Subgroups — #35321813

Order 21248.

470 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24920566	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19127514	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909179	C₂	1	1	yes	maximal, characteristic, complemented
4	#24907517	C₂	1	1	yes	maximal, characteristic, complemented
5	#24905577	C₂	1	1	yes	maximal, characteristic, complemented
6	#24905566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#472×C₈₃	C₂	1	1	yes	maximal, characteristic
8	#19128118	—	2	2	no	—
9	#19128118	—	2	2	no	—
10	C₂×#14581732	—	2	2	no	—
11	C₂×#14581732	C₂²	1	1	yes	characteristic
12	C₂×#14581732	C₂²	1	1	yes	characteristic
13	C₂×#14581728	C₂²	1	1	yes	characteristic
14	#19127739	C₂²	1	1	yes	characteristic
15	#19127718	—	2	1	no	—
16	#19127514	—	2	2	no	—
17	#19127502	—	2	2	no	—
18	#19127473	—	2	1	no	—
19	C₂×AGL(1,ℤ/8)×C₈₃	—	2	1	no	—
20	C₂²×D₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×#131×C₈₃	C₂²	1	1	yes	characteristic
22	#472×C₈₃	—	2	2	no	—
23	#472×C₈₃	C₄	1	4	yes	complemented
24	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
25	C₂×#100×C₈₃	—	2	1	no	—
26	C₂×#13050574	—	2	1	no	—
27	#14581751	—	2	1	no	—
28	#14581732	—	2	4	no	—
29	#14581732	—	2	1	no	—
30	#14581732	—	2	1	no	—
31	#14581732	—	2	1	no	—
32	#14581732	D₄	1	2	yes	—
33	C₂×C₈₃⋊C₁₆	—	2	4	no	—
34	C₂×C₈₃⋊C₁₆	—	2	1	no	—
35	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
36	#14581728	—	2	1	no	—
37	#14581728	—	2	1	no	—
38	#14581726	—	2	1	no	—
39	#14581724	—	2	1	no	—
40	#14581719	—	4	2	no	—
41	C₂×#41×C₈₃	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₈₃	—	2	1	no	—
43	AGL(1,ℤ/8)×C₈₃	—	2	4	no	—
44	AGL(1,ℤ/8)×C₈₃	—	4	2	no	—
45	C₂×SD₁₆×C₈₃	—	4	1	no	—
46	C₂×D₈×C₈₃	—	2	1	no	—
47	C₂×D₈×C₈₃	—	2	1	no	—
48	C₂×D₈×C₈₃	—	2	1	no	—
49	C₂×D₈×C₈₃	—	2	1	no	—
50	C₂×D₈×C₈₃	—	2	1	no	—
51	C₂×D₈×C₈₃	—	4	1	no	—
52	C₂×D₈×C₈₃	C₂×C₄	1	2	yes	—
53	#131×C₈₃	—	2	1	no	—
54	#131×C₈₃	—	2	1	no	—
55	#131×C₈₃	C₂×C₄	1	2	yes	—
56	#131×C₈₃	D₄	1	2	yes	—
57	C₂×M₁₆×C₈₃	—	2	1	no	—
58	C₂×M₁₆×C₈₃	—	2	1	no	—
59	C₂×M₁₆×C₈₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₈₃	—	2	1	no	—
61	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
62	#108×C₈₃	—	2	1	no	—
63	#108×C₈₃	C₂×C₄	1	2	yes	—
64	#100×C₈₃	—	2	4	no	—
65	#13050574	—	4	1	no	—
66	#13050574	—	4	1	no	—
67	C₂×C₈₃⋊C₈	—	4	1	no	—
68	C₄⋊Dic₈₃	—	4	1	no	—
69	C₂²⋊Dic₈₃	—	4	1	no	—
70	C₄×Dic₈₃	—	4	1	no	—
71	C₈₃⋊C₁₆	—	2	2	no	—
72	C₈₃⋊C₁₆	—	2	2	no	—
73	C₂⁴×C₈₃	—	4	1	no	—
74	#41×C₈₃	—	2	1	no	—
75	#41×C₈₃	—	2	1	no	—
76	#41×C₈₃	C₂²⋊C₄	1	2	yes	—
77	#41×C₈₃	D₈	1	2	yes	—
78	C₂×Q₈×C₈₃	SD₁₆	1	1	yes	characteristic
79	C₂×D₄×C₈₃	—	2	1	no	—
80	C₂×D₄×C₈₃	—	2	2	no	—
81	C₂×D₄×C₈₃	—	4	1	no	—
82	C₂×D₄×C₈₃	—	4	1	no	—
83	C₂×D₄×C₈₃	—	4	1	no	—
84	C₂×D₄×C₈₃	—	4	1	no	—
85	C₂×D₄×C₈₃	—	4	1	no	—
86	C₂×D₄×C₈₃	SD₁₆	1	1	yes	characteristic
87	C₂²×C₄×C₈₃	—	2	1	no	—
88	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
89	SD₁₆×C₈₃	—	4	4	no	—
90	D₈×C₈₃	—	2	2	no	—
91	D₈×C₈₃	—	2	2	no	—
92	D₈×C₈₃	—	4	4	no	—
93	D₈×C₈₃	—	4	1	no	—
94	D₈×C₈₃	—	4	1	no	—
95	M₁₆×C₈₃	—	2	1	no	—
96	M₁₆×C₈₃	—	2	2	no	—
97	M₁₆×C₈₃	—	2	2	no	—
98	M₁₆×C₈₃	—	2	2	no	—
99	M₁₆×C₈₃	—	4	1	no	—
100	M₁₆×C₈₃	C₂²⋊C₄	1	2	yes	—
101	C₂×C₈×C₈₃	—	2	1	no	—
102	C₂×C₈×C₈₃	—	2	2	no	—
103	C₂×C₈×C₈₃	—	2	2	no	—
104	C₂×C₈×C₈₃	—	4	1	no	—
105	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
111	C₂³×C₈₃	—	2	1	no	—
112	C₂³×C₈₃	—	4	2	no	—
113	C₂³×C₈₃	—	8	1	no	—
114	C₂³×C₈₃	—	8	1	no	—
115	C₂³×C₈₃	—	16	2	no	—
116	C₂³×C₈₃	#135	1	1	yes	characteristic
117	Q₈×C₈₃	—	2	1	no	—
118	Q₈×C₈₃	#102	1	2	yes	—
119	D₄×C₈₃	—	2	2	no	—
120	D₄×C₈₃	—	2	2	no	—
121	D₄×C₈₃	—	2	1	no	—
122	D₄×C₈₃	—	4	2	no	—
123	D₄×C₈₃	—	4	2	no	—
124	D₄×C₈₃	—	8	1	no	—
125	D₄×C₈₃	—	8	1	no	—
126	D₄×C₈₃	#102	1	2	yes	—
127	C₂×C₄×C₈₃	—	2	1	no	—
128	C₂×C₄×C₈₃	—	2	2	no	—
129	C₂×C₄×C₈₃	—	2	2	no	—
130	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
136	C₂×Dic₈₃	—	8	1	no	—
137	C₈×C₈₃	—	2	2	no	—
138	C₈×C₈₃	—	4	2	no	—
139	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
140	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
141	C₈×C₈₃	#120	1	1	yes	characteristic
142	C₈×C₈₃	#123	1	1	yes	characteristic
143	C₈₃⋊C₈	—	8	1	no	—
144	C₂²×C₈₃	—	2	2	no	—
145	C₂²×C₈₃	—	4	1	no	—
146	C₂²×C₈₃	—	8	1	no	—
147	C₂²×C₈₃	—	8	2	no	—
148	C₂²×C₈₃	—	16	2	no	—
149	C₂²×C₈₃	—	16	2	no	—
150	C₂²×C₈₃	—	16	2	no	—
151	C₂²×C₈₃	#416	1	1	yes	characteristic
152	C₂²×C₈₃	#465	1	1	yes	characteristic
153	C₂²×C₈₃	#466	1	1	yes	characteristic
154	C₄×C₈₃	—	2	2	no	—
155	C₄×C₈₃	#386	1	1	yes	characteristic
156	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
157	C₄×C₈₃	#465	1	1	yes	characteristic
158	C₄×C₈₃	#466	1	1	yes	characteristic
159	Dic₈₃	—	16	1	no	—
160	#13068	—	83	1	no	maximal, Sylow
161	C₂×C₈₃	—	2	1	no	—
162	C₂×C₈₃	—	2	1	no	—
163	C₂×C₈₃	—	2	1	no	—
164	C₂×C₈₃	—	4	2	no	—
165	C₂×C₈₃	—	16	2	no	—
166	C₂×C₈₃	#1893	1	1	yes	characteristic
167	#3407	—	83	1	no	—
168	C₂×#472	D₈₃	1	1	yes	characteristic
169	C₂×#361	—	83	1	no	—
170	#2140	—	83	1	no	—
171	#1814	—	83	1	no	—
172	#1389	—	83	1	no	—
173	#1385	—	83	1	no	—
174	C₈₃	#13068	1	1	yes	characteristic, complemented, Sylow
175	C₂×AGL(1,ℤ/8)	—	2	1	no	—
176	C₂²×D₈	C₂×D₈₃	1	1	yes	characteristic
177	C₂×#131	C₂×D₈₃	1	1	yes	characteristic
178	#504	—	166	2	no	—
179	#504	—	166	2	no	—
180	C₂×M₃₂	—	83	1	no	—
181	C₂×M₃₂	—	83	1	no	—
182	C₂×M₃₂	—	166	2	no	—
183	#472	—	2	2	no	—
184	#472	Dic₈₃	1	4	yes	complemented
185	C₂×#108	—	83	1	no	—
186	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
187	#427	—	83	1	no	—
188	#421	—	166	1	no	—
189	C₂×#100	—	2	1	no	—
190	#361	—	166	2	no	—
191	#357	—	166	2	no	—
192	#344	—	166	1	no	—
193	C₂×#41	C₂²⋊D₈₃	1	1	yes	characteristic
194	C₂²×D₄	—	2	1	no	—
195	AGL(1,ℤ/8)	—	2	4	no	—
196	AGL(1,ℤ/8)	—	4	2	no	—
197	C₂×SD₁₆	—	4	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	4	1	no	—
204	C₂×D₈	C₂×Dic₈₃	1	2	yes	—
205	#131	—	2	1	no	—
206	#131	—	2	1	no	—
207	#131	C₂×Dic₈₃	1	2	yes	—
208	#131	C₂²⋊D₈₃	1	2	yes	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	166	1	no	—
212	C₂×M₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
215	#117	—	166	1	no	—
216	M₃₂	—	83	2	no	—
217	M₃₂	—	166	4	no	—
218	M₃₂	—	166	1	no	—
219	M₃₂	—	166	1	no	—
220	M₃₂	—	166	1	no	—
221	C₂×C₁₆	—	83	2	no	—
222	C₂×C₁₆	—	166	4	no	—
223	C₂×C₁₆	—	166	1	no	—
224	#108	—	2	1	no	—
225	#108	—	166	1	no	—
226	#108	—	166	1	no	—
227	#108	C₂×Dic₈₃	1	2	yes	—
228	#107	—	166	1	no	—
229	#106	—	166	1	no	—
230	#104	—	332	2	no	—
231	#100	—	2	4	no	—
232	C₂⁴	—	4	1	no	—
233	#41	—	2	1	no	—
234	#41	—	2	1	no	—
235	#41	C₂²⋊Dic₈₃	1	2	yes	—
236	#41	#13050575	1	2	yes	—
237	C₂×Q₈	#13050578	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	#13050578	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
248	SD₁₆	—	4	4	no	—
249	D₈	—	2	2	no	—
250	D₈	—	2	2	no	—
251	D₈	—	4	4	no	—
252	D₈	—	4	1	no	—
253	D₈	—	4	1	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	332	1	no	—
260	M₁₆	—	332	1	no	—
261	M₁₆	C₂²⋊Dic₈₃	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	332	1	no	—
267	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
269	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
271	C₂×C₈	#13050589	1	1	yes	characteristic
272	C₂×C₈	#13050589	1	1	yes	characteristic
273	C₄⋊C₄	—	332	1	no	—
274	C₂²⋊C₄	—	332	1	no	—
275	C₄²	—	332	1	no	—
276	C₁₆	—	166	2	no	—
277	C₁₆	—	166	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	8	1	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	16	2	no	—
283	C₂³	#14581812	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	#14581713	1	2	yes	—
286	D₄	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	1	no	—
292	D₄	—	8	1	no	—
293	D₄	#14581713	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	664	1	no	—
298	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
299	C₂×C₄	#14581764	1	1	yes	characteristic
300	C₂×C₄	#14581764	1	1	yes	characteristic
301	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
302	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
303	C₂×C₄	#14581812	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	664	1	no	—
307	C₈	D₄×Dic₈₃	1	1	yes	characteristic
308	C₈	D₄×Dic₈₃	1	1	yes	characteristic
309	C₈	#14581760	1	1	yes	characteristic
310	C₈	#14581773	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	16	2	no	—
316	C₂²	—	16	2	no	—
317	C₂²	—	16	2	no	—
318	C₂²	#19127697	1	1	yes	characteristic
319	C₂²	#19127928	1	1	yes	characteristic
320	C₂²	#19127935	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	1328	1	no	—
323	C₄	#19127581	1	1	yes	characteristic
324	C₄	C₂×#14581713	1	1	yes	characteristic
325	C₄	#19127928	1	1	yes	characteristic
326	C₄	#19127935	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	4	2	no	—
331	C₂	—	16	2	no	—
332	C₂	#24907892	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#24907892
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting, prime radical	C₂×#472×C₈₃	C₂
socle	C₂×C₈₃	#1893
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#472	D₈₃
hypocenter	C₈₃	#13068
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#472

Complements

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

N	H ≅ G/N	type
1. #24920566	C₂	semidirect
2. C₂×#19127514	C₂	semidirect
3. #24909179	C₂	semidirect
4. #24907517	C₂	semidirect
5. #24905577	C₂	semidirect
6. #24905566	C₂	semidirect
23. #472×C₈₃	C₄	semidirect
174. C₈₃	#13068	semidirect
184. #472	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24920566	2
C₂×#19127514	2
#24909179	2
#24907517	2
#24905577	2
#24905566	2
C₂×#472×C₈₃	2
#13068	83