Subgroups — #36524560

Order 22784.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029333	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029328	C₂	1	1	yes	maximal, characteristic
3	#26019131	C₂	1	2	yes	maximal, complemented
4	#26019096	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊#10623494	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#10623494	C₂	1	2	yes	maximal, complemented
7	#26019032	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018699	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018669	C₂	1	1	yes	maximal, characteristic
10	#26018547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3103	C₂	1	1	yes	maximal, characteristic
12	Q₈×#10623494	—	2	2	no	—
13	Q₈×#10623494	C₂²	1	2	yes	—
14	Q₈×#10623494	C₂²	1	2	yes	—
15	#19151676	C₂²	1	2	yes	—
16	#19151670	—	2	2	no	—
17	D₄×#10623494	C₂²	1	2	yes	—
18	#19151655	C₂²	1	1	yes	characteristic
19	#19151651	—	2	2	no	—
20	#19151651	C₂²	1	1	yes	characteristic
21	#19151651	C₂²	1	1	yes	characteristic
22	#19151565	—	2	2	no	—
23	C₄×#13083558	C₂²	1	1	yes	characteristic
24	#19151351	C₂²	1	2	yes	—
25	#19151349	C₂²	1	2	yes	—
26	Q₈⋊#10623494	C₂²	1	2	yes	—
27	Q₈⋊#10623494	C₂²	1	2	yes	—
28	D₄⋊#10623494	C₂²	1	2	yes	—
29	D₄⋊#10623494	C₂²	1	2	yes	—
30	#19151320	C₂²	1	1	yes	characteristic
31	#19151320	C₂²	1	1	yes	characteristic
32	#19151319	C₂²	1	1	yes	characteristic
33	C₈×#10623494	C₂²	1	1	yes	characteristic
34	#532×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
36	#481×C₈₉	C₄	1	2	yes	—
37	#475×C₈₉	C₄	1	2	yes	—
38	#460×C₈₉	C₄	1	2	yes	—
39	#448×C₈₉	C₄	1	2	yes	—
40	C₄×SD₁₆×C₈₉	C₂²	1	2	yes	—
41	C₄×SD₁₆×C₈₉	C₂²	1	2	yes	—
42	#431×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#13083558	—	2	2	no	—
44	C₂×#13083558	—	2	1	no	—
45	C₂²×#10623494	—	2	2	no	—
46	C₄⋊#10623494	—	2	2	no	—
47	C₄⋊#10623494	—	2	2	no	—
48	C₄⋊#10623494	D₄	1	2	yes	—
49	C₄⋊#10623494	C₂³	1	2	yes	—
50	C₄⋊#10623494	C₂³	1	2	yes	—
51	C₂²⋊#10623494	—	2	2	no	—
52	C₂²⋊#10623494	—	2	2	no	—
53	C₂²⋊#10623494	—	2	1	no	—
54	C₂²⋊#10623494	—	2	1	no	—
55	C₂²⋊#10623494	D₄	1	2	yes	—
56	#15667923	—	2	1	no	—
57	#15667923	—	2	1	no	—
58	C₄×#10623494	—	2	2	no	—
59	C₄×#10623494	—	2	2	no	—
60	C₄×#10623494	—	2	1	no	—
61	C₄×#10623494	C₂³	1	1	yes	characteristic
62	C₄×#10623494	C₂³	1	1	yes	characteristic
63	C₄×#10623494	C₂³	1	1	yes	characteristic
64	C₄×#10623494	C₂³	1	1	yes	characteristic
65	C₂×#41×C₈₉	—	2	1	no	—
66	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂×SD₁₆×C₈₉	C₂×C₄	1	2	yes	—
68	C₂×SD₁₆×C₈₉	C₂×C₄	1	2	yes	—
69	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
71	#124×C₈₉	—	2	1	no	—
72	#121×C₈₉	—	2	1	no	—
73	#121×C₈₉	C₂×C₄	1	2	yes	—
74	C₄×Q₈×C₈₉	—	2	1	no	—
75	C₄×Q₈×C₈₉	—	2	1	no	—
76	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
77	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
78	C₄×D₄×C₈₉	—	2	1	no	—
79	C₄×D₄×C₈₉	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₈₉	—	2	1	no	—
83	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
84	#106×C₈₉	C₂×C₄	1	2	yes	—
85	#106×C₈₉	C₂×C₄	1	2	yes	—
86	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
87	#103×C₈₉	C₂×C₄	1	2	yes	—
88	#103×C₈₉	C₂×C₄	1	2	yes	—
89	#102×C₈₉	C₂×C₄	1	2	yes	—
90	#102×C₈₉	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	#13083558	—	4	2	no	—
95	#13083558	—	8	2	no	—
96	C₂×#10623494	—	2	1	no	—
97	C₂×#10623494	—	2	2	no	—
98	C₂×#10623494	—	2	2	no	—
99	C₂×#10623494	—	2	1	no	—
100	C₂×#10623494	—	8	2	no	—
101	C₂×#10623494	C₂×D₄	1	2	yes	—
102	C₂×#10623494	C₂×D₄	1	2	yes	—
103	#41×C₈₉	—	8	2	no	—
104	C₂×Q₈×C₈₉	—	2	4	no	—
105	C₂×Q₈×C₈₉	—	2	1	no	—
106	C₂×Q₈×C₈₉	—	2	1	no	—
107	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₈₉	—	2	2	no	—
109	C₂×D₄×C₈₉	—	2	1	no	—
110	C₂×D₄×C₈₉	—	2	1	no	—
111	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₈₉	—	2	1	no	—
113	C₂²×C₄×C₈₉	—	2	1	no	—
114	C₂²×C₄×C₈₉	—	2	2	no	—
115	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
118	SD₁₆×C₈₉	—	2	4	no	—
119	SD₁₆×C₈₉	—	2	4	no	—
120	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	—	2	1	no	—
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₈₉	—	2	2	no	—
129	C₂²⋊C₄×C₈₉	—	2	2	no	—
130	C₄²×C₈₉	—	2	2	no	—
131	C₄²×C₈₉	—	2	2	no	—
132	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₈₉	#41	1	1	yes	characteristic
135	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₉	—	2	2	no	—
137	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₈₉	—	2	4	no	—
139	Q₈×C₈₉	—	2	2	no	—
140	Q₈×C₈₉	—	8	1	no	—
141	Q₈×C₈₉	#131	1	4	yes	—
142	D₄×C₈₉	—	2	2	no	—
143	D₄×C₈₉	—	2	2	no	—
144	D₄×C₈₉	—	8	1	no	—
145	D₄×C₈₉	—	8	1	no	—
146	D₄×C₈₉	—	8	2	no	—
147	C₂×C₄×C₈₉	—	2	4	no	—
148	C₂×C₄×C₈₉	—	2	2	no	—
149	C₂×C₄×C₈₉	—	2	2	no	—
150	C₂×C₄×C₈₉	—	2	2	no	—
151	C₂×C₄×C₈₉	—	2	2	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	4	1	no	—
154	C₂×C₄×C₈₉	—	4	2	no	—
155	C₂×C₄×C₈₉	—	4	2	no	—
156	C₂×C₄×C₈₉	—	8	1	no	—
157	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₈×C₈₉	—	2	2	no	—
164	C₈×C₈₉	—	2	2	no	—
165	#10623494	—	2	2	no	—
166	#10623494	—	2	2	no	—
167	#10623494	—	2	2	no	—
168	#10623494	—	4	1	no	—
169	#10623494	—	4	2	no	—
170	#10623494	—	4	2	no	—
171	C₂²×C₈₉	—	2	1	no	—
172	C₂²×C₈₉	—	2	1	no	—
173	C₂²×C₈₉	—	2	1	no	—
174	C₂²×C₈₉	—	4	2	no	—
175	C₂²×C₈₉	—	4	2	no	—
176	C₂²×C₈₉	—	4	2	no	—
177	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₈₉	—	2	4	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	—	4	2	no	—
184	C₄×C₈₉	—	4	2	no	—
185	C₄×C₈₉	C₂×#131	1	2	yes	—
186	C₄×C₈₉	#577	1	2	yes	—
187	#17923	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	—	4	1	no	—
189	C₂×C₈₉	—	8	2	no	—
190	C₂×C₈₉	#2929	1	1	yes	characteristic
191	C₂×C₈₉	#2949	1	1	yes	characteristic
192	C₂×C₈₉	#2956	1	1	yes	characteristic
193	#3103	D₈₉	1	1	yes	characteristic
194	#2982	—	89	1	no	—
195	#2981	—	89	1	no	—
196	#1588	—	89	2	no	—
197	#1583	—	89	2	no	—
198	SD₁₆⋊C₈	—	89	2	no	—
199	C₈×SD₁₆	—	89	2	no	—
200	#1572	—	89	1	no	—
201	#1491	—	89	1	no	—
202	#1484	—	89	1	no	—
203	#1453	—	89	1	no	—
204	C₈₉	#17923	1	1	yes	characteristic, complemented, Sylow
205	#532	C₂×D₈₉	1	1	yes	characteristic
206	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
207	#481	Dic₈₉	1	2	yes	—
208	#475	Dic₈₉	1	2	yes	—
209	#460	Dic₈₉	1	2	yes	—
210	#448	Dic₈₉	1	2	yes	—
211	C₈×Q₈	—	89	2	no	—
212	C₈×Q₈	—	89	2	no	—
213	C₈×Q₈	—	178	2	no	—
214	C₄×SD₁₆	C₂×D₈₉	1	2	yes	—
215	C₄×SD₁₆	C₂×D₈₉	1	2	yes	—
216	#436	—	89	2	no	—
217	#435	—	178	2	no	—
218	C₈×D₄	—	89	2	no	—
219	#432	—	89	1	no	—
220	#431	—	89	1	no	—
221	#431	—	89	1	no	—
222	#431	—	178	2	no	—
223	#431	C₂×D₈₉	1	1	yes	characteristic
224	#408	—	178	2	no	—
225	C₄×M₁₆	—	89	1	no	—
226	#335	—	89	2	no	—
227	#334	—	89	2	no	—
228	Q₈⋊C₈	—	89	2	no	—
229	Q₈⋊C₈	—	89	2	no	—
230	D₄⋊C₈	—	89	2	no	—
231	D₄⋊C₈	—	89	2	no	—
232	#322	—	89	1	no	—
233	#322	—	89	1	no	—
234	#322	—	89	1	no	—
235	C₈²	—	89	1	no	—
236	C₂×#41	—	2	1	no	—
237	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
238	C₂×SD₁₆	C₂×Dic₈₉	1	2	yes	—
239	C₂×SD₁₆	C₂×Dic₈₉	1	2	yes	—
240	C₂×M₁₆	—	178	2	no	—
241	C₂×M₁₆	—	178	1	no	—
242	C₂²×C₈	—	178	2	no	—
243	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
244	#127	C₂×Dic₈₉	1	1	yes	characteristic
245	#124	—	2	1	no	—
246	#121	—	2	1	no	—
247	#121	C₂×Dic₈₉	1	2	yes	—
248	C₄×Q₈	—	2	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	C₂×Dic₈₉	1	2	yes	—
251	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	C₂²×D₈₉	1	2	yes	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
256	C₂×C₄²	—	2	1	no	—
257	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
258	#106	C₂×Dic₈₉	1	2	yes	—
259	#106	C₂×Dic₈₉	1	2	yes	—
260	C₄⋊C₈	—	89	2	no	—
261	C₄⋊C₈	—	89	2	no	—
262	C₄⋊C₈	—	89	2	no	—
263	C₄⋊C₈	—	178	2	no	—
264	C₄⋊C₈	—	178	2	no	—
265	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
266	#103	C₂×Dic₈₉	1	2	yes	—
267	#103	C₂×Dic₈₉	1	2	yes	—
268	#102	C₂×Dic₈₉	1	2	yes	—
269	#102	C₂×Dic₈₉	1	2	yes	—
270	C₂²⋊C₈	—	89	2	no	—
271	C₂²⋊C₈	—	178	2	no	—
272	C₂²⋊C₈	—	178	2	no	—
273	C₂²⋊C₈	—	178	1	no	—
274	C₂²⋊C₈	—	178	1	no	—
275	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
276	#97	—	178	1	no	—
277	#97	—	178	1	no	—
278	C₄×C₈	—	89	1	no	—
279	C₄×C₈	—	89	1	no	—
280	C₄×C₈	—	89	1	no	—
281	C₄×C₈	—	89	1	no	—
282	C₄×C₈	—	178	2	no	—
283	C₄×C₈	—	178	2	no	—
284	C₄×C₈	—	178	1	no	—
285	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	#41	—	8	2	no	—
288	C₂×Q₈	—	2	4	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	—	2	1	no	—
291	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
300	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
301	C₂²×C₄	#13083573	1	1	yes	characteristic
302	SD₁₆	—	2	4	no	—
303	SD₁₆	—	2	4	no	—
304	M₁₆	—	356	2	no	—
305	M₁₆	—	712	2	no	—
306	C₂×C₈	—	89	2	no	—
307	C₂×C₈	—	89	2	no	—
308	C₂×C₈	—	178	2	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	—	178	1	no	—
311	C₂×C₈	—	178	1	no	—
312	C₂×C₈	—	712	2	no	—
313	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
314	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
320	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
326	C₄²	D₄×D₈₉	1	1	yes	characteristic
327	C₄²	#13083573	1	1	yes	characteristic
328	C₄²	C₂³×D₈₉	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
331	Q₈	—	2	4	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	8	1	no	—
334	Q₈	#15668019	1	4	yes	—
335	D₄	—	2	2	no	—
336	D₄	—	2	2	no	—
337	D₄	—	8	1	no	—
338	D₄	—	8	1	no	—
339	D₄	—	8	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	1	no	—
350	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
351	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
352	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
353	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
355	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	2	no	—
358	C₈	—	356	1	no	—
359	C₈	—	356	2	no	—
360	C₈	—	356	2	no	—
361	C₈	—	178	2	no	—
362	C₈	—	178	2	no	—
363	C₈	—	178	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	4	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	C₂×#15668019	1	2	yes	—
379	C₄	#19152404	1	2	yes	—
380	C₂	—	4	1	no	—
381	C₂	—	8	2	no	—
382	C₂	#26028947	1	1	yes	characteristic
383	C₂	#26029112	1	1	yes	characteristic
384	C₂	#26029162	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3103	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#532×C₈₉	C₂²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#3103	D₈₉
hypocenter	C₈₉	#17923
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3103

Complements

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

N	H ≅ G/N	type
1. #26029333	C₂	semidirect
3. #26019131	C₂	semidirect
4. #26019096	C₂	semidirect
5. SD₁₆⋊#10623494	C₂	semidirect
6. SD₁₆×#10623494	C₂	semidirect
7. #26019032	C₂	semidirect
8. #26018699	C₂	semidirect
10. #26018547	C₂	semidirect
204. C₈₉	#17923	semidirect

Maximal subgroups

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

subgroup	index
#26029333	2
#26029328	2
#26019131	2
#26019096	2
SD₁₆⋊#10623494	2
SD₁₆×#10623494	2
#26019032	2
#26018699	2
#26018669	2
#26018547	2
C₈₉×#3103	2
#17923	89