Subgroups — #35303648

Order 21248.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128078	C₂	1	1	yes	maximal, characteristic
2	#24908389	C₂	1	2	yes	maximal, complemented
3	C₈×C₄⋊Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19127444	C₂	1	2	yes	maximal, complemented
5	C₂×#335×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₆₄	—	2	1	no	—
7	C₂²×D₆₆₄	—	2	1	no	—
8	C₂×#14581789	C₂²	1	2	yes	—
9	#19128078	—	2	2	no	—
10	#19128078	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14581714	—	2	4	no	—
13	C₂×C₈×Dic₈₃	—	2	1	no	—
14	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
15	#19127455	—	2	1	no	—
16	#19127444	—	2	4	no	—
17	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	#335×C₈₃	—	2	2	no	—
20	C₂²×D₃₃₂	—	2	2	no	—
21	C₂²×D₃₃₂	—	2	2	no	—
22	C₂²×D₃₃₂	—	2	2	no	—
23	C₂×D₆₆₄	—	2	4	no	—
24	C₂×D₆₆₄	—	2	4	no	—
25	C₂×D₆₆₄	—	4	2	no	—
26	C₂×D₆₆₄	—	4	2	no	—
27	C₂×D₆₆₄	—	4	2	no	—
28	C₂×D₆₆₄	—	4	2	no	—
29	C₂²×C₈₃⋊C₈	—	2	1	no	—
30	#14581789	—	2	2	no	—
31	#14581789	—	2	2	no	—
32	#14581789	—	2	2	no	—
33	#14581789	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊Dic₈₃	D₄	1	2	yes	—
35	C₂×C₄×Dic₈₃	—	2	1	no	—
36	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
37	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
38	#14581714	—	4	8	no	—
39	C₈×Dic₈₃	—	4	2	no	—
40	C₈₃⋊#95	—	2	1	no	—
41	C₂²×C₈×C₈₃	—	2	2	no	—
42	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₈₃	—	2	2	no	—
46	C₄⋊C₈×C₈₃	—	2	2	no	—
47	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₈₃	D₄	1	2	yes	—
49	C₂³×D₈₃	—	4	2	no	—
50	C₂×D₃₃₂	—	2	4	no	—
51	C₂×D₃₃₂	—	2	4	no	—
52	C₂×D₃₃₂	—	2	4	no	—
53	C₂×D₃₃₂	—	4	2	no	—
54	C₂×D₃₃₂	—	4	4	no	—
55	C₂×D₃₃₂	—	4	2	no	—
56	C₂×D₃₃₂	—	4	4	no	—
57	C₂×D₃₃₂	—	4	2	no	—
58	C₂×D₃₃₂	—	4	2	no	—
59	C₂×D₃₃₂	—	4	2	no	—
60	C₂×D₃₃₂	—	4	2	no	—
61	C₂×D₃₃₂	—	4	2	no	—
62	C₂×D₃₃₂	—	8	4	no	—
63	C₂²×Dic₈₃	—	2	2	no	—
64	D₆₆₄	—	4	8	no	—
65	D₆₆₄	—	4	8	no	—
66	C₂×C₈₃⋊C₈	—	2	2	no	—
67	C₂×C₈₃⋊C₈	—	4	1	no	—
68	C₂×C₈₃⋊C₈	—	4	1	no	—
69	C₄⋊Dic₈₃	—	2	2	no	—
70	C₄⋊Dic₈₃	D₈	1	4	yes	complemented
71	C₄×Dic₈₃	—	4	1	no	—
72	C₄×Dic₈₃	—	4	1	no	—
73	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₈₃	—	2	1	no	—
77	C₂×C₈×C₈₃	—	2	1	no	—
78	C₂×C₈×C₈₃	—	4	2	no	—
79	C₂×C₈×C₈₃	—	4	4	no	—
80	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₈₃	D₈	1	2	yes	—
83	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
84	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₈₃	#41	1	1	yes	characteristic
88	C₂³×C₈₃	#121	1	1	yes	characteristic
89	C₂²×D₈₃	—	4	4	no	—
90	C₂²×D₈₃	—	8	2	no	—
91	C₂²×D₈₃	—	8	2	no	—
92	C₂²×D₈₃	—	8	4	no	—
93	C₂²×D₈₃	—	8	4	no	—
94	C₂×C₄×C₈₃	—	2	2	no	—
95	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
100	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₈₃	#135	1	2	yes	—
105	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×Dic₈₃	—	2	4	no	—
108	C₂×Dic₈₃	—	4	2	no	—
109	C₂×Dic₈₃	—	4	2	no	—
110	D₃₃₂	—	4	4	no	—
111	D₃₃₂	—	4	4	no	—
112	D₃₃₂	—	4	8	no	—
113	D₃₃₂	—	8	4	no	—
114	D₃₃₂	—	8	4	no	—
115	D₃₃₂	—	8	2	no	—
116	D₃₃₂	—	8	2	no	—
117	D₃₃₂	—	8	2	no	—
118	D₃₃₂	—	8	2	no	—
119	C₈×C₈₃	—	2	2	no	—
120	C₈×C₈₃	—	8	2	no	—
121	C₈×C₈₃	#102	1	4	yes	complemented
122	C₈₃⋊C₈	—	4	2	no	—
123	C₂²×C₈₃	#390	1	1	yes	characteristic
124	C₂²×C₈₃	#459	1	1	yes	characteristic
125	C₂²×C₈₃	#460	1	1	yes	characteristic
126	C₂²×C₈₃	#466	1	2	yes	—
127	C₂²×C₈₃	#472	1	2	yes	—
128	C₂×D₈₃	—	8	4	no	—
129	C₂×D₈₃	—	8	4	no	—
130	C₂×D₈₃	—	8	4	no	—
131	C₂×D₈₃	—	16	4	no	—
132	C₂×D₈₃	—	16	4	no	—
133	C₄×C₈₃	—	2	1	no	—
134	C₄×C₈₃	—	2	1	no	—
135	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
136	C₄×C₈₃	#418	1	1	yes	characteristic
137	C₄×C₈₃	C₄×D₈	1	2	yes	—
138	C₄×C₈₃	#459	1	1	yes	characteristic
139	C₄×C₈₃	#460	1	1	yes	characteristic
140	C₄×C₈₃	#493	1	1	yes	characteristic
141	C₄×C₈₃	#495	1	1	yes	characteristic
142	Dic₈₃	—	4	4	no	—
143	#9734	—	83	1	no	maximal, Sylow
144	C₂×C₈₃	#1667	1	2	yes	—
145	C₂×C₈₃	#1668	1	2	yes	—
146	C₂×C₈₃	#1970	1	1	yes	characteristic
147	C₂×C₈₃	#1978	1	1	yes	characteristic
148	C₂×C₈₃	#1982	1	1	yes	characteristic
149	D₈₃	—	16	4	no	—
150	C₂×#493	—	83	1	no	—
151	#1982	—	83	2	no	—
152	C₈×C₄⋊C₄	—	83	1	no	—
153	C₂×#335	D₈₃	1	1	yes	characteristic, complemented
154	C₂×#331	—	83	2	no	—
155	C₈₃	#9734	1	1	yes	characteristic, complemented, Sylow
156	C₂²×D₈	—	166	1	no	—
157	C₂²×D₈	—	166	1	no	—
158	C₂×#127	—	83	2	no	—
159	#493	—	83	4	no	—
160	#493	—	166	2	no	—
161	C₂×C₄⋊C₈	—	83	1	no	—
162	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
163	C₂×#102	—	166	4	no	—
164	C₂×C₄×C₈	—	166	1	no	—
165	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
166	C₄×C₄⋊C₄	—	83	1	no	—
167	#336	—	166	1	no	—
168	#335	—	2	2	no	—
169	#331	—	166	4	no	—
170	C₂²×D₄	—	166	2	no	—
171	C₂²×D₄	—	166	2	no	—
172	C₂²×D₄	—	166	2	no	—
173	C₂×D₈	—	332	2	no	—
174	C₂×D₈	—	332	2	no	—
175	C₂×D₈	—	332	2	no	—
176	C₂×D₈	—	332	2	no	—
177	C₂×D₈	—	166	4	no	—
178	C₂×D₈	—	166	4	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	166	1	no	—
181	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
182	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
183	#127	—	83	4	no	—
184	#127	—	166	2	no	—
185	#127	—	166	2	no	—
186	#127	—	166	2	no	—
187	C₂×C₄⋊C₄	—	83	2	no	—
188	C₂×C₄²	—	166	1	no	—
189	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	83	2	no	—
193	C₄⋊C₈	—	166	1	no	—
194	#102	—	332	8	no	—
195	C₄×C₈	—	332	2	no	—
196	C₄×C₈	C₄×D₈₃	1	2	yes	—
197	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
198	#95	—	166	1	no	—
199	C₂⁴	—	332	2	no	—
200	C₂×D₄	—	332	2	no	—
201	C₂×D₄	—	332	2	no	—
202	C₂×D₄	—	332	2	no	—
203	C₂×D₄	—	332	4	no	—
204	C₂×D₄	—	332	2	no	—
205	C₂×D₄	—	332	2	no	—
206	C₂×D₄	—	332	4	no	—
207	C₂×D₄	—	332	2	no	—
208	C₂×D₄	—	332	2	no	—
209	C₂×D₄	—	664	4	no	—
210	C₂×D₄	—	166	4	no	—
211	C₂×D₄	—	166	4	no	—
212	C₂×D₄	—	166	4	no	—
213	C₂²×C₄	—	166	2	no	—
214	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
216	C₂²×C₄	#13050591	1	1	yes	characteristic
217	D₈	—	332	8	no	—
218	D₈	—	332	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	4	no	—
223	C₂×C₈	—	332	1	no	—
224	C₂×C₈	—	332	1	no	—
225	C₂×C₈	—	166	2	no	—
226	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
227	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
228	C₂×C₈	D₆₆₄	1	2	yes	—
229	C₂×C₈	#13050579	1	2	yes	—
230	C₄⋊C₄	—	83	4	no	—
231	C₄⋊C₄	—	166	2	no	—
232	C₄²	—	332	1	no	—
233	C₄²	—	332	1	no	—
234	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
235	C₄²	D₄×D₈₃	1	1	yes	characteristic
236	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
237	C₄²	#13050591	1	1	yes	characteristic
238	C₂³	—	332	4	no	—
239	C₂³	—	664	2	no	—
240	C₂³	—	664	2	no	—
241	C₂³	—	664	4	no	—
242	C₂³	—	664	4	no	—
243	C₂³	#14581767	1	1	yes	characteristic
244	D₄	—	332	4	no	—
245	D₄	—	332	4	no	—
246	D₄	—	332	8	no	—
247	D₄	—	664	4	no	—
248	D₄	—	664	4	no	—
249	D₄	—	664	2	no	—
250	D₄	—	664	2	no	—
251	D₄	—	664	2	no	—
252	D₄	—	664	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	332	2	no	—
255	C₂×C₄	—	332	2	no	—
256	C₂×C₄	—	166	4	no	—
257	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	#14581754	1	1	yes	characteristic
259	C₂×C₄	#14581754	1	1	yes	characteristic
260	C₂×C₄	#14581767	1	1	yes	characteristic
261	C₂×C₄	#14581781	1	1	yes	characteristic
262	C₂×C₄	#14581788	1	1	yes	characteristic
263	C₂×C₄	D₈×D₈₃	1	2	yes	—
264	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
265	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
266	C₂×C₄	#14581815	1	2	yes	—
267	C₂×C₄	#14581820	1	1	yes	characteristic
268	C₂×C₄	#14581820	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	8	2	no	—
271	C₈	—	332	2	no	—
272	C₈	#14581714	1	4	yes	complemented
273	C₂²	—	1328	4	no	—
274	C₂²	—	1328	4	no	—
275	C₂²	—	664	4	no	—
276	C₂²	—	664	4	no	—
277	C₂²	—	664	4	no	—
278	C₂²	#19127602	1	1	yes	characteristic
279	C₂²	#19127897	1	1	yes	characteristic
280	C₂²	#19127902	1	1	yes	characteristic
281	C₂²	#19127938	1	2	yes	—
282	C₂²	#19127972	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	332	4	no	—
286	C₄	C₂×#14581714	1	1	yes	characteristic
287	C₄	#19127707	1	1	yes	characteristic
288	C₄	#19127783	1	2	yes	—
289	C₄	#19127897	1	1	yes	characteristic
290	C₄	#19127902	1	1	yes	characteristic
291	C₄	#19128076	1	1	yes	characteristic
292	C₄	#19128086	1	1	yes	characteristic
293	C₂	—	1328	4	no	—
294	C₂	#24906819	1	2	yes	—
295	C₂	#24906824	1	2	yes	—
296	C₂	#24908327	1	1	yes	characteristic
297	C₂	#24908371	1	1	yes	characteristic
298	C₂	#24908389	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581767
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#335×C₈₃	C₂
socle	C₂³×C₈₃	#121
prime radical	C₂×#19128078	C₂
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#335	D₈₃
hypocenter	C₈₃	#9734
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. #24908389	C₂	semidirect
3. C₈×C₄⋊Dic₈₃	C₂	semidirect
4. C₂×#19127444	C₂	semidirect
5. C₂×#335×C₈₃	C₂	semidirect
10. #19128078	C₄	semidirect
70. C₄⋊Dic₈₃	D₈	semidirect
121. C₈×C₈₃	#102	semidirect
153. C₂×#335	D₈₃	semidirect
155. C₈₃	#9734	semidirect
272. C₈	#14581714	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128078	2
#24908389	2
C₈×C₄⋊Dic₈₃	2
C₂×#19127444	2
C₂×#335×C₈₃	2
#9734	83