Subgroups — #35557646

Order 21248.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128117	C₂	1	2	yes	maximal, complemented
2	#24916367	C₂	1	1	yes	maximal, characteristic
3	#24909306	C₂	1	4	yes	maximal, complemented
4	#24909301	C₂	1	4	yes	maximal, complemented
5	#24909229	C₂	1	1	yes	maximal, characteristic, complemented
6	#24909090	C₂	1	2	yes	maximal, complemented
7	C₈₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128501	C₄	1	1	yes	characteristic
9	C₂×#14581798	C₂²	1	2	yes	—
10	#19128117	—	2	2	no	—
11	#19128117	—	2	2	no	—
12	#19128117	C₂²	1	4	yes	complemented
13	#19128117	C₂²	1	4	yes	complemented
14	C₂×#14581732	C₂²	1	2	yes	—
15	C₂×#14581731	—	2	2	no	—
16	C₂×C₁₆×D₈₃	—	2	2	no	—
17	#19127824	C₄	1	4	yes	—
18	#19127819	C₂²	1	4	yes	—
19	#19127750	C₄	1	1	yes	characteristic
20	#19127719	C₄	1	2	yes	—
21	#19127522	C₂²	1	4	yes	complemented
22	#19127484	C₂²	1	8	yes	complemented
23	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
25	#405×C₈₃	C₂²	1	1	yes	characteristic
26	#346×C₈₃	C₂²	1	2	yes	complemented
27	C₄×C₁₆×C₈₃	C₂²	1	2	yes	complemented
28	C₂×#13050590	C₂×C₄	1	2	yes	—
29	#14581825	C₈	1	4	yes	—
30	#14581820	C₈	1	4	yes	—
31	#14581814	C₂×C₄	1	4	yes	—
32	#14581798	—	2	2	no	—
33	#14581798	—	2	2	no	—
34	#14581798	C₂×C₄	1	4	yes	—
35	#14581798	C₂³	1	4	yes	—
36	C₂×#13050574	C₂×C₄	1	2	yes	—
37	C₂×#13050573	—	2	2	no	—
38	C₂×C₈×D₈₃	—	2	2	no	—
39	#14581732	—	2	2	no	—
40	#14581732	C₂³	1	4	yes	—
41	#14581731	—	2	4	no	—
42	#14581731	—	2	4	no	—
43	C₂×C₈₃⋊C₁₆	—	2	2	no	—
44	C₁₆×D₈₃	—	2	4	no	—
45	C₁₆×D₈₃	—	2	4	no	—
46	#14581728	C₂×C₄	1	4	yes	—
47	#14581720	C₂×C₄	1	8	yes	—
48	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
50	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₈₃	—	2	2	no	—
52	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
53	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
54	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
55	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
56	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
57	#97×C₈₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₃	C₂³	1	2	yes	—
59	#13050590	—	2	2	no	—
60	#13050590	—	2	2	no	—
61	#13050590	C₂×C₈	1	4	yes	—
62	#13050590	C₂²×C₄	1	4	yes	—
63	C₂×Dic₁₆₆	C₂×C₈	1	2	yes	—
64	C₂×D₃₃₂	C₂×C₈	1	2	yes	—
65	C₂×C₂²⋊D₈₃	—	2	2	no	—
66	C₂×C₄×D₈₃	—	2	2	no	—
67	Dic₃₃₂	C₂×C₈	1	4	yes	—
68	#13050579	C₂×C₈	1	8	yes	—
69	D₆₆₄	C₂×C₈	1	4	yes	—
70	#13050574	—	2	2	no	—
71	#13050574	C₂²×C₄	1	4	yes	—
72	#13050573	—	2	4	no	—
73	#13050573	—	2	4	no	—
74	C₂×C₈₃⋊C₈	—	2	2	no	—
75	C₈×D₈₃	—	2	4	no	—
76	C₈×D₈₃	—	2	4	no	—
77	C₈₃⋊C₁₆	—	2	4	no	—
78	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
79	M₁₆×C₈₃	C₂×C₈	1	4	yes	—
80	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₈₃	#41	1	2	yes	—
84	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₈₃	—	2	1	no	—
86	C₂²⋊C₄×C₈₃	—	2	1	no	—
87	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₈₃	—	2	2	no	—
89	C₁₆×C₈₃	C₂×D₄	1	2	yes	—
90	C₁₆×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₁₆×C₈₃	C₂×D₄	1	1	yes	characteristic
92	C₂³×C₈₃	#131	1	1	yes	characteristic
93	C₂²×D₈₃	—	2	2	no	—
94	C₂×C₄×C₈₃	—	2	2	no	—
95	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₈₃	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²⋊D₈₃	—	2	4	no	—
100	C₂²⋊D₈₃	—	2	4	no	—
101	C₂×Dic₈₃	—	2	2	no	—
102	D₃₃₂	—	2	2	no	—
103	D₃₃₂	C₂²×C₈	1	4	yes	—
104	C₄×D₈₃	—	2	4	no	—
105	C₄×D₈₃	—	2	4	no	—
106	Dic₁₆₆	—	2	2	no	—
107	Dic₁₆₆	C₂²×C₈	1	4	yes	—
108	C₈×C₈₃	C₄×D₄	1	2	yes	—
109	C₈×C₈₃	C₂²×C₈	1	4	yes	—
110	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
112	C₈₃⋊C₈	—	2	4	no	—
113	C₂²×C₈₃	—	4	1	no	—
114	C₂²×C₈₃	C₈×D₄	1	2	yes	—
115	C₂²×C₈₃	C₂×#131	1	1	yes	characteristic
116	C₂×D₈₃	—	2	4	no	—
117	C₂×D₈₃	—	4	4	no	—
118	C₄×C₈₃	—	4	2	no	—
119	C₄×C₈₃	C₈×D₄	1	2	yes	—
120	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₈₃	C₂³×C₈	1	1	yes	characteristic
122	Dic₈₃	—	2	4	no	—
123	#33441	—	83	1	no	maximal, Sylow
124	C₂×C₈₃	—	2	1	no	—
125	C₂×C₈₃	—	2	2	no	—
126	C₂×C₈₃	C₂×C₈×D₄	1	1	yes	characteristic
127	D₈₃	—	4	4	no	—
128	C₂×#504	—	83	2	no	—
129	#2955	—	83	1	no	—
130	#2171	—	83	4	no	—
131	#2170	—	83	4	no	—
132	#2153	—	83	1	no	—
133	#2117	—	83	2	no	—
134	#2108	D₈₃	1	1	yes	characteristic, complemented
135	C₈₃	#33441	1	1	yes	characteristic, complemented, Sylow
136	#575	—	83	1	no	—
137	C₂×#131	—	83	2	no	—
138	#504	—	83	4	no	—
139	#504	—	83	4	no	—
140	#504	—	166	2	no	—
141	#504	—	166	2	no	—
142	C₂×M₃₂	—	83	2	no	—
143	C₂×M₃₂	—	166	2	no	—
144	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
145	C₂²×C₁₆	—	166	2	no	—
146	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
147	#444	—	83	4	no	—
148	#443	—	83	4	no	—
149	#430	—	83	1	no	—
150	#421	—	83	2	no	—
151	#405	C₂×D₈₃	1	1	yes	characteristic
152	#364	—	83	4	no	—
153	#350	—	83	8	no	—
154	#346	C₂×D₈₃	1	2	yes	complemented
155	C₄×C₁₆	C₂×D₈₃	1	2	yes	complemented
156	C₂×#41	—	83	2	no	—
157	#137	—	83	4	no	—
158	AGL(1,ℤ/8)	—	83	4	no	—
159	#135	—	83	4	no	—
160	#131	—	83	4	no	—
161	#131	—	83	4	no	—
162	#131	—	166	2	no	—
163	#131	—	166	2	no	—
164	C₂×M₁₆	—	83	2	no	—
165	C₂×M₁₆	—	166	2	no	—
166	C₂×M₁₆	C₄×D₈₃	1	1	yes	characteristic
167	C₂²×C₈	—	166	2	no	—
168	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
169	#117	C₄×D₈₃	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	83	4	no	—
172	M₃₂	—	166	2	no	—
173	M₃₂	—	166	4	no	—
174	M₃₂	—	166	4	no	—
175	C₂×C₁₆	—	166	2	no	—
176	C₂×C₁₆	—	166	4	no	—
177	C₂×C₁₆	—	166	4	no	—
178	C₂×C₁₆	D₃₃₂	1	2	yes	—
179	C₂×C₁₆	D₃₃₂	1	2	yes	—
180	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
181	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
182	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
183	#108	—	83	4	no	—
184	#104	—	83	8	no	—
185	#97	C₄×D₈₃	1	2	yes	—
186	C₄×C₈	C₂²×D₈₃	1	2	yes	—
187	#41	—	83	4	no	—
188	#41	—	83	4	no	—
189	#41	—	166	2	no	—
190	#41	—	166	2	no	—
191	C₂×Q₈	—	83	2	no	—
192	C₂×D₄	—	83	2	no	—
193	C₂×D₄	—	166	2	no	—
194	C₂²×C₄	—	166	2	no	—
195	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
196	Q₁₆	—	83	4	no	—
197	SD₁₆	—	83	8	no	—
198	D₈	—	83	4	no	—
199	M₁₆	—	83	4	no	—
200	M₁₆	—	166	2	no	—
201	M₁₆	—	166	4	no	—
202	M₁₆	—	166	4	no	—
203	M₁₆	C₈×D₈₃	1	4	yes	—
204	C₂×C₈	—	166	2	no	—
205	C₂×C₈	—	166	4	no	—
206	C₂×C₈	—	166	4	no	—
207	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
210	C₂×C₈	#13050590	1	2	yes	—
211	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂×C₄×D₈₃	1	2	yes	—
215	C₁₆	—	2	2	no	—
216	C₁₆	—	166	4	no	—
217	C₁₆	C₂×D₃₃₂	1	2	yes	—
218	C₁₆	C₂×D₃₃₂	1	1	yes	characteristic
219	C₁₆	C₂×D₃₃₂	1	1	yes	characteristic
220	C₂³	—	166	2	no	—
221	C₂³	#14581798	1	1	yes	characteristic
222	Q₈	—	83	4	no	—
223	Q₈	—	166	2	no	—
224	D₄	—	83	4	no	—
225	D₄	—	166	4	no	—
226	D₄	—	166	4	no	—
227	D₄	—	166	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	166	2	no	—
230	C₂×C₄	—	166	4	no	—
231	C₂×C₄	—	166	4	no	—
232	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₈₃	1	2	yes	—
234	C₂×C₄	#14581798	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
236	C₈	—	166	4	no	—
237	C₈	C₄×D₃₃₂	1	2	yes	—
238	C₈	C₂×C₈×D₈₃	1	4	yes	—
239	C₈	C₂²×D₃₃₂	1	1	yes	characteristic
240	C₈	C₂×#13050590	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	332	4	no	—
243	C₂²	—	166	4	no	—
244	C₂²	C₈×D₃₃₂	1	2	yes	—
245	C₂²	C₂×#14581798	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	166	4	no	—
248	C₄	C₈×D₃₃₂	1	2	yes	—
249	C₄	C₂×C₄×D₃₃₂	1	1	yes	characteristic
250	C₄	C₂²×C₈×D₈₃	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	332	4	no	—
254	C₂	C₂×C₈×D₃₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×D₃₃₂
commutator	C₄×C₈₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2108	C₂
socle	C₂×C₈₃	C₂×C₈×D₄
prime radical	#19128501	C₄
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2108	D₈₃
hypocenter	C₈₃	#33441
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
1. C₂×#19128117	C₂	semidirect
3. #24909306	C₂	semidirect
4. #24909301	C₂	semidirect
5. #24909229	C₂	semidirect
6. #24909090	C₂	semidirect
7. C₈₃×#2108	C₂	semidirect
12. #19128117	C₂²	semidirect
13. #19128117	C₂²	semidirect
21. #19127522	C₂²	semidirect
22. #19127484	C₂²	semidirect
26. #346×C₈₃	C₂²	semidirect
27. C₄×C₁₆×C₈₃	C₂²	semidirect
134. #2108	D₈₃	semidirect
135. C₈₃	#33441	semidirect
154. #346	C₂×D₈₃	semidirect
155. C₄×C₁₆	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128117	2
#24916367	2
#24909306	2
#24909301	2
#24909229	2
#24909090	2
C₈₃×#2108	2
#33441	83