Subgroups — #36707893

Order 22784.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152039	C₂	1	1	yes	maximal, characteristic
2	#26029170	C₂	1	1	yes	maximal, characteristic
3	#26022141	C₂	1	8	yes	maximal, complemented
4	#26022024	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151414	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021987	C₂	1	1	yes	maximal, characteristic
7	#26021857	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2950	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	1	yes	characteristic
10	#19152364	—	2	1	no	—
11	#19152039	C₄	1	4	yes	complemented
12	#19152039	C₂²	1	4	yes	—
13	C₂×#15667958	C₄	1	2	yes	complemented
14	C₂×#15667955	C₄	1	2	yes	complemented
15	C₂×#15667952	C₂²	1	1	yes	characteristic
16	C₂×#15667952	C₂²	1	1	yes	characteristic
17	#19151721	C₂²	1	4	yes	—
18	#19151716	—	2	2	no	—
19	C₂×#15667948	C₂²	1	1	yes	characteristic
20	#19151616	—	2	1	no	—
21	#19151420	C₂²	1	4	yes	complemented
22	#19151414	C₂²	1	4	yes	complemented
23	#19151407	C₂²	1	4	yes	—
24	#19151376	C₂²	1	4	yes	complemented
25	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	#519×C₈₉	—	2	1	no	—
27	#441×C₈₉	C₂²	1	4	yes	complemented
28	C₄×Q₁₆×C₈₉	—	2	2	no	—
29	#427×C₈₉	C₂²	1	1	yes	characteristic
30	#419×C₈₉	—	2	1	no	—
31	#405×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#13083574	—	2	1	no	—
33	#15668034	—	2	1	no	—
34	#15668034	—	2	1	no	—
35	#15668034	C₂×C₄	1	2	yes	—
36	#15668034	C₂³	1	2	yes	—
37	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083563	—	2	1	no	—
39	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
40	#15668020	—	4	2	no	—
41	#15668018	—	2	4	no	—
42	C₂×#13083558	—	2	1	no	—
43	M₁₆×D₈₉	—	4	2	no	—
44	#15667958	C₂×C₄	1	8	yes	—
45	#15667955	C₂×C₄	1	8	yes	—
46	#15667952	—	2	1	no	—
47	#15667952	C₂×C₄	1	2	yes	—
48	#15667952	C₂³	1	2	yes	—
49	#15667952	C₂³	1	2	yes	—
50	C₂×#13083549	—	2	1	no	—
51	C₂×#13083549	C₂×C₄	1	2	yes	—
52	#15667948	—	2	1	no	—
53	#15667948	C₂³	1	2	yes	—
54	#15667940	—	2	4	no	—
55	C₂²×Q₈×C₈₉	—	2	1	no	—
56	C₂×Q₁₆×C₈₉	—	2	1	no	—
57	C₂×Q₁₆×C₈₉	—	2	1	no	—
58	C₂×Q₁₆×C₈₉	—	2	1	no	—
59	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
60	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
61	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
62	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₄×Q₈×C₈₉	—	2	4	no	—
65	C₄×Q₈×C₈₉	—	4	2	no	—
66	#117×C₈₉	—	2	1	no	—
67	#117×C₈₉	—	4	2	no	—
68	#117×C₈₉	D₄	1	1	yes	characteristic
69	#107×C₈₉	—	2	1	no	—
70	#106×C₈₉	C₂³	1	2	yes	—
71	#103×C₈₉	—	2	4	no	—
72	#97×C₈₉	C₂³	1	2	yes	—
73	C₄×C₈×C₈₉	D₄	1	2	yes	—
74	#13083574	—	2	2	no	—
75	#13083574	—	2	2	no	—
76	#13083574	—	4	1	no	—
77	#13083574	—	4	1	no	—
78	C₂×Dic₁₇₈	—	2	1	no	—
79	C₂×D₃₅₆	—	2	1	no	—
80	C₂×C₂²⋊D₈₉	—	4	1	no	—
81	C₂×C₄×D₈₉	—	4	1	no	—
82	Dic₃₅₆	—	2	1	no	—
83	Dic₃₅₆	C₂²×C₄	1	2	yes	—
84	#13083563	—	2	2	no	—
85	#13083563	—	2	2	no	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	C₂²×C₄	1	2	yes	—
88	#13083558	—	2	2	no	—
89	#13083558	—	4	1	no	—
90	#13083557	—	4	4	no	—
91	C₂×#10623494	—	4	1	no	—
92	C₈×D₈₉	—	4	4	no	—
93	#13083549	—	2	2	no	—
94	#13083549	C₂²×C₄	1	4	yes	—
95	C₂×Q₈×C₈₉	—	2	2	no	—
96	C₂×Q₈×C₈₉	—	2	2	no	—
97	C₂×Q₈×C₈₉	—	2	2	no	—
98	C₂×Q₈×C₈₉	—	4	1	no	—
99	C₂×Q₈×C₈₉	—	4	1	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂²×C₄×C₈₉	—	4	1	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₈₉	—	2	2	no	—
104	Q₁₆×C₈₉	—	2	2	no	—
105	Q₁₆×C₈₉	—	2	2	no	—
106	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
107	M₁₆×C₈₉	—	2	1	no	—
108	M₁₆×C₈₉	—	2	1	no	—
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	—	2	1	no	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	4	4	no	—
119	C₂²⋊C₄×C₈₉	—	2	1	no	—
120	C₂²⋊C₄×C₈₉	—	4	1	no	—
121	C₄²×C₈₉	—	4	4	no	—
122	C₄²×C₈₉	—	4	1	no	—
123	C₄²×C₈₉	C₂×D₄	1	2	yes	—
124	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₈₉	—	4	1	no	—
126	Q₈×C₈₉	—	2	4	no	—
127	Q₈×C₈₉	—	4	2	no	—
128	Q₈×C₈₉	—	4	2	no	—
129	Q₈×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	4	2	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂²⋊D₈₉	—	4	2	no	—
142	C₂²⋊D₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	4	1	no	—
144	D₃₅₆	—	2	2	no	—
145	D₃₅₆	—	4	1	no	—
146	C₄×D₈₉	—	4	2	no	—
147	C₄×D₈₉	—	4	2	no	—
148	Dic₁₇₈	—	2	2	no	—
149	Dic₁₇₈	—	4	1	no	—
150	C₈×C₈₉	—	2	2	no	—
151	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
152	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
153	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
154	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
155	#10623494	—	4	2	no	—
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	#442	1	1	yes	characteristic
158	C₂²×C₈₉	#442	1	1	yes	characteristic
159	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×D₈₉	—	4	2	no	—
161	C₂×D₈₉	—	16	1	no	—
162	C₄×C₈₉	—	4	2	no	—
163	C₄×C₈₉	—	4	4	no	—
164	C₄×C₈₉	—	8	2	no	—
165	C₄×C₈₉	#442	1	1	yes	characteristic
166	C₄×C₈₉	#442	1	1	yes	characteristic
167	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	Dic₈₉	—	4	2	no	—
170	#33470	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	—	2	1	no	—
172	C₂×C₈₉	—	2	1	no	—
173	C₂×C₈₉	—	2	1	no	—
174	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
175	D₈₉	—	8	2	no	—
176	C₂×#510	—	89	1	no	—
177	#2957	—	89	1	no	—
178	#2950	D₈₉	1	1	yes	characteristic, complemented
179	#2180	—	89	8	no	—
180	#2155	—	89	1	no	—
181	C₂×#362	—	89	1	no	—
182	#2145	—	89	1	no	—
183	#2110	—	89	1	no	—
184	C₈₉	#33470	1	1	yes	characteristic, complemented, Sylow
185	C₂×#135	—	89	1	no	—
186	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
187	#568	—	178	1	no	—
188	#519	—	2	1	no	—
189	#510	—	89	4	no	—
190	#510	—	89	4	no	—
191	C₂×Q₃₂	—	89	2	no	—
192	C₂×SD₃₂	—	89	2	no	—
193	C₂×M₃₂	—	89	1	no	—
194	C₂×M₃₂	—	89	1	no	—
195	#444	—	89	4	no	—
196	#443	—	178	2	no	—
197	#441	C₂×D₈₉	1	4	yes	complemented
198	C₄×Q₁₆	—	2	2	no	—
199	C₂×#108	—	89	1	no	—
200	#427	C₂×D₈₉	1	1	yes	characteristic
201	#421	—	178	1	no	—
202	#419	—	2	1	no	—
203	#405	C₂×D₈₉	1	1	yes	characteristic
204	#365	—	89	4	no	—
205	#362	—	89	4	no	—
206	#360	—	89	4	no	—
207	#347	—	89	4	no	—
208	C₂×#41	—	178	1	no	—
209	C₂²×Q₈	—	2	1	no	—
210	#135	—	89	2	no	—
211	#135	—	89	2	no	—
212	#135	—	178	1	no	—
213	#135	—	178	1	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×Q₁₆	—	89	1	no	—
218	C₂×Q₁₆	C₄×D₈₉	1	2	yes	—
219	C₂×Q₁₆	C₄×D₈₉	1	2	yes	—
220	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
221	C₂×SD₁₆	—	178	1	no	—
222	C₂×D₈	—	89	1	no	—
223	#131	—	356	2	no	—
224	#131	—	178	4	no	—
225	C₂×M₁₆	—	356	2	no	—
226	C₂×M₁₆	—	178	1	no	—
227	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
229	C₄×Q₈	—	2	4	no	—
230	C₄×Q₈	—	4	2	no	—
231	#117	—	2	1	no	—
232	#117	—	4	2	no	—
233	#117	C₂²⋊D₈₉	1	1	yes	characteristic
234	Q₃₂	—	89	8	no	—
235	SD₃₂	—	89	8	no	—
236	M₃₂	—	89	2	no	—
237	M₃₂	—	89	2	no	—
238	M₃₂	—	89	2	no	—
239	M₃₂	—	178	1	no	—
240	C₂×C₁₆	—	89	2	no	—
241	C₂×C₁₆	—	178	1	no	—
242	#108	—	89	2	no	—
243	#108	—	178	1	no	—
244	#107	—	2	1	no	—
245	#106	C₂²×D₈₉	1	2	yes	—
246	#104	—	178	4	no	—
247	#103	—	2	4	no	—
248	#97	C₂²×D₈₉	1	2	yes	—
249	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
250	#41	—	356	1	no	—
251	#41	—	356	1	no	—
252	#41	—	178	2	no	—
253	#41	—	178	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	178	1	no	—
261	C₂×D₄	—	356	1	no	—
262	C₂×D₄	—	178	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	356	1	no	—
265	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	89	2	no	—
270	Q₁₆	—	178	1	no	—
271	Q₁₆	C₂×C₄×D₈₉	1	4	yes	—
272	SD₁₆	—	178	2	no	—
273	SD₁₆	—	178	2	no	—
274	D₈	—	89	2	no	—
275	D₈	—	178	1	no	—
276	M₁₆	—	2	1	no	—
277	M₁₆	—	2	1	no	—
278	M₁₆	—	356	4	no	—
279	M₁₆	—	356	1	no	—
280	M₁₆	—	178	2	no	—
281	C₂×C₈	—	356	4	no	—
282	C₂×C₈	—	356	1	no	—
283	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
284	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
285	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
286	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₂×C₈	#13083574	1	1	yes	characteristic
288	C₂×C₈	#13083574	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	4	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₄²	—	4	4	no	—
296	C₄²	—	4	1	no	—
297	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
298	C₁₆	—	89	4	no	—
299	C₁₆	—	178	2	no	—
300	C₂³	—	356	1	no	—
301	C₂³	#15668038	1	1	yes	characteristic
302	Q₈	—	2	4	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	356	1	no	—
307	Q₈	—	178	2	no	—
308	D₄	—	356	2	no	—
309	D₄	—	356	2	no	—
310	D₄	—	356	1	no	—
311	D₄	—	178	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	356	1	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	356	2	no	—
320	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
321	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
322	C₂×C₄	#15668037	1	1	yes	characteristic
323	C₂×C₄	#15668037	1	1	yes	characteristic
324	C₂×C₄	#15668038	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	356	2	no	—
328	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
329	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
331	C₈	C₂×#13083574	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	356	2	no	—
334	C₂²	—	1424	1	no	—
335	C₂²	#19151708	1	1	yes	characteristic
336	C₂²	#19151708	1	1	yes	characteristic
337	C₂²	C₂×#15668038	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	4	4	no	—
340	C₄	—	8	2	no	—
341	C₄	—	356	2	no	—
342	C₄	#19151708	1	1	yes	characteristic
343	C₄	#19151708	1	1	yes	characteristic
344	C₄	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
345	C₄	C₂×#15668037	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	712	2	no	—
350	C₂	C₂×#19151708	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19151708
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2950	C₂
socle	C₂×C₈₉	C₂×#442
prime radical	C₂×#15668034	C₂²
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2950	D₈₉
hypocenter	C₈₉	#33470
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #26022141	C₂	semidirect
4. #26022024	C₂	semidirect
5. C₂×#19151414	C₂	semidirect
7. #26021857	C₂	semidirect
8. C₈₉×#2950	C₂	semidirect
11. #19152039	C₄	semidirect
13. C₂×#15667958	C₄	semidirect
14. C₂×#15667955	C₄	semidirect
21. #19151420	C₂²	semidirect
22. #19151414	C₂²	semidirect
24. #19151376	C₂²	semidirect
27. #441×C₈₉	C₂²	semidirect
178. #2950	D₈₉	semidirect
184. C₈₉	#33470	semidirect
197. #441	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152039	2
#26029170	2
#26022141	2
#26022024	2
C₂×#19151414	2
#26021987	2
#26021857	2
C₈₉×#2950	2
#33470	89