Subgroups — #35322907

Order 21248.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128085	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127522	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19127516	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19127513	C₂	1	1	yes	maximal, characteristic
5	#24909182	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909178	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₃₂	C₂²	1	1	yes	characteristic
9	C₂²×#13050579	—	2	1	no	—
10	C₂²×D₆₆₄	C₂²	1	1	yes	characteristic
11	C₂×#14581780	—	2	1	no	—
12	C₂×#14581732	C₂²	1	1	yes	characteristic
13	C₂×#14581732	C₂²	1	1	yes	characteristic
14	#19128085	—	2	2	no	—
15	#19128085	C₄	1	4	yes	complemented
16	#19127522	—	2	2	no	—
17	#19127516	—	2	2	no	—
18	#19127513	—	2	2	no	—
19	#19127504	—	2	2	no	—
20	#19127501	—	2	2	no	—
21	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
22	#427×C₈₃	C₂²	1	1	yes	characteristic
23	#405×C₈₃	—	2	1	no	—
24	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	#339×C₈₃	—	2	1	no	—
26	C₂²×Dic₁₆₆	—	2	1	no	—
27	C₂²×D₃₃₂	—	2	1	no	—
28	C₂×Dic₃₃₂	—	2	1	no	—
29	C₂×Dic₃₃₂	—	2	1	no	—
30	C₂×Dic₃₃₂	—	2	1	no	—
31	C₂×Dic₃₃₂	—	2	1	no	—
32	C₂×Dic₃₃₂	—	2	1	no	—
33	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
34	C₂×#13050579	—	2	4	no	—
35	C₂×#13050579	—	4	1	no	—
36	C₂×#13050579	—	4	1	no	—
37	C₂×#13050579	—	4	2	no	—
38	C₂×D₆₆₄	—	2	1	no	—
39	C₂×D₆₆₄	—	2	1	no	—
40	C₂×D₆₆₄	—	2	1	no	—
41	C₂×D₆₆₄	—	2	1	no	—
42	C₂×D₆₆₄	—	2	1	no	—
43	C₂×D₆₆₄	C₂×C₄	1	2	yes	—
44	#14581780	—	2	4	no	—
45	#14581780	—	4	2	no	—
46	C₂×(C₂×D₈₃)⋊C₄	—	4	2	no	—
47	#14581732	—	2	1	no	—
48	#14581732	—	2	1	no	—
49	#14581732	—	2	1	no	—
50	#14581732	—	2	1	no	—
51	C₂×C₈₃⋊C₁₆	—	2	1	no	—
52	C₂×C₈₃⋊C₁₆	—	2	1	no	—
53	C₂×M₁₆×C₈₃	—	2	1	no	—
54	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
56	#117×C₈₃	—	2	1	no	—
57	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
58	#108×C₈₃	—	2	1	no	—
59	#108×C₈₃	D₄	1	2	yes	—
60	#107×C₈₃	D₄	1	2	yes	—
61	#106×C₈₃	—	2	1	no	—
62	#97×C₈₃	—	4	1	no	—
63	C₄×C₈×C₈₃	—	4	1	no	—
64	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₈₃	D₄	1	2	yes	—
66	C₂³×D₈₃	—	4	1	no	—
67	C₂×Dic₁₆₆	—	2	2	no	—
68	C₂×Dic₁₆₆	—	4	1	no	—
69	C₂×Dic₁₆₆	—	4	1	no	—
70	C₂×Dic₁₆₆	—	4	1	no	—
71	C₂×Dic₁₆₆	—	4	1	no	—
72	C₂×Dic₁₆₆	—	4	1	no	—
73	C₂×D₃₃₂	—	2	2	no	—
74	C₂×D₃₃₂	—	4	1	no	—
75	C₂×D₃₃₂	—	4	1	no	—
76	C₂×D₃₃₂	—	4	1	no	—
77	C₂×D₃₃₂	—	4	1	no	—
78	C₂×D₃₃₂	—	4	1	no	—
79	C₂²×Dic₈₃	—	4	1	no	—
80	Dic₃₃₂	—	2	2	no	—
81	Dic₃₃₂	—	2	2	no	—
82	Dic₃₃₂	—	4	1	no	—
83	Dic₃₃₂	—	4	1	no	—
84	#13050579	—	4	4	no	—
85	#13050579	—	4	4	no	—
86	D₆₆₄	—	2	2	no	—
87	D₆₆₄	—	2	2	no	—
88	D₆₆₄	—	4	1	no	—
89	D₆₆₄	—	4	1	no	—
90	(C₂×D₈₃)⋊C₄	—	4	8	no	—
91	C₈₃⋊C₁₆	—	4	1	no	—
92	C₈₃⋊C₁₆	—	4	1	no	—
93	C₂²×C₄×C₈₃	—	2	1	no	—
94	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
95	M₁₆×C₈₃	—	2	2	no	—
96	M₁₆×C₈₃	—	4	1	no	—
97	C₂×C₈×C₈₃	—	4	1	no	—
98	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
100	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
101	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
102	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
107	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
108	C₄⋊C₄×C₈₃	—	2	2	no	—
109	C₂²⋊C₄×C₈₃	—	4	1	no	—
110	C₄²×C₈₃	—	2	1	no	—
111	C₄²×C₈₃	—	4	1	no	—
112	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
113	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂²×D₈₃	—	4	2	no	—
115	C₂²×D₈₃	—	8	4	no	—
116	C₂²×D₈₃	—	8	1	no	—
117	C₂²×D₈₃	—	8	1	no	—
118	C₂×C₄×C₈₃	—	2	2	no	—
119	C₂×C₄×C₈₃	—	4	2	no	—
120	C₂×C₄×C₈₃	—	4	2	no	—
121	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×Dic₈₃	—	4	2	no	—
128	C₂×Dic₈₃	—	8	1	no	—
129	C₂×Dic₈₃	—	8	1	no	—
130	D₃₃₂	—	4	2	no	—
131	D₃₃₂	—	4	2	no	—
132	D₃₃₂	—	8	1	no	—
133	D₃₃₂	—	8	1	no	—
134	Dic₁₆₆	—	4	2	no	—
135	Dic₁₆₆	—	4	2	no	—
136	Dic₁₆₆	—	8	1	no	—
137	Dic₁₆₆	—	8	1	no	—
138	C₈×C₈₃	—	4	2	no	—
139	C₈×C₈₃	#102	1	2	yes	—
140	C₈×C₈₃	#102	1	2	yes	—
141	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
142	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
143	C₈×C₈₃	#124	1	1	yes	characteristic
144	C₈×C₈₃	#127	1	1	yes	characteristic
145	C₂²×C₈₃	—	2	2	no	—
146	C₂²×C₈₃	#418	1	1	yes	characteristic
147	C₂²×C₈₃	#459	1	1	yes	characteristic
148	C₂²×C₈₃	#460	1	1	yes	characteristic
149	C₂×D₈₃	—	8	4	no	—
150	C₂×D₈₃	—	8	2	no	—
151	C₂×D₈₃	—	16	2	no	—
152	C₂×D₈₃	—	16	2	no	—
153	C₄×C₈₃	—	4	2	no	—
154	C₄×C₈₃	—	8	2	no	—
155	C₄×C₈₃	#390	1	1	yes	characteristic
156	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
157	C₄×C₈₃	#459	1	1	yes	characteristic
158	C₄×C₈₃	#460	1	1	yes	characteristic
159	Dic₈₃	—	8	2	no	—
160	#13260	—	83	1	no	maximal, Sylow
161	C₂×C₈₃	—	2	1	no	—
162	C₂×C₈₃	—	2	1	no	—
163	C₂×C₈₃	#1982	1	1	yes	characteristic
164	C₂×C₈₃	#2212	1	2	yes	—
165	D₈₃	—	16	2	no	—
166	C₂×#495	—	83	1	no	—
167	C₂×#364	—	83	1	no	—
168	C₂×#362	—	83	1	no	—
169	C₂×#361	—	83	1	no	—
170	#2141	—	83	1	no	—
171	#2140	—	83	1	no	—
172	#1772	D₈₃	1	1	yes	characteristic, complemented
173	C₈₃	#13260	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₁₆	—	83	1	no	—
175	C₂²×SD₁₆	—	166	1	no	—
176	C₂²×D₈	—	83	1	no	—
177	C₂×#124	—	166	1	no	—
178	C₂×M₃₂	—	83	1	no	—
179	C₂×M₃₂	—	83	1	no	—
180	#495	—	83	4	no	—
181	#495	—	166	2	no	—
182	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
183	#427	C₂×D₈₃	1	1	yes	characteristic
184	#405	—	2	1	no	—
185	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
186	#364	—	166	2	no	—
187	#362	—	166	2	no	—
188	#361	—	166	2	no	—
189	#358	—	166	2	no	—
190	#357	—	166	2	no	—
191	#339	—	2	1	no	—
192	C₂²×Q₈	—	166	1	no	—
193	C₂²×D₄	—	166	1	no	—
194	C₂×Q₁₆	—	83	2	no	—
195	C₂×Q₁₆	—	166	1	no	—
196	C₂×Q₁₆	—	166	1	no	—
197	C₂×Q₁₆	—	166	1	no	—
198	C₂×Q₁₆	—	166	1	no	—
199	C₂×Q₁₆	—	166	1	no	—
200	C₂×SD₁₆	—	332	1	no	—
201	C₂×SD₁₆	—	332	1	no	—
202	C₂×SD₁₆	—	332	2	no	—
203	C₂×SD₁₆	—	166	4	no	—
204	C₂×D₈	—	83	2	no	—
205	C₂×D₈	—	166	1	no	—
206	C₂×D₈	—	166	1	no	—
207	C₂×D₈	—	166	1	no	—
208	C₂×D₈	—	166	1	no	—
209	C₂×D₈	—	166	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
213	#124	—	332	2	no	—
214	#124	—	166	4	no	—
215	#117	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	332	2	no	—
217	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
218	M₃₂	—	166	1	no	—
219	M₃₂	—	166	1	no	—
220	M₃₂	—	166	1	no	—
221	M₃₂	—	166	1	no	—
222	C₂×C₁₆	—	166	1	no	—
223	C₂×C₁₆	—	166	1	no	—
224	#108	—	2	1	no	—
225	#108	D₃₃₂	1	2	yes	—
226	#107	D₃₃₂	1	2	yes	—
227	#106	—	2	1	no	—
228	#97	—	4	1	no	—
229	C₄×C₈	—	4	1	no	—
230	C₄×C₈	C₄×D₈₃	1	2	yes	—
231	C₄×C₈	D₃₃₂	1	2	yes	—
232	C₂⁴	—	332	1	no	—
233	C₂×Q₈	—	332	1	no	—
234	C₂×Q₈	—	332	1	no	—
235	C₂×Q₈	—	332	1	no	—
236	C₂×Q₈	—	332	1	no	—
237	C₂×Q₈	—	332	1	no	—
238	C₂×Q₈	—	166	2	no	—
239	C₂×D₄	—	332	1	no	—
240	C₂×D₄	—	332	1	no	—
241	C₂×D₄	—	332	1	no	—
242	C₂×D₄	—	332	1	no	—
243	C₂×D₄	—	332	1	no	—
244	C₂×D₄	—	166	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	332	1	no	—
247	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
248	Q₁₆	—	332	1	no	—
249	Q₁₆	—	332	1	no	—
250	Q₁₆	—	166	2	no	—
251	Q₁₆	—	166	2	no	—
252	SD₁₆	—	332	4	no	—
253	SD₁₆	—	332	4	no	—
254	D₈	—	332	1	no	—
255	D₈	—	332	1	no	—
256	D₈	—	166	2	no	—
257	D₈	—	166	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
262	C₂×C₈	#13050575	1	1	yes	characteristic
263	C₂×C₈	#13050575	1	1	yes	characteristic
264	C₂×C₈	#13050577	1	2	yes	—
265	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
267	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
268	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
269	C₂×C₈	#13050590	1	1	yes	characteristic
270	C₂×C₈	#13050590	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	332	8	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	(C₂×D₈₃)⋊C₄	1	2	yes	—
277	C₁₆	—	332	1	no	—
278	C₁₆	—	332	1	no	—
279	C₂³	—	332	2	no	—
280	C₂³	—	664	4	no	—
281	C₂³	—	664	1	no	—
282	C₂³	—	664	1	no	—
283	C₂³	#14581817	1	1	yes	characteristic
284	Q₈	—	332	2	no	—
285	Q₈	—	332	2	no	—
286	Q₈	—	664	1	no	—
287	Q₈	—	664	1	no	—
288	D₄	—	332	2	no	—
289	D₄	—	332	2	no	—
290	D₄	—	664	1	no	—
291	D₄	—	664	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	332	2	no	—
296	C₂×C₄	—	664	1	no	—
297	C₂×C₄	—	664	1	no	—
298	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#14581766	1	1	yes	characteristic
300	C₂×C₄	#14581766	1	1	yes	characteristic
301	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
302	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
303	C₂×C₄	#14581817	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	#14581712	1	2	yes	—
306	C₈	#14581712	1	2	yes	—
307	C₈	C₄×D₃₃₂	1	1	yes	characteristic
308	C₈	C₄×D₃₃₂	1	1	yes	characteristic
309	C₈	#14581780	1	1	yes	characteristic
310	C₈	#14581789	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	1328	2	no	—
313	C₂²	—	1328	2	no	—
314	C₂²	—	664	4	no	—
315	C₂²	—	664	2	no	—
316	C₂²	#19127705	1	1	yes	characteristic
317	C₂²	#19127896	1	1	yes	characteristic
318	C₂²	#19127901	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	664	2	no	—
322	C₄	#19127603	1	1	yes	characteristic
323	C₄	C₂×#14581712	1	1	yes	characteristic
324	C₄	#19127896	1	1	yes	characteristic
325	C₄	#19127901	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	1328	2	no	—
329	C₂	#24908390	1	1	yes	characteristic
330	C₂	#24909532	1	2	yes	—

	subgroup	G/H
center	C₂²	#19127896
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1772	C₂
socle	C₂²×C₈₃	#459
prime radical	C₂²×D₆₆₄	C₂²
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1772	D₈₃
hypocenter	C₈₃	#13260
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
2. C₂×#19127522	C₂	semidirect
3. C₂×#19127516	C₂	semidirect
5. #24909182	C₂	semidirect
7. C₈₃×#1772	C₂	semidirect
15. #19128085	C₄	semidirect
172. #1772	D₈₃	semidirect
173. C₈₃	#13260	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128085	2
C₂×#19127522	2
C₂×#19127516	2
C₂×#19127513	2
#24909182	2
#24909178	2
C₈₃×#1772	2
#13260	83