Subgroups — #36524004

Order 22784.

688 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029218	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#13083574	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#15668034	C₂	1	1	yes	maximal, characteristic
4	#26019132	C₂	1	2	yes	maximal, complemented
5	#26019125	C₂	1	2	yes	maximal, complemented
6	#26019118	C₂	1	2	yes	maximal, complemented
7	#26019111	C₂	1	2	yes	maximal, complemented
8	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018635	C₂	1	1	yes	maximal, characteristic, complemented
10	#26018635	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668034	C₄	1	1	yes	characteristic
13	C₄×#13083574	C₂²	1	1	yes	characteristic
14	C₄×#13083574	C₂²	1	1	yes	characteristic
15	#19151728	C₂²	1	2	yes	—
16	C₈×Dic₁₇₈	C₂²	1	2	yes	—
17	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
18	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
19	C₄×#13083563	C₄	1	1	yes	characteristic
20	C₄×#13083563	C₄	1	1	yes	characteristic
21	C₄×#13083563	C₂²	1	1	yes	characteristic
22	C₄×#13083563	C₂²	1	1	yes	characteristic
23	C₄×D₇₁₂	C₄	1	1	yes	characteristic
24	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
25	#19151677	C₂²	1	2	yes	—
26	#19151669	—	2	2	no	—
27	C₈×D₃₅₆	C₂²	1	2	yes	—
28	C₈×C₂²⋊D₈₉	—	2	2	no	—
29	#19151651	C₂²	1	1	yes	characteristic
30	#19151651	C₂²	1	1	yes	characteristic
31	#19151649	—	2	1	no	—
32	#19151649	—	2	1	no	—
33	#19151636	C₄	1	1	yes	characteristic
34	#19151595	C₄	1	1	yes	characteristic
35	#19151595	C₄	1	1	yes	characteristic
36	C₄×#13083557	—	2	1	no	—
37	C₄×C₈×D₈₉	—	2	1	no	—
38	#19151351	C₂²	1	2	yes	—
39	#19151349	C₂²	1	2	yes	—
40	#19151330	C₂²	1	2	yes	—
41	#19151330	C₂²	1	2	yes	—
42	#19151327	C₂²	1	2	yes	—
43	#19151327	C₂²	1	2	yes	—
44	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
47	#322×C₈₉	C₂²	1	2	yes	—
48	#322×C₈₉	C₂²	1	2	yes	—
49	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
50	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
51	#15668034	—	2	4	no	—
52	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
53	C₂×#13083563	C₂×C₄	1	1	yes	characteristic
54	C₂×#13083563	C₂×C₄	1	1	yes	characteristic
55	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
56	C₂×#13083557	—	2	2	no	—
57	C₂×C₈×D₈₉	—	2	2	no	—
58	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
63	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
64	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₄×C₂²⋊D₈₉	—	2	1	no	—
67	C₄×C₂²⋊D₈₉	—	2	1	no	—
68	C₄×C₂²⋊D₈₉	—	2	1	no	—
69	C₄×C₂²⋊D₈₉	—	2	1	no	—
70	#15667971	C₂×C₄	1	1	yes	characteristic
71	#15667971	C₂×C₄	1	1	yes	characteristic
72	#15667969	—	2	1	no	—
73	#15667969	—	2	1	no	—
74	C₄²×D₈₉	—	2	1	no	—
75	C₄²×D₈₉	—	2	1	no	—
76	#15667946	C₂×C₄	1	1	yes	characteristic
77	#15667946	C₂×C₄	1	1	yes	characteristic
78	#15667944	C₂×C₄	1	1	yes	characteristic
79	#15667944	C₂×C₄	1	1	yes	characteristic
80	C₄⋊#10623494	C₂³	1	2	yes	—
81	C₄⋊#10623494	C₂³	1	2	yes	—
82	#15667941	—	2	2	no	—
83	#15667941	—	2	2	no	—
84	#15667937	C₂×C₄	1	1	yes	characteristic
85	#15667937	C₂×C₄	1	1	yes	characteristic
86	#15667937	C₂×C₄	1	1	yes	characteristic
87	#15667937	C₂×C₄	1	1	yes	characteristic
88	#15667934	C₂×C₄	1	1	yes	characteristic
89	#15667934	C₂×C₄	1	1	yes	characteristic
90	#15667934	C₂×C₄	1	1	yes	characteristic
91	#15667934	C₂×C₄	1	1	yes	characteristic
92	C₂²⋊#10623494	—	2	1	no	—
93	C₂²⋊#10623494	—	2	1	no	—
94	#15667924	—	2	2	no	—
95	#15667924	—	2	2	no	—
96	#15667922	—	2	2	no	—
97	C₄×#10623494	—	2	1	no	—
98	C₄×#10623494	—	2	1	no	—
99	C₈×Dic₈₉	—	2	2	no	—
100	C₂²×C₈×C₈₉	—	2	1	no	—
101	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
103	C₂²×C₈×C₈₉	D₄	1	2	yes	—
104	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
105	C₄×C₈×C₈₉	—	2	1	no	—
106	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
107	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
108	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
109	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
110	C₄×C₈×C₈₉	C₂³	1	2	yes	—
111	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₈₉	C₂³	1	2	yes	—
114	#13083574	—	2	2	no	—
115	#13083574	—	2	2	no	—
116	#13083574	—	2	2	no	—
117	#13083574	M₁₆	1	4	yes	complemented
118	C₂×Dic₁₇₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₇₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₃₅₆	C₂²×C₄	1	1	yes	characteristic
121	C₂×D₃₅₆	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₂²⋊D₈₉	—	2	1	no	—
123	C₂×C₂²⋊D₈₉	—	2	1	no	—
124	C₂×C₄×D₈₉	—	2	1	no	—
125	C₂×C₄×D₈₉	—	2	1	no	—
126	C₂×C₄×D₈₉	—	2	1	no	—
127	C₂×C₄×D₈₉	—	2	1	no	—
128	C₂×C₄×D₈₉	—	2	1	no	—
129	C₂×C₄×D₈₉	—	2	1	no	—
130	Dic₃₅₆	—	2	2	no	—
131	#13083563	—	2	2	no	—
132	#13083563	—	2	2	no	—
133	D₇₁₂	—	2	2	no	—
134	#13083557	—	4	4	no	—
135	C₂×#10623494	—	2	2	no	—
136	C₂×#10623494	—	2	2	no	—
137	C₈×D₈₉	—	4	4	no	—
138	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
139	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
140	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
141	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
142	#13083553	—	2	1	no	—
143	#13083553	—	2	1	no	—
144	#13083553	—	2	1	no	—
145	#13083553	—	2	1	no	—
146	C₂²⋊Dic₈₉	—	2	1	no	—
147	C₂²⋊Dic₈₉	—	2	1	no	—
148	#13083551	—	2	1	no	—
149	#13083551	—	2	1	no	—
150	#13083551	—	2	1	no	—
151	#13083551	—	2	1	no	—
152	C₄×Dic₈₉	—	2	1	no	—
153	C₄×Dic₈₉	—	2	1	no	—
154	C₄×Dic₈₉	—	2	1	no	—
155	C₄×Dic₈₉	—	2	1	no	—
156	C₄×Dic₈₉	—	2	1	no	—
157	C₄×Dic₈₉	—	2	1	no	—
158	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
161	C₂×C₈×C₈₉	—	2	4	no	—
162	C₂×C₈×C₈₉	—	2	2	no	—
163	C₂×C₈×C₈₉	—	2	2	no	—
164	C₂×C₈×C₈₉	—	2	2	no	—
165	C₂×C₈×C₈₉	—	4	2	no	—
166	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
172	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₈₉	#41	1	1	yes	characteristic
175	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
176	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
177	C₂²×D₈₉	—	2	1	no	—
178	C₂²×D₈₉	—	2	1	no	—
179	C₂×C₄×C₈₉	—	2	2	no	—
180	C₂×C₄×C₈₉	—	2	2	no	—
181	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
185	C₂×C₄×C₈₉	#131	1	2	yes	—
186	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
189	C₂²⋊D₈₉	—	2	4	no	—
190	C₂²⋊D₈₉	—	4	2	no	—
191	C₂×Dic₈₉	—	2	1	no	—
192	C₂×Dic₈₉	—	2	1	no	—
193	C₂×Dic₈₉	—	2	1	no	—
194	C₂×Dic₈₉	—	2	1	no	—
195	C₂×Dic₈₉	—	2	1	no	—
196	C₂×Dic₈₉	—	2	1	no	—
197	D₃₅₆	—	2	1	no	—
198	D₃₅₆	—	2	1	no	—
199	D₃₅₆	—	2	1	no	—
200	D₃₅₆	C₂×M₁₆	1	2	yes	—
201	C₄×D₈₉	—	2	4	no	—
202	C₄×D₈₉	—	4	2	no	—
203	C₄×D₈₉	—	4	2	no	—
204	C₄×D₈₉	—	4	2	no	—
205	C₄×D₈₉	—	4	2	no	—
206	C₄×D₈₉	—	4	2	no	—
207	Dic₁₇₈	—	2	1	no	—
208	Dic₁₇₈	—	2	1	no	—
209	Dic₁₇₈	—	2	1	no	—
210	Dic₁₇₈	C₂×M₁₆	1	2	yes	—
211	C₈×C₈₉	—	2	4	no	—
212	C₈×C₈₉	—	2	1	no	—
213	C₈×C₈₉	—	2	1	no	—
214	C₈×C₈₉	—	2	1	no	—
215	C₈×C₈₉	—	2	1	no	—
216	C₈×C₈₉	—	2	2	no	—
217	C₈×C₈₉	—	2	2	no	—
218	#10623494	—	4	2	no	—
219	#10623494	—	4	2	no	—
220	C₂²×C₈₉	—	2	1	no	—
221	C₂²×C₈₉	—	2	1	no	—
222	C₂²×C₈₉	#435	1	2	yes	—
223	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂×D₈₉	—	2	2	no	—
225	C₂×D₈₉	—	4	1	no	—
226	C₂×D₈₉	—	4	1	no	—
227	C₂×D₈₉	—	4	1	no	—
228	C₂×D₈₉	—	4	1	no	—
229	C₂×D₈₉	—	4	1	no	—
230	C₄×C₈₉	—	2	1	no	—
231	C₄×C₈₉	—	2	1	no	—
232	C₄×C₈₉	#435	1	2	yes	—
233	C₄×C₈₉	#443	1	2	yes	—
234	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
235	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
236	C₄×C₈₉	#575	1	2	yes	—
237	Dic₈₉	—	2	2	no	—
238	Dic₈₉	—	4	1	no	—
239	Dic₈₉	—	4	1	no	—
240	Dic₈₉	—	4	1	no	—
241	Dic₈₉	—	4	1	no	—
242	Dic₈₉	—	4	1	no	—
243	#17868	—	89	1	no	maximal, Sylow
244	C₂×C₈₉	—	2	2	no	—
245	C₂×C₈₉	C₂×#435	1	1	yes	characteristic
246	C₂×C₈₉	#2943	1	1	yes	characteristic
247	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
248	D₈₉	—	4	2	no	—
249	D₈₉	—	8	1	no	—
250	#2965	—	89	1	no	—
251	C₈×#41	—	89	1	no	—
252	C₄×#135	—	89	1	no	—
253	#1588	—	89	2	no	—
254	#1587	—	89	2	no	—
255	#1586	—	89	2	no	—
256	#1585	—	89	2	no	—
257	#1564	—	89	1	no	—
258	#1476	—	89	1	no	—
259	#1476	—	89	1	no	—
260	C₂×#322	D₈₉	1	1	yes	characteristic, complemented
261	C₈₉	#17868	1	1	yes	characteristic, complemented, Sylow
262	C₂×#135	—	89	1	no	—
263	C₄×#41	—	89	1	no	—
264	C₄×#41	—	89	1	no	—
265	#446	—	89	2	no	—
266	C₈×Q₈	—	89	2	no	—
267	C₄×Q₁₆	—	89	1	no	—
268	C₄×Q₁₆	—	89	1	no	—
269	C₄×SD₁₆	—	89	1	no	—
270	C₄×SD₁₆	—	89	1	no	—
271	C₄×SD₁₆	—	89	1	no	—
272	C₄×SD₁₆	—	89	1	no	—
273	C₄×D₈	—	89	1	no	—
274	C₄×D₈	—	89	1	no	—
275	#436	—	89	2	no	—
276	#435	—	178	2	no	—
277	C₈×D₄	—	89	2	no	—
278	C₈×D₄	—	178	2	no	—
279	#431	—	89	1	no	—
280	#431	—	89	1	no	—
281	#431	—	178	1	no	—
282	#431	—	178	1	no	—
283	#427	—	89	1	no	—
284	#416	—	89	1	no	—
285	#416	—	89	1	no	—
286	C₄×M₁₆	—	178	1	no	—
287	C₂×C₄×C₈	—	178	1	no	—
288	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
289	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
290	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
291	#335	—	89	2	no	—
292	#334	—	89	2	no	—
293	Q₈⋊C₈	—	89	2	no	—
294	Q₈⋊C₈	—	89	2	no	—
295	D₄⋊C₈	—	89	2	no	—
296	D₄⋊C₈	—	89	2	no	—
297	#322	C₂×D₈₉	1	2	yes	—
298	#322	C₂×D₈₉	1	2	yes	—
299	C₂×#41	—	89	1	no	—
300	C₂×#41	—	89	1	no	—
301	#135	—	178	4	no	—
302	C₂×Q₁₆	—	89	1	no	—
303	C₂×SD₁₆	—	89	1	no	—
304	C₂×SD₁₆	—	89	1	no	—
305	C₂×D₈	—	89	1	no	—
306	C₂×M₁₆	—	178	2	no	—
307	C₂²×C₈	—	2	1	no	—
308	C₂²×C₈	—	178	2	no	—
309	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
310	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
311	C₂²×C₈	D₃₅₆	1	2	yes	—
312	C₄×Q₈	—	89	1	no	—
313	C₄×Q₈	—	89	1	no	—
314	C₄×Q₈	—	89	1	no	—
315	C₄×Q₈	—	89	1	no	—
316	C₄×D₄	—	89	1	no	—
317	C₄×D₄	—	89	1	no	—
318	C₄×D₄	—	89	1	no	—
319	C₄×D₄	—	89	1	no	—
320	C₄×D₄	—	178	1	no	—
321	C₄×D₄	—	178	1	no	—
322	C₄×D₄	—	178	1	no	—
323	C₄×D₄	—	178	1	no	—
324	#117	—	89	1	no	—
325	#117	—	89	1	no	—
326	#117	—	178	1	no	—
327	#117	—	178	1	no	—
328	C₂×C₄²	—	178	1	no	—
329	C₂×C₄²	—	178	1	no	—
330	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
331	#107	—	89	1	no	—
332	#107	—	89	1	no	—
333	#106	—	89	1	no	—
334	#106	—	89	1	no	—
335	C₄⋊C₈	—	89	2	no	—
336	C₄⋊C₈	—	89	2	no	—
337	C₄⋊C₈	—	178	2	no	—
338	C₄⋊C₈	—	178	2	no	—
339	#103	—	89	1	no	—
340	#103	—	89	1	no	—
341	#103	—	89	1	no	—
342	#103	—	89	1	no	—
343	#102	—	89	1	no	—
344	#102	—	89	1	no	—
345	#102	—	89	1	no	—
346	#102	—	89	1	no	—
347	C₂²⋊C₈	—	178	1	no	—
348	C₂²⋊C₈	—	178	1	no	—
349	C₂²⋊C₈	—	178	2	no	—
350	C₂²⋊C₈	—	178	2	no	—
351	#97	—	178	2	no	—
352	C₄×C₈	—	2	1	no	—
353	C₄×C₈	—	178	2	no	—
354	C₄×C₈	—	178	1	no	—
355	C₄×C₈	—	178	1	no	—
356	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
357	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
358	C₄×C₈	D₃₅₆	1	1	yes	characteristic
359	C₄×C₈	D₃₅₆	1	1	yes	characteristic
360	C₄×C₈	C₂²×D₈₉	1	2	yes	—
361	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
362	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
363	C₄×C₈	C₂²×D₈₉	1	2	yes	—
364	#41	—	89	4	no	—
365	#41	—	178	2	no	—
366	#41	—	178	2	no	—
367	#41	—	178	2	no	—
368	C₂×Q₈	—	89	1	no	—
369	C₂×Q₈	—	89	1	no	—
370	C₂×D₄	—	89	1	no	—
371	C₂×D₄	—	89	1	no	—
372	C₂×D₄	—	178	1	no	—
373	C₂×D₄	—	178	1	no	—
374	C₂²×C₄	—	178	1	no	—
375	C₂²×C₄	—	178	1	no	—
376	C₂²×C₄	—	178	1	no	—
377	C₂²×C₄	—	178	1	no	—
378	C₂²×C₄	—	178	1	no	—
379	C₂²×C₄	—	178	1	no	—
380	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
382	C₂²×C₄	#13083574	1	1	yes	characteristic
383	Q₁₆	—	178	2	no	—
384	SD₁₆	—	178	2	no	—
385	SD₁₆	—	178	2	no	—
386	D₈	—	178	2	no	—
387	M₁₆	—	356	4	no	—
388	C₂×C₈	—	2	4	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	4	2	no	—
393	C₂×C₈	—	356	4	no	—
394	C₂×C₈	—	178	2	no	—
395	C₂×C₈	—	178	2	no	—
396	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
397	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
398	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
399	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
400	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
401	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
402	C₄⋊C₄	—	89	1	no	—
403	C₄⋊C₄	—	89	1	no	—
404	C₄⋊C₄	—	89	1	no	—
405	C₄⋊C₄	—	89	1	no	—
406	C₄⋊C₄	—	178	1	no	—
407	C₄⋊C₄	—	178	1	no	—
408	C₄⋊C₄	—	178	1	no	—
409	C₄⋊C₄	—	178	1	no	—
410	C₂²⋊C₄	—	178	1	no	—
411	C₂²⋊C₄	—	178	1	no	—
412	C₂²⋊C₄	—	178	1	no	—
413	C₂²⋊C₄	—	178	1	no	—
414	C₂²⋊C₄	—	178	1	no	—
415	C₂²⋊C₄	—	178	1	no	—
416	C₄²	—	178	1	no	—
417	C₄²	—	178	1	no	—
418	C₄²	—	178	1	no	—
419	C₄²	—	178	1	no	—
420	C₄²	—	178	1	no	—
421	C₄²	—	178	1	no	—
422	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
423	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
424	C₄²	#13083574	1	1	yes	characteristic
425	C₄²	C₂³×D₈₉	1	1	yes	characteristic
426	C₂³	—	178	1	no	—
427	C₂³	—	178	1	no	—
428	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
429	Q₈	—	89	2	no	—
430	Q₈	—	178	1	no	—
431	Q₈	—	178	1	no	—
432	Q₈	—	178	1	no	—
433	D₄	—	89	2	no	—
434	D₄	—	356	2	no	—
435	D₄	—	178	4	no	—
436	D₄	—	178	1	no	—
437	D₄	—	178	1	no	—
438	D₄	—	178	1	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	356	2	no	—
442	C₂×C₄	—	356	2	no	—
443	C₂×C₄	—	356	2	no	—
444	C₂×C₄	—	356	2	no	—
445	C₂×C₄	—	356	2	no	—
446	C₂×C₄	—	178	1	no	—
447	C₂×C₄	—	178	1	no	—
448	C₂×C₄	—	178	1	no	—
449	C₂×C₄	—	178	1	no	—
450	C₂×C₄	—	178	1	no	—
451	C₂×C₄	—	178	1	no	—
452	C₂×C₄	—	178	4	no	—
453	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
454	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
455	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
456	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
457	C₂×C₄	#15668020	1	2	yes	—
458	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
460	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
461	C₈	—	2	4	no	—
462	C₈	—	2	2	no	—
463	C₈	—	2	2	no	—
464	C₈	—	2	1	no	—
465	C₈	—	2	1	no	—
466	C₈	—	2	1	no	—
467	C₈	—	2	1	no	—
468	C₈	—	356	2	no	—
469	C₈	—	356	2	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	356	1	no	—
473	C₂²	—	356	1	no	—
474	C₂²	—	356	1	no	—
475	C₂²	—	356	1	no	—
476	C₂²	—	356	1	no	—
477	C₂²	—	178	2	no	—
478	C₂²	#19151671	1	2	yes	—
479	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	356	1	no	—
483	C₄	—	356	1	no	—
484	C₄	—	356	1	no	—
485	C₄	—	356	1	no	—
486	C₄	—	356	1	no	—
487	C₄	—	178	2	no	—
488	C₄	#19151671	1	2	yes	—
489	C₄	#19151716	1	2	yes	—
490	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
491	C₄	C₂×#15668020	1	1	yes	characteristic
492	C₄	#19152398	1	2	yes	—
493	C₂	—	2	2	no	—
494	C₂	—	356	2	no	—
495	C₂	—	712	1	no	—
496	C₂	C₂×#19151671	1	1	yes	characteristic
497	C₂	#26029054	1	1	yes	characteristic
498	C₂	C₂×#19151716	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂×#15668034	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#17868
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
1. #26029218	C₂	semidirect
2. C₈×#13083574	C₂	semidirect
4. #26019132	C₂	semidirect
5. #26019125	C₂	semidirect
6. #26019118	C₂	semidirect
7. #26019111	C₂	semidirect
8. #26019000	C₂	semidirect
9. #26018635	C₂	semidirect
10. #26018635	C₂	semidirect
11. C₂×#322×C₈₉	C₂	semidirect
117. #13083574	M₁₆	semidirect
260. C₂×#322	D₈₉	semidirect
261. C₈₉	#17868	semidirect

Maximal subgroups

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

subgroup	index
#26029218	2
C₈×#13083574	2
C₄×#15668034	2
#26019132	2
#26019125	2
#26019118	2
#26019111	2
#26019000	2
#26018635	2
C₂×#322×C₈₉	2
#17868	89