Subgroups — #36613679

Order 22784.

702 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26025287	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26021813	C₂	1	2	yes	maximal
4	#26021755	C₂	1	2	yes	maximal
5	#26021015	C₂	1	1	yes	maximal, characteristic
6	#26020970	C₂	1	1	yes	maximal, characteristic
7	#26020518	C₂	1	2	yes	maximal
8	#26020380	C₂	1	1	yes	maximal, characteristic
9	#26020257	C₂	1	1	yes	maximal, characteristic
10	#26020255	C₂	1	2	yes	maximal
11	C₄×#103×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#15668004	—	2	1	no	—
13	C₂×Q₈×Dic₈₉	C₂²	1	2	yes	—
14	C₂×#15667943	C₂²	1	2	yes	—
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	C₂×#15667941	C₂²	1	2	yes	—
17	C₂×#15667936	C₂²	1	2	yes	—
18	#19151532	—	2	2	no	—
19	#19151497	—	2	2	no	—
20	#19151497	C₂²	1	2	yes	—
21	#19151496	—	2	1	no	—
22	#19151494	C₂²	1	2	yes	—
23	#19151468	—	2	1	no	—
24	#19151468	C₂²	1	1	yes	characteristic
25	#19151468	C₂²	1	1	yes	characteristic
26	#19151467	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₈₉	—	2	1	no	—
28	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
29	#19151361	C₂²	1	2	yes	—
30	#19151361	C₂²	1	2	yes	—
31	#19151360	C₂²	1	2	yes	—
32	#19151360	C₂²	1	2	yes	—
33	#19151352	C₂²	1	2	yes	—
34	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
39	#340×C₈₉	C₂²	1	1	yes	characteristic
40	#340×C₈₉	C₂²	1	1	yes	characteristic
41	C₂²×#10623494	C₂³	1	2	yes	—
42	#15668004	—	4	4	no	—
43	Q₈×Dic₈₉	—	2	4	no	—
44	Q₈×Dic₈₉	—	4	2	no	—
45	C₂×C₄⋊Dic₈₉	—	2	2	no	—
46	C₂×C₄⋊Dic₈₉	—	2	2	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×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
54	C₂×C₄⋊Dic₈₉	C₂³	1	2	yes	—
55	C₂×#13083553	—	2	2	no	—
56	C₂×#13083553	—	2	2	no	—
57	C₂×#13083553	—	2	2	no	—
58	C₂×C₄×Dic₈₉	—	2	2	no	—
59	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
60	#15667943	—	4	2	no	—
61	C₄⋊#10623494	—	2	2	no	—
62	#15667941	—	4	2	no	—
63	#15667936	—	2	4	no	—
64	#15667919	—	2	2	no	—
65	#15667919	—	2	2	no	—
66	#15667919	D₄	1	2	yes	—
67	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₄×Q₈×C₈₉	—	2	2	no	—
71	C₄×Q₈×C₈₉	Q₈	1	4	yes	—
72	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₈₉	—	2	1	no	—
77	C₂×C₄²×C₈₉	—	2	1	no	—
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	#103×C₈₉	—	2	2	no	—
80	#103×C₈₉	—	2	2	no	—
81	C₄×C₈×C₈₉	—	2	2	no	—
82	#95×C₈₉	—	2	1	no	—
83	C₂²×Dic₈₉	—	2	2	no	—
84	C₂²×Dic₈₉	—	2	2	no	—
85	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
86	C₂²×Dic₈₉	C₂×D₄	1	2	yes	—
87	C₂×#10623494	—	2	2	no	—
88	C₂×#10623494	—	2	2	no	—
89	C₂×#10623494	—	2	2	no	—
90	C₄⋊Dic₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	2	2	no	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	2	2	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	4	4	no	—
96	C₄⋊Dic₈₉	—	4	2	no	—
97	C₄⋊Dic₈₉	—	4	2	no	—
98	C₄⋊Dic₈₉	—	8	2	no	—
99	C₄⋊Dic₈₉	—	8	1	no	—
100	C₄⋊Dic₈₉	—	8	1	no	—
101	C₄⋊Dic₈₉	SD₁₆	1	4	yes	—
102	#13083553	—	4	4	no	—
103	#13083553	—	4	4	no	—
104	#13083553	—	8	2	no	—
105	C₄×Dic₈₉	—	2	2	no	—
106	C₄×Dic₈₉	—	2	2	no	—
107	C₄×Dic₈₉	—	4	4	no	—
108	C₂×Q₈×C₈₉	—	2	1	no	—
109	C₂×Q₈×C₈₉	—	2	1	no	—
110	C₂×Q₈×C₈₉	—	2	1	no	—
111	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
112	C₂×Q₈×C₈₉	#41	1	2	yes	—
113	C₂×Q₈×C₈₉	#41	1	2	yes	—
114	C₂²×C₄×C₈₉	—	2	1	no	—
115	C₂²×C₄×C₈₉	—	2	1	no	—
116	C₂²×C₄×C₈₉	—	2	1	no	—
117	C₂²×C₄×C₈₉	—	2	1	no	—
118	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
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	1	no	—
123	C₂×C₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	—	2	1	no	—
125	C₂×C₈×C₈₉	—	2	1	no	—
126	C₂×C₈×C₈₉	—	2	1	no	—
127	C₄⋊C₄×C₈₉	—	2	4	no	—
128	C₄⋊C₄×C₈₉	—	2	1	no	—
129	C₄⋊C₄×C₈₉	—	2	1	no	—
130	C₄⋊C₄×C₈₉	—	2	1	no	—
131	C₄⋊C₄×C₈₉	—	2	1	no	—
132	C₄⋊C₄×C₈₉	—	2	1	no	—
133	C₄⋊C₄×C₈₉	C₂×Q₈	1	2	yes	—
134	C₄²×C₈₉	—	2	4	no	—
135	C₄²×C₈₉	—	4	2	no	—
136	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
137	C₄²×C₈₉	#41	1	2	yes	—
138	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	Q₈×C₈₉	—	2	2	no	—
140	Q₈×C₈₉	—	2	2	no	—
141	Q₈×C₈₉	—	2	2	no	—
142	Q₈×C₈₉	#125	1	4	yes	complemented
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	2	1	no	—
147	C₂×C₄×C₈₉	—	2	1	no	—
148	C₂×C₄×C₈₉	—	2	2	no	—
149	C₂×C₄×C₈₉	—	4	1	no	—
150	C₂×C₄×C₈₉	—	4	1	no	—
151	C₂×C₄×C₈₉	—	4	1	no	—
152	C₂×C₄×C₈₉	—	4	1	no	—
153	C₂×C₄×C₈₉	—	4	1	no	—
154	C₂×C₄×C₈₉	—	4	1	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	—	4	1	no	—
157	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
158	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
159	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
160	C₂×C₄×C₈₉	#137	1	2	yes	—
161	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂×Dic₈₉	—	2	2	no	—
168	C₂×Dic₈₉	—	2	2	no	—
169	C₂×Dic₈₉	—	2	2	no	—
170	C₂×Dic₈₉	—	2	2	no	—
171	C₂×Dic₈₉	—	2	2	no	—
172	C₂×Dic₈₉	—	4	2	no	—
173	C₂×Dic₈₉	—	4	2	no	—
174	C₂×Dic₈₉	—	4	2	no	—
175	C₂×Dic₈₉	—	4	2	no	—
176	C₂×Dic₈₉	—	4	2	no	—
177	C₂×Dic₈₉	—	4	2	no	—
178	C₂×Dic₈₉	C₂×SD₁₆	1	4	yes	—
179	C₈×C₈₉	—	4	1	no	—
180	C₈×C₈₉	—	4	1	no	—
181	#10623494	—	4	2	no	—
182	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂²×C₈₉	#546	1	1	yes	characteristic
184	C₂²×C₈₉	#548	1	1	yes	characteristic
185	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
186	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
187	C₂²×C₈₉	#576	1	1	yes	characteristic
188	C₂²×C₈₉	#578	1	1	yes	characteristic
189	C₄×C₈₉	—	2	2	no	—
190	C₄×C₈₉	—	2	4	no	—
191	C₄×C₈₉	—	4	2	no	—
192	C₄×C₈₉	—	8	1	no	—
193	C₄×C₈₉	—	8	1	no	—
194	C₄×C₈₉	#475	1	4	yes	—
195	C₄×C₈₉	C₂×#125	1	1	yes	characteristic
196	C₄×C₈₉	#544	1	1	yes	characteristic
197	C₄×C₈₉	#552	1	1	yes	characteristic
198	C₄×C₈₉	#553	1	1	yes	characteristic
199	Dic₈₉	—	2	4	no	—
200	Dic₈₉	—	4	2	no	—
201	Dic₈₉	—	8	2	no	—
202	Dic₈₉	—	8	2	no	—
203	#26108	—	89	1	no	maximal, Sylow
204	C₂×C₈₉	#2442	1	1	yes	characteristic
205	C₂×C₈₉	C₂×#475	1	1	yes	characteristic
206	C₂×C₈₉	#3083	1	1	yes	characteristic
207	C₂×C₈₉	#3298	1	1	yes	characteristic
208	C₂×C₈₉	#3333	1	1	yes	characteristic
209	C₂×C₈₉	#3362	1	1	yes	characteristic
210	C₂×C₈₉	#3374	1	1	yes	characteristic
211	#2552	—	89	1	no	—
212	Q₈×C₄⋊C₄	—	89	1	no	—
213	#2099	—	89	2	no	—
214	#2088	—	89	2	no	—
215	#1948	—	89	1	no	—
216	#1939	—	89	1	no	—
217	#1865	—	89	2	no	—
218	#1834	—	89	1	no	—
219	#1803	—	89	2	no	—
220	#1803	—	89	1	no	—
221	C₄×#103	D₈₉	1	1	yes	characteristic
222	C₈₉	#26108	1	1	yes	characteristic, complemented, Sylow
223	C₂×#125	—	178	1	no	—
224	C₂×C₄×Q₈	—	89	2	no	—
225	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
226	C₂×#106	—	89	2	no	—
227	C₂×C₄⋊C₈	—	89	2	no	—
228	C₂×C₄⋊C₈	—	89	1	no	—
229	C₂×#103	—	89	2	no	—
230	C₂×#103	C₂×D₈₉	1	1	yes	characteristic
231	C₂×#103	C₂×D₈₉	1	1	yes	characteristic
232	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
233	#398	—	178	2	no	—
234	#389	—	89	2	no	—
235	#389	—	89	2	no	—
236	#389	—	178	2	no	—
237	#389	—	178	1	no	—
238	#384	—	89	2	no	—
239	#384	—	89	1	no	—
240	#384	—	89	1	no	—
241	#384	—	178	1	no	—
242	C₄×C₄⋊C₄	—	89	1	no	—
243	C₄×C₄⋊C₄	—	178	1	no	—
244	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
245	#340	—	89	2	no	—
246	#340	—	89	2	no	—
247	#340	—	89	2	no	—
248	#340	—	89	2	no	—
249	#340	C₂×D₈₉	1	1	yes	characteristic
250	#340	C₂×D₈₉	1	1	yes	characteristic
251	#336	—	89	2	no	—
252	C₂²×Q₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₂²×C₈	—	89	2	no	—
254	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
256	#125	—	356	4	no	—
257	C₄×Q₈	—	2	2	no	—
258	C₄×Q₈	—	356	2	no	—
259	C₄×Q₈	—	178	4	no	—
260	C₄×Q₈	Dic₁₇₈	1	4	yes	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	89	1	no	—
263	C₂×C₄⋊C₄	—	89	1	no	—
264	C₂×C₄⋊C₄	—	89	2	no	—
265	C₂×C₄⋊C₄	—	89	2	no	—
266	C₂×C₄⋊C₄	—	178	2	no	—
267	C₂×C₄⋊C₄	—	178	2	no	—
268	C₂×C₄⋊C₄	—	178	2	no	—
269	C₂×C₄⋊C₄	—	178	1	no	—
270	C₂×C₄⋊C₄	—	178	1	no	—
271	C₂×C₄⋊C₄	—	178	2	no	—
272	C₂×C₄⋊C₄	—	178	2	no	—
273	C₂×C₄⋊C₄	—	178	1	no	—
274	C₂×C₄⋊C₄	—	178	1	no	—
275	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₂×C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
278	C₂×C₄²	—	2	1	no	—
279	C₂×C₄²	—	2	1	no	—
280	C₂×C₄²	—	89	2	no	—
281	C₂×C₄²	—	178	2	no	—
282	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
283	#106	—	356	2	no	—
284	C₄⋊C₈	—	356	2	no	—
285	C₄⋊C₈	—	178	2	no	—
286	#103	—	2	2	no	—
287	#103	—	2	2	no	—
288	#103	—	178	4	no	—
289	C₄×C₈	—	2	2	no	—
290	#95	—	2	1	no	—
291	#95	—	89	2	no	—
292	#95	—	178	2	no	—
293	#95	—	178	2	no	—
294	C₂×Q₈	—	2	1	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×Q₈	—	2	1	no	—
297	C₂×Q₈	C₂×Dic₁₇₈	1	2	yes	—
298	C₂×Q₈	#13083574	1	2	yes	—
299	C₂×Q₈	#13083574	1	2	yes	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	89	2	no	—
305	C₂²×C₄	—	89	2	no	—
306	C₂²×C₄	—	178	2	no	—
307	C₂²×C₄	—	178	2	no	—
308	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
309	C₂²×C₄	D₄×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	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	178	2	no	—
318	C₂×C₈	—	178	2	no	—
319	C₂×C₈	—	178	2	no	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	89	4	no	—
327	C₄⋊C₄	—	356	4	no	—
328	C₄⋊C₄	—	356	4	no	—
329	C₄⋊C₄	—	356	4	no	—
330	C₄⋊C₄	—	356	2	no	—
331	C₄⋊C₄	—	356	2	no	—
332	C₄⋊C₄	—	178	2	no	—
333	C₄⋊C₄	—	178	2	no	—
334	C₄⋊C₄	—	178	2	no	—
335	C₄⋊C₄	—	178	2	no	—
336	C₄⋊C₄	—	178	2	no	—
337	C₄⋊C₄	—	712	1	no	—
338	C₄⋊C₄	—	712	1	no	—
339	C₄⋊C₄	—	712	2	no	—
340	C₄⋊C₄	—	712	2	no	—
341	C₄⋊C₄	C₂×Dic₁₇₈	1	2	yes	—
342	C₄²	—	2	4	no	—
343	C₄²	—	4	2	no	—
344	C₄²	—	356	4	no	—
345	C₄²	—	178	2	no	—
346	C₄²	—	178	2	no	—
347	C₄²	C₂×Dic₁₇₈	1	2	yes	—
348	C₄²	#13083573	1	2	yes	—
349	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
350	Q₈	—	2	2	no	—
351	Q₈	—	2	2	no	—
352	Q₈	—	2	2	no	—
353	Q₈	#15668004	1	4	yes	complemented
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	89	4	no	—
369	C₂×C₄	—	356	2	no	—
370	C₂×C₄	—	356	2	no	—
371	C₂×C₄	—	356	2	no	—
372	C₂×C₄	—	356	2	no	—
373	C₂×C₄	—	356	2	no	—
374	C₂×C₄	—	356	2	no	—
375	C₂×C₄	—	178	2	no	—
376	C₂×C₄	—	178	2	no	—
377	C₂×C₄	—	178	2	no	—
378	C₂×C₄	—	178	2	no	—
379	C₂×C₄	—	178	2	no	—
380	C₂×C₄	#15667989	1	2	yes	—
381	C₂×C₄	#15667989	1	2	yes	—
382	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
383	C₂×C₄	#15668044	1	2	yes	—
384	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
385	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
386	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
387	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
388	C₂×C₄	#15668058	1	1	yes	characteristic
389	C₂×C₄	#15668059	1	1	yes	characteristic
390	C₈	—	4	1	no	—
391	C₈	—	4	1	no	—
392	C₈	—	356	2	no	—
393	C₂²	C₂×#15667989	1	1	yes	characteristic
394	C₂²	#19152244	1	1	yes	characteristic
395	C₂²	#19152257	1	1	yes	characteristic
396	C₂²	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
397	C₂²	C₂×#15668044	1	1	yes	characteristic
398	C₂²	#19152400	1	1	yes	characteristic
399	C₂²	#19152412	1	1	yes	characteristic
400	C₄	—	2	2	no	—
401	C₄	—	2	4	no	—
402	C₄	—	4	2	no	—
403	C₄	—	8	1	no	—
404	C₄	—	8	1	no	—
405	C₄	—	356	2	no	—
406	C₄	—	178	4	no	—
407	C₄	—	712	2	no	—
408	C₄	—	712	2	no	—
409	C₄	#19151881	1	4	yes	—
410	C₄	C₂×#15668004	1	1	yes	characteristic
411	C₄	#19152226	1	1	yes	characteristic
412	C₄	#19152288	1	1	yes	characteristic
413	C₄	#19152294	1	1	yes	characteristic
414	C₂	#26023976	1	1	yes	characteristic
415	C₂	C₂×#19151881	1	1	yes	characteristic
416	C₂	#26030101	1	1	yes	characteristic
417	C₂	#26032119	1	1	yes	characteristic
418	C₂	#26032592	1	1	yes	characteristic
419	C₂	#26032921	1	1	yes	characteristic
420	C₂	#26033053	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄×#103×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
hypercenter	C₄×#103	D₈₉
hypocenter	C₈₉	#26108
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#103

Complements

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

N	H ≅ G/N	type
142. Q₈×C₈₉	#125	semidirect
222. C₈₉	#26108	semidirect
353. Q₈	#15668004	semidirect

Maximal subgroups

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

subgroup	index
#26025287	2
Q₈×C₄⋊Dic₈₉	2
#26021813	2
#26021755	2
#26021015	2
#26020970	2
#26020518	2
#26020380	2
#26020257	2
#26020255	2
C₄×#103×C₈₉	2
#26108	89