Subgroups — #36472690

Order 22784.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151983	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151414	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151412	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151410	C₂	1	1	yes	maximal, characteristic
5	M₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020422	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#362×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083561	—	2	1	no	—
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×#15668001	—	2	1	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	#19151983	—	2	1	no	—
13	#19151983	—	2	1	no	—
14	#19151983	C₄	1	2	yes	—
15	#19151983	C₄	1	2	yes	—
16	C₂×#15667948	C₂²	1	1	yes	characteristic
17	C₂×#15667947	—	2	1	no	—
18	#19151621	—	2	1	no	—
19	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
20	#19151414	—	2	2	no	—
21	#19151412	—	2	2	no	—
22	#19151410	—	2	2	no	—
23	#19151373	—	2	1	no	—
24	C₁₆×Dic₈₉	—	2	1	no	—
25	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
28	#362×C₈₉	—	2	2	no	—
29	C₂²×D₃₅₆	—	2	1	no	—
30	C₂×#13083561	—	2	2	no	—
31	C₂×#13083561	—	2	2	no	—
32	C₂×#13083561	—	4	2	no	—
33	C₂×#13083561	—	4	2	no	—
34	C₂×D₇₁₂	—	2	1	no	—
35	C₂×D₇₁₂	—	2	1	no	—
36	C₂×D₇₁₂	—	2	1	no	—
37	C₂×D₇₁₂	—	2	1	no	—
38	C₂×D₇₁₂	—	2	1	no	—
39	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
40	C₂×#13083558	—	2	1	no	—
41	C₂²×#10623494	D₄	1	1	yes	characteristic
42	#15668001	—	2	2	no	—
43	#15668001	—	2	2	no	—
44	#15668001	—	4	2	no	—
45	C₂×#13083551	—	4	2	no	—
46	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
47	#15667952	—	2	1	no	—
48	#15667952	D₄	1	2	yes	—
49	C₂×#13083549	—	2	1	no	—
50	#15667948	—	2	1	no	—
51	#15667948	—	2	1	no	—
52	#15667947	—	4	2	no	—
53	#15667941	—	4	2	no	—
54	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₈₉	D₄	1	1	yes	characteristic
57	C₈×Dic₈₉	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₈₉	—	2	1	no	—
59	C₂×Q₁₆×C₈₉	—	2	1	no	—
60	C₂×Q₁₆×C₈₉	—	2	1	no	—
61	C₂×Q₁₆×C₈₉	—	2	1	no	—
62	C₂×Q₁₆×C₈₉	—	2	1	no	—
63	C₂×Q₁₆×C₈₉	—	2	1	no	—
64	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₈₉	—	2	1	no	—
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	M₃₂×C₈₉	—	2	1	no	—
68	M₃₂×C₈₉	D₄	1	2	yes	—
69	C₂×C₁₆×C₈₉	—	2	1	no	—
70	#108×C₈₉	—	2	1	no	—
71	#108×C₈₉	—	2	1	no	—
72	C₂³×D₈₉	—	4	1	no	—
73	C₂×D₃₅₆	—	2	2	no	—
74	C₂×D₃₅₆	—	4	1	no	—
75	C₂×D₃₅₆	—	4	1	no	—
76	C₂×D₃₅₆	—	4	1	no	—
77	C₂×D₃₅₆	—	4	1	no	—
78	C₂×D₃₅₆	—	4	1	no	—
79	C₂²×Dic₈₉	SD₁₆	1	2	yes	—
80	#13083561	—	4	4	no	—
81	#13083561	—	4	4	no	—
82	D₇₁₂	—	2	2	no	—
83	D₇₁₂	—	2	2	no	—
84	D₇₁₂	—	4	1	no	—
85	D₇₁₂	—	4	1	no	—
86	#13083558	—	4	1	no	—
87	#13083558	—	4	1	no	—
88	C₂×#10623494	—	4	1	no	—
89	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#10623494	D₈	1	2	yes	—
92	C₂×#10623494	SD₁₆	1	2	yes	—
93	#13083551	—	4	4	no	—
94	#13083551	—	4	4	no	—
95	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×Dic₈₉	D₈	1	2	yes	—
98	#13083549	—	2	2	no	—
99	C₂×Q₈×C₈₉	—	2	2	no	—
100	C₂×Q₈×C₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	—	4	1	no	—
102	C₂×Q₈×C₈₉	—	4	1	no	—
103	C₂×Q₈×C₈₉	—	4	1	no	—
104	C₂×Q₈×C₈₉	—	4	1	no	—
105	C₂²×C₄×C₈₉	—	4	1	no	—
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	Q₁₆×C₈₉	—	2	2	no	—
108	Q₁₆×C₈₉	—	2	2	no	—
109	Q₁₆×C₈₉	—	4	1	no	—
110	Q₁₆×C₈₉	—	4	1	no	—
111	M₁₆×C₈₉	—	4	1	no	—
112	M₁₆×C₈₉	—	4	1	no	—
113	C₂×C₈×C₈₉	—	4	1	no	—
114	C₂×C₈×C₈₉	C₂²×C₄	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	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
119	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
120	C₁₆×C₈₉	—	2	2	no	—
121	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×D₈₉	—	4	2	no	—
123	C₂²×D₈₉	—	8	4	no	—
124	C₂²×D₈₉	—	8	1	no	—
125	C₂²×D₈₉	—	8	1	no	—
126	Q₈×C₈₉	—	4	2	no	—
127	Q₈×C₈₉	—	4	2	no	—
128	Q₈×C₈₉	—	8	1	no	—
129	Q₈×C₈₉	—	8	1	no	—
130	C₂×C₄×C₈₉	—	4	2	no	—
131	C₂×C₄×C₈₉	—	8	1	no	—
132	C₂×C₄×C₈₉	—	8	1	no	—
133	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	#102	1	2	yes	—
141	C₂×Dic₈₉	#102	1	2	yes	—
142	D₃₅₆	—	4	2	no	—
143	D₃₅₆	—	4	2	no	—
144	D₃₅₆	—	8	1	no	—
145	D₃₅₆	—	8	1	no	—
146	C₈×C₈₉	—	8	1	no	—
147	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
148	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
149	C₈×C₈₉	#121	1	1	yes	characteristic
150	C₈×C₈₉	#121	1	1	yes	characteristic
151	#10623494	—	8	1	no	—
152	#10623494	#102	1	2	yes	—
153	#10623494	#102	1	2	yes	—
154	C₂²×C₈₉	—	2	2	no	—
155	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
156	C₂²×C₈₉	#486	1	1	yes	characteristic
157	C₂²×C₈₉	#492	1	1	yes	characteristic
158	C₂×D₈₉	—	8	4	no	—
159	C₂×D₈₉	—	8	2	no	—
160	C₂×D₈₉	—	16	2	no	—
161	C₂×D₈₉	—	16	2	no	—
162	C₄×C₈₉	—	8	2	no	—
163	C₄×C₈₉	#390	1	1	yes	characteristic
164	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
165	C₄×C₈₉	#486	1	1	yes	characteristic
166	C₄×C₈₉	#493	1	1	yes	characteristic
167	Dic₈₉	—	2	2	no	—
168	Dic₈₉	—	2	2	no	—
169	#13214	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	—	2	1	no	—
171	C₂×C₈₉	—	2	1	no	—
172	C₂×C₈₉	#1957	1	1	yes	characteristic
173	C₂×C₈₉	#2148	1	2	yes	—
174	D₈₉	—	16	2	no	—
175	C₂×#495	—	89	1	no	—
176	C₂×#362	—	89	1	no	—
177	C₂×#362	D₈₉	1	1	yes	characteristic, complemented
178	C₂×#361	—	89	1	no	—
179	C₂×#361	—	89	1	no	—
180	C₄×M₃₂	—	89	1	no	—
181	#1848	—	89	1	no	—
182	C₈₉	#13214	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
184	C₂²×SD₁₆	—	178	1	no	—
185	C₂²×D₈	—	89	1	no	—
186	C₂×#124	—	178	1	no	—
187	C₂×M₃₂	—	89	1	no	—
188	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
189	#495	—	89	2	no	—
190	#495	—	89	2	no	—
191	#495	—	178	1	no	—
192	#495	—	178	1	no	—
193	C₂×#108	—	89	1	no	—
194	C₂×#108	—	178	1	no	—
195	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
196	#423	—	178	1	no	—
197	C₂×C₄×C₈	—	89	1	no	—
198	#362	—	2	2	no	—
199	#362	—	178	2	no	—
200	#361	—	178	2	no	—
201	#361	—	178	2	no	—
202	#346	—	178	1	no	—
203	C₄×C₁₆	—	178	1	no	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×D₄	—	178	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₄×D₈₉	1	2	yes	—
212	C₂×SD₁₆	—	356	2	no	—
213	C₂×SD₁₆	—	356	2	no	—
214	C₂×SD₁₆	—	178	2	no	—
215	C₂×SD₁₆	—	178	2	no	—
216	C₂×D₈	—	89	2	no	—
217	C₂×D₈	—	178	1	no	—
218	C₂×D₈	—	178	1	no	—
219	C₂×D₈	—	178	1	no	—
220	C₂×D₈	—	178	1	no	—
221	C₂×D₈	—	178	1	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	—	178	1	no	—
224	C₂²×C₈	—	89	1	no	—
225	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
226	#124	—	356	2	no	—
227	#124	—	178	2	no	—
228	#124	—	178	2	no	—
229	C₂×C₂²⋊C₄	—	356	2	no	—
230	C₂×C₄²	—	89	1	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	89	2	no	—
233	M₃₂	—	178	1	no	—
234	M₃₂	C₂²⋊D₈₉	1	2	yes	—
235	C₂×C₁₆	—	2	1	no	—
236	C₂×C₁₆	—	178	1	no	—
237	#108	—	2	1	no	—
238	#108	—	2	1	no	—
239	#108	—	356	2	no	—
240	#108	—	178	1	no	—
241	#108	—	178	1	no	—
242	C₄⋊C₈	—	356	2	no	—
243	C₄×C₈	—	89	1	no	—
244	C₄×C₈	—	89	1	no	—
245	C₄×C₈	—	89	1	no	—
246	C₄×C₈	—	89	1	no	—
247	C₂⁴	—	356	1	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×D₄	—	356	1	no	—
255	C₂×D₄	—	356	1	no	—
256	C₂×D₄	—	356	1	no	—
257	C₂×D₄	—	356	1	no	—
258	C₂×D₄	—	356	1	no	—
259	C₂×D₄	—	178	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	89	2	no	—
262	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	4	1	no	—
266	Q₁₆	—	4	1	no	—
267	SD₁₆	—	356	4	no	—
268	SD₁₆	—	356	4	no	—
269	D₈	—	356	1	no	—
270	D₈	—	356	1	no	—
271	D₈	—	178	2	no	—
272	D₈	—	178	2	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	356	1	no	—
276	M₁₆	—	356	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	89	2	no	—
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	—	89	1	no	—
281	C₂×C₈	—	89	1	no	—
282	C₂×C₈	—	356	1	no	—
283	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
284	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
285	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₂×C₈	#13083573	1	1	yes	characteristic
288	C₂×C₈	#13083574	1	1	yes	characteristic
289	C₂²⋊C₄	—	356	4	no	—
290	C₂²⋊C₄	—	356	4	no	—
291	C₄²	—	89	2	no	—
292	C₄²	—	89	1	no	—
293	C₄²	—	89	1	no	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	178	2	no	—
296	C₂³	—	356	2	no	—
297	C₂³	—	712	4	no	—
298	C₂³	—	712	1	no	—
299	C₂³	—	712	1	no	—
300	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	1	no	—
304	Q₈	—	8	1	no	—
305	D₄	—	356	2	no	—
306	D₄	—	356	2	no	—
307	D₄	—	712	1	no	—
308	D₄	—	712	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	89	2	no	—
313	C₂×C₄	—	89	2	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
316	C₂×C₄	#15667998	1	1	yes	characteristic
317	C₂×C₄	#15668008	1	1	yes	characteristic
318	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
319	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
320	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
321	C₈	—	8	1	no	—
322	C₈	—	89	2	no	—
323	C₈	—	89	2	no	—
324	C₈	—	712	1	no	—
325	C₈	#15667972	1	1	yes	characteristic
326	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
327	C₈	#15667984	1	1	yes	characteristic
328	C₈	#15667985	1	1	yes	characteristic
329	C₂²	—	2	2	no	—
330	C₂²	—	1424	2	no	—
331	C₂²	—	1424	2	no	—
332	C₂²	—	712	4	no	—
333	C₂²	—	712	2	no	—
334	C₂²	#102×D₈₉	1	1	yes	characteristic
335	C₂²	#19151943	1	1	yes	characteristic
336	C₂²	#19151969	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	178	2	no	—
339	C₄	—	178	2	no	—
340	C₄	#19151498	1	1	yes	characteristic
341	C₄	#102×D₈₉	1	1	yes	characteristic
342	C₄	#19151943	1	1	yes	characteristic
343	C₄	#19151973	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	1424	2	no	—
347	C₂	#26021056	1	1	yes	characteristic
348	C₂	#26022000	1	2	yes	—

	subgroup	G/H
center	C₂²	#102×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#362×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#362	D₈₉
hypocenter	C₈₉	#13214
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
2. C₂×#19151414	C₂	semidirect
5. M₃₂×Dic₈₉	C₂	semidirect
6. #26020422	C₂	semidirect
7. C₂×#362×C₈₉	C₂	semidirect
177. C₂×#362	D₈₉	semidirect
182. C₈₉	#13214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151983	2
C₂×#19151414	2
C₂×#19151412	2
C₂×#19151410	2
M₃₂×Dic₈₉	2
#26020422	2
C₂×#362×C₈₉	2
#13214	89