Subgroups — #36537507

Order 22784.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031375	C₂	1	2	yes	maximal
2	C₂×#19151978	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028965	C₂	1	1	yes	maximal, characteristic
4	#26019742	C₂	1	4	yes	maximal, complemented
5	#26019618	C₂	1	4	yes	maximal, complemented
6	#26018923	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₅₆	—	2	1	no	—
9	C₂²×Dic₃₅₆	—	2	1	no	—
10	C₂×#15668018	—	2	1	no	—
11	#19152148	C₂²	1	1	yes	characteristic
12	C₂×#15668012	C₂²	1	2	yes	—
13	#19152003	—	2	2	no	—
14	#19151978	—	2	1	no	—
15	#19151978	—	2	1	no	—
16	#19151978	C₂²	1	2	yes	—
17	#19151978	C₂²	1	2	yes	—
18	#19151945	—	2	2	no	—
19	#19151912	—	2	4	no	—
20	#19151879	C₂²	1	8	yes	—
21	#19151677	—	2	1	no	—
22	#19151677	C₂²	1	2	yes	—
23	C₈×D₃₅₆	—	2	1	no	—
24	C₈×D₃₅₆	C₂²	1	2	yes	—
25	#19151622	C₂²	1	1	yes	characteristic
26	#19151564	—	2	1	no	—
27	#19151343	C₂²	1	8	yes	—
28	#431×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	#335×C₈₉	C₂²	1	2	yes	—
31	#334×C₈₉	C₂²	1	2	yes	—
32	C₂×#13083574	—	2	1	no	—
33	C₂²×Dic₁₇₈	—	2	2	no	—
34	C₂×Dic₃₅₆	—	2	2	no	—
35	C₂×Dic₃₅₆	—	2	2	no	—
36	C₂×Dic₃₅₆	—	2	2	no	—
37	C₂×Dic₃₅₆	—	2	2	no	—
38	C₂×Dic₃₅₆	—	4	4	no	—
39	C₂×Dic₃₅₆	—	4	4	no	—
40	#15668018	—	4	4	no	—
41	C₂×#13083558	—	2	1	no	—
42	C₂×#13083557	—	2	2	no	—
43	C₂×C₈×D₈₉	—	2	2	no	—
44	#15668012	—	2	2	no	—
45	#15668012	—	2	2	no	—
46	#15668012	—	2	2	no	—
47	#15668012	D₄	1	1	yes	characteristic
48	#15668012	C₂³	1	4	yes	—
49	#15668009	D₄	1	1	yes	characteristic
50	#15668000	—	2	1	no	—
51	#15667986	—	2	1	no	—
52	#15667986	D₄	1	2	yes	—
53	C₄×D₃₅₆	—	2	1	no	—
54	C₄×D₃₅₆	C₂³	1	2	yes	—
55	C₂×C₄⋊Dic₈₉	—	2	2	no	—
56	C₂×C₄⋊Dic₈₉	—	2	2	no	—
57	#15667946	—	2	8	no	—
58	C₄⋊#10623494	—	2	1	no	—
59	C₄⋊#10623494	C₂³	1	2	yes	—
60	#15667934	—	2	8	no	—
61	#15667924	—	2	2	no	—
62	#15667924	—	2	2	no	—
63	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
67	C₂²⋊C₈×C₈₉	—	2	2	no	—
68	C₄×C₈×C₈₉	—	2	2	no	—
69	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	#13083574	—	4	2	no	—
74	#13083574	—	4	2	no	—
75	C₂×Dic₁₇₈	—	2	1	no	—
76	C₂×Dic₁₇₈	—	2	4	no	—
77	C₂×Dic₁₇₈	—	4	4	no	—
78	C₂×Dic₁₇₈	—	4	4	no	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
83	C₂×C₂²⋊D₈₉	—	2	2	no	—
84	C₂²×Dic₈₉	—	4	2	no	—
85	C₂×C₄×D₈₉	—	2	2	no	—
86	Dic₃₅₆	—	4	4	no	—
87	Dic₃₅₆	—	4	4	no	—
88	Dic₃₅₆	—	4	4	no	—
89	Dic₃₅₆	—	4	4	no	—
90	#13083558	—	4	2	no	—
91	#13083557	—	4	4	no	—
92	C₂×#10623494	—	2	2	no	—
93	C₈×D₈₉	—	4	4	no	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	2	4	no	—
96	C₄⋊Dic₈₉	—	2	4	no	—
97	C₄⋊Dic₈₉	—	4	2	no	—
98	C₄⋊Dic₈₉	—	4	2	no	—
99	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
100	#13083551	—	2	2	no	—
101	#13083551	—	2	2	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	—	2	4	no	—
107	C₂×C₈×C₈₉	D₈	1	4	yes	—
108	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₈₉	#41	1	1	yes	characteristic
115	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₂³×C₈₉	#121	1	1	yes	characteristic
117	C₂²×D₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	4	4	no	—
129	C₂×Dic₈₉	—	2	2	no	—
130	C₂×Dic₈₉	—	4	4	no	—
131	C₂×Dic₈₉	—	8	4	no	—
132	D₃₅₆	—	2	4	no	—
133	D₃₅₆	—	4	2	no	—
134	D₃₅₆	—	4	2	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	—	4	2	no	—
137	Dic₁₇₈	—	4	2	no	—
138	Dic₁₇₈	—	4	4	no	—
139	Dic₁₇₈	—	4	4	no	—
140	Dic₁₇₈	—	8	4	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	4	4	no	—
144	C₈×C₈₉	C₂×D₈	1	2	yes	—
145	C₈×C₈₉	C₂×D₈	1	2	yes	—
146	#10623494	—	4	2	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	#459	1	2	yes	—
150	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
151	C₂×D₈₉	—	4	2	no	—
152	C₂×D₈₉	—	4	2	no	—
153	C₂×D₈₉	—	4	2	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	#459	1	2	yes	—
157	C₄×C₈₉	#473	1	2	yes	—
158	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₄×C₈₉	#578	1	2	yes	—
161	Dic₈₉	—	4	2	no	—
162	Dic₈₉	—	8	4	no	—
163	#19188	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	—	2	2	no	—
165	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
166	C₂×C₈₉	#3059	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
168	D₈₉	—	8	2	no	—
169	#3226	—	89	2	no	—
170	C₂×#494	—	89	1	no	—
171	#2931	—	89	1	no	—
172	#1689	—	89	4	no	—
173	#1669	—	89	4	no	—
174	#1564	D₈₉	1	1	yes	characteristic, complemented
175	#1546	—	89	2	no	—
176	C₈₉	#19188	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₁₆	—	178	1	no	—
178	C₂²×Q₁₆	—	178	1	no	—
179	C₂×#131	—	178	1	no	—
180	#532	—	89	1	no	—
181	C₂×C₄⋊Q₈	—	89	2	no	—
182	#500	—	178	2	no	—
183	#494	—	89	2	no	—
184	#494	—	89	2	no	—
185	#494	—	178	1	no	—
186	#494	—	178	1	no	—
187	#486	—	178	2	no	—
188	#480	—	178	4	no	—
189	#474	—	89	8	no	—
190	#436	—	89	2	no	—
191	#436	—	178	1	no	—
192	C₈×D₄	—	89	2	no	—
193	C₈×D₄	—	178	1	no	—
194	#431	C₂×D₈₉	1	2	yes	—
195	#423	—	89	1	no	—
196	#408	—	178	1	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	#335	C₂×D₈₉	1	2	yes	—
199	#334	C₂×D₈₉	1	2	yes	—
200	#332	—	89	8	no	—
201	C₂×#41	—	178	1	no	—
202	C₂²×Q₈	—	178	2	no	—
203	C₂×Q₁₆	—	356	4	no	—
204	C₂×Q₁₆	—	356	4	no	—
205	C₂×Q₁₆	—	178	2	no	—
206	C₂×Q₁₆	—	178	2	no	—
207	C₂×Q₁₆	—	178	2	no	—
208	C₂×Q₁₆	—	178	2	no	—
209	#131	—	356	4	no	—
210	C₂×M₁₆	—	178	1	no	—
211	C₂×M₁₆	—	178	2	no	—
212	C₂²×C₈	—	178	2	no	—
213	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
214	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
215	C₄⋊Q₈	—	89	1	no	—
216	C₄⋊Q₈	—	89	4	no	—
217	C₄⋊Q₈	—	178	2	no	—
218	C₄⋊Q₈	—	178	2	no	—
219	C₄⋊Q₈	—	178	2	no	—
220	#127	—	89	1	no	—
221	#124	—	178	1	no	—
222	#121	—	89	2	no	—
223	#121	—	178	1	no	—
224	C₄×D₄	—	89	2	no	—
225	C₄×D₄	—	178	1	no	—
226	C₂×C₄⋊C₄	—	178	2	no	—
227	C₂×C₄⋊C₄	—	178	2	no	—
228	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
229	#107	—	178	8	no	—
230	C₄⋊C₈	—	89	2	no	—
231	C₄⋊C₈	—	178	1	no	—
232	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
233	#102	—	178	8	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	178	2	no	—
236	C₂²⋊C₈	—	178	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	D₃₅₆	1	1	yes	characteristic
239	C₄×C₈	D₃₅₆	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	#41	—	356	2	no	—
243	#41	—	356	2	no	—
244	C₂×Q₈	—	356	2	no	—
245	C₂×Q₈	—	356	4	no	—
246	C₂×Q₈	—	356	4	no	—
247	C₂×Q₈	—	178	4	no	—
248	C₂×Q₈	—	178	1	no	—
249	C₂×D₄	—	89	2	no	—
250	C₂×D₄	—	178	2	no	—
251	C₂×D₄	—	178	1	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂²×C₄	—	356	2	no	—
254	C₂²×C₄	—	178	2	no	—
255	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
257	C₂²×C₄	#13083575	1	1	yes	characteristic
258	Q₁₆	—	356	4	no	—
259	Q₁₆	—	356	4	no	—
260	Q₁₆	—	356	4	no	—
261	Q₁₆	—	356	4	no	—
262	M₁₆	—	356	2	no	—
263	M₁₆	—	356	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	356	4	no	—
267	C₂×C₈	—	178	2	no	—
268	C₂×C₈	D₇₁₂	1	4	yes	—
269	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
270	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
271	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
272	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
273	C₄⋊C₄	—	89	2	no	—
274	C₄⋊C₄	—	356	2	no	—
275	C₄⋊C₄	—	356	2	no	—
276	C₄⋊C₄	—	178	4	no	—
277	C₄⋊C₄	—	178	4	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	2	no	—
280	C₂²⋊C₄	—	178	2	no	—
281	C₄²	D₄×D₈₉	1	1	yes	characteristic
282	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
283	C₄²	#13083575	1	1	yes	characteristic
284	C₄²	C₂³×D₈₉	1	1	yes	characteristic
285	C₂³	—	178	2	no	—
286	C₂³	#15667987	1	1	yes	characteristic
287	Q₈	—	356	2	no	—
288	Q₈	—	356	4	no	—
289	Q₈	—	356	4	no	—
290	Q₈	—	712	4	no	—
291	D₄	—	356	4	no	—
292	D₄	—	356	2	no	—
293	D₄	—	356	2	no	—
294	D₄	—	178	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	356	2	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	356	4	no	—
300	C₂×C₄	—	178	2	no	—
301	C₂×C₄	—	712	4	no	—
302	C₂×C₄	#15667987	1	1	yes	characteristic
303	C₂×C₄	#15667987	1	1	yes	characteristic
304	C₂×C₄	#15667987	1	1	yes	characteristic
305	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
306	C₂×C₄	#15668040	1	2	yes	—
307	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
309	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	356	2	no	—
314	C₈	C₂×D₇₁₂	1	2	yes	—
315	C₈	C₂×D₇₁₂	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	356	2	no	—
319	C₂²	—	356	2	no	—
320	C₂²	—	356	2	no	—
321	C₂²	#19151794	1	2	yes	—
322	C₂²	C₂×#15667987	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	356	2	no	—
326	C₄	—	712	4	no	—
327	C₄	#19151794	1	2	yes	—
328	C₄	#19151874	1	2	yes	—
329	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
330	C₄	C₂×#15668040	1	1	yes	characteristic
331	C₄	#19152412	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	712	2	no	—
334	C₂	C₂×#19151794	1	1	yes	characteristic
335	C₂	#26029908	1	1	yes	characteristic
336	C₂	C₂×#19151874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#19152148	C₂²
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#19188
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. C₂×#19151978	C₂	semidirect
4. #26019742	C₂	semidirect
5. #26019618	C₂	semidirect
6. #26018923	C₂	semidirect
7. C₈₉×#1564	C₂	semidirect
174. #1564	D₈₉	semidirect
176. C₈₉	#19188	semidirect

Maximal subgroups

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

subgroup	index
#26031375	2
C₂×#19151978	2
#26028965	2
#26019742	2
#26019618	2
#26018923	2
C₈₉×#1564	2
#19188	89