Subgroups — #36574319

Order 22784.

784 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×Q₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	#26021189	C₂	1	1	yes	maximal, characteristic
3	#26020810	C₂	1	2	yes	maximal
4	#26020808	C₂	1	2	yes	maximal
5	#26020403	C₂	1	1	yes	maximal, characteristic
6	#26020367	C₂	1	2	yes	maximal
7	#26020117	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
9	C₄×#15667941	C₂	1	1	yes	maximal, characteristic
10	#26020100	C₂	1	2	yes	maximal
11	C₈₉×#1984	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×Dic₈₉	C₄	1	1	yes	characteristic
13	C₂×Q₈×Dic₈₉	C₄	1	1	yes	characteristic
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×#15667941	C₂²	1	2	yes	—
16	C₂×#15667941	C₂²	1	1	yes	characteristic
17	C₂×#15667941	C₂²	1	2	yes	—
18	C₂×#15667941	C₂²	1	1	yes	characteristic
19	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
21	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₈₉	C₄	1	2	yes	—
23	C₄⋊C₄×Dic₈₉	C₄	1	1	yes	characteristic
24	C₄⋊C₄×Dic₈₉	C₄	1	1	yes	characteristic
25	C₄⋊C₄×Dic₈₉	C₄	1	2	yes	—
26	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
28	C₄²×Dic₈₉	C₂²	1	2	yes	—
29	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
30	#19151353	C₂²	1	2	yes	—
31	#19151353	C₂²	1	2	yes	—
32	#19151352	C₂²	1	1	yes	characteristic
33	#19151352	C₂²	1	1	yes	characteristic
34	#19151352	C₂²	1	2	yes	—
35	#19151352	C₂²	1	2	yes	—
36	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
39	#336×C₈₉	C₂²	1	2	yes	—
40	#336×C₈₉	C₂²	1	2	yes	—
41	C₂²×#10623494	C₂³	1	2	yes	—
42	C₂²×#10623494	C₂³	1	1	yes	characteristic
43	C₂²×#10623494	C₂³	1	1	yes	characteristic
44	Q₈×Dic₈₉	—	2	4	no	—
45	Q₈×Dic₈₉	C₈	1	8	yes	complemented
46	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
60	C₂×C₄×Dic₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊#10623494	—	2	2	no	—
62	#15667941	—	2	4	no	—
63	#15667941	—	2	2	no	—
64	#15667941	—	4	2	no	—
65	#15667941	Q₈	1	4	yes	complemented
66	C₄×#10623494	—	2	2	no	—
67	C₈×Dic₈₉	—	4	2	no	—
68	C₈×Dic₈₉	—	4	2	no	—
69	#15667919	C₂×C₄	1	2	yes	—
70	#15667919	C₂×C₄	1	2	yes	—
71	#15667919	C₂×C₄	1	2	yes	—
72	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₈₉	—	2	2	no	—
77	C₄×Q₈×C₈₉	—	2	2	no	—
78	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
81	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₈₉	—	2	2	no	—
83	C₄×C₈×C₈₉	—	2	2	no	—
84	C₂²×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
88	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
89	C₂²×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×#10623494	—	2	1	no	—
91	C₂×#10623494	—	2	1	no	—
92	C₂×#10623494	—	2	1	no	—
93	C₂×#10623494	—	2	1	no	—
94	C₂×#10623494	—	2	2	no	—
95	C₂×#10623494	—	2	2	no	—
96	C₂×#10623494	—	2	2	no	—
97	C₂×#10623494	—	2	1	no	—
98	C₂×#10623494	C₂×Q₈	1	2	yes	—
99	C₄⋊Dic₈₉	—	2	4	no	—
100	C₄⋊Dic₈₉	C₂×C₈	1	8	yes	—
101	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
102	C₄⋊Dic₈₉	M₁₆	1	4	yes	—
103	C₄×Dic₈₉	—	2	2	no	—
104	C₄×Dic₈₉	—	2	2	no	—
105	C₄×Dic₈₉	—	2	1	no	—
106	C₄×Dic₈₉	—	2	2	no	—
107	C₄×Dic₈₉	—	2	1	no	—
108	C₄×Dic₈₉	—	2	4	no	—
109	C₄×Dic₈₉	—	2	4	no	—
110	C₄×Dic₈₉	—	2	2	no	—
111	C₄×Dic₈₉	—	4	1	no	—
112	C₄×Dic₈₉	—	4	2	no	—
113	C₄×Dic₈₉	C₂×C₈	1	8	yes	—
114	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
115	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
116	C₄×Dic₈₉	#41	1	2	yes	—
117	C₂×Q₈×C₈₉	—	2	2	no	—
118	C₂×Q₈×C₈₉	—	2	2	no	—
119	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
120	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂×C₈×C₈₉	—	2	1	no	—
126	C₂×C₈×C₈₉	—	2	1	no	—
127	C₂×C₈×C₈₉	—	2	1	no	—
128	C₂×C₈×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	—	2	2	no	—
131	C₂×C₈×C₈₉	—	2	2	no	—
132	C₂×C₈×C₈₉	—	2	1	no	—
133	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
134	C₄⋊C₄×C₈₉	—	2	4	no	—
135	C₄⋊C₄×C₈₉	—	2	2	no	—
136	C₄²×C₈₉	—	2	2	no	—
137	C₄²×C₈₉	—	2	2	no	—
138	C₄²×C₈₉	#41	1	2	yes	—
139	C₄²×C₈₉	#41	1	2	yes	—
140	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
141	Q₈×C₈₉	—	2	8	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	1	no	—
145	C₂×C₄×C₈₉	—	2	1	no	—
146	C₂×C₄×C₈₉	—	2	2	no	—
147	C₂×C₄×C₈₉	—	2	2	no	—
148	C₂×C₄×C₈₉	—	2	2	no	—
149	C₂×C₄×C₈₉	#117	1	2	yes	—
150	C₂×C₄×C₈₉	#117	1	2	yes	—
151	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
152	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
153	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
154	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
160	C₂×Dic₈₉	—	2	2	no	—
161	C₂×Dic₈₉	—	2	1	no	—
162	C₂×Dic₈₉	—	2	2	no	—
163	C₂×Dic₈₉	—	2	1	no	—
164	C₂×Dic₈₉	—	2	2	no	—
165	C₂×Dic₈₉	—	2	1	no	—
166	C₂×Dic₈₉	—	2	1	no	—
167	C₂×Dic₈₉	—	2	2	no	—
168	C₂×Dic₈₉	—	2	1	no	—
169	C₂×Dic₈₉	—	2	1	no	—
170	C₂×Dic₈₉	—	2	1	no	—
171	C₂×Dic₈₉	C₄×Q₈	1	2	yes	—
172	C₂×Dic₈₉	C₄×Q₈	1	2	yes	—
173	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
174	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
175	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
176	C₂×Dic₈₉	C₂×M₁₆	1	4	yes	—
177	C₂×Dic₈₉	#131	1	2	yes	—
178	C₈×C₈₉	—	2	2	no	—
179	C₈×C₈₉	—	4	1	no	—
180	C₈×C₈₉	—	4	2	no	—
181	#10623494	—	2	2	no	—
182	#10623494	—	4	1	no	—
183	#10623494	—	4	2	no	—
184	C₂²×C₈₉	C₂×#117	1	1	yes	characteristic
185	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
186	C₂²×C₈₉	#520	1	1	yes	characteristic
187	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
188	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
189	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
190	C₂²×C₈₉	#568	1	1	yes	characteristic
191	C₄×C₈₉	—	2	4	no	—
192	C₄×C₈₉	—	2	2	no	—
193	C₄×C₈₉	—	4	2	no	—
194	C₄×C₈₉	#431	1	4	yes	—
195	C₄×C₈₉	#554	1	1	yes	characteristic
196	C₄×C₈₉	#556	1	1	yes	characteristic
197	C₄×C₈₉	#557	1	1	yes	characteristic
198	C₄×C₈₉	#559	1	1	yes	characteristic
199	Dic₈₉	—	2	2	no	—
200	Dic₈₉	—	2	4	no	—
201	Dic₈₉	—	2	2	no	—
202	Dic₈₉	—	2	4	no	—
203	Dic₈₉	—	4	1	no	—
204	Dic₈₉	C₈×Q₈	1	4	yes	—
205	#22821	—	89	1	no	maximal, Sylow
206	C₂×C₈₉	#2355	1	1	yes	characteristic
207	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
208	C₂×C₈₉	#2925	1	1	yes	characteristic
209	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
210	C₂×C₈₉	Q₈×M₁₆	1	1	yes	characteristic
211	C₂×C₈₉	#2988	1	1	yes	characteristic
212	C₂×C₈₉	#2990	1	1	yes	characteristic
213	C₄²×Q₈	—	89	1	no	—
214	#1984	—	89	1	no	—
215	#1984	D₈₉	1	1	yes	characteristic
216	#1916	—	89	2	no	—
217	#1916	—	89	2	no	—
218	#1842	—	89	1	no	—
219	#1831	—	89	2	no	—
220	C₈×C₄⋊C₄	—	89	1	no	—
221	C₈×C₄⋊C₄	—	89	1	no	—
222	C₄×C₄⋊C₈	—	89	1	no	—
223	C₄×C₄⋊C₈	—	89	2	no	—
224	C₈₉	#22821	1	1	yes	characteristic, complemented, Sylow
225	C₂×C₄×Q₈	—	89	1	no	—
226	C₂×C₄×Q₈	—	89	1	no	—
227	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
228	C₂×C₄⋊C₈	—	89	1	no	—
229	C₂×C₄⋊C₈	—	89	2	no	—
230	C₂×C₄⋊C₈	—	89	1	no	—
231	C₂×C₄⋊C₈	—	89	2	no	—
232	C₂×C₄⋊C₈	—	89	1	no	—
233	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
234	C₂×C₄×C₈	—	89	2	no	—
235	C₂×C₄×C₈	—	89	2	no	—
236	C₂×C₄×C₈	—	89	1	no	—
237	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
238	C₄×C₄⋊C₄	—	89	1	no	—
239	C₄×C₄⋊C₄	—	89	2	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	—	89	1	no	—
244	C₄³	—	89	2	no	—
245	C₄³	—	89	1	no	—
246	#336	—	89	2	no	—
247	#336	—	89	2	no	—
248	#336	—	89	1	no	—
249	#336	—	89	1	no	—
250	#336	—	89	2	no	—
251	#336	—	89	2	no	—
252	#336	C₂×D₈₉	1	2	yes	—
253	#336	C₂×D₈₉	1	2	yes	—
254	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
255	C₂²×C₈	—	89	1	no	—
256	C₂²×C₈	—	89	1	no	—
257	C₂²×C₈	—	89	2	no	—
258	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
260	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₄×Q₈	—	2	2	no	—
262	C₄×Q₈	—	2	2	no	—
263	C₄×Q₈	—	89	8	no	—
264	C₄×Q₈	—	178	4	no	—
265	C₂×C₄⋊C₄	—	89	2	no	—
266	C₂×C₄⋊C₄	—	89	1	no	—
267	C₂×C₄⋊C₄	—	89	1	no	—
268	C₂×C₄⋊C₄	—	89	2	no	—
269	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
270	C₂×C₄⋊C₄	C₄×D₈₉	1	2	yes	—
271	C₂×C₄²	—	89	2	no	—
272	C₂×C₄²	—	89	1	no	—
273	C₂×C₄²	—	89	1	no	—
274	C₂×C₄²	—	89	2	no	—
275	C₂×C₄²	—	89	1	no	—
276	C₂×C₄²	—	89	2	no	—
277	C₂×C₄²	—	89	1	no	—
278	C₂×C₄²	—	89	2	no	—
279	C₂×C₄²	—	89	1	no	—
280	C₂×C₄²	—	89	2	no	—
281	C₂×C₄²	—	89	1	no	—
282	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
283	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
284	C₄⋊C₈	—	2	2	no	—
285	C₄⋊C₈	—	89	4	no	—
286	C₄⋊C₈	—	356	2	no	—
287	C₄⋊C₈	—	178	2	no	—
288	C₄⋊C₈	—	178	4	no	—
289	C₄⋊C₈	—	178	2	no	—
290	C₄×C₈	—	2	2	no	—
291	C₄×C₈	—	356	2	no	—
292	C₄×C₈	—	356	2	no	—
293	C₄×C₈	—	178	2	no	—
294	#95	—	89	2	no	—
295	#95	—	89	2	no	—
296	#95	—	89	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	C₈×D₈₉	1	4	yes	—
300	C₂²×C₄	—	89	1	no	—
301	C₂²×C₄	—	89	2	no	—
302	C₂²×C₄	—	89	1	no	—
303	C₂²×C₄	—	89	2	no	—
304	C₂²×C₄	—	89	1	no	—
305	C₂²×C₄	—	89	1	no	—
306	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
307	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
308	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
309	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
310	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
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	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	89	2	no	—
320	C₂×C₈	—	178	1	no	—
321	C₂×C₈	—	178	2	no	—
322	C₂×C₈	—	178	1	no	—
323	C₂×C₈	—	178	2	no	—
324	C₂×C₈	—	178	2	no	—
325	C₂×C₈	—	178	1	no	—
326	C₂×C₈	—	178	1	no	—
327	C₂×C₈	—	178	1	no	—
328	C₂×C₈	Q₈×D₈₉	1	2	yes	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	89	8	no	—
332	C₄⋊C₄	—	89	4	no	—
333	C₄⋊C₄	—	89	4	no	—
334	C₄⋊C₄	—	178	4	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	89	8	no	—
338	C₄²	—	89	4	no	—
339	C₄²	—	89	2	no	—
340	C₄²	—	89	2	no	—
341	C₄²	—	356	1	no	—
342	C₄²	—	356	2	no	—
343	C₄²	—	178	4	no	—
344	C₄²	—	178	2	no	—
345	C₄²	—	178	4	no	—
346	C₄²	—	178	2	no	—
347	C₄²	—	178	2	no	—
348	C₄²	—	178	1	no	—
349	C₄²	—	178	1	no	—
350	C₄²	—	178	2	no	—
351	C₄²	#13083573	1	2	yes	—
352	C₄²	#13083573	1	2	yes	—
353	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
354	Q₈	—	2	8	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	89	2	no	—
363	C₂×C₄	—	89	2	no	—
364	C₂×C₄	—	89	2	no	—
365	C₂×C₄	—	89	2	no	—
366	C₂×C₄	—	89	4	no	—
367	C₂×C₄	—	89	2	no	—
368	C₂×C₄	—	89	4	no	—
369	C₂×C₄	—	178	1	no	—
370	C₂×C₄	—	178	1	no	—
371	C₂×C₄	—	178	1	no	—
372	C₂×C₄	—	178	2	no	—
373	C₂×C₄	—	178	1	no	—
374	C₂×C₄	—	178	2	no	—
375	C₂×C₄	—	178	1	no	—
376	C₂×C₄	—	178	2	no	—
377	C₂×C₄	—	178	1	no	—
378	C₂×C₄	—	178	1	no	—
379	C₂×C₄	—	178	2	no	—
380	C₂×C₄	#15667968	1	2	yes	—
381	C₂×C₄	#15667968	1	2	yes	—
382	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
383	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
384	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
385	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
386	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
387	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
388	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
389	C₂×C₄	#15668058	1	1	yes	characteristic
390	C₂×C₄	#15668060	1	1	yes	characteristic
391	C₈	—	2	2	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	2	no	—
394	C₈	—	356	1	no	—
395	C₈	—	356	2	no	—
396	C₈	—	178	2	no	—
397	C₂²	C₂×#15667968	1	1	yes	characteristic
398	C₂²	C₄×Q₈×D₈₉	1	1	yes	characteristic
399	C₂²	#19152080	1	1	yes	characteristic
400	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
401	C₂²	C₂×M₁₆×D₈₉	1	1	yes	characteristic
402	C₂²	#131×D₈₉	1	1	yes	characteristic
403	C₂²	#19152363	1	1	yes	characteristic
404	C₄	—	2	4	no	—
405	C₄	—	2	2	no	—
406	C₄	—	4	2	no	—
407	C₄	—	89	4	no	—
408	C₄	—	356	1	no	—
409	C₄	—	178	4	no	—
410	C₄	—	178	2	no	—
411	C₄	—	178	4	no	—
412	C₄	—	178	2	no	—
413	C₄	#19151648	1	4	yes	—
414	C₄	#19152306	1	1	yes	characteristic
415	C₄	#19152319	1	1	yes	characteristic
416	C₄	#19152323	1	1	yes	characteristic
417	C₄	#19152330	1	1	yes	characteristic
418	C₂	#26023208	1	1	yes	characteristic
419	C₂	C₂×#19151648	1	1	yes	characteristic
420	C₂	#26028914	1	1	yes	characteristic
421	C₂	C₈×Q₈×D₈₉	1	1	yes	characteristic
422	C₂	#26029204	1	1	yes	characteristic
423	C₂	#26029362	1	1	yes	characteristic
424	C₂	#26029376	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#1984	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1984	D₈₉
hypocenter	C₈₉	#22821
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1984

Complements

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

N	H ≅ G/N	type
45. Q₈×Dic₈₉	C₈	semidirect
65. #15667941	Q₈	semidirect
224. C₈₉	#22821	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×Dic₈₉	2
#26021189	2
#26020810	2
#26020808	2
#26020403	2
#26020367	2
#26020117	2
C₈×C₄⋊Dic₈₉	2
C₄×#15667941	2
#26020100	2
C₈₉×#1984	2
#22821	89