Subgroups — #36523421

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029361	C₂	1	1	yes	maximal, characteristic
2	#26029360	C₂	1	1	yes	maximal, characteristic
3	C₄×#15668020	C₂	1	1	yes	maximal, characteristic
4	#26019004	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019003	C₂	1	2	yes	maximal, complemented
6	#26019001	C₂	1	2	yes	maximal, complemented
7	#26018998	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018997	C₂	1	2	yes	maximal, complemented
9	C₂×#19151350	C₂	1	1	yes	maximal, characteristic, complemented
10	#335×D₈₉	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668020	C₄	1	1	yes	characteristic
13	C₂×#15668020	C₄	1	1	yes	characteristic
14	C₄×#13083574	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₇₈	—	2	1	no	—
16	#19151722	—	2	1	no	—
17	C₈×D₃₅₆	—	2	1	no	—
18	#19151663	—	2	1	no	—
19	C₈×C₂²⋊D₈₉	—	2	2	no	—
20	#19151661	—	2	2	no	—
21	#19151651	C₂²	1	1	yes	characteristic
22	#19151650	C₂²	1	2	yes	—
23	#19151650	C₂²	1	2	yes	—
24	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×D₈₉	C₂²	1	2	yes	—
26	C₄⋊C₈×D₈₉	C₂²	1	2	yes	—
27	#19151556	C₄	1	1	yes	characteristic
28	#19151556	C₄	1	1	yes	characteristic
29	#19151555	C₄	1	2	yes	—
30	#19151555	C₄	1	2	yes	—
31	C₄×#13083557	C₂²	1	2	yes	—
32	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₈₉	C₂²	1	2	yes	—
34	#19151351	C₂²	1	2	yes	—
35	#19151350	C₂²	1	1	yes	characteristic
36	#19151350	C₂²	1	2	yes	—
37	#19151350	C₂²	1	2	yes	—
38	#19151350	C₂²	1	1	yes	characteristic
39	#19151349	C₂²	1	2	yes	—
40	#19151348	C₂²	1	1	yes	characteristic
41	#19151348	C₂²	1	1	yes	characteristic
42	#431×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
45	#335×C₈₉	C₂²	1	2	yes	—
46	#334×C₈₉	C₂²	1	2	yes	—
47	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
48	#15668020	—	2	4	no	—
49	#15668020	C₈	1	8	yes	complemented
50	C₂×#13083557	C₂×C₄	1	2	yes	—
51	C₂×#13083557	C₂×C₄	1	2	yes	—
52	C₂²×#10623494	—	2	1	no	—
53	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
54	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₈₉	—	2	2	no	—
56	C₂×C₈×D₈₉	—	2	2	no	—
57	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
58	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
60	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
61	C₄×D₃₅₆	D₄	1	1	yes	characteristic
62	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
63	C₄×C₂²⋊D₈₉	D₄	1	2	yes	—
64	C₄×C₂²⋊D₈₉	Q₈	1	2	yes	—
65	#15667971	C₂×C₄	1	1	yes	characteristic
66	#15667969	C₂×C₄	1	2	yes	—
67	C₄²×D₈₉	C₂³	1	2	yes	—
68	C₄⋊#10623494	—	2	1	no	—
69	C₄⋊#10623494	C₂³	1	1	yes	characteristic
70	C₄⋊#10623494	C₂³	1	2	yes	—
71	C₄⋊#10623494	C₂³	1	1	yes	characteristic
72	#15667941	—	2	2	no	—
73	#15667941	—	2	2	no	—
74	C₂²⋊#10623494	—	2	1	no	—
75	#15667924	—	2	2	no	—
76	#15667924	—	2	2	no	—
77	#15667922	C₂×C₄	1	2	yes	—
78	#15667922	C₂×C₄	1	2	yes	—
79	C₄×#10623494	—	2	1	no	—
80	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
81	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
82	C₄×#10623494	C₂³	1	1	yes	characteristic
83	C₄×#10623494	C₂³	1	1	yes	characteristic
84	C₈×Dic₈₉	—	2	2	no	—
85	C₈×Dic₈₉	—	2	2	no	—
86	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
87	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
88	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
89	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₈×C₈₉	—	2	1	no	—
91	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₈₉	—	2	1	no	—
93	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
95	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₈₉	—	2	1	no	—
97	#97×C₈₉	C₂×C₄	1	2	yes	—
98	C₄×C₈×C₈₉	—	2	1	no	—
99	C₄×C₈×C₈₉	C₂³	1	2	yes	—
100	#13083574	C₂×C₈	1	4	yes	—
101	#13083574	M₁₆	1	4	yes	—
102	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
103	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₂²⋊D₈₉	C₄⋊C₄	1	2	yes	—
105	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
106	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
107	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
108	#13083557	—	2	4	no	—
109	#13083557	C₂×C₈	1	8	yes	complemented
110	C₂×#10623494	—	2	2	no	—
111	C₂×#10623494	—	2	1	no	—
112	C₂×#10623494	—	2	2	no	—
113	C₂×#10623494	—	2	1	no	—
114	C₂×#10623494	—	8	1	no	—
115	C₂×#10623494	C₂×C₈	1	4	yes	complemented
116	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
117	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
118	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
119	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
120	C₈×D₈₉	—	2	4	no	—
121	C₈×D₈₉	—	8	2	no	—
122	C₈×D₈₉	—	8	2	no	—
123	C₈×D₈₉	C₂×C₈	1	8	yes	complemented
124	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
125	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
126	#13083553	C₄⋊C₄	1	2	yes	—
127	#13083553	C₄⋊C₄	1	2	yes	—
128	C₂²⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
129	#13083551	C₄⋊C₄	1	2	yes	—
130	#13083551	C₄⋊C₄	1	2	yes	—
131	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
132	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
133	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
134	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
137	M₁₆×C₈₉	—	2	2	no	—
138	M₁₆×C₈₉	C₂×C₈	1	4	yes	complemented
139	C₂×C₈×C₈₉	—	2	1	no	—
140	C₂×C₈×C₈₉	—	2	2	no	—
141	C₂×C₈×C₈₉	—	2	1	no	—
142	C₂×C₈×C₈₉	—	8	1	no	—
143	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
145	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
146	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
148	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
149	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	1	no	—
153	C₂×C₄×C₈₉	—	4	1	no	—
154	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
158	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
159	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
162	C₂²⋊D₈₉	C₄⋊C₈	1	8	yes	—
163	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
166	D₃₅₆	C₄⋊C₈	1	4	yes	—
167	C₄×D₈₉	—	2	2	no	—
168	C₄×D₈₉	—	2	2	no	—
169	C₄×D₈₉	—	4	2	no	—
170	C₄×D₈₉	C₂²×C₈	1	4	yes	—
171	C₄×D₈₉	C₂×M₁₆	1	4	yes	—
172	Dic₁₇₈	C₄⋊C₈	1	4	yes	—
173	C₈×C₈₉	—	2	2	no	—
174	C₈×C₈₉	—	4	2	no	—
175	C₈×C₈₉	—	4	1	no	—
176	C₈×C₈₉	—	4	1	no	—
177	C₈×C₈₉	C₂²×C₈	1	4	yes	—
178	#10623494	—	2	1	no	—
179	#10623494	—	2	1	no	—
180	#10623494	—	4	1	no	—
181	#10623494	—	4	2	no	—
182	#10623494	—	4	1	no	—
183	#10623494	C₂²×C₈	1	2	yes	—
184	#10623494	C₂²×C₈	1	2	yes	—
185	C₂²×C₈₉	—	2	1	no	—
186	C₂²×C₈₉	—	2	1	no	—
187	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
188	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
189	C₂×D₈₉	—	2	2	no	—
190	C₂×D₈₉	—	2	2	no	—
191	C₂×D₈₉	C₂×C₄⋊C₈	1	4	yes	—
192	C₄×C₈₉	—	2	1	no	—
193	C₄×C₈₉	—	2	1	no	—
194	C₄×C₈₉	—	2	1	no	—
195	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
196	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
197	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
198	C₄×C₈₉	#576	1	1	yes	characteristic
199	C₄×C₈₉	#578	1	1	yes	characteristic
200	Dic₈₉	—	2	2	no	—
201	Dic₈₉	—	2	2	no	—
202	Dic₈₉	C₂×C₄⋊C₈	1	4	yes	—
203	#17789	—	89	1	no	maximal, Sylow
204	C₂×C₈₉	—	2	2	no	—
205	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
206	C₂×C₈₉	#2913	1	1	yes	characteristic
207	C₂×C₈₉	#2916	1	1	yes	characteristic
208	D₈₉	—	2	4	no	—
209	#2988	—	89	1	no	—
210	#2988	—	89	1	no	—
211	C₄×#131	—	89	1	no	—
212	#1565	—	89	1	no	—
213	#1565	—	89	2	no	—
214	#1565	—	89	2	no	—
215	#1565	D₈₉	1	1	yes	characteristic, complemented
216	#1564	—	89	1	no	—
217	#1564	—	89	2	no	—
218	C₂×#335	—	89	1	no	—
219	C₂×#335	—	89	2	no	—
220	C₈₉	#17789	1	1	yes	characteristic, complemented, Sylow
221	C₂×#131	—	89	1	no	—
222	C₂×#131	—	89	1	no	—
223	C₄×#41	—	89	1	no	—
224	C₈×Q₈	—	178	1	no	—
225	C₈×Q₈	—	178	1	no	—
226	C₈×D₄	—	178	1	no	—
227	C₈×D₄	—	178	2	no	—
228	C₈×D₄	—	178	1	no	—
229	C₈×D₄	—	178	2	no	—
230	#431	—	89	1	no	—
231	#431	—	89	2	no	—
232	#431	—	89	2	no	—
233	#431	C₂×D₈₉	1	1	yes	characteristic
234	C₂×C₄⋊C₈	—	89	1	no	—
235	C₂×C₄⋊C₈	—	89	2	no	—
236	C₂×C₄⋊C₈	—	89	2	no	—
237	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
238	#405	—	89	1	no	—
239	#405	—	89	1	no	—
240	#405	—	89	2	no	—
241	#405	—	89	2	no	—
242	C₄×M₁₆	—	89	2	no	—
243	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
244	C₂×C₄×C₈	—	89	1	no	—
245	C₂×C₄×C₈	—	89	2	no	—
246	#335	—	89	1	no	—
247	#335	—	89	2	no	—
248	#335	—	89	2	no	—
249	#335	—	89	1	no	—
250	#335	—	89	2	no	—
251	#335	C₂×D₈₉	1	2	yes	—
252	#334	—	89	1	no	—
253	#334	—	89	1	no	—
254	#334	—	89	2	no	—
255	#334	C₂×D₈₉	1	2	yes	—
256	C₂×#41	—	89	1	no	—
257	#131	—	89	8	no	—
258	#131	—	178	4	no	—
259	C₂×M₁₆	—	89	2	no	—
260	C₂×M₁₆	—	89	2	no	—
261	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
262	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
263	C₂²×C₈	—	2	1	no	—
264	C₂²×C₈	—	89	1	no	—
265	C₂²×C₈	—	89	2	no	—
266	C₂²×C₈	—	89	1	no	—
267	C₂²×C₈	—	89	2	no	—
268	C₂²×C₈	—	178	2	no	—
269	C₂²×C₈	—	178	2	no	—
270	C₂²×C₈	—	178	1	no	—
271	C₄×Q₈	—	89	1	no	—
272	C₄×Q₈	—	89	1	no	—
273	C₄×D₄	—	89	2	no	—
274	C₄×D₄	—	89	2	no	—
275	C₄×D₄	—	89	1	no	—
276	C₄×D₄	—	89	1	no	—
277	#117	—	89	1	no	—
278	#117	—	89	2	no	—
279	C₂×C₄²	—	89	2	no	—
280	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	89	1	no	—
283	C₄⋊C₈	—	89	1	no	—
284	C₄⋊C₈	—	89	2	no	—
285	C₄⋊C₈	—	178	2	no	—
286	C₄⋊C₈	—	178	2	no	—
287	C₄⋊C₈	—	178	1	no	—
288	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
290	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
291	C₂²⋊C₈	—	2	1	no	—
292	C₂²⋊C₈	—	178	2	no	—
293	C₂²⋊C₈	—	178	2	no	—
294	C₂²⋊C₈	—	178	1	no	—
295	#97	—	89	2	no	—
296	#97	—	89	2	no	—
297	#97	C₄×D₈₉	1	2	yes	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	—	89	2	no	—
300	C₄×C₈	—	89	2	no	—
301	C₄×C₈	—	89	1	no	—
302	C₄×C₈	—	89	1	no	—
303	C₄×C₈	—	89	1	no	—
304	C₄×C₈	—	89	1	no	—
305	C₄×C₈	—	178	2	no	—
306	C₄×C₈	—	178	2	no	—
307	C₄×C₈	—	178	1	no	—
308	C₄×C₈	C₂²×D₈₉	1	2	yes	—
309	#41	—	89	4	no	—
310	#41	—	89	4	no	—
311	C₂×Q₈	—	89	1	no	—
312	C₂×D₄	—	89	2	no	—
313	C₂×D₄	—	89	1	no	—
314	C₂²×C₄	—	89	2	no	—
315	C₂²×C₄	—	89	2	no	—
316	C₂²×C₄	—	89	2	no	—
317	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
318	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
319	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
320	M₁₆	—	2	2	no	—
321	M₁₆	—	89	8	no	—
322	M₁₆	—	178	4	no	—
323	M₁₆	C₈×D₈₉	1	4	yes	complemented
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	8	1	no	—
328	C₂×C₈	—	89	4	no	—
329	C₂×C₈	—	89	1	no	—
330	C₂×C₈	—	89	1	no	—
331	C₂×C₈	—	89	8	no	—
332	C₂×C₈	—	89	1	no	—
333	C₂×C₈	—	89	1	no	—
334	C₂×C₈	—	178	4	no	—
335	C₂×C₈	—	178	1	no	—
336	C₂×C₈	—	178	2	no	—
337	C₂×C₈	—	178	1	no	—
338	C₂×C₈	—	178	2	no	—
339	C₂×C₈	—	712	2	no	—
340	C₂×C₈	—	712	2	no	—
341	C₂×C₈	—	712	1	no	—
342	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
343	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
344	C₄⋊C₄	—	89	2	no	—
345	C₄⋊C₄	—	89	2	no	—
346	C₄⋊C₄	—	89	1	no	—
347	C₄⋊C₄	—	89	1	no	—
348	C₂²⋊C₄	—	89	2	no	—
349	C₂²⋊C₄	—	89	2	no	—
350	C₂²⋊C₄	—	89	2	no	—
351	C₄²	—	89	2	no	—
352	C₄²	—	89	2	no	—
353	C₄²	—	89	2	no	—
354	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
355	C₄²	D₄×D₈₉	1	1	yes	characteristic
356	C₄²	Q₈×D₈₉	1	1	yes	characteristic
357	C₄²	C₂³×D₈₉	1	1	yes	characteristic
358	C₂³	—	89	2	no	—
359	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
360	Q₈	—	89	4	no	—
361	D₄	—	89	8	no	—
362	D₄	—	89	4	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	89	2	no	—
367	C₂×C₄	—	89	4	no	—
368	C₂×C₄	—	89	4	no	—
369	C₂×C₄	—	89	2	no	—
370	C₂×C₄	—	89	2	no	—
371	C₂×C₄	—	356	2	no	—
372	C₂×C₄	—	178	2	no	—
373	C₂×C₄	—	178	2	no	—
374	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
375	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
376	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
377	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
378	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
379	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
380	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
381	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	89	2	no	—
387	C₈	—	89	2	no	—
388	C₈	—	356	1	no	—
389	C₈	—	356	2	no	—
390	C₈	—	356	1	no	—
391	C₈	—	178	1	no	—
392	C₈	—	178	1	no	—
393	C₈	C₂×C₈×D₈₉	1	4	yes	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	89	4	no	—
397	C₂²	—	178	2	no	—
398	C₂²	—	178	2	no	—
399	C₂²	C₄⋊C₈×D₈₉	1	2	yes	—
400	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	89	4	no	—
405	C₄	—	178	2	no	—
406	C₄	—	178	2	no	—
407	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
408	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
409	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
410	C₄	#19152402	1	1	yes	characteristic
411	C₄	#19152414	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	178	4	no	—
414	C₂	C₂×C₄⋊C₈×D₈₉	1	1	yes	characteristic
415	C₂	#26028851	1	1	yes	characteristic
416	C₂	#26028862	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1565	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1565	D₈₉
hypocenter	C₈₉	#17789
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #26019004	C₂	semidirect
5. #26019003	C₂	semidirect
6. #26019001	C₂	semidirect
7. #26018998	C₂	semidirect
8. #26018997	C₂	semidirect
9. C₂×#19151350	C₂	semidirect
10. #335×D₈₉	C₂	semidirect
11. C₈₉×#1565	C₂	semidirect
49. #15668020	C₈	semidirect
109. #13083557	C₂×C₈	semidirect
115. C₂×#10623494	C₂×C₈	semidirect
123. C₈×D₈₉	C₂×C₈	semidirect
138. M₁₆×C₈₉	C₂×C₈	semidirect
215. #1565	D₈₉	semidirect
220. C₈₉	#17789	semidirect
323. M₁₆	C₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029361	2
#26029360	2
C₄×#15668020	2
#26019004	2
#26019003	2
#26019001	2
#26018998	2
#26018997	2
C₂×#19151350	2
#335×D₈₉	2
C₈₉×#1565	2
#17789	89