Subgroups — #36573293

Order 22784.

774 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26023209	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26021022	C₂	1	2	yes	maximal
4	#26021016	C₂	1	2	yes	maximal
5	#26021016	C₂	1	1	yes	maximal, characteristic
6	#26021011	C₂	1	1	yes	maximal, characteristic
7	#26020420	C₂	1	2	yes	maximal
8	#26020409	C₂	1	1	yes	maximal, characteristic
9	#26020149	C₂	1	2	yes	maximal
10	C₄×#15667946	C₂	1	1	yes	maximal, characteristic
11	C₂×#446×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×Dic₈₉	—	2	1	no	—
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667946	C₂²	1	2	yes	—
15	C₂×#15667946	C₂²	1	1	yes	characteristic
16	C₂×#15667944	C₂²	1	2	yes	—
17	C₂×#15667944	C₂²	1	2	yes	—
18	C₂×#15667944	C₂²	1	2	yes	—
19	#19151505	—	2	1	no	—
20	#19151497	—	2	1	no	—
21	#19151497	—	2	2	no	—
22	#19151484	—	2	2	no	—
23	#19151468	C₂²	1	1	yes	characteristic
24	#19151468	C₂²	1	2	yes	—
25	#19151468	C₂²	1	1	yes	characteristic
26	#19151464	C₂²	1	2	yes	—
27	C₄⋊C₄×Dic₈₉	—	2	1	no	—
28	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
30	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
32	#19151361	C₂²	1	1	yes	characteristic
33	#19151361	C₂²	1	1	yes	characteristic
34	#19151361	C₂²	1	2	yes	—
35	#19151361	C₂²	1	2	yes	—
36	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	#446×C₈₉	C₄	1	8	yes	complemented
38	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
40	C₂×#97×C₈₉	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
42	Q₈×Dic₈₉	—	4	4	no	—
43	C₂×C₄⋊Dic₈₉	—	2	1	no	—
44	C₂×C₄⋊Dic₈₉	—	2	2	no	—
45	C₂×C₄⋊Dic₈₉	—	2	1	no	—
46	C₂×C₄⋊Dic₈₉	—	2	1	no	—
47	C₂×C₄⋊Dic₈₉	—	2	2	no	—
48	C₂×C₄⋊Dic₈₉	—	2	1	no	—
49	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₈₉	—	2	1	no	—
56	C₂×C₄×Dic₈₉	—	2	2	no	—
57	C₂×C₄×Dic₈₉	—	2	1	no	—
58	C₂×C₄×Dic₈₉	—	2	2	no	—
59	#15667946	—	2	2	no	—
60	#15667946	—	4	2	no	—
61	#15667946	Q₈	1	4	yes	complemented
62	#15667944	—	2	4	no	—
63	#15667944	—	4	2	no	—
64	#15667944	—	4	2	no	—
65	#15667919	—	2	1	no	—
66	#15667919	—	2	2	no	—
67	#15667919	—	2	2	no	—
68	#15667919	—	2	1	no	—
69	#15667919	—	2	2	no	—
70	#15667919	—	2	2	no	—
71	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
76	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
77	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
81	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
82	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
83	#97×C₈₉	C₂×C₄	1	8	yes	—
84	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
85	C₂²×Dic₈₉	—	2	1	no	—
86	C₂²×Dic₈₉	—	2	1	no	—
87	C₂²×Dic₈₉	—	2	1	no	—
88	C₂²×Dic₈₉	—	2	2	no	—
89	C₂²×Dic₈₉	—	2	2	no	—
90	C₂²×Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	—	2	1	no	—
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	—	2	1	no	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	2	1	no	—
99	C₄⋊Dic₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	2	1	no	—
101	C₄⋊Dic₈₉	—	4	4	no	—
102	C₄⋊Dic₈₉	—	4	2	no	—
103	C₄⋊Dic₈₉	—	4	2	no	—
104	C₄⋊Dic₈₉	—	8	1	no	—
105	C₄⋊Dic₈₉	—	8	2	no	—
106	C₄⋊Dic₈₉	—	8	1	no	—
107	C₄⋊Dic₈₉	C₂×Q₈	1	2	yes	—
108	C₄⋊Dic₈₉	C₂×Q₈	1	2	yes	—
109	C₄×Dic₈₉	—	4	4	no	—
110	C₄×Dic₈₉	—	4	2	no	—
111	C₄×Dic₈₉	—	4	2	no	—
112	C₄×Dic₈₉	—	8	2	no	—
113	C₂×Q₈×C₈₉	—	2	2	no	—
114	C₂×Q₈×C₈₉	—	2	2	no	—
115	C₂×Q₈×C₈₉	C₄⋊C₄	1	4	yes	—
116	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₈₉	—	2	1	no	—
122	C₂×C₈×C₈₉	—	2	2	no	—
123	C₂×C₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
127	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
129	C₂×C₈×C₈₉	#41	1	2	yes	—
130	C₄⋊C₄×C₈₉	C₄⋊C₄	1	8	yes	—
131	C₄⋊C₄×C₈₉	C₄⋊C₄	1	4	yes	—
132	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
133	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
134	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
135	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
136	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	Q₈×C₈₉	—	2	8	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	1	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
145	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
146	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
147	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₈₉	#137	1	2	yes	—
150	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×Dic₈₉	—	2	2	no	—
157	C₂×Dic₈₉	—	2	2	no	—
158	C₂×Dic₈₉	—	4	1	no	—
159	C₂×Dic₈₉	—	4	1	no	—
160	C₂×Dic₈₉	—	4	1	no	—
161	C₂×Dic₈₉	—	4	1	no	—
162	C₂×Dic₈₉	—	4	2	no	—
163	C₂×Dic₈₉	—	4	1	no	—
164	C₂×Dic₈₉	—	4	1	no	—
165	C₂×Dic₈₉	—	4	2	no	—
166	C₂×Dic₈₉	—	4	2	no	—
167	C₂×Dic₈₉	—	4	1	no	—
168	C₂×Dic₈₉	—	4	1	no	—
169	C₂×Dic₈₉	—	4	2	no	—
170	C₂×Dic₈₉	—	4	2	no	—
171	C₂×Dic₈₉	—	4	2	no	—
172	C₂×Dic₈₉	—	4	1	no	—
173	C₂×Dic₈₉	—	4	1	no	—
174	C₈×C₈₉	—	2	2	no	—
175	C₈×C₈₉	—	2	4	no	—
176	C₈×C₈₉	C₄×Q₈	1	4	yes	complemented
177	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
179	C₂²×C₈₉	#550	1	1	yes	characteristic
180	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
182	C₂²×C₈₉	#577	1	1	yes	characteristic
183	C₂²×C₈₉	#577	1	1	yes	characteristic
184	C₄×C₈₉	—	2	4	no	—
185	C₄×C₈₉	—	2	4	no	—
186	C₄×C₈₉	—	2	2	no	—
187	C₄×C₈₉	#428	1	4	yes	—
188	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
189	C₄×C₈₉	#520	1	1	yes	characteristic
190	C₄×C₈₉	#557	1	1	yes	characteristic
191	C₄×C₈₉	Q₈²	1	1	yes	characteristic
192	Dic₈₉	—	4	2	no	—
193	Dic₈₉	—	4	2	no	—
194	Dic₈₉	—	8	2	no	—
195	Dic₈₉	—	8	1	no	—
196	Dic₈₉	—	8	1	no	—
197	Dic₈₉	—	8	2	no	—
198	#22729	—	89	1	no	maximal, Sylow
199	C₂×C₈₉	Q₈×C₄⋊C₄	1	1	yes	characteristic
200	C₂×C₈₉	C₂×#428	1	1	yes	characteristic
201	C₂×C₈₉	#2912	1	1	yes	characteristic
202	C₂×C₈₉	#3384	1	1	yes	characteristic
203	C₂×C₈₉	#3387	1	1	yes	characteristic
204	C₂×C₈₉	#3397	1	1	yes	characteristic
205	C₂×C₈₉	#3400	1	1	yes	characteristic
206	C₂×#446	D₈₉	1	1	yes	characteristic
207	#2355	—	89	1	no	—
208	Q₈×C₄⋊C₄	—	89	1	no	—
209	#1949	—	89	2	no	—
210	#1948	—	89	2	no	—
211	#1948	—	89	1	no	—
212	#1947	—	89	1	no	—
213	#1847	—	89	2	no	—
214	#1844	—	89	1	no	—
215	#1776	—	89	2	no	—
216	C₄×#107	—	89	1	no	—
217	C₈₉	#22729	1	1	yes	characteristic, complemented, Sylow
218	C₂×C₄×Q₈	—	178	1	no	—
219	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
220	#446	Dic₈₉	1	8	yes	complemented
221	C₂×#107	—	89	1	no	—
222	C₂×#107	—	89	2	no	—
223	C₂×#107	—	89	1	no	—
224	C₂×#106	—	89	2	no	—
225	C₂×#106	—	89	2	no	—
226	C₂×#106	—	89	2	no	—
227	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
228	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
229	C₂×#97	C₂×D₈₉	1	2	yes	—
230	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
231	#391	—	178	1	no	—
232	#389	—	178	1	no	—
233	#389	—	178	2	no	—
234	#387	—	178	2	no	—
235	#384	—	89	1	no	—
236	#384	—	89	2	no	—
237	#384	—	89	1	no	—
238	#382	—	89	2	no	—
239	C₄×C₄⋊C₄	—	89	1	no	—
240	C₄×C₄⋊C₄	—	89	2	no	—
241	C₄×C₄⋊C₄	—	89	1	no	—
242	C₄×C₄⋊C₄	—	89	2	no	—
243	C₄×C₄⋊C₄	—	178	1	no	—
244	#340	—	89	2	no	—
245	#340	—	89	1	no	—
246	#340	—	89	2	no	—
247	#340	—	89	1	no	—
248	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
249	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
251	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₄×Q₈	—	356	4	no	—
253	C₄×Q₈	Dic₁₇₈	1	4	yes	—
254	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
255	C₂×C₄⋊C₄	—	89	1	no	—
256	C₂×C₄⋊C₄	—	89	1	no	—
257	C₂×C₄⋊C₄	—	89	1	no	—
258	C₂×C₄⋊C₄	—	89	2	no	—
259	C₂×C₄⋊C₄	—	89	2	no	—
260	C₂×C₄⋊C₄	—	89	1	no	—
261	C₂×C₄⋊C₄	—	178	2	no	—
262	C₂×C₄⋊C₄	—	178	1	no	—
263	C₂×C₄⋊C₄	—	178	1	no	—
264	C₂×C₄⋊C₄	—	178	1	no	—
265	C₂×C₄⋊C₄	—	178	2	no	—
266	C₂×C₄⋊C₄	—	178	1	no	—
267	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
268	C₂×C₄⋊C₄	D₃₅₆	1	2	yes	—
269	C₂×C₄²	—	178	2	no	—
270	C₂×C₄²	—	178	1	no	—
271	C₂×C₄²	—	178	1	no	—
272	C₂×C₄²	—	178	2	no	—
273	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
274	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
275	#107	—	89	4	no	—
276	#107	—	356	2	no	—
277	#107	—	178	2	no	—
278	#106	—	356	2	no	—
279	#106	—	356	2	no	—
280	#106	—	178	4	no	—
281	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
282	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
283	#97	C₂×Dic₈₉	1	8	yes	—
284	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
285	#95	—	178	1	no	—
286	#95	—	178	2	no	—
287	#95	—	178	2	no	—
288	#95	—	178	1	no	—
289	#95	—	178	2	no	—
290	#95	—	178	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	C₄⋊Dic₈₉	1	4	yes	—
294	C₂²×C₄	—	178	1	no	—
295	C₂²×C₄	—	178	2	no	—
296	C₂²×C₄	—	178	2	no	—
297	C₂²×C₄	—	178	1	no	—
298	C₂²×C₄	—	178	1	no	—
299	C₂²×C₄	—	178	1	no	—
300	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
301	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
302	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
304	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
310	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
311	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
312	C₂×C₈	Q₈×D₈₉	1	2	yes	—
313	C₂×C₈	#13083575	1	2	yes	—
314	C₄⋊C₄	—	89	2	no	—
315	C₄⋊C₄	—	89	2	no	—
316	C₄⋊C₄	—	356	4	no	—
317	C₄⋊C₄	—	356	2	no	—
318	C₄⋊C₄	—	356	2	no	—
319	C₄⋊C₄	—	178	2	no	—
320	C₄⋊C₄	—	178	2	no	—
321	C₄⋊C₄	—	178	1	no	—
322	C₄⋊C₄	—	178	1	no	—
323	C₄⋊C₄	—	178	2	no	—
324	C₄⋊C₄	—	178	1	no	—
325	C₄⋊C₄	—	178	2	no	—
326	C₄⋊C₄	—	178	1	no	—
327	C₄⋊C₄	—	178	1	no	—
328	C₄⋊C₄	—	178	1	no	—
329	C₄⋊C₄	—	712	1	no	—
330	C₄⋊C₄	—	712	2	no	—
331	C₄⋊C₄	—	712	1	no	—
332	C₄⋊C₄	C₄⋊Dic₈₉	1	8	yes	—
333	C₄⋊C₄	C₄⋊Dic₈₉	1	4	yes	—
334	C₄²	—	356	4	no	—
335	C₄²	—	356	2	no	—
336	C₄²	—	356	2	no	—
337	C₄²	—	712	2	no	—
338	C₄²	C₂²×Dic₈₉	1	2	yes	—
339	C₄²	C₂²×Dic₈₉	1	4	yes	—
340	C₄²	C₂×Dic₁₇₈	1	4	yes	—
341	C₄²	C₂×Dic₁₇₈	1	2	yes	—
342	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
343	Q₈	—	2	8	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	356	1	no	—
351	C₂×C₄	—	356	1	no	—
352	C₂×C₄	—	356	1	no	—
353	C₂×C₄	—	356	1	no	—
354	C₂×C₄	—	356	2	no	—
355	C₂×C₄	—	356	1	no	—
356	C₂×C₄	—	356	1	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	2	no	—
363	C₂×C₄	—	356	1	no	—
364	C₂×C₄	—	356	1	no	—
365	C₂×C₄	—	356	2	no	—
366	C₂×C₄	—	178	2	no	—
367	C₂×C₄	—	178	2	no	—
368	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
369	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
370	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
371	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
372	C₂×C₄	#15668040	1	2	yes	—
373	C₂×C₄	#15668045	1	2	yes	—
374	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
375	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
376	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
377	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
378	C₂×C₄	#15668058	1	1	yes	characteristic
379	C₂×C₄	#15668059	1	1	yes	characteristic
380	C₈	—	2	2	no	—
381	C₈	—	2	4	no	—
382	C₈	Q₈×Dic₈₉	1	4	yes	complemented
383	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
384	C₂²	Q₈×D₃₅₆	1	1	yes	characteristic
385	C₂²	#19152270	1	1	yes	characteristic
386	C₂²	C₂×#15668040	1	1	yes	characteristic
387	C₂²	C₂×#15668045	1	1	yes	characteristic
388	C₂²	#19152410	1	1	yes	characteristic
389	C₂²	#19152410	1	1	yes	characteristic
390	C₄	—	2	4	no	—
391	C₄	—	2	4	no	—
392	C₄	—	2	2	no	—
393	C₄	—	356	2	no	—
394	C₄	—	356	2	no	—
395	C₄	—	712	2	no	—
396	C₄	—	712	1	no	—
397	C₄	—	712	1	no	—
398	C₄	—	712	2	no	—
399	C₄	#19151640	1	4	yes	—
400	C₄	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
401	C₄	#19152081	1	1	yes	characteristic
402	C₄	#19152327	1	1	yes	characteristic
403	C₄	Q₈×Dic₁₇₈	1	1	yes	characteristic
404	C₂	Q₈×C₄⋊Dic₈₉	1	1	yes	characteristic
405	C₂	C₂×#19151640	1	1	yes	characteristic
406	C₂	#26028847	1	1	yes	characteristic
407	C₂	#26033153	1	1	yes	characteristic
408	C₂	#26033186	1	1	yes	characteristic
409	C₂	#26033274	1	1	yes	characteristic
410	C₂	#26033307	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#446×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×#446	D₈₉
hypocenter	C₈₉	#22729
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
37. #446×C₈₉	C₄	semidirect
61. #15667946	Q₈	semidirect
176. C₈×C₈₉	C₄×Q₈	semidirect
217. C₈₉	#22729	semidirect
220. #446	Dic₈₉	semidirect
382. C₈	Q₈×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26023209	2
Q₈×C₄⋊Dic₈₉	2
#26021022	2
#26021016	2
#26021011	2
#26020420	2
#26020409	2
#26020149	2
C₄×#15667946	2
C₂×#446×C₈₉	2
#22729	89