Subgroups — C₂×C₄×Dic₅₆₈ #32150728

Order 18176.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₈₄	C₂	1	2	yes	maximal
3	C₄×Dic₅₆₈	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17985825	C₂	1	1	yes	maximal, characteristic
5	C₂×#17985804	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₈₄	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₄₂	—	2	2	no	—
9	C₂×Dic₅₆₈	C₄	1	8	yes	direct product factor
10	C₂×Dic₅₆₈	C₂²	1	4	yes	—
11	C₄×Dic₂₈₄	—	2	4	no	—
12	C₄×Dic₂₈₄	C₂²	1	8	yes	—
13	C₂×#14556062	C₂²	1	2	yes	—
14	C₂×#14556053	—	2	2	no	—
15	#17985825	C₂²	1	4	yes	—
16	#17985804	C₂²	1	8	yes	—
17	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₇₁	C₂²	1	4	yes	—
21	C₂²×Dic₁₄₂	—	2	2	no	—
22	C₂×Dic₂₈₄	—	2	2	no	—
23	C₂×Dic₂₈₄	—	2	4	no	—
24	C₂×Dic₂₈₄	C₂×C₄	1	8	yes	—
25	C₂×Dic₂₈₄	C₂³	1	4	yes	—
26	C₄×Dic₁₄₂	—	2	8	no	—
27	C₄×Dic₁₄₂	—	4	4	no	—
28	C₂×C₄⋊Dic₇₁	—	2	2	no	—
29	C₂×Dic₇₁⋊C₄	—	4	2	no	—
30	C₂×C₄×Dic₇₁	—	4	2	no	—
31	Dic₅₆₈	C₂×C₄	1	16	yes	direct product factor
32	#14556062	—	2	2	no	—
33	#14556062	C₂³	1	4	yes	—
34	#14556053	—	2	8	no	—
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
41	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
42	C₄×C₈×C₇₁	D₄	1	2	yes	—
43	C₄×C₈×C₇₁	C₂³	1	2	yes	—
44	C₂×Dic₁₄₂	—	2	8	no	—
45	C₂×Dic₁₄₂	—	2	4	no	—
46	C₂×Dic₁₄₂	—	4	2	no	—
47	C₂×Dic₁₄₂	—	4	4	no	—
48	C₂²×Dic₇₁	—	4	2	no	—
49	C₂²×Dic₇₁	—	4	2	no	—
50	Dic₂₈₄	—	2	8	no	—
51	Dic₂₈₄	—	2	4	no	—
52	Dic₂₈₄	C₂²×C₄	1	8	yes	—
53	C₄⋊Dic₇₁	—	2	4	no	—
54	C₄⋊Dic₇₁	—	4	2	no	—
55	Dic₇₁⋊C₄	—	4	8	no	—
56	C₄×Dic₇₁	—	4	8	no	—
57	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	—	2	2	no	—
61	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	#41	1	2	yes	—
65	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
66	C₄²×C₇₁	D₈	1	2	yes	—
67	C₄²×C₇₁	C₂×D₄	1	2	yes	—
68	C₁₆×C₇₁	—	2	2	no	—
69	C₁₆×C₇₁	C₂²×C₄	1	4	yes	—
70	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	Q₃₂	1	4	yes	complemented
73	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₇₁	#135	1	2	yes	—
78	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
79	C₂×Dic₇₁	—	4	4	no	—
80	C₂×Dic₇₁	—	4	8	no	—
81	C₂×Dic₇₁	—	4	4	no	—
82	C₂×Dic₇₁	—	8	4	no	—
83	Dic₁₄₂	—	2	8	no	—
84	Dic₁₄₂	—	4	4	no	—
85	Dic₁₄₂	—	4	8	no	—
86	C₈×C₇₁	—	2	2	no	—
87	C₈×C₇₁	C₄×D₄	1	2	yes	—
88	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
90	C₂²×C₇₁	C₄×D₈	1	2	yes	—
91	C₂²×C₇₁	C₂×Q₃₂	1	2	yes	—
92	C₂²×C₇₁	#508	1	2	yes	—
93	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₇₁	—	2	2	no	—
95	C₄×C₇₁	C₄×D₈	1	2	yes	—
96	C₄×C₇₁	C₂×Q₃₂	1	4	yes	complemented
97	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
98	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
99	Dic₇₁	—	4	8	no	—
100	Dic₇₁	—	8	4	no	—
101	C₂×C₄×Q₃₂	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	C₄×Q₃₂	1	4	yes	complemented
103	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
104	C₂×C₇₁	C₂²×Q₃₂	1	1	yes	characteristic
105	C₂×C₇₁	C₂×#508	1	1	yes	characteristic
106	C₂²×Q₃₂	—	71	1	no	—
107	C₂×C₄×Q₁₆	—	71	2	no	—
108	C₄×Q₃₂	—	71	8	no	—
109	C₂×#366	—	71	1	no	—
110	C₂×#358	—	71	2	no	—
111	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
112	C₇₁	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₁₆	—	71	2	no	—
114	C₂×C₄×Q₈	—	142	2	no	—
115	C₂×Q₃₂	—	71	8	no	—
116	C₂×Q₃₂	—	71	4	no	—
117	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
118	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
119	C₄×Q₁₆	—	71	8	no	—
120	C₄×Q₁₆	—	142	4	no	—
121	C₂×#107	—	71	2	no	—
122	C₂×#103	—	142	2	no	—
123	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
124	#366	—	71	4	no	—
125	#358	—	71	8	no	—
126	C₄×C₁₆	C₂×D₇₁	1	4	yes	—
127	C₂²×Q₈	—	142	2	no	—
128	C₂×Q₁₆	—	71	8	no	—
129	C₂×Q₁₆	—	71	4	no	—
130	C₂×Q₁₆	—	142	2	no	—
131	C₂×Q₁₆	—	142	4	no	—
132	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
134	C₄×Q₈	—	284	4	no	—
135	C₄×Q₈	—	142	8	no	—
136	C₂×C₄⋊C₄	—	284	2	no	—
137	C₂×C₄⋊C₄	—	142	2	no	—
138	C₂×C₄²	—	284	2	no	—
139	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
140	Q₃₂	—	71	16	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	C₄×D₇₁	1	4	yes	—
143	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
144	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
145	#107	—	71	4	no	—
146	#107	—	142	2	no	—
147	#103	—	142	8	no	—
148	C₄×C₈	D₂₈₄	1	2	yes	—
149	C₄×C₈	C₂²×D₇₁	1	2	yes	—
150	C₂×Q₈	—	284	2	no	—
151	C₂×Q₈	—	284	4	no	—
152	C₂×Q₈	—	142	4	no	—
153	C₂×Q₈	—	142	8	no	—
154	C₂²×C₄	—	284	2	no	—
155	C₂²×C₄	—	284	2	no	—
156	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
157	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
159	Q₁₆	—	71	8	no	—
160	Q₁₆	—	142	4	no	—
161	Q₁₆	—	142	8	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
164	C₂×C₈	C₂×D₂₈₄	1	2	yes	—
165	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
166	C₂×C₈	#11765865	1	2	yes	—
167	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
168	C₄⋊C₄	—	284	8	no	—
169	C₄⋊C₄	—	284	2	no	—
170	C₄⋊C₄	—	142	4	no	—
171	C₄²	—	284	8	no	—
172	C₄²	D₅₆₈	1	2	yes	—
173	C₄²	C₂×D₂₈₄	1	2	yes	—
174	C₁₆	—	2	2	no	—
175	C₁₆	C₂×C₄×D₇₁	1	4	yes	—
176	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
177	Q₈	—	284	8	no	—
178	Q₈	—	284	4	no	—
179	Q₈	—	142	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	284	8	no	—
182	C₂×C₄	—	284	4	no	—
183	C₂×C₄	—	284	4	no	—
184	C₂×C₄	—	568	4	no	—
185	C₂×C₄	Dic₅₆₈	1	4	yes	direct product factor
186	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
187	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
188	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
189	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
190	C₂×C₄	#14556150	1	2	yes	—
191	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₄×D₂₈₄	1	2	yes	—
194	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
195	C₈	C₂×#11765865	1	1	yes	characteristic
196	C₂²	C₄×D₅₆₈	1	2	yes	—
197	C₂²	C₂×Dic₅₆₈	1	2	yes	—
198	C₂²	#17986431	1	2	yes	—
199	C₂²	C₂²×D₅₆₈	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	284	8	no	—
202	C₄	—	568	4	no	—
203	C₄	C₄×D₅₆₈	1	2	yes	—
204	C₄	C₂×Dic₅₆₈	1	4	yes	direct product factor
205	C₄	C₂×C₄×D₂₈₄	1	1	yes	characteristic
206	C₄	C₂×#14556150	1	1	yes	characteristic
207	C₂	C₄×Dic₅₆₈	1	4	yes	direct product factor
208	C₂	C₂×C₄×D₅₆₈	1	1	yes	characteristic
209	C₂	C₂²×Dic₅₆₈	1	1	yes	characteristic
210	C₂	C₂×#17986431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₅₆₈
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₅₆₈	C₂	direct (N × H)
9. C₂×Dic₅₆₈	C₄	direct (N × H)
31. Dic₅₆₈	C₂×C₄	direct (N × H)
72. C₂×C₄×C₇₁	Q₃₂	semidirect
96. C₄×C₇₁	C₂×Q₃₂	semidirect
102. C₂×C₇₁	C₄×Q₃₂	semidirect
112. C₇₁	C₂×C₄×Q₃₂	semidirect
185. C₂×C₄	Dic₅₆₈	direct (N × H)
204. C₄	C₂×Dic₅₆₈	direct (N × H)
207. C₂	C₄×Dic₅₆₈	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×Dic₅₆₈	2
C₂×C₄×Dic₂₈₄	2
C₄×Dic₅₆₈	2
C₂×#17985825	2
C₂×#17985804	2
C₂×C₄×C₁₆×C₇₁	2
C₂×C₄×Q₃₂	71

Subgroups — C2×C4×Dic568 #32150728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₅₆₈ #32150728