Subgroups — #36472857

Order 22784.

422 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667957	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151415	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151412	C₂	1	1	yes	maximal, characteristic
5	#26021167	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021162	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×#15667957	—	2	2	no	—
11	C₂×#15667957	—	2	1	no	—
12	C₂×#15667957	—	2	1	no	—
13	C₂×#15667957	C₄	1	4	yes	complemented
14	C₂×#15667948	C₂²	1	1	yes	characteristic
15	C₂×#15667948	C₂²	1	1	yes	characteristic
16	#19151606	—	2	1	no	—
17	#19151602	—	2	1	no	—
18	C₂×#15667930	—	2	1	no	—
19	C₂×#15667928	—	2	1	no	—
20	#19151428	—	2	2	no	—
21	#19151415	—	2	2	no	—
22	#19151412	—	2	2	no	—
23	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	#405×C₈₉	C₂²	1	1	yes	characteristic
26	#346×C₈₉	—	2	1	no	—
27	C₄×C₁₆×C₈₉	—	2	1	no	—
28	C₂²×Dic₁₇₈	—	2	1	no	—
29	C₂²×D₃₅₆	—	2	1	no	—
30	C₂×Dic₃₅₆	—	2	1	no	—
31	C₂×Dic₃₅₆	—	2	1	no	—
32	C₂×Dic₃₅₆	—	2	1	no	—
33	C₂×Dic₃₅₆	—	2	1	no	—
34	C₂×Dic₃₅₆	—	2	1	no	—
35	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
36	C₂×D₇₁₂	—	2	1	no	—
37	C₂×D₇₁₂	—	2	1	no	—
38	C₂×D₇₁₂	—	2	1	no	—
39	C₂×D₇₁₂	—	2	1	no	—
40	C₂×D₇₁₂	—	2	1	no	—
41	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
42	C₂×#13083558	—	2	1	no	—
43	C₂×#13083558	—	2	1	no	—
44	#15667957	—	2	4	no	—
45	#15667957	—	2	4	no	—
46	#15667948	—	2	1	no	—
47	#15667948	—	2	1	no	—
48	#15667948	—	2	1	no	—
49	#15667948	—	2	1	no	—
50	#15667935	—	4	2	no	—
51	#15667932	—	4	2	no	—
52	#15667930	—	4	2	no	—
53	#15667928	—	4	2	no	—
54	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	#117×C₈₉	D₄	1	1	yes	characteristic
57	M₃₂×C₈₉	—	2	1	no	—
58	M₃₂×C₈₉	D₄	1	2	yes	—
59	C₂×C₁₆×C₈₉	—	2	1	no	—
60	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
61	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
62	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
63	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
64	#97×C₈₉	—	2	1	no	—
65	C₄×C₈×C₈₉	—	2	1	no	—
66	C₂³×D₈₉	—	4	1	no	—
67	C₂×Dic₁₇₈	—	2	2	no	—
68	C₂×Dic₁₇₈	—	4	1	no	—
69	C₂×Dic₁₇₈	—	4	1	no	—
70	C₂×Dic₁₇₈	—	4	1	no	—
71	C₂×Dic₁₇₈	—	4	1	no	—
72	C₂×Dic₁₇₈	—	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	—	4	1	no	—
80	Dic₃₅₆	—	2	2	no	—
81	Dic₃₅₆	—	2	2	no	—
82	Dic₃₅₆	—	4	1	no	—
83	Dic₃₅₆	—	4	1	no	—
84	D₇₁₂	—	2	2	no	—
85	D₇₁₂	—	2	2	no	—
86	D₇₁₂	—	4	1	no	—
87	D₇₁₂	—	4	1	no	—
88	#13083558	—	4	1	no	—
89	#13083558	—	4	1	no	—
90	#13083558	—	4	1	no	—
91	#13083558	—	4	1	no	—
92	C₂×#10623494	—	4	1	no	—
93	C₂×#10623494	—	4	1	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	M₁₆×C₈₉	—	2	1	no	—
96	M₁₆×C₈₉	D₈	1	2	yes	—
97	C₂×C₈×C₈₉	—	2	1	no	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
103	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₄²×C₈₉	—	2	1	no	—
107	C₁₆×C₈₉	—	2	2	no	—
108	C₁₆×C₈₉	C₂²⋊C₄	1	2	yes	—
109	C₁₆×C₈₉	C₂²⋊C₄	1	2	yes	—
110	C₂³×C₈₉	#135	1	1	yes	characteristic
111	C₂²×D₈₉	—	4	2	no	—
112	C₂²×D₈₉	—	8	1	no	—
113	C₂²×D₈₉	—	8	1	no	—
114	C₂²×D₈₉	—	16	2	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
122	C₂×Dic₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	8	1	no	—
124	C₂×Dic₈₉	—	8	1	no	—
125	D₃₅₆	—	4	2	no	—
126	D₃₅₆	—	4	2	no	—
127	D₃₅₆	—	8	1	no	—
128	D₃₅₆	—	8	1	no	—
129	Dic₁₇₈	—	4	2	no	—
130	Dic₁₇₈	—	4	2	no	—
131	Dic₁₇₈	—	8	1	no	—
132	Dic₁₇₈	—	8	1	no	—
133	C₈×C₈₉	—	2	2	no	—
134	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
136	C₈×C₈₉	#121	1	1	yes	characteristic
137	C₈×C₈₉	#124	1	1	yes	characteristic
138	#10623494	—	8	1	no	—
139	#10623494	—	8	1	no	—
140	C₂²×C₈₉	—	2	2	no	—
141	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
142	C₂²×C₈₉	#466	1	1	yes	characteristic
143	C₂²×C₈₉	#495	1	1	yes	characteristic
144	C₂×D₈₉	—	8	2	no	—
145	C₂×D₈₉	—	16	2	no	—
146	C₂×D₈₉	—	16	2	no	—
147	C₂×D₈₉	—	16	2	no	—
148	C₄×C₈₉	—	4	2	no	—
149	C₄×C₈₉	#390	1	1	yes	characteristic
150	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
151	C₄×C₈₉	#466	1	1	yes	characteristic
152	C₄×C₈₉	#493	1	1	yes	characteristic
153	Dic₈₉	—	8	2	no	—
154	#13246	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	2	1	no	—
156	C₂×C₈₉	—	2	1	no	—
157	C₂×C₈₉	#1970	1	1	yes	characteristic
158	C₂×C₈₉	#2223	1	2	yes	—
159	D₈₉	—	16	2	no	—
160	C₂²×SD₃₂	—	89	1	no	—
161	C₂×#368	—	89	1	no	—
162	C₂×#362	—	89	1	no	—
163	C₂×#361	—	89	1	no	—
164	#2108	D₈₉	1	1	yes	characteristic, complemented
165	#1979	—	89	1	no	—
166	#1978	—	89	1	no	—
167	C₈₉	#13246	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₁₆	—	89	1	no	—
169	C₂²×D₈	—	89	1	no	—
170	C₂×SD₃₂	—	89	4	no	—
171	C₂×SD₃₂	—	178	2	no	—
172	C₂×SD₃₂	—	178	1	no	—
173	C₂×SD₃₂	—	178	1	no	—
174	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
175	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
176	C₂×#108	—	89	1	no	—
177	C₂×#108	—	89	1	no	—
178	#419	—	178	1	no	—
179	#418	—	178	1	no	—
180	C₂×#101	—	178	1	no	—
181	C₂×#100	—	178	1	no	—
182	#405	C₂×D₈₉	1	1	yes	characteristic
183	#368	—	178	2	no	—
184	#362	—	178	2	no	—
185	#361	—	178	2	no	—
186	#346	—	2	1	no	—
187	C₄×C₁₆	—	2	1	no	—
188	C₂²×Q₈	—	178	1	no	—
189	C₂²×D₄	—	178	1	no	—
190	C₂×Q₁₆	—	89	2	no	—
191	C₂×Q₁₆	—	178	1	no	—
192	C₂×Q₁₆	—	178	1	no	—
193	C₂×Q₁₆	—	178	1	no	—
194	C₂×Q₁₆	—	178	1	no	—
195	C₂×Q₁₆	—	178	1	no	—
196	C₂×D₈	—	89	2	no	—
197	C₂×D₈	—	178	1	no	—
198	C₂×D₈	—	178	1	no	—
199	C₂×D₈	—	178	1	no	—
200	C₂×D₈	—	178	1	no	—
201	C₂×D₈	—	178	1	no	—
202	C₂×M₁₆	—	178	1	no	—
203	C₂×M₁₆	—	178	1	no	—
204	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
206	#117	D₃₅₆	1	1	yes	characteristic
207	SD₃₂	—	178	4	no	—
208	SD₃₂	—	178	4	no	—
209	M₃₂	—	2	1	no	—
210	M₃₂	D₃₅₆	1	2	yes	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
213	C₂×C₁₆	D₃₅₆	1	2	yes	—
214	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
215	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
216	#108	—	178	1	no	—
217	#108	—	178	1	no	—
218	#108	—	178	1	no	—
219	#108	—	178	1	no	—
220	#103	—	356	2	no	—
221	#102	—	356	2	no	—
222	#101	—	356	2	no	—
223	#100	—	356	2	no	—
224	#97	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₂⁴	—	356	1	no	—
227	C₂×Q₈	—	356	1	no	—
228	C₂×Q₈	—	356	1	no	—
229	C₂×Q₈	—	356	1	no	—
230	C₂×Q₈	—	356	1	no	—
231	C₂×Q₈	—	356	1	no	—
232	C₂×Q₈	—	178	2	no	—
233	C₂×D₄	—	356	1	no	—
234	C₂×D₄	—	356	1	no	—
235	C₂×D₄	—	356	1	no	—
236	C₂×D₄	—	356	1	no	—
237	C₂×D₄	—	356	1	no	—
238	C₂×D₄	—	178	2	no	—
239	C₂²×C₄	—	356	1	no	—
240	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
241	Q₁₆	—	356	1	no	—
242	Q₁₆	—	356	1	no	—
243	Q₁₆	—	178	2	no	—
244	Q₁₆	—	178	2	no	—
245	D₈	—	356	1	no	—
246	D₈	—	356	1	no	—
247	D₈	—	178	2	no	—
248	D₈	—	178	2	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	356	1	no	—
251	M₁₆	—	356	1	no	—
252	M₁₆	—	356	1	no	—
253	M₁₆	—	356	1	no	—
254	M₁₆	D₇₁₂	1	2	yes	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	356	1	no	—
257	C₂×C₈	—	356	1	no	—
258	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
260	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
261	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
262	C₂×C₈	#13083574	1	1	yes	characteristic
263	C₂×C₈	#13083574	1	1	yes	characteristic
264	C₄⋊C₄	D₇₁₂	1	2	yes	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	#13083551	1	2	yes	—
269	C₁₆	#13083551	1	2	yes	—
270	C₂³	—	356	2	no	—
271	C₂³	—	1424	2	no	—
272	C₂³	—	712	1	no	—
273	C₂³	—	712	1	no	—
274	C₂³	#15668034	1	1	yes	characteristic
275	Q₈	—	356	2	no	—
276	Q₈	—	356	2	no	—
277	Q₈	—	712	1	no	—
278	Q₈	—	712	1	no	—
279	D₄	—	356	2	no	—
280	D₄	—	356	2	no	—
281	D₄	—	712	1	no	—
282	D₄	—	712	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	356	2	no	—
285	C₂×C₄	—	712	1	no	—
286	C₂×C₄	—	712	1	no	—
287	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
288	C₂×C₄	#15667986	1	1	yes	characteristic
289	C₂×C₄	#15668009	1	1	yes	characteristic
290	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
291	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
292	C₂×C₄	#15668034	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	712	1	no	—
295	C₈	—	712	1	no	—
296	C₈	C₂×#13083551	1	1	yes	characteristic
297	C₈	C₄×D₃₅₆	1	1	yes	characteristic
298	C₈	#15667986	1	1	yes	characteristic
299	C₈	#15668000	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	1424	2	no	—
302	C₂²	—	1424	2	no	—
303	C₂²	—	1424	2	no	—
304	C₂²	—	712	2	no	—
305	C₂²	C₄×D₇₁₂	1	1	yes	characteristic
306	C₂²	#19151834	1	1	yes	characteristic
307	C₂²	#19151982	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	712	2	no	—
310	C₄	#19151500	1	1	yes	characteristic
311	C₄	C₄×D₇₁₂	1	1	yes	characteristic
312	C₄	#19151834	1	1	yes	characteristic
313	C₄	#19151975	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	1424	2	no	—
317	C₂	#26021118	1	1	yes	characteristic
318	C₂	#26022382	1	2	yes	—

	subgroup	G/H
center	C₂²	#19151834
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂²×C₈₉	#466
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#13246
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. C₂×#19151428	C₂	semidirect
3. C₂×#19151415	C₂	semidirect
5. #26021167	C₂	semidirect
7. C₈₉×#2108	C₂	semidirect
13. C₂×#15667957	C₄	semidirect
164. #2108	D₈₉	semidirect
167. C₈₉	#13246	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667957	2
C₂×#19151428	2
C₂×#19151415	2
C₂×#19151412	2
#26021167	2
#26021162	2
C₈₉×#2108	2
#13246	89