Subgroups — #36450580

Order 22784.

610 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151804	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26021213	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020253	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151345	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018136	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083563	—	2	1	no	—
9	C₂²×C₈×D₈₉	—	2	1	no	—
10	C₂×#15668012	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
12	#19151804	—	2	2	no	—
13	#19151804	C₄	1	4	yes	complemented
14	C₈×D₃₅₆	—	2	2	no	—
15	C₈×D₃₅₆	—	2	2	no	—
16	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
17	C₂×#15667937	—	2	1	no	—
18	C₂×#15667934	—	2	1	no	—
19	C₂×#15667924	—	2	1	no	—
20	#19151468	C₂²	1	1	yes	characteristic
21	#19151361	—	2	1	no	—
22	#19151361	—	2	1	no	—
23	#19151361	—	2	1	no	—
24	#19151345	—	2	2	no	—
25	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂²×Dic₁₇₈	—	2	1	no	—
29	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₈₉	—	2	1	no	—
31	C₂×#13083563	—	2	4	no	—
32	C₂×#13083563	—	4	2	no	—
33	C₂×#13083563	—	4	2	no	—
34	C₂²×#10623494	—	2	1	no	—
35	C₂×C₈×D₈₉	—	2	4	no	—
36	C₂×C₈×D₈₉	—	4	2	no	—
37	C₂×C₈×D₈₉	—	4	2	no	—
38	#15668012	—	2	1	no	—
39	#15668012	—	2	1	no	—
40	#15668012	—	2	1	no	—
41	#15668012	C₂×C₄	1	2	yes	—
42	C₄×D₃₅₆	—	2	1	no	—
43	C₄×D₃₅₆	—	2	1	no	—
44	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
45	C₄×D₃₅₆	D₄	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	—	2	1	no	—
47	C₂×C₄⋊Dic₈₉	—	2	1	no	—
48	C₂×C₄⋊Dic₈₉	—	2	1	no	—
49	C₂×C₄⋊Dic₈₉	—	2	1	no	—
50	C₂×C₄⋊Dic₈₉	—	2	1	no	—
51	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
52	C₂×#13083551	—	2	1	no	—
53	C₄⋊#10623494	—	2	1	no	—
54	C₄⋊#10623494	D₄	1	2	yes	—
55	#15667937	—	2	4	no	—
56	#15667934	—	4	2	no	—
57	#15667924	—	2	4	no	—
58	C₂²×C₈×C₈₉	—	2	1	no	—
59	C₂²×C₈×C₈₉	—	2	1	no	—
60	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₈₉	—	2	1	no	—
64	C₄⋊C₈×C₈₉	—	2	1	no	—
65	C₄⋊C₈×C₈₉	—	2	1	no	—
66	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₈₉	—	2	1	no	—
68	C₄×C₈×C₈₉	—	2	1	no	—
69	C₂³×D₈₉	—	2	1	no	—
70	C₂×Dic₁₇₈	—	2	2	no	—
71	C₂×Dic₁₇₈	—	4	1	no	—
72	C₂×Dic₁₇₈	—	4	1	no	—
73	C₂×Dic₁₇₈	—	4	1	no	—
74	C₂×Dic₁₇₈	—	4	1	no	—
75	C₂×Dic₁₇₈	—	4	1	no	—
76	C₂×D₃₅₆	—	2	1	no	—
77	C₂×D₃₅₆	—	2	1	no	—
78	C₂×D₃₅₆	—	2	1	no	—
79	C₂×D₃₅₆	C₂²⋊C₄	1	2	yes	—
80	C₂×D₃₅₆	D₈	1	2	yes	—
81	C₂×D₃₅₆	SD₁₆	1	2	yes	—
82	C₂²×Dic₈₉	—	2	1	no	—
83	C₂²×Dic₈₉	—	4	1	no	—
84	C₂²×Dic₈₉	—	4	1	no	—
85	C₂×C₄×D₈₉	—	2	2	no	—
86	C₂×C₄×D₈₉	—	2	2	no	—
87	C₂×C₄×D₈₉	—	4	1	no	—
88	C₂×C₄×D₈₉	—	4	1	no	—
89	C₂×C₄×D₈₉	—	4	1	no	—
90	C₂×C₄×D₈₉	—	4	1	no	—
91	#13083563	—	4	8	no	—
92	C₂×#10623494	—	2	2	no	—
93	C₂×#10623494	—	4	1	no	—
94	C₂×#10623494	—	4	1	no	—
95	C₈×D₈₉	—	4	8	no	—
96	C₄⋊Dic₈₉	—	2	1	no	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	4	1	no	—
100	C₄⋊Dic₈₉	—	4	1	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	C₄⋊Dic₈₉	—	4	1	no	—
103	C₄⋊Dic₈₉	—	4	1	no	—
104	C₄⋊Dic₈₉	—	4	1	no	—
105	C₄⋊Dic₈₉	—	8	1	no	—
106	C₄⋊Dic₈₉	C₂²⋊C₄	1	2	yes	—
107	#13083551	—	2	2	no	—
108	#13083551	—	2	2	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₉	—	2	1	no	—
113	C₂×C₈×C₈₉	—	2	1	no	—
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	2	1	no	—
116	C₂×C₈×C₈₉	—	2	1	no	—
117	C₂×C₈×C₈₉	—	2	1	no	—
118	C₂×C₈×C₈₉	—	4	1	no	—
119	C₂×C₈×C₈₉	—	4	1	no	—
120	C₂×C₈×C₈₉	—	4	1	no	—
121	C₂×C₈×C₈₉	—	4	1	no	—
122	C₂×C₈×C₈₉	—	4	1	no	—
123	C₂×C₈×C₈₉	SD₁₆	1	2	yes	complemented
124	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	#41	1	1	yes	characteristic
128	C₂³×C₈₉	#121	1	1	yes	characteristic
129	C₂²×D₈₉	—	2	2	no	—
130	C₂²×D₈₉	—	2	2	no	—
131	C₂²×D₈₉	—	2	2	no	—
132	C₂²×D₈₉	—	4	1	no	—
133	C₂²×D₈₉	—	4	1	no	—
134	C₂²×D₈₉	—	4	1	no	—
135	C₂²×D₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	2	1	no	—
138	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
148	C₂×C₄×C₈₉	#135	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	#137	1	1	yes	characteristic
152	C₂×Dic₈₉	—	2	2	no	—
153	C₂×Dic₈₉	—	4	1	no	—
154	C₂×Dic₈₉	—	4	1	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	C₂×Dic₈₉	—	8	1	no	—
157	C₂×Dic₈₉	—	8	1	no	—
158	C₂×Dic₈₉	—	8	1	no	—
159	C₂×Dic₈₉	—	8	1	no	—
160	C₂×Dic₈₉	—	8	1	no	—
161	D₃₅₆	—	2	2	no	—
162	D₃₅₆	—	2	2	no	—
163	D₃₅₆	—	2	2	no	—
164	D₃₅₆	#102	1	4	yes	—
165	C₄×D₈₉	—	4	2	no	—
166	C₄×D₈₉	—	4	2	no	—
167	C₄×D₈₉	—	4	2	no	—
168	C₄×D₈₉	—	4	2	no	—
169	Dic₁₇₈	—	4	2	no	—
170	Dic₁₇₈	—	4	2	no	—
171	Dic₁₇₈	—	8	1	no	—
172	Dic₁₇₈	—	8	1	no	—
173	C₈×C₈₉	—	2	2	no	—
174	C₈×C₈₉	—	4	2	no	—
175	C₈×C₈₉	—	4	1	no	—
176	C₈×C₈₉	—	8	1	no	—
177	#10623494	—	4	2	no	—
178	C₂²×C₈₉	#386	1	1	yes	characteristic
179	C₂²×C₈₉	#461	1	1	yes	characteristic
180	C₂²×C₈₉	#463	1	1	yes	characteristic
181	C₂²×C₈₉	#465	1	1	yes	characteristic
182	C₂²×C₈₉	#466	1	1	yes	characteristic
183	C₂²×C₈₉	#471	1	1	yes	characteristic
184	C₂²×C₈₉	#473	1	1	yes	characteristic
185	C₂×D₈₉	—	2	4	no	—
186	C₂×D₈₉	—	4	2	no	—
187	C₂×D₈₉	—	4	2	no	—
188	C₂×D₈₉	—	4	2	no	—
189	C₂×D₈₉	—	4	2	no	—
190	C₂×D₈₉	—	4	2	no	—
191	C₂×D₈₉	—	4	2	no	—
192	C₄×C₈₉	—	2	1	no	—
193	C₄×C₈₉	—	2	1	no	—
194	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
195	C₄×C₈₉	#416	1	1	yes	characteristic
196	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
197	C₄×C₈₉	#440	1	1	yes	characteristic
198	C₄×C₈₉	#450	1	1	yes	characteristic
199	C₄×C₈₉	#452	1	1	yes	characteristic
200	C₄×C₈₉	#465	1	1	yes	characteristic
201	C₄×C₈₉	#466	1	1	yes	characteristic
202	Dic₈₉	—	4	2	no	—
203	Dic₈₉	—	8	2	no	—
204	Dic₈₉	—	16	1	no	—
205	#9217	—	89	1	no	maximal, Sylow
206	C₂×C₈₉	#1671	1	1	yes	characteristic
207	C₂×C₈₉	#1673	1	1	yes	characteristic
208	C₂×C₈₉	#1679	1	1	yes	characteristic
209	C₂×C₈₉	#1681	1	1	yes	characteristic
210	C₂×C₈₉	#1877	1	1	yes	characteristic
211	C₂×C₈₉	#1893	1	1	yes	characteristic
212	C₂×C₈₉	#1908	1	1	yes	characteristic
213	D₈₉	—	4	4	no	—
214	C₂×#461	—	89	1	no	—
215	C₂×C₈×D₄	—	89	1	no	—
216	#1990	—	89	1	no	—
217	#1802	—	89	1	no	—
218	C₂×#332	—	89	1	no	—
219	#1296	—	89	1	no	—
220	#1277	D₈₉	1	1	yes	characteristic, complemented
221	C₈₉	#9217	1	1	yes	characteristic, complemented, Sylow
222	C₂²×SD₁₆	—	178	1	no	—
223	C₂³×C₈	—	178	1	no	—
224	C₂×C₄⋊Q₈	—	89	1	no	—
225	C₂×C₄×D₄	—	89	1	no	—
226	#461	—	89	4	no	—
227	#461	—	178	2	no	—
228	C₈×D₄	—	178	2	no	—
229	C₈×D₄	—	178	2	no	—
230	C₂×C₄⋊C₈	—	89	1	no	—
231	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
232	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
233	C₂×#103	—	178	1	no	—
234	C₂×#102	—	178	1	no	—
235	C₂×C₂²⋊C₈	—	178	1	no	—
236	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
237	#384	—	89	1	no	—
238	#340	—	178	1	no	—
239	#340	—	178	1	no	—
240	#340	—	178	1	no	—
241	#332	—	178	2	no	—
242	C₂²×Q₈	—	178	1	no	—
243	C₂²×D₄	—	89	1	no	—
244	C₂³×C₄	—	178	1	no	—
245	C₂×SD₁₆	—	356	2	no	—
246	C₂×SD₁₆	—	356	2	no	—
247	C₂×SD₁₆	—	178	4	no	—
248	C₂²×C₈	—	2	1	no	—
249	C₂²×C₈	—	2	1	no	—
250	C₂²×C₈	—	356	2	no	—
251	C₂²×C₈	—	356	2	no	—
252	C₂²×C₈	—	178	4	no	—
253	C₂²×C₈	—	178	1	no	—
254	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
255	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
256	C₄⋊Q₈	—	89	2	no	—
257	C₄⋊Q₈	—	178	1	no	—
258	C₄⋊Q₈	—	178	1	no	—
259	C₄⋊Q₈	—	178	1	no	—
260	C₄×D₄	—	89	2	no	—
261	C₄×D₄	—	89	2	no	—
262	C₄×D₄	—	178	1	no	—
263	C₄×D₄	—	178	1	no	—
264	C₂×C₄⋊C₄	—	89	1	no	—
265	C₂×C₄⋊C₄	—	178	1	no	—
266	C₂×C₄⋊C₄	—	178	1	no	—
267	C₂×C₄⋊C₄	—	178	1	no	—
268	C₂×C₄⋊C₄	—	178	1	no	—
269	C₂×C₄⋊C₄	—	178	1	no	—
270	C₂×C₂²⋊C₄	—	178	1	no	—
271	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	89	2	no	—
276	C₄⋊C₈	—	178	1	no	—
277	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
278	#103	—	178	4	no	—
279	#102	—	356	2	no	—
280	C₂²⋊C₈	—	178	4	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	2	1	no	—
283	C₂⁴	—	178	1	no	—
284	C₂×Q₈	—	356	1	no	—
285	C₂×Q₈	—	356	1	no	—
286	C₂×Q₈	—	356	1	no	—
287	C₂×Q₈	—	356	1	no	—
288	C₂×Q₈	—	356	1	no	—
289	C₂×Q₈	—	178	2	no	—
290	C₂×D₄	—	89	2	no	—
291	C₂×D₄	—	89	2	no	—
292	C₂×D₄	—	89	2	no	—
293	C₂×D₄	—	178	1	no	—
294	C₂×D₄	—	178	1	no	—
295	C₂×D₄	—	178	1	no	—
296	C₂²×C₄	—	356	1	no	—
297	C₂²×C₄	—	356	1	no	—
298	C₂²×C₄	—	356	1	no	—
299	C₂²×C₄	—	356	1	no	—
300	C₂²×C₄	—	356	1	no	—
301	C₂²×C₄	—	356	1	no	—
302	C₂²×C₄	—	178	1	no	—
303	C₂²×C₄	—	178	2	no	—
304	C₂²×C₄	—	178	2	no	—
305	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
307	C₂²×C₄	#13083575	1	1	yes	characteristic
308	SD₁₆	—	356	8	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	1	no	—
318	C₂×C₈	—	4	1	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	356	8	no	—
321	C₂×C₈	—	356	1	no	—
322	C₂×C₈	—	356	1	no	—
323	C₂×C₈	—	178	2	no	—
324	C₂×C₈	#13083563	1	2	yes	complemented
325	C₄⋊C₄	—	89	2	no	—
326	C₄⋊C₄	—	356	1	no	—
327	C₄⋊C₄	—	356	1	no	—
328	C₄⋊C₄	—	356	1	no	—
329	C₄⋊C₄	—	356	1	no	—
330	C₄⋊C₄	—	356	1	no	—
331	C₄⋊C₄	—	356	1	no	—
332	C₄⋊C₄	—	178	2	no	—
333	C₄⋊C₄	—	178	2	no	—
334	C₄⋊C₄	—	178	1	no	—
335	C₄⋊C₄	—	712	1	no	—
336	C₂²⋊C₄	—	178	2	no	—
337	C₂²⋊C₄	—	178	2	no	—
338	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
339	C₄²	D₄×D₈₉	1	1	yes	characteristic
340	C₄²	D₄×D₈₉	1	1	yes	characteristic
341	C₄²	#13083574	1	1	yes	characteristic
342	C₂³	—	356	1	no	—
343	C₂³	—	356	1	no	—
344	C₂³	—	356	1	no	—
345	C₂³	—	356	1	no	—
346	C₂³	—	178	2	no	—
347	C₂³	—	178	2	no	—
348	C₂³	—	178	2	no	—
349	C₂³	#15667987	1	1	yes	characteristic
350	Q₈	—	356	2	no	—
351	Q₈	—	356	2	no	—
352	Q₈	—	712	1	no	—
353	Q₈	—	712	1	no	—
354	D₄	—	89	4	no	—
355	D₄	—	178	2	no	—
356	D₄	—	178	2	no	—
357	D₄	—	178	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	356	1	no	—
361	C₂×C₄	—	356	1	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	—	356	2	no	—
367	C₂×C₄	—	178	2	no	—
368	C₂×C₄	—	712	1	no	—
369	C₂×C₄	—	712	1	no	—
370	C₂×C₄	—	712	1	no	—
371	C₂×C₄	—	712	1	no	—
372	C₂×C₄	—	712	1	no	—
373	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
374	C₂×C₄	#15667974	1	1	yes	characteristic
375	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
376	C₂×C₄	#15667982	1	1	yes	characteristic
377	C₂×C₄	#15667985	1	1	yes	characteristic
378	C₂×C₄	#15667997	1	1	yes	characteristic
379	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
380	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
381	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
382	C₂×C₄	#15668033	1	1	yes	characteristic
383	C₂×C₄	#15668034	1	1	yes	characteristic
384	C₂×C₄	#15668035	1	1	yes	characteristic
385	C₂×C₄	#15668040	1	1	yes	characteristic
386	C₂×C₄	#15668045	1	1	yes	characteristic
387	C₈	—	2	2	no	—
388	C₈	—	4	1	no	—
389	C₈	—	4	2	no	—
390	C₈	—	8	1	no	—
391	C₈	—	356	2	no	—
392	C₂²	—	356	2	no	—
393	C₂²	—	356	2	no	—
394	C₂²	—	356	2	no	—
395	C₂²	—	356	2	no	—
396	C₂²	—	356	2	no	—
397	C₂²	—	356	2	no	—
398	C₂²	—	178	4	no	—
399	C₂²	#19151476	1	1	yes	characteristic
400	C₂²	#19151804	1	1	yes	characteristic
401	C₂²	#19151814	1	1	yes	characteristic
402	C₂²	#19151828	1	1	yes	characteristic
403	C₂²	#19151835	1	1	yes	characteristic
404	C₂²	#19151862	1	1	yes	characteristic
405	C₂²	#19151874	1	1	yes	characteristic
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	356	2	no	—
409	C₄	—	1424	1	no	—
410	C₄	—	712	2	no	—
411	C₄	#102×D₈₉	1	1	yes	characteristic
412	C₄	#19151593	1	1	yes	characteristic
413	C₄	C₄×#13083563	1	1	yes	characteristic
414	C₄	#19151700	1	1	yes	characteristic
415	C₄	#19151751	1	1	yes	characteristic
416	C₄	#19151762	1	1	yes	characteristic
417	C₄	#19151830	1	1	yes	characteristic
418	C₄	#19151833	1	1	yes	characteristic
419	C₂	—	356	4	no	—
420	C₂	#26019628	1	1	yes	characteristic
421	C₂	#26019642	1	1	yes	characteristic
422	C₂	#26019676	1	1	yes	characteristic
423	C₂	#26019690	1	1	yes	characteristic
424	C₂	#26020585	1	1	yes	characteristic
425	C₂	#26020679	1	1	yes	characteristic
426	C₂	#26020766	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667987
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1277	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₃₅₆	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1277	D₈₉
hypocenter	C₈₉	#9217
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
3. #26021213	C₂	semidirect
5. C₂×#19151345	C₂	semidirect
6. #26018136	C₂	semidirect
7. C₈₉×#1277	C₂	semidirect
13. #19151804	C₄	semidirect
123. C₂×C₈×C₈₉	SD₁₆	semidirect
220. #1277	D₈₉	semidirect
221. C₈₉	#9217	semidirect
324. C₂×C₈	#13083563	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151804	2
C₂×C₈×D₃₅₆	2
#26021213	2
#26020253	2
C₂×#19151345	2
#26018136	2
C₈₉×#1277	2
#9217	89