Subgroups — #36523468

Order 22784.

666 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	#26029369	C₂	1	1	yes	maximal, characteristic
2	C₄×#15668019	C₂	1	1	yes	maximal, characteristic
3	#26019020	C₂	1	2	yes	maximal, complemented
4	#26019017	C₂	1	2	yes	maximal, complemented
5	#26019013	C₂	1	2	yes	maximal, complemented
6	#26019007	C₂	1	2	yes	maximal, complemented
7	#26019005	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019002	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018998	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151348	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2989	C₂	1	1	yes	maximal, characteristic
12	C₂×#15668019	C₄	1	1	yes	characteristic
13	C₂×#15668019	C₄	1	1	yes	characteristic
14	#19151729	—	2	1	no	—
15	#19151676	—	2	1	no	—
16	#19151670	—	2	2	no	—
17	#19151655	C₂²	1	2	yes	—
18	#19151651	C₂²	1	1	yes	characteristic
19	#19151622	C₂²	1	2	yes	—
20	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
21	#19151554	C₄	1	2	yes	—
22	#19151554	C₄	1	2	yes	—
23	#19151554	C₄	1	1	yes	characteristic
24	#19151554	C₄	1	1	yes	characteristic
25	C₄×#13083558	C₂²	1	1	yes	characteristic
26	C₄×#13083558	C₂²	1	2	yes	—
27	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
28	C₂×C₄×#10623494	C₂²	1	2	yes	—
29	#19151350	C₂²	1	1	yes	characteristic
30	#19151350	C₂²	1	1	yes	characteristic
31	#19151350	C₂²	1	2	yes	—
32	#19151350	C₂²	1	2	yes	—
33	#19151348	C₂²	1	2	yes	—
34	#19151348	C₂²	1	2	yes	—
35	#19151348	C₂²	1	1	yes	characteristic
36	#19151348	C₂²	1	1	yes	characteristic
37	#19151348	C₂²	1	2	yes	—
38	#19151348	C₂²	1	2	yes	—
39	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
40	#446×C₈₉	—	2	1	no	—
41	#436×C₈₉	—	2	1	no	—
42	#435×C₈₉	—	2	2	no	—
43	#432×C₈₉	C₂²	1	2	yes	—
44	#431×C₈₉	C₂²	1	1	yes	characteristic
45	#423×C₈₉	C₂²	1	2	yes	—
46	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
47	#15668019	—	2	4	no	—
48	#15668019	—	2	4	no	—
49	C₂×#13083558	—	2	2	no	—
50	C₂×#13083558	C₂×C₄	1	2	yes	—
51	C₂×#13083558	C₂×C₄	1	2	yes	—
52	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
53	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
54	C₂²×#10623494	—	2	1	no	—
55	C₂²×#10623494	C₂×C₄	1	2	yes	—
56	C₂²×#10623494	C₂×C₄	1	2	yes	—
57	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
58	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#10623494	—	2	1	no	—
60	C₄⋊#10623494	—	2	2	no	—
61	C₄⋊#10623494	C₂³	1	2	yes	—
62	C₄⋊#10623494	C₂³	1	1	yes	characteristic
63	C₄⋊#10623494	C₂³	1	1	yes	characteristic
64	C₂²⋊#10623494	—	2	1	no	—
65	C₂²⋊#10623494	—	2	2	no	—
66	#15667923	—	2	2	no	—
67	#15667923	C₂×C₄	1	2	yes	—
68	#15667923	C₂×C₄	1	2	yes	—
69	#15667923	C₂×C₄	1	2	yes	—
70	C₄×#10623494	—	2	1	no	—
71	C₄×#10623494	C₂×C₄	1	2	yes	—
72	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
73	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
74	C₄×#10623494	C₂×C₄	1	2	yes	—
75	C₄×#10623494	C₂³	1	1	yes	characteristic
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₄×#10623494	C₂³	1	2	yes	—
78	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
79	C₂×M₁₆×C₈₉	—	2	2	no	—
80	C₂²×C₈×C₈₉	—	2	1	no	—
81	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₈₉	Q₈	1	1	yes	characteristic
83	C₄×D₄×C₈₉	D₄	1	2	yes	—
84	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
85	C₄×D₄×C₈₉	Q₈	1	2	yes	—
86	C₄×D₄×C₈₉	Q₈	1	1	yes	characteristic
87	#117×C₈₉	C₂×C₄	1	2	yes	—
88	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
91	C₄⋊C₈×C₈₉	—	2	1	no	—
92	C₄⋊C₈×C₈₉	—	2	2	no	—
93	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
94	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₈₉	—	2	1	no	—
97	C₂²⋊C₈×C₈₉	—	2	2	no	—
98	#97×C₈₉	—	2	2	no	—
99	C₄×C₈×C₈₉	—	2	1	no	—
100	#13083558	—	2	2	no	—
101	#13083558	—	2	4	no	—
102	#13083558	—	2	4	no	—
103	#13083558	—	8	2	no	—
104	#13083558	M₁₆	1	4	yes	complemented
105	C₂×#10623494	—	2	2	no	—
106	C₂×#10623494	—	2	4	no	—
107	C₂×#10623494	—	2	4	no	—
108	C₂×#10623494	—	2	2	no	—
109	C₂×#10623494	—	2	1	no	—
110	C₂×#10623494	—	2	1	no	—
111	C₂×#10623494	—	8	1	no	—
112	C₂×#10623494	M₁₆	1	4	yes	complemented
113	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
114	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
115	C₂×#10623494	C₂²×C₄	1	2	yes	—
116	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
117	C₂×#10623494	C₂²×C₄	1	2	yes	—
118	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
119	#41×C₈₉	—	2	2	no	—
120	#41×C₈₉	—	2	2	no	—
121	C₂×Q₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
122	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
123	C₂×D₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
129	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
130	M₁₆×C₈₉	—	8	2	no	—
131	C₂×C₈×C₈₉	—	2	1	no	—
132	C₂×C₈×C₈₉	—	2	1	no	—
133	C₂×C₈×C₈₉	—	2	2	no	—
134	C₂×C₈×C₈₉	—	8	1	no	—
135	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
136	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
137	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
138	C₄⋊C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
140	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
141	C₂²⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
142	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
144	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₈₉	C₂×D₄	1	2	yes	—
146	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
147	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
148	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
149	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
150	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
151	Q₈×C₈₉	—	2	2	no	—
152	D₄×C₈₉	—	2	4	no	—
153	D₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	2	2	no	—
156	C₂×C₄×C₈₉	—	2	1	no	—
157	C₂×C₄×C₈₉	—	2	2	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	2	2	no	—
160	C₂×C₄×C₈₉	—	4	1	no	—
161	C₂×C₄×C₈₉	—	4	2	no	—
162	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
169	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
170	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
173	C₈×C₈₉	—	4	1	no	—
174	C₈×C₈₉	—	4	2	no	—
175	C₈×C₈₉	—	4	1	no	—
176	#10623494	—	2	1	no	—
177	#10623494	—	2	1	no	—
178	#10623494	—	2	2	no	—
179	#10623494	—	4	1	no	—
180	#10623494	—	4	2	no	—
181	#10623494	—	4	1	no	—
182	#10623494	C₂×M₁₆	1	2	yes	—
183	#10623494	C₂×M₁₆	1	4	yes	—
184	#10623494	C₂×M₁₆	1	2	yes	—
185	C₂²×C₈₉	—	2	1	no	—
186	C₂²×C₈₉	—	2	1	no	—
187	C₂²×C₈₉	—	2	2	no	—
188	C₂²×C₈₉	—	2	2	no	—
189	C₂²×C₈₉	—	2	2	no	—
190	C₂²×C₈₉	#423	1	2	yes	—
191	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
192	C₄×C₈₉	—	2	2	no	—
193	C₄×C₈₉	—	2	1	no	—
194	C₄×C₈₉	—	2	2	no	—
195	C₄×C₈₉	—	2	1	no	—
196	C₄×C₈₉	—	2	1	no	—
197	C₄×C₈₉	—	2	2	no	—
198	C₄×C₈₉	#423	1	2	yes	—
199	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
200	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
201	C₄×C₈₉	#577	1	1	yes	characteristic
202	C₄×C₈₉	#577	1	1	yes	characteristic
203	#17798	—	89	1	no	maximal, Sylow
204	C₂×C₈₉	—	2	2	no	—
205	C₂×C₈₉	—	4	2	no	—
206	C₂×C₈₉	C₂×#423	1	1	yes	characteristic
207	C₂×C₈₉	#2912	1	1	yes	characteristic
208	C₂×C₈₉	#2916	1	1	yes	characteristic
209	#2989	—	89	1	no	—
210	#2989	D₈₉	1	1	yes	characteristic
211	C₄×#131	—	89	1	no	—
212	#1569	—	89	2	no	—
213	#1569	—	89	2	no	—
214	#1568	—	89	2	no	—
215	#1566	—	89	2	no	—
216	#1565	—	89	1	no	—
217	#1565	—	89	1	no	—
218	#1564	—	89	1	no	—
219	C₂×#334	—	89	1	no	—
220	C₈₉	#17798	1	1	yes	characteristic, complemented, Sylow
221	C₂×#131	—	89	1	no	—
222	C₂×#131	—	89	1	no	—
223	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
224	#446	—	2	1	no	—
225	#446	—	178	1	no	—
226	#436	—	2	1	no	—
227	#436	—	178	1	no	—
228	#435	—	2	2	no	—
229	#435	—	178	2	no	—
230	#432	—	89	2	no	—
231	#432	C₂×D₈₉	1	2	yes	—
232	#431	—	89	1	no	—
233	#431	C₂×D₈₉	1	1	yes	characteristic
234	#423	—	89	2	no	—
235	#423	C₂×D₈₉	1	2	yes	—
236	C₂×C₄⋊C₈	—	89	1	no	—
237	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
238	#405	—	89	1	no	—
239	#405	—	89	2	no	—
240	#405	—	89	2	no	—
241	#405	—	89	1	no	—
242	C₄×M₁₆	—	89	1	no	—
243	C₄×M₁₆	—	89	2	no	—
244	C₂×C₄×C₈	—	89	1	no	—
245	C₂×C₄×C₈	—	89	2	no	—
246	#335	—	89	1	no	—
247	#335	—	89	1	no	—
248	#335	—	89	2	no	—
249	#335	—	89	2	no	—
250	#334	—	89	2	no	—
251	#334	—	89	2	no	—
252	#334	—	89	1	no	—
253	#334	—	89	1	no	—
254	#334	—	89	2	no	—
255	#334	—	89	2	no	—
256	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
257	#131	—	178	4	no	—
258	#131	—	178	4	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	89	1	no	—
261	C₂×M₁₆	—	89	2	no	—
262	C₂×M₁₆	—	89	2	no	—
263	C₂×M₁₆	—	89	1	no	—
264	C₂×M₁₆	—	178	2	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	89	1	no	—
267	C₂²×C₈	—	89	2	no	—
268	C₂²×C₈	—	89	2	no	—
269	C₂²×C₈	—	89	1	no	—
270	C₂²×C₈	—	178	1	no	—
271	C₄×Q₈	Dic₁₇₈	1	1	yes	characteristic
272	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
273	C₄×D₄	Dic₁₇₈	1	2	yes	—
274	C₄×D₄	Dic₁₇₈	1	1	yes	characteristic
275	C₄×D₄	C₂²⋊D₈₉	1	2	yes	—
276	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
277	#117	C₄×D₈₉	1	2	yes	—
278	#117	C₄×D₈₉	1	1	yes	characteristic
279	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
280	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	89	2	no	—
284	C₄⋊C₈	—	89	1	no	—
285	C₄⋊C₈	—	89	1	no	—
286	C₄⋊C₈	—	178	1	no	—
287	C₄⋊C₈	—	178	2	no	—
288	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
289	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
290	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
291	C₂²⋊C₈	—	2	1	no	—
292	C₂²⋊C₈	—	2	2	no	—
293	C₂²⋊C₈	—	178	1	no	—
294	C₂²⋊C₈	—	178	2	no	—
295	#97	—	2	2	no	—
296	#97	—	89	2	no	—
297	#97	—	89	2	no	—
298	#97	—	89	2	no	—
299	#97	—	178	2	no	—
300	C₄×C₈	—	2	1	no	—
301	C₄×C₈	—	89	2	no	—
302	C₄×C₈	—	89	1	no	—
303	C₄×C₈	—	89	1	no	—
304	C₄×C₈	—	89	2	no	—
305	C₄×C₈	—	89	1	no	—
306	C₄×C₈	—	89	1	no	—
307	C₄×C₈	—	89	2	no	—
308	C₄×C₈	—	178	1	no	—
309	#41	—	2	2	no	—
310	#41	—	2	2	no	—
311	C₂×Q₈	#13083553	1	1	yes	characteristic
312	C₂×D₄	#13083553	1	2	yes	—
313	C₂×D₄	#13083553	1	1	yes	characteristic
314	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
315	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
316	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
317	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
318	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
319	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
320	M₁₆	—	8	2	no	—
321	M₁₆	—	89	4	no	—
322	M₁₆	—	178	4	no	—
323	M₁₆	—	178	4	no	—
324	M₁₆	—	178	2	no	—
325	M₁₆	—	712	2	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	8	1	no	—
330	C₂×C₈	—	89	4	no	—
331	C₂×C₈	—	89	1	no	—
332	C₂×C₈	—	89	1	no	—
333	C₂×C₈	—	89	2	no	—
334	C₂×C₈	—	89	1	no	—
335	C₂×C₈	—	89	2	no	—
336	C₂×C₈	—	89	1	no	—
337	C₂×C₈	—	178	4	no	—
338	C₂×C₈	—	178	4	no	—
339	C₂×C₈	—	178	1	no	—
340	C₂×C₈	—	178	1	no	—
341	C₂×C₈	—	178	2	no	—
342	C₂×C₈	—	178	2	no	—
343	C₂×C₈	—	712	1	no	—
344	C₄⋊C₄	#13083553	1	2	yes	—
345	C₄⋊C₄	#13083553	1	1	yes	characteristic
346	C₄⋊C₄	#13083553	1	2	yes	—
347	C₄⋊C₄	#13083553	1	1	yes	characteristic
348	C₂²⋊C₄	#13083553	1	2	yes	—
349	C₂²⋊C₄	#13083553	1	2	yes	—
350	C₂²⋊C₄	#13083553	1	2	yes	—
351	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
352	C₄²	C₂×C₄×D₈₉	1	2	yes	—
353	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
354	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
355	C₄²	C₂×Dic₁₇₈	1	2	yes	—
356	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
357	C₄²	C₂³×D₈₉	1	1	yes	characteristic
358	C₂³	C₂×#13083553	1	2	yes	—
359	C₂³	C₂×#13083553	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	2	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
372	C₂×C₄	C₂×#13083553	1	2	yes	—
373	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
374	C₂×C₄	C₂×#13083553	1	2	yes	—
375	C₂×C₄	C₂×#13083553	1	2	yes	—
376	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
377	C₂×C₄	M₁₆×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	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
381	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
382	C₈	—	4	1	no	—
383	C₈	—	4	2	no	—
384	C₈	—	4	1	no	—
385	C₈	—	89	2	no	—
386	C₈	—	89	2	no	—
387	C₈	—	89	4	no	—
388	C₈	—	356	2	no	—
389	C₈	—	356	1	no	—
390	C₈	—	356	1	no	—
391	C₈	—	178	1	no	—
392	C₈	—	178	2	no	—
393	C₈	—	178	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	2	no	—
399	C₂²	#19151621	1	2	yes	—
400	C₂²	C₂²×#13083553	1	1	yes	characteristic
401	C₄	—	2	2	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	2	no	—
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	2	2	no	—
407	C₄	#19151621	1	2	yes	—
408	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
409	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
410	C₄	#19152406	1	1	yes	characteristic
411	C₄	#19152409	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	4	2	no	—
414	C₂	C₂×#19151621	1	1	yes	characteristic
415	C₂	#26028845	1	1	yes	characteristic
416	C₂	#26028861	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2989	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2989	D₈₉
hypocenter	C₈₉	#17798
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2989

Complements

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

N	H ≅ G/N	type
3. #26019020	C₂	semidirect
4. #26019017	C₂	semidirect
5. #26019013	C₂	semidirect
6. #26019007	C₂	semidirect
7. #26019005	C₂	semidirect
8. #26019002	C₂	semidirect
9. #26018998	C₂	semidirect
10. C₂×#19151348	C₂	semidirect
104. #13083558	M₁₆	semidirect
112. C₂×#10623494	M₁₆	semidirect
220. C₈₉	#17798	semidirect

Maximal subgroups

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

subgroup	index
#26029369	2
C₄×#15668019	2
#26019020	2
#26019017	2
#26019013	2
#26019007	2
#26019005	2
#26019002	2
#26018998	2
C₂×#19151348	2
C₈₉×#2989	2
#17798	89