Subgroups — C₂×C₄×2₋¹⁺⁴×C₈₃ #35290474

Order 21248.

11210 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₄×2₋¹⁺⁴×C₈₃	C₂	1	32	yes	maximal, direct product factor
3	C₂×#520×C₈₃	C₂	1	10	yes	maximal, complemented
4	C₂×#519×C₈₃	C₂	1	5	yes	maximal, complemented
5	C₂×C₄×#41×C₈₃	C₂	1	10	yes	maximal, complemented
6	C₂²×C₄×Q₈×C₈₃	C₂	1	5	yes	maximal, complemented
7	C₂×2₋¹⁺⁴×C₈₃	C₄	1	32	yes	direct product factor
8	C₂×2₋¹⁺⁴×C₈₃	C₂²	1	16	yes	—
9	C₂²×#41×C₈₃	C₂²	1	10	yes	—
10	C₂³×Q₈×C₈₃	C₂²	1	5	yes	—
11	#520×C₈₃	C₂²	1	160	yes	complemented
12	#519×C₈₃	C₂²	1	80	yes	complemented
13	C₄×#41×C₈₃	C₂²	1	160	yes	complemented
14	C₂×C₄×Q₈×C₈₃	C₂²	1	10	yes	—
15	C₂×C₄×Q₈×C₈₃	C₂²	1	20	yes	—
16	C₂×C₄×Q₈×C₈₃	C₂²	1	10	yes	—
17	C₂×C₄×Q₈×C₈₃	C₂²	1	80	yes	complemented
18	C₂×C₄×D₄×C₈₃	C₂²	1	10	yes	—
19	C₂×C₄×D₄×C₈₃	C₂²	1	30	yes	—
20	C₂×#117×C₈₃	C₂²	1	30	yes	—
21	C₂²×C₄⋊C₄×C₈₃	C₂²	1	15	yes	—
22	C₂²×C₄²×C₈₃	C₂²	1	15	yes	—
23	2₋¹⁺⁴×C₈₃	C₂×C₄	1	256	yes	direct product factor
24	C₂×#41×C₈₃	C₂×C₄	1	160	yes	complemented
25	C₂×#41×C₈₃	C₂³	1	80	yes	—
26	C₂²×Q₈×C₈₃	C₂×C₄	1	80	yes	complemented
27	C₂²×Q₈×C₈₃	C₂³	1	10	yes	—
28	C₂²×Q₈×C₈₃	C₂³	1	40	yes	—
29	C₂²×D₄×C₈₃	C₂³	1	10	yes	—
30	C₂³×C₄×C₈₃	C₂³	1	5	yes	—
31	C₂³×C₄×C₈₃	C₂³	1	15	yes	—
32	C₂³×C₄×C₈₃	C₂³	1	15	yes	—
33	C₄×Q₈×C₈₃	C₂³	1	80	yes	—
34	C₄×Q₈×C₈₃	C₂³	1	160	yes	—
35	C₄×Q₈×C₈₃	C₂³	1	80	yes	—
36	C₄×D₄×C₈₃	C₂³	1	80	yes	—
37	C₄×D₄×C₈₃	C₂³	1	240	yes	—
38	#117×C₈₃	C₂³	1	240	yes	—
39	C₂×C₄⋊C₄×C₈₃	C₂³	1	30	yes	—
40	C₂×C₄⋊C₄×C₈₃	C₂³	1	120	yes	—
41	C₂×C₄⋊C₄×C₈₃	C₂³	1	10	yes	—
42	C₂×C₂²⋊C₄×C₈₃	C₂³	1	20	yes	—
43	C₂×C₄²×C₈₃	C₂³	1	30	yes	—
44	C₂×C₄²×C₈₃	C₂³	1	120	yes	—
45	C₂×C₄²×C₈₃	C₂³	1	10	yes	—
46	C₂⁴×C₈₃	C₂⁴	1	5	yes	—
47	#41×C₈₃	C₂²×C₄	1	640	yes	complemented
48	C₂×Q₈×C₈₃	C₂²×C₄	1	80	yes	—
49	C₂×Q₈×C₈₃	C₂²×C₄	1	320	yes	complemented
50	C₂×Q₈×C₈₃	C₂⁴	1	40	yes	—
51	C₂×D₄×C₈₃	C₂²×C₄	1	80	yes	—
52	C₂×D₄×C₈₃	C₂⁴	1	40	yes	—
53	C₂²×C₄×C₈₃	—	2	20	no	—
54	C₂²×C₄×C₈₃	—	4	30	no	—
55	C₂²×C₄×C₈₃	C₂²×C₄	1	120	yes	—
56	C₂²×C₄×C₈₃	C₂⁴	1	5	yes	—
57	C₂²×C₄×C₈₃	C₂⁴	1	20	yes	—
58	C₂²×C₄×C₈₃	C₂⁴	1	10	yes	—
59	C₂²×C₄×C₈₃	C₂⁴	1	60	yes	—
60	C₂²×C₄×C₈₃	C₂⁴	1	10	yes	—
61	C₂²×C₄×C₈₃	C₂⁴	1	60	yes	—
62	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₈₃	C₂⁴	1	120	yes	—
64	C₄⋊C₄×C₈₃	C₂⁴	1	40	yes	—
65	C₂²⋊C₄×C₈₃	C₂⁴	1	80	yes	—
66	C₄²×C₈₃	C₂⁴	1	40	yes	—
67	C₄²×C₈₃	C₂⁴	1	120	yes	—
68	C₂³×C₈₃	—	2	10	no	—
69	C₂³×C₈₃	—	2	10	no	—
70	C₂³×C₈₃	C₂³×C₄	1	20	yes	—
71	C₂³×C₈₃	C₂⁵	1	10	yes	—
72	C₂³×C₈₃	C₂⁵	1	1	yes	characteristic
73	Q₈×C₈₃	C₂³×C₄	1	160	yes	—
74	D₄×C₈₃	C₂³×C₄	1	160	yes	—
75	C₂×C₄×C₈₃	—	2	10	no	—
76	C₂×C₄×C₈₃	—	2	20	no	—
77	C₂×C₄×C₈₃	—	2	40	no	—
78	C₂×C₄×C₈₃	—	4	60	no	—
79	C₂×C₄×C₈₃	C₂³×C₄	1	40	yes	—
80	C₂×C₄×C₈₃	C₂³×C₄	1	240	yes	—
81	C₂×C₄×C₈₃	2₋¹⁺⁴	1	4	yes	direct product factor
82	C₂×C₄×C₈₃	C₂⁵	1	10	yes	—
83	C₂×C₄×C₈₃	C₂⁵	1	20	yes	—
84	C₂×C₄×C₈₃	C₂⁵	1	20	yes	—
85	C₂×C₄×C₈₃	C₂⁵	1	2	yes	—
86	C₂²×C₈₃	—	2	10	no	—
87	C₂²×C₈₃	—	2	10	no	—
88	C₂²×C₈₃	—	2	40	no	—
89	C₂²×C₈₃	C₂⁴×C₄	1	20	yes	—
90	C₂²×C₈₃	C₂⁴×C₄	1	2	yes	—
91	C₂²×C₈₃	C₂×2₋¹⁺⁴	1	2	yes	—
92	C₂²×C₈₃	#585	1	2	yes	—
93	C₂²×C₈₃	C₂⁶	1	1	yes	characteristic
94	C₄×C₈₃	—	2	10	no	—
95	C₄×C₈₃	—	2	20	no	—
96	C₄×C₈₃	C₂⁴×C₄	1	40	yes	—
97	C₄×C₈₃	C₂×2₋¹⁺⁴	1	4	yes	direct product factor
98	C₂×C₄×2₋¹⁺⁴	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₈₃	—	2	20	no	—
100	C₂×C₈₃	C₄×2₋¹⁺⁴	1	4	yes	direct product factor
101	C₂×C₈₃	C₂⁵×C₄	1	1	yes	characteristic
102	C₂×C₈₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
103	C₂×C₈₃	C₂×#585	1	1	yes	characteristic
104	C₂²×2₋¹⁺⁴	C₂×C₈₃	1	1	yes	characteristic
105	C₄×2₋¹⁺⁴	C₂×C₈₃	1	32	yes	direct product factor
106	C₂×#520	C₂×C₈₃	1	10	yes	complemented
107	C₂×#519	C₂×C₈₃	1	5	yes	complemented
108	C₂×C₄×#41	C₂×C₈₃	1	10	yes	complemented
109	C₂²×C₄×Q₈	C₂×C₈₃	1	5	yes	complemented
110	C₈₃	C₂×C₄×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×2₋¹⁺⁴	C₄×C₈₃	1	32	yes	direct product factor
112	C₂×2₋¹⁺⁴	C₂²×C₈₃	1	16	yes	—
113	C₂²×#41	C₂²×C₈₃	1	10	yes	—
114	C₂³×Q₈	C₂²×C₈₃	1	5	yes	—
115	#520	C₂²×C₈₃	1	160	yes	complemented
116	#519	C₂²×C₈₃	1	80	yes	complemented
117	C₄×#41	C₂²×C₈₃	1	160	yes	complemented
118	C₂×C₄×Q₈	C₂²×C₈₃	1	10	yes	—
119	C₂×C₄×Q₈	C₂²×C₈₃	1	20	yes	—
120	C₂×C₄×Q₈	C₂²×C₈₃	1	10	yes	—
121	C₂×C₄×Q₈	C₂²×C₈₃	1	80	yes	complemented
122	C₂×C₄×D₄	C₂²×C₈₃	1	10	yes	—
123	C₂×C₄×D₄	C₂²×C₈₃	1	30	yes	—
124	C₂×#117	C₂²×C₈₃	1	30	yes	—
125	C₂²×C₄⋊C₄	C₂²×C₈₃	1	15	yes	—
126	C₂²×C₄²	C₂²×C₈₃	1	15	yes	—
127	2₋¹⁺⁴	C₂×C₄×C₈₃	1	256	yes	direct product factor
128	C₂×#41	C₂×C₄×C₈₃	1	160	yes	complemented
129	C₂×#41	C₂³×C₈₃	1	80	yes	—
130	C₂²×Q₈	C₂×C₄×C₈₃	1	80	yes	complemented
131	C₂²×Q₈	C₂³×C₈₃	1	10	yes	—
132	C₂²×Q₈	C₂³×C₈₃	1	40	yes	—
133	C₂²×D₄	C₂³×C₈₃	1	10	yes	—
134	C₂³×C₄	C₂³×C₈₃	1	5	yes	—
135	C₂³×C₄	C₂³×C₈₃	1	15	yes	—
136	C₂³×C₄	C₂³×C₈₃	1	15	yes	—
137	C₄×Q₈	C₂³×C₈₃	1	80	yes	—
138	C₄×Q₈	C₂³×C₈₃	1	160	yes	—
139	C₄×Q₈	C₂³×C₈₃	1	80	yes	—
140	C₄×D₄	C₂³×C₈₃	1	80	yes	—
141	C₄×D₄	C₂³×C₈₃	1	240	yes	—
142	#117	C₂³×C₈₃	1	240	yes	—
143	C₂×C₄⋊C₄	C₂³×C₈₃	1	30	yes	—
144	C₂×C₄⋊C₄	C₂³×C₈₃	1	120	yes	—
145	C₂×C₄⋊C₄	C₂³×C₈₃	1	10	yes	—
146	C₂×C₂²⋊C₄	C₂³×C₈₃	1	20	yes	—
147	C₂×C₄²	C₂³×C₈₃	1	30	yes	—
148	C₂×C₄²	C₂³×C₈₃	1	10	yes	—
149	C₂×C₄²	C₂³×C₈₃	1	120	yes	—
150	C₂⁴	C₂⁴×C₈₃	1	5	yes	—
151	#41	C₂²×C₄×C₈₃	1	640	yes	complemented
152	C₂×Q₈	C₂²×C₄×C₈₃	1	80	yes	—
153	C₂×Q₈	C₂²×C₄×C₈₃	1	320	yes	complemented
154	C₂×Q₈	C₂⁴×C₈₃	1	40	yes	—
155	C₂×D₄	C₂²×C₄×C₈₃	1	80	yes	—
156	C₂×D₄	C₂⁴×C₈₃	1	40	yes	—
157	C₂²×C₄	—	2	20	no	—
158	C₂²×C₄	—	4	30	no	—
159	C₂²×C₄	C₂²×C₄×C₈₃	1	120	yes	—
160	C₂²×C₄	C₂⁴×C₈₃	1	5	yes	—
161	C₂²×C₄	C₂⁴×C₈₃	1	20	yes	—
162	C₂²×C₄	C₂⁴×C₈₃	1	10	yes	—
163	C₂²×C₄	C₂⁴×C₈₃	1	60	yes	—
164	C₂²×C₄	C₂⁴×C₈₃	1	10	yes	—
165	C₂²×C₄	C₂⁴×C₈₃	1	60	yes	—
166	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
167	C₄⋊C₄	C₂⁴×C₈₃	1	120	yes	—
168	C₄⋊C₄	C₂⁴×C₈₃	1	40	yes	—
169	C₂²⋊C₄	C₂⁴×C₈₃	1	80	yes	—
170	C₄²	C₂⁴×C₈₃	1	40	yes	—
171	C₄²	C₂⁴×C₈₃	1	120	yes	—
172	C₂³	—	2	10	no	—
173	C₂³	—	2	10	no	—
174	C₂³	C₂³×C₄×C₈₃	1	20	yes	—
175	C₂³	C₂⁵×C₈₃	1	10	yes	—
176	C₂³	C₂⁵×C₈₃	1	1	yes	characteristic
177	Q₈	C₂³×C₄×C₈₃	1	160	yes	—
178	D₄	C₂³×C₄×C₈₃	1	160	yes	—
179	C₂×C₄	—	2	10	no	—
180	C₂×C₄	—	2	20	no	—
181	C₂×C₄	—	2	40	no	—
182	C₂×C₄	—	4	60	no	—
183	C₂×C₄	C₂³×C₄×C₈₃	1	40	yes	—
184	C₂×C₄	C₂³×C₄×C₈₃	1	240	yes	—
185	C₂×C₄	2₋¹⁺⁴×C₈₃	1	4	yes	direct product factor
186	C₂×C₄	C₂⁵×C₈₃	1	10	yes	—
187	C₂×C₄	C₂⁵×C₈₃	1	20	yes	—
188	C₂×C₄	C₂⁵×C₈₃	1	20	yes	—
189	C₂×C₄	C₂⁵×C₈₃	1	2	yes	—
190	C₂²	—	2	10	no	—
191	C₂²	—	2	10	no	—
192	C₂²	—	2	40	no	—
193	C₂²	C₂⁴×C₄×C₈₃	1	20	yes	—
194	C₂²	C₂⁴×C₄×C₈₃	1	2	yes	—
195	C₂²	C₂×2₋¹⁺⁴×C₈₃	1	2	yes	—
196	C₂²	#585×C₈₃	1	2	yes	—
197	C₂²	C₂⁶×C₈₃	1	1	yes	characteristic
198	C₄	—	2	10	no	—
199	C₄	—	2	20	no	—
200	C₄	C₂⁴×C₄×C₈₃	1	40	yes	—
201	C₄	C₂×2₋¹⁺⁴×C₈₃	1	4	yes	direct product factor
202	C₂	—	2	20	no	—
203	C₂	C₄×2₋¹⁺⁴×C₈₃	1	4	yes	direct product factor
204	C₂	C₂⁵×C₄×C₈₃	1	1	yes	characteristic
205	C₂	C₂²×2₋¹⁺⁴×C₈₃	1	1	yes	characteristic
206	C₂	C₂×#585×C₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₈₃	C₂⁴
commutator	C₂	C₂⁵×C₄×C₈₃
Frattini, cosocle	C₂²	C₂⁶×C₈₃
socle	C₂³×C₈₃	C₂⁵
prime radical	C₂²×2₋¹⁺⁴×C₈₃	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×2₋¹⁺⁴×C₈₃	C₂	direct (N × H)
3. C₂×#520×C₈₃	C₂	semidirect
4. C₂×#519×C₈₃	C₂	semidirect
5. C₂×C₄×#41×C₈₃	C₂	semidirect
6. C₂²×C₄×Q₈×C₈₃	C₂	semidirect
7. C₂×2₋¹⁺⁴×C₈₃	C₄	direct (N × H)
11. #520×C₈₃	C₂²	semidirect
12. #519×C₈₃	C₂²	semidirect
13. C₄×#41×C₈₃	C₂²	semidirect
17. C₂×C₄×Q₈×C₈₃	C₂²	semidirect
23. 2₋¹⁺⁴×C₈₃	C₂×C₄	direct (N × H)
24. C₂×#41×C₈₃	C₂×C₄	semidirect
26. C₂²×Q₈×C₈₃	C₂×C₄	semidirect
47. #41×C₈₃	C₂²×C₄	semidirect
49. C₂×Q₈×C₈₃	C₂²×C₄	semidirect
81. C₂×C₄×C₈₃	2₋¹⁺⁴	direct (N × H)
97. C₄×C₈₃	C₂×2₋¹⁺⁴	direct (N × H)
98. C₂×C₄×2₋¹⁺⁴	C₈₃	direct (N × H)
100. C₂×C₈₃	C₄×2₋¹⁺⁴	direct (N × H)
105. C₄×2₋¹⁺⁴	C₂×C₈₃	direct (N × H)
106. C₂×#520	C₂×C₈₃	semidirect
107. C₂×#519	C₂×C₈₃	semidirect
108. C₂×C₄×#41	C₂×C₈₃	semidirect
109. C₂²×C₄×Q₈	C₂×C₈₃	semidirect
110. C₈₃	C₂×C₄×2₋¹⁺⁴	direct (N × H)
111. C₂×2₋¹⁺⁴	C₄×C₈₃	direct (N × H)
115. #520	C₂²×C₈₃	semidirect
116. #519	C₂²×C₈₃	semidirect
117. C₄×#41	C₂²×C₈₃	semidirect
121. C₂×C₄×Q₈	C₂²×C₈₃	semidirect
127. 2₋¹⁺⁴	C₂×C₄×C₈₃	direct (N × H)
128. C₂×#41	C₂×C₄×C₈₃	semidirect
130. C₂²×Q₈	C₂×C₄×C₈₃	semidirect
151. #41	C₂²×C₄×C₈₃	semidirect
153. C₂×Q₈	C₂²×C₄×C₈₃	semidirect
185. C₂×C₄	2₋¹⁺⁴×C₈₃	direct (N × H)
201. C₄	C₂×2₋¹⁺⁴×C₈₃	direct (N × H)
203. C₂	C₄×2₋¹⁺⁴×C₈₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₈₃	2
C₄×2₋¹⁺⁴×C₈₃	2
C₂×#520×C₈₃	2
C₂×#519×C₈₃	2
C₂×C₄×#41×C₈₃	2
C₂²×C₄×Q₈×C₈₃	2
C₂×C₄×2₋¹⁺⁴	83

Subgroups — C2×C4×2−1+4×C83 #35290474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×2₋¹⁺⁴×C₈₃ #35290474