Subgroups — #35322643

Order 21248.

490 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917891	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127507	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14581814	—	2	1	no	—
6	C₂×#14581814	C₂²	1	2	yes	—
7	#19128251	—	2	1	no	—
8	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
9	#19128085	—	2	1	no	—
10	#19128085	C₄	1	2	yes	—
11	#19128081	C₄	1	1	yes	characteristic
12	#19128078	C₄	1	1	yes	characteristic
13	#19128074	—	2	1	no	—
14	#19127507	—	2	8	no	—
15	C₄×C₈₃⋊C₁₆	—	2	1	no	—
16	C₄×C₈₃⋊C₁₆	—	2	1	no	—
17	C₂×#108×C₈₃	—	2	1	no	—
18	C₂×#108×C₈₃	C₂²	1	2	yes	—
19	#423×C₈₃	—	2	1	no	—
20	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×#13050590	—	2	2	no	—
22	#14581814	—	2	4	no	—
23	#14581814	—	2	4	no	—
24	#14581814	—	4	2	no	—
25	#14581814	—	4	2	no	—
26	C₂×Dic₃₃₂	—	2	1	no	—
27	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
28	C₂×#13050579	—	2	2	no	—
29	C₂×#13050579	—	2	2	no	—
30	C₂×D₆₆₄	—	2	1	no	—
31	C₂×D₆₆₄	C₂×C₄	1	2	yes	—
32	#14581792	—	2	1	no	—
33	#14581789	—	2	1	no	—
34	#14581780	—	2	2	no	—
35	#14581766	—	4	2	no	—
36	C₄×D₃₃₂	—	4	2	no	—
37	C₂×C₈₃⋊C₁₆	—	2	2	no	—
38	C₂×C₈₃⋊C₁₆	—	2	2	no	—
39	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	complemented
40	C₂×M₁₆×C₈₃	—	2	2	no	—
41	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
44	#108×C₈₃	—	2	4	no	—
45	#108×C₈₃	—	4	2	no	—
46	C₄⋊C₈×C₈₃	—	4	2	no	—
47	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
51	#13050590	—	4	2	no	—
52	#13050590	—	4	2	no	—
53	#13050590	—	4	2	no	—
54	#13050590	—	4	2	no	—
55	C₂×Dic₁₆₆	—	2	2	no	—
56	C₂×D₃₃₂	—	2	2	no	—
57	C₂×D₃₃₂	—	4	2	no	—
58	C₂×C₂²⋊D₈₃	—	4	2	no	—
59	C₂×C₄×D₈₃	—	4	2	no	—
60	Dic₃₃₂	—	2	4	no	—
61	Dic₃₃₂	—	4	2	no	—
62	#13050579	—	4	4	no	—
63	#13050579	—	4	4	no	—
64	D₆₆₄	—	2	4	no	—
65	D₆₆₄	—	4	2	no	—
66	C₄⋊Dic₈₃	—	4	2	no	—
67	(C₂×D₈₃)⋊C₄	—	4	4	no	—
68	C₈₃⋊C₁₆	—	2	4	no	—
69	C₂²×C₄×C₈₃	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
71	M₁₆×C₈₃	—	4	2	no	—
72	M₁₆×C₈₃	—	4	2	no	—
73	C₂×C₈×C₈₃	—	4	2	no	—
74	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₃	D₈	1	2	yes	—
77	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
78	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₃	#41	1	2	yes	—
82	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₈₃	D₈	1	2	yes	—
85	C₂³×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂²×D₈₃	—	4	2	no	—
87	C₂×C₄×C₈₃	—	2	4	no	—
88	C₂×C₄×C₈₃	#102	1	2	yes	—
89	C₂×C₄×C₈₃	#102	1	2	yes	—
90	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
92	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂²⋊D₈₃	—	8	2	no	—
97	C₂²⋊D₈₃	—	8	2	no	—
98	C₂×Dic₈₃	—	4	2	no	—
99	D₃₃₂	—	4	2	no	—
100	D₃₃₂	—	4	2	no	—
101	D₃₃₂	—	8	4	no	—
102	C₄×D₈₃	—	8	2	no	—
103	C₄×D₈₃	—	8	2	no	—
104	Dic₁₆₆	—	4	2	no	—
105	Dic₁₆₆	—	4	2	no	—
106	C₈×C₈₃	—	8	2	no	—
107	C₈×C₈₃	#102	1	2	yes	—
108	C₈×C₈₃	#102	1	2	yes	—
109	C₈×C₈₃	C₄×D₄	1	2	yes	—
110	C₈×C₈₃	#121	1	2	yes	—
111	C₂²×C₈₃	—	2	2	no	—
112	C₂²×C₈₃	C₂×#102	1	1	yes	characteristic
113	C₂²×C₈₃	#486	1	1	yes	characteristic
114	C₂²×C₈₃	#492	1	1	yes	characteristic
115	C₂×D₈₃	—	8	4	no	—
116	C₂×D₈₃	—	8	2	no	—
117	C₄×C₈₃	—	2	2	no	—
118	C₄×C₈₃	#359	1	4	yes	complemented
119	C₄×C₈₃	#390	1	1	yes	characteristic
120	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
121	C₄×C₈₃	#486	1	1	yes	characteristic
122	C₄×C₈₃	#493	1	1	yes	characteristic
123	Dic₈₃	—	8	2	no	—
124	#13208	—	83	1	no	maximal, Sylow
125	C₂×C₈₃	—	2	1	no	—
126	C₂×C₈₃	—	2	1	no	—
127	C₂×C₈₃	#1957	1	1	yes	characteristic
128	C₂×C₈₃	C₂×#359	1	2	yes	—
129	D₈₃	—	16	2	no	—
130	#3147	—	83	1	no	—
131	C₂×#359	—	83	4	no	—
132	C₂×C₄×C₁₆	—	83	1	no	—
133	#1848	D₈₃	1	1	yes	characteristic, complemented
134	C₈₃	#13208	1	1	yes	characteristic, complemented, Sylow
135	C₂×#135	—	83	2	no	—
136	C₂×#135	—	166	1	no	—
137	#532	—	166	1	no	—
138	C₂²×C₁₆	—	83	2	no	—
139	#495	—	83	2	no	—
140	#495	—	166	1	no	—
141	#494	—	83	1	no	—
142	#493	—	83	1	no	—
143	#492	—	166	1	no	—
144	C₂×#108	—	2	1	no	—
145	C₂×#108	C₂×D₈₃	1	2	yes	—
146	#423	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
148	#359	—	166	8	no	—
149	C₄×C₁₆	—	166	1	no	—
150	C₄×C₁₆	—	166	1	no	—
151	C₂×#41	—	166	2	no	—
152	#135	—	332	2	no	—
153	#135	—	332	2	no	—
154	#135	—	166	4	no	—
155	#135	—	166	4	no	—
156	C₂×Q₁₆	—	83	2	no	—
157	C₂×Q₁₆	—	166	1	no	—
158	C₂×SD₁₆	—	166	2	no	—
159	C₂×SD₁₆	—	166	2	no	—
160	C₂×D₈	—	83	2	no	—
161	C₂×D₈	—	166	1	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	166	1	no	—
166	#127	—	166	1	no	—
167	#124	—	166	2	no	—
168	#121	—	332	2	no	—
169	C₄×D₄	—	332	2	no	—
170	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
171	C₂×C₁₆	—	83	4	no	—
172	C₂×C₁₆	—	166	2	no	—
173	C₂×C₁₆	—	166	2	no	—
174	#108	—	2	4	no	—
175	#108	—	4	2	no	—
176	C₄⋊C₈	—	4	2	no	—
177	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
178	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
179	C₄×C₈	D₃₃₂	1	1	yes	characteristic
180	C₄×C₈	D₃₃₂	1	1	yes	characteristic
181	#41	—	332	2	no	—
182	#41	—	332	2	no	—
183	#41	—	332	2	no	—
184	#41	—	332	2	no	—
185	C₂×Q₈	—	166	2	no	—
186	C₂×D₄	—	332	2	no	—
187	C₂×D₄	—	332	2	no	—
188	C₂×D₄	—	166	2	no	—
189	C₂²×C₄	—	332	2	no	—
190	C₂²×C₄	#13050577	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
192	Q₁₆	—	332	2	no	—
193	Q₁₆	—	166	4	no	—
194	SD₁₆	—	332	4	no	—
195	SD₁₆	—	332	4	no	—
196	D₈	—	332	2	no	—
197	D₈	—	166	4	no	—
198	M₁₆	—	4	2	no	—
199	M₁₆	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
202	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	#13050575	1	2	yes	—
204	C₂×C₈	#13050577	1	2	yes	—
205	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
206	C₂×C₈	D₄×D₈₃	1	2	yes	—
207	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
208	C₂×C₈	#13050591	1	2	yes	—
209	C₄⋊C₄	—	332	2	no	—
210	C₂²⋊C₄	—	332	4	no	—
211	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
212	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
213	C₄²	#13050575	1	2	yes	—
214	C₁₆	—	166	4	no	—
215	C₂³	—	332	2	no	—
216	C₂³	C₂×#13050577	1	1	yes	characteristic
217	Q₈	—	332	2	no	—
218	Q₈	—	332	2	no	—
219	D₄	—	332	2	no	—
220	D₄	—	332	2	no	—
221	D₄	—	664	2	no	—
222	D₄	—	664	2	no	—
223	D₄	—	664	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	332	2	no	—
226	C₂×C₄	—	664	2	no	—
227	C₂×C₄	—	664	2	no	—
228	C₂×C₄	#14581712	1	2	yes	—
229	C₂×C₄	#14581712	1	2	yes	—
230	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
231	C₂×C₄	#14581781	1	1	yes	characteristic
232	C₂×C₄	#14581788	1	1	yes	characteristic
233	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
234	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
235	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
236	C₈	—	8	2	no	—
237	C₈	#14581712	1	2	yes	—
238	C₈	#14581712	1	2	yes	—
239	C₈	#14581754	1	2	yes	—
240	C₈	#14581767	1	2	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	664	4	no	—
243	C₂²	—	664	2	no	—
244	C₂²	C₂×#14581712	1	1	yes	characteristic
245	C₂²	#19128049	1	1	yes	characteristic
246	C₂²	#19128075	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	664	2	no	—
249	C₄	#19127507	1	4	yes	complemented
250	C₄	#19127602	1	1	yes	characteristic
251	C₄	C₂×#14581712	1	1	yes	characteristic
252	C₄	#19128049	1	1	yes	characteristic
253	C₄	#19128077	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	1328	2	no	—
257	C₂	#24908268	1	1	yes	characteristic
258	C₂	C₂×#19127507	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1848	C₂
socle	C₂²×C₈₃	C₂×#102
prime radical	#24917891	C₂
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1848	D₈₃
hypocenter	C₈₃	#13208
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
2. C₂×#19127507	C₂	semidirect
3. C₂×C₄×C₈₃⋊C₁₆	C₂	semidirect
4. C₈₃×#1848	C₂	semidirect
39. C₂×C₈₃⋊C₁₆	D₄	semidirect
118. C₄×C₈₃	#359	semidirect
133. #1848	D₈₃	semidirect
134. C₈₃	#13208	semidirect
249. C₄	#19127507	semidirect

Maximal subgroups

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

subgroup	index
#24917891	2
C₂×#19127507	2
C₂×C₄×C₈₃⋊C₁₆	2
C₈₃×#1848	2
#13208	83