Subgroups — #36538240

Order 22784.

738 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030339	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019658	C₂	1	4	yes	maximal, complemented
4	#26019648	C₂	1	2	yes	maximal, complemented
5	#26019604	C₂	1	2	yes	maximal, complemented
6	#26019015	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018947	C₂	1	2	yes	maximal, complemented
8	C₈₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₈₉	—	2	1	no	—
10	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
11	#19151914	—	2	2	no	—
12	#19151885	C₂²	1	4	yes	—
13	#19151790	C₂²	1	4	yes	—
14	#19151731	—	2	2	no	—
15	#19151708	—	2	4	no	—
16	C₈×D₃₅₆	—	2	1	no	—
17	C₈×D₃₅₆	—	2	1	no	—
18	C₈×D₃₅₆	C₂²	1	2	yes	—
19	C₈×D₃₅₆	C₂²	1	2	yes	—
20	#19151655	C₂²	1	2	yes	—
21	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
22	C₂×#15667924	—	2	1	no	—
23	#19151351	C₂²	1	2	yes	complemented
24	#19151349	C₂²	1	2	yes	complemented
25	#19151347	C₂²	1	4	yes	complemented
26	#19151342	C₂²	1	4	yes	complemented
27	#532×C₈₉	C₂²	1	2	yes	—
28	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#493×C₈₉	C₂²	1	1	yes	characteristic, complemented
30	#486×C₈₉	C₂²	1	2	yes	complemented
31	#466×C₈₉	—	2	1	no	—
32	#466×C₈₉	—	2	1	no	—
33	C₄×D₈×C₈₉	—	2	1	no	—
34	C₄×D₈×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₈₉	—	2	1	no	—
38	C₂×#13083559	—	2	4	no	—
39	C₂²×#10623494	—	2	1	no	—
40	C₂×C₈×D₈₉	—	2	2	no	—
41	C₂×C₈×D₈₉	—	2	2	no	—
42	C₂×C₈×D₈₉	—	4	4	no	—
43	C₄×D₃₅₆	—	2	1	no	—
44	C₄×D₃₅₆	—	2	1	no	—
45	C₄×D₃₅₆	D₄	1	2	yes	—
46	C₄×D₃₅₆	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
48	C₂×#13083551	—	2	1	no	—
49	#15667943	—	2	4	no	—
50	C₄⋊#10623494	—	2	1	no	—
51	C₄⋊#10623494	C₂³	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	4	yes	—
53	#15667933	—	2	4	no	—
54	#15667932	—	2	4	no	—
55	C₂²⋊#10623494	—	2	2	no	—
56	#15667924	—	2	2	no	—
57	#15667924	—	2	2	no	—
58	#15667923	—	2	2	no	—
59	C₂×#41×C₈₉	—	2	2	no	—
60	C₂×D₈×C₈₉	—	2	1	no	—
61	C₂×D₈×C₈₉	—	2	1	no	—
62	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
65	#127×C₈₉	C₂³	1	2	yes	—
66	#124×C₈₉	—	2	2	no	—
67	#121×C₈₉	—	2	2	no	—
68	#121×C₈₉	—	2	2	no	—
69	C₄×D₄×C₈₉	—	2	2	no	—
70	C₄×D₄×C₈₉	—	2	2	no	—
71	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	—	2	1	no	—
74	#102×C₈₉	—	2	2	no	—
75	#102×C₈₉	—	2	2	no	—
76	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂³×D₈₉	—	2	1	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×D₃₅₆	—	2	4	no	—
83	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
84	C₂²×Dic₈₉	—	2	1	no	—
85	C₂×C₄×D₈₉	—	2	2	no	—
86	C₂×C₄×D₈₉	—	2	2	no	—
87	C₂×C₄×D₈₉	—	4	2	no	—
88	C₂×C₄×D₈₉	—	4	2	no	—
89	#13083559	—	4	8	no	—
90	C₂×#10623494	—	2	2	no	—
91	C₂×#10623494	—	2	4	no	—
92	C₂×#10623494	—	4	2	no	—
93	C₈×D₈₉	—	4	4	no	—
94	C₈×D₈₉	—	4	4	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
97	#13083551	—	2	2	no	—
98	#13083551	—	2	2	no	—
99	#41×C₈₉	—	8	4	no	—
100	C₂×Q₈×C₈₉	—	2	2	no	—
101	C₂×D₄×C₈₉	—	2	2	no	—
102	C₂×D₄×C₈₉	—	2	2	no	—
103	C₂×D₄×C₈₉	—	2	2	no	—
104	C₂×D₄×C₈₉	—	4	2	no	—
105	C₂²×C₄×C₈₉	—	4	2	no	—
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
109	D₈×C₈₉	—	4	2	no	—
110	D₈×C₈₉	—	4	2	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	4	1	no	—
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₂²⋊C₄×C₈₉	—	4	2	no	—
120	C₂²⋊C₄×C₈₉	—	4	2	no	—
121	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₉	#41	1	1	yes	characteristic
124	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂³×C₈₉	—	4	2	no	—
126	C₂³×C₈₉	#121	1	1	yes	characteristic
127	C₂²×D₈₉	—	2	4	no	—
128	C₂²×D₈₉	—	2	2	no	—
129	C₂²×D₈₉	—	4	2	no	—
130	C₂²×D₈₉	—	4	2	no	—
131	Q₈×C₈₉	—	8	2	no	—
132	D₄×C₈₉	—	4	4	no	—
133	D₄×C₈₉	—	4	2	no	—
134	D₄×C₈₉	—	4	2	no	—
135	D₄×C₈₉	—	8	2	no	—
136	D₄×C₈₉	—	8	4	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	1	no	—
140	C₂×C₄×C₈₉	—	2	1	no	—
141	C₂×C₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	8	4	no	—
143	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₂×Dic₈₉	—	2	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	D₃₅₆	—	2	4	no	—
152	D₃₅₆	—	2	2	no	—
153	D₃₅₆	#135	1	4	yes	—
154	C₄×D₈₉	—	4	4	no	—
155	C₄×D₈₉	—	4	2	no	—
156	C₄×D₈₉	—	4	2	no	—
157	C₈×C₈₉	—	2	1	no	—
158	C₈×C₈₉	—	2	1	no	—
159	C₈×C₈₉	—	2	1	no	—
160	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
161	#10623494	—	4	4	no	—
162	#10623494	—	4	2	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	8	2	no	—
166	C₂²×C₈₉	—	8	2	no	—
167	C₂²×C₈₉	—	8	2	no	—
168	C₂²×C₈₉	#472	1	2	yes	—
169	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
170	C₂×D₈₉	—	2	4	no	—
171	C₂×D₈₉	—	4	8	no	—
172	C₂×D₈₉	—	4	2	no	—
173	C₂×D₈₉	—	4	2	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	8	2	no	—
179	C₄×C₈₉	C₂×#135	1	2	yes	—
180	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₄×C₈₉	#575	1	1	yes	characteristic
182	Dic₈₉	—	4	2	no	—
183	#19258	—	89	1	no	maximal, Sylow
184	C₂×C₈₉	—	2	2	no	—
185	C₂×C₈₉	—	16	2	no	—
186	C₂×C₈₉	#3036	1	1	yes	characteristic
187	C₂×C₈₉	#3050	1	1	yes	characteristic
188	C₂×C₈₉	C₂×#472	1	1	yes	characteristic
189	D₈₉	—	4	4	no	—
190	#3169	D₈₉	1	1	yes	characteristic, complemented
191	#3109	—	89	2	no	—
192	C₂×C₈×D₄	—	89	1	no	—
193	#1677	—	89	4	no	—
194	#1675	—	89	2	no	—
195	#1667	—	89	2	no	—
196	#1568	—	89	1	no	—
197	#1551	—	89	2	no	—
198	C₈₉	#19258	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₈	—	178	1	no	—
200	#532	C₂×D₈₉	1	2	yes	—
201	C₂×C₄×D₄	—	89	1	no	—
202	#500	C₂×D₈₉	1	1	yes	characteristic, complemented
203	#493	C₂×D₈₉	1	1	yes	characteristic, complemented
204	#486	C₂×D₈₉	1	2	yes	complemented
205	#481	—	178	2	no	—
206	#476	—	89	4	no	—
207	#466	—	2	1	no	—
208	#466	—	2	1	no	—
209	#459	—	89	4	no	—
210	#447	—	178	2	no	—
211	#442	—	178	4	no	—
212	C₄×D₈	—	2	1	no	—
213	C₄×D₈	—	2	1	no	—
214	C₈×D₄	—	89	2	no	—
215	C₈×D₄	—	89	2	no	—
216	C₈×D₄	—	178	1	no	—
217	C₈×D₄	—	178	1	no	—
218	#432	—	89	2	no	—
219	C₂×C₄⋊C₈	—	89	1	no	—
220	C₂×C₂²⋊C₈	—	178	1	no	—
221	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
222	#335	—	89	2	no	—
223	#334	—	89	2	no	—
224	#333	—	89	4	no	—
225	#331	—	89	4	no	—
226	C₂×#41	—	2	2	no	—
227	C₂²×D₄	—	89	1	no	—
228	C₂³×C₄	—	178	1	no	—
229	C₂×D₈	—	2	1	no	—
230	C₂×D₈	—	2	1	no	—
231	C₂×D₈	—	178	4	no	—
232	C₂²×C₈	—	356	4	no	—
233	C₂²×C₈	—	178	1	no	—
234	C₂²×C₈	—	178	2	no	—
235	C₂²×C₈	—	178	2	no	—
236	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
237	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
238	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
239	#127	C₂²×D₈₉	1	2	yes	—
240	#124	—	2	2	no	—
241	#121	—	2	2	no	—
242	#121	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	2	2	no	—
245	C₄×D₄	—	89	2	no	—
246	C₄×D₄	—	89	2	no	—
247	C₄×D₄	—	178	1	no	—
248	C₄×D₄	—	178	1	no	—
249	C₂×C₄⋊C₄	—	89	1	no	—
250	C₂×C₂²⋊C₄	—	178	1	no	—
251	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
252	#107	—	2	1	no	—
253	#107	—	2	1	no	—
254	#106	—	178	4	no	—
255	C₄⋊C₈	—	89	2	no	—
256	C₄⋊C₈	—	89	4	no	—
257	C₄⋊C₈	—	178	1	no	—
258	#102	—	2	2	no	—
259	#102	—	2	2	no	—
260	#102	—	178	4	no	—
261	#102	—	178	4	no	—
262	C₂²⋊C₈	—	178	2	no	—
263	C₂²⋊C₈	—	178	2	no	—
264	C₂²⋊C₈	—	178	2	no	—
265	#97	—	178	2	no	—
266	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
267	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₂⁴	—	178	1	no	—
271	#41	—	8	4	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	89	2	no	—
278	C₂×D₄	—	178	4	no	—
279	C₂×D₄	—	178	1	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	—	356	2	no	—
282	C₂²×C₄	—	356	2	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	—	178	2	no	—
285	C₂²×C₄	—	178	2	no	—
286	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₂²×C₄	#13083573	1	1	yes	characteristic
289	D₈	—	4	2	no	—
290	D₈	—	4	2	no	—
291	D₈	—	356	8	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	356	4	no	—
295	C₂×C₈	—	356	2	no	—
296	C₂×C₈	—	356	4	no	—
297	C₂×C₈	—	178	4	no	—
298	C₂×C₈	—	178	2	no	—
299	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
300	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
301	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
302	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	89	2	no	—
306	C₄⋊C₄	—	178	1	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₂²⋊C₄	—	178	2	no	—
310	C₂²⋊C₄	—	178	2	no	—
311	C₄²	D₄×D₈₉	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
313	C₄²	#13083573	1	1	yes	characteristic
314	C₄²	C₂³×D₈₉	1	1	yes	characteristic
315	C₂³	—	4	2	no	—
316	C₂³	—	356	2	no	—
317	C₂³	—	356	2	no	—
318	C₂³	—	178	4	no	—
319	C₂³	—	178	2	no	—
320	C₂³	#15667983	1	1	yes	characteristic
321	Q₈	—	8	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	4	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	4	no	—
327	D₄	—	89	4	no	—
328	D₄	—	178	4	no	—
329	D₄	—	178	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	8	4	no	—
336	C₂×C₄	—	356	4	no	—
337	C₂×C₄	—	356	2	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	178	2	no	—
341	C₂×C₄	#15667983	1	1	yes	characteristic
342	C₂×C₄	#15667983	1	1	yes	characteristic
343	C₂×C₄	#15667983	1	1	yes	characteristic
344	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
345	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
346	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	2	1	no	—
350	C₈	—	356	2	no	—
351	C₈	—	356	4	no	—
352	C₈	#15668037	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	356	8	no	—
359	C₂²	—	356	2	no	—
360	C₂²	—	356	2	no	—
361	C₂²	—	178	4	no	—
362	C₂²	#19151865	1	2	yes	—
363	C₂²	C₂×#15667983	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	8	2	no	—
369	C₄	—	356	2	no	—
370	C₄	#135×D₈₉	1	2	yes	—
371	C₄	C₂×#15668037	1	1	yes	characteristic
372	C₄	#19152396	1	1	yes	characteristic
373	C₂	—	2	2	no	—
374	C₂	—	16	2	no	—
375	C₂	—	356	4	no	—
376	C₂	#26029712	1	1	yes	characteristic
377	C₂	#26029829	1	1	yes	characteristic
378	C₂	C₂×#19151865	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3169	C₂
socle	C₂²×C₈₉	C₂×#121
hypercenter	#3169	D₈₉
hypocenter	C₈₉	#19258
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #26030339	C₂	semidirect
2. C₂×C₈×D₃₅₆	C₂	semidirect
3. #26019658	C₂	semidirect
4. #26019648	C₂	semidirect
5. #26019604	C₂	semidirect
6. #26019015	C₂	semidirect
7. #26018947	C₂	semidirect
8. C₈₉×#3169	C₂	semidirect
23. #19151351	C₂²	semidirect
24. #19151349	C₂²	semidirect
25. #19151347	C₂²	semidirect
26. #19151342	C₂²	semidirect
28. #500×C₈₉	C₂²	semidirect
29. #493×C₈₉	C₂²	semidirect
30. #486×C₈₉	C₂²	semidirect
190. #3169	D₈₉	semidirect
198. C₈₉	#19258	semidirect
202. #500	C₂×D₈₉	semidirect
203. #493	C₂×D₈₉	semidirect
204. #486	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030339	2
C₂×C₈×D₃₅₆	2
#26019658	2
#26019648	2
#26019604	2
#26019015	2
#26018947	2
C₈₉×#3169	2
#19258	89