C₄⋊C₄	—	356	1	no	—
325	C₄⋊C₄	—	356	1	no	—
326	C₄⋊C₄	—	178	1	no	—
327	C₄⋊C₄	#13083563	1	2	yes	—
328	C₂²⋊C₄	—	356	4	no	—
329	C₂²⋊C₄	—	178	2	no	—
330	C₂²⋊C₄	—	178	2	no	—
331	C₄²	—	4	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
334	C₄²	Q₈×D₈₉	1	1	yes	characteristic
335	C₄²	#13083573	1	1	yes	characteristic
336	C₄²	#13083574	1	1	yes	characteristic
337	C₂³	—	356	1	no	—
338	C₂³	—	356	1	no	—
339	C₂³	—	356	1	no	—
340	C₂³	—	356	1	no	—
341	C₂³	—	178	2	no	—
342	C₂³	—	178	2	no	—
343	C₂³	—	178	2	no	—
344	C₂³	#15667982	1	1	yes	characteristic
345	D₄	—	89	4	no	—
346	D₄	—	178	2	no	—
347	D₄	—	178	2	no	—
348	D₄	—	178	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	356	1	no	—
358	C₂×C₄	—	356	1	no	—
359	C₂×C₄	—	356	2	no	—
360	C₂×C₄	—	356	2	no	—
361	C₂×C₄	—	356	2	no	—
362	C₂×C₄	—	356	2	no	—
363	C₂×C₄	—	356	2	no	—
364	C₂×C₄	—	356	2	no	—
365	C₂×C₄	—	356	2	no	—
366	C₂×C₄	—	178	2	no	—
367	C₂×C₄	—	712	8	no	—
368	C₂×C₄	—	712	1	no	—
369	C₂×C₄	—	712	1	no	—
370	C₂×C₄	—	712	1	no	—
371	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
372	C₂×C₄	#15667972	1	1	yes	characteristic
373	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
374	C₂×C₄	#15667990	1	1	yes	characteristic
375	C₂×C₄	#15667992	1	1	yes	characteristic
376	C₂×C₄	#15667994	1	1	yes	characteristic
377	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
378	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
379	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
380	C₂×C₄	#15668034	1	1	yes	characteristic
381	C₂×C₄	#15668036	1	1	yes	characteristic
382	C₂×C₄	#15668040	1	1	yes	characteristic
383	C₂×C₄	#15668040	1	1	yes	characteristic
384	C₂×C₄	#15668044	1	1	yes	characteristic
385	C₈	—	2	4	no	—
386	C₈	—	8	1	no	—
387	C₈	—	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	—	356	2	no	—
393	C₂²	—	356	2	no	—
394	C₂²	—	178	4	no	—
395	C₂²	#19151472	1	1	yes	characteristic
396	C₂²	#19151738	1	1	yes	characteristic
397	C₂²	#19151745	1	1	yes	characteristic
398	C₂²	#19151750	1	1	yes	characteristic
399	C₂²	#19151751	1	1	yes	characteristic
400	C₂²	#19151767	1	1	yes	characteristic
401	C₂²	#19151782	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	4	2	no	—
405	C₄	—	8	1	no	—
406	C₄	—	356	2	no	—
407	C₄	—	1424	1	no	—
408	C₄	—	712	2	no	—
409	C₄	#106×D₈₉	1	1	yes	characteristic
410	C₄	#19151639	1	1	yes	characteristic
411	C₄	C₄×#13083563	1	1	yes	characteristic
412	C₄	#19151711	1	1	yes	characteristic
413	C₄	#19151889	1	1	yes	characteristic
414	C₄	#19151899	1	1	yes	characteristic
415	C₄	#19151915	1	1	yes	characteristic
416	C₄	#19151927	1	1	yes	characteristic
417	C₂	—	356	4	no	—
418	C₂	#26019327	1	1	yes	characteristic
419	C₂	#26019370	1	1	yes	characteristic
420	C₂	#26019518	1	1	yes	characteristic
421	C₂	#26019538	1	1	yes	characteristic
422	C₂	#26020446	1	1	yes	characteristic
423	C₂	#26020479	1	1	yes	characteristic
424	C₂	#26020516	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667982
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1279	C₂
socle	C₂³×C₈₉	#120
prime radical	C₂²×D₃₅₆	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1279	D₈₉
hypocenter	C₈₉	#8886
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
2. #26021015	C₂	semidirect
3. #26020409	C₂	semidirect
6. #26018136	C₂	semidirect
7. C₈₉×#1279	C₂	semidirect
22. #19151327	C₄	semidirect
51. C₄⋊#10623494	C₂×C₄	semidirect
63. C₄×C₈×C₈₉	C₂×C₄	semidirect
166. D₃₅₆	#106	semidirect
219. #1279	D₈₉	semidirect
220. C₈₉	#8886	semidirect
275. 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
C₄⋊C₄×D₃₅₆	2
#26021015	2
#26020409	2
#26020253	2
C₂×#19151327	2
#26018136	2
C₈₉×#1279	2
#8886	89