Subgroups — C₂×C₈×D₅₅₆ #31815078

Order 17792.

616 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₃₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₅₅₆	C₂	1	1	yes	maximal, characteristic
3	C₈×D₅₅₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₁₃₉)⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₅₆	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₁₃₉	C₂²	1	2	yes	—
9	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
10	C₂×C₈×D₁₃₉	—	2	8	no	—
11	C₂×C₈×D₁₃₉	C₂²	1	8	yes	complemented
12	C₄×D₅₅₆	C₄	1	4	yes	—
13	C₄×D₅₅₆	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₁₃₉	C₄	1	1	yes	characteristic
15	C₂×(C₂×D₁₃₉)⋊C₄	C₄	1	2	yes	—
16	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	4	yes	complemented
17	(C₂×D₁₃₉)⋊C₈	C₂²	1	8	yes	complemented
18	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₁₃₉	C₂²	1	4	yes	complemented
22	C₂³×D₁₃₉	C₂×C₄	1	2	yes	—
23	C₂×D₅₅₆	C₈	1	8	yes	direct product factor
24	C₂×D₅₅₆	C₂×C₄	1	4	yes	—
25	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₁₃₉	—	2	4	no	—
27	C₂×C₄×D₁₃₉	—	2	4	no	—
28	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₃₉	C₂³	1	4	yes	—
30	C₂×C₁₃₉⋊C₈	—	2	4	no	—
31	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
32	C₈×D₁₃₉	—	2	16	no	—
33	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
34	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	8	yes	—
35	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₃₉	—	2	2	no	—
39	C₂×C₈×C₁₃₉	D₄	1	4	yes	complemented
40	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
41	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
42	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
43	C₄²×C₁₃₉	C₂³	1	2	yes	—
44	C₂²×D₁₃₉	—	2	4	no	—
45	C₂²×D₁₃₉	—	2	4	no	—
46	C₂²×D₁₃₉	C₂×C₈	1	8	yes	complemented
47	C₂²×D₁₃₉	C₂²×C₄	1	4	yes	—
48	D₅₅₆	C₂×C₈	1	16	yes	direct product factor
49	C₂×Dic₁₃₉	—	2	4	no	—
50	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
51	C₄×D₁₃₉	—	2	8	no	—
52	C₄×D₁₃₉	—	2	4	no	—
53	C₄×D₁₃₉	—	2	4	no	—
54	C₁₃₉⋊C₈	—	2	4	no	—
55	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	—	2	2	no	—
57	C₂×C₄×C₁₃₉	C₂×C₈	1	4	yes	complemented
58	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₃₉	#41	1	2	yes	—
63	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
64	C₈×C₁₃₉	—	2	2	no	—
65	C₈×C₁₃₉	C₂×D₄	1	4	yes	complemented
66	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
67	C₂²×C₁₃₉	C₂²×C₈	1	2	yes	—
68	C₂²×C₁₃₉	#131	1	2	yes	—
69	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
70	C₂×D₁₃₉	—	2	4	no	—
71	C₂×D₁₃₉	—	2	4	no	—
72	C₂×D₁₃₉	—	2	16	no	—
73	C₂×D₁₃₉	C₂²×C₈	1	8	yes	complemented
74	C₄×C₁₃₉	—	2	2	no	—
75	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
76	C₄×C₁₃₉	C₂²×C₈	1	4	yes	complemented
77	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
79	Dic₁₃₉	—	2	4	no	—
80	C₂×C₁₃₉	C₈×D₄	1	4	yes	complemented
81	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
82	C₂×C₁₃₉	C₂³×C₈	1	1	yes	characteristic
83	C₂×C₁₃₉	C₂×#131	1	1	yes	characteristic
84	D₁₃₉	—	2	8	no	—
85	C₁₃₉	C₂×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₈×D₄	—	139	1	no	maximal, Sylow
87	C₂³×C₈	—	139	2	no	—
88	C₂×C₄×D₄	—	139	1	no	—
89	C₈×D₄	—	139	8	no	—
90	C₂×C₄⋊C₈	—	139	1	no	—
91	C₂×C₂²⋊C₈	—	139	2	no	—
92	C₂×C₄×C₈	D₁₃₉	1	1	yes	characteristic, complemented
93	C₂²×D₄	—	139	1	no	—
94	C₂³×C₄	—	139	2	no	—
95	C₂²×C₈	—	278	8	no	—
96	C₂²×C₈	—	139	2	no	—
97	C₂²×C₈	—	139	8	no	—
98	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
99	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
100	C₄×D₄	—	139	4	no	—
101	C₄×D₄	—	139	4	no	—
102	C₂×C₄⋊C₄	—	139	1	no	—
103	C₂×C₂²⋊C₄	—	139	2	no	—
104	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
105	C₄⋊C₈	—	139	4	no	—
106	C₂²⋊C₈	—	139	8	no	—
107	C₄×C₈	C₂×D₁₃₉	1	4	yes	complemented
108	C₂⁴	—	139	2	no	—
109	C₂×D₄	—	139	8	no	—
110	C₂×D₄	—	139	4	no	—
111	C₂²×C₄	—	278	4	no	—
112	C₂²×C₄	—	278	4	no	—
113	C₂²×C₄	—	139	2	no	—
114	C₂²×C₄	—	139	4	no	—
115	C₂²×C₄	—	139	4	no	—
116	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
117	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	278	4	no	—
121	C₂×C₈	—	278	16	no	—
122	C₂×C₈	—	139	4	no	—
123	C₂×C₈	D₅₅₆	1	4	yes	direct product factor
124	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
125	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
126	C₄⋊C₄	—	139	4	no	—
127	C₂²⋊C₄	—	139	8	no	—
128	C₄²	C₄×D₁₃₉	1	2	yes	—
129	C₄²	C₂²×D₁₃₉	1	2	yes	—
130	C₂³	—	278	4	no	—
131	C₂³	—	278	4	no	—
132	C₂³	—	139	8	no	—
133	C₂³	—	139	4	no	—
134	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
135	D₄	—	139	16	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	278	4	no	—
138	C₂×C₄	—	278	8	no	—
139	C₂×C₄	—	278	4	no	—
140	C₂×C₄	—	278	4	no	—
141	C₂×C₄	—	139	4	no	—
142	C₂×C₄	C₈×D₁₃₉	1	4	yes	complemented
143	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
144	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
146	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
147	C₂×C₄	#14555785	1	2	yes	—
148	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	278	4	no	—
151	C₈	C₂×D₅₅₆	1	4	yes	direct product factor
152	C₂²	—	278	16	no	—
153	C₂²	—	278	4	no	—
154	C₂²	—	278	4	no	—
155	C₂²	—	139	8	no	—
156	C₂²	C₄×D₅₅₆	1	2	yes	—
157	C₂²	C₂×C₈×D₁₃₉	1	2	yes	—
158	C₂²	#17984848	1	2	yes	—
159	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	278	4	no	—
162	C₄	C₄×D₅₅₆	1	2	yes	—
163	C₄	C₂×C₈×D₁₃₉	1	4	yes	complemented
164	C₄	C₂²×D₅₅₆	1	1	yes	characteristic
165	C₄	C₂×#14555785	1	1	yes	characteristic
166	C₂	—	278	8	no	—
167	C₂	C₈×D₅₅₆	1	4	yes	direct product factor
168	C₂	C₂×C₄×D₅₅₆	1	1	yes	characteristic
169	C₂	C₂²×C₈×D₁₃₉	1	1	yes	characteristic
170	C₂	C₂×#17984848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₃₉
commutator	C₂×C₁₃₉	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×C₄×C₈×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂²×C₄
prime radical	C₂²×D₅₅₆	C₄
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₃₉
hypocenter	C₁₃₉	C₂×C₈×D₄
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₈×D₁₃₉	C₂	semidirect
3. C₈×D₅₅₆	C₂	direct (N × H)
4. C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	semidirect
5. C₂×(C₂×D₁₃₉)⋊C₈	C₂	semidirect
6. C₂×C₄×C₈×C₁₃₉	C₂	semidirect
11. C₂×C₈×D₁₃₉	C₂²	semidirect
16. C₄⋊(C₁₃₉⋊C₈)	C₂²	semidirect
17. (C₂×D₁₃₉)⋊C₈	C₂²	semidirect
21. C₄×C₈×C₁₃₉	C₂²	semidirect
23. C₂×D₅₅₆	C₈	direct (N × H)
39. C₂×C₈×C₁₃₉	D₄	semidirect
46. C₂²×D₁₃₉	C₂×C₈	semidirect
48. D₅₅₆	C₂×C₈	direct (N × H)
57. C₂×C₄×C₁₃₉	C₂×C₈	semidirect
65. C₈×C₁₃₉	C₂×D₄	semidirect
73. C₂×D₁₃₉	C₂²×C₈	semidirect
76. C₄×C₁₃₉	C₂²×C₈	semidirect
80. C₂×C₁₃₉	C₈×D₄	semidirect
85. C₁₃₉	C₂×C₈×D₄	semidirect
92. C₂×C₄×C₈	D₁₃₉	semidirect
107. C₄×C₈	C₂×D₁₃₉	semidirect
123. C₂×C₈	D₅₅₆	direct (N × H)
142. C₂×C₄	C₈×D₁₃₉	semidirect
151. C₈	C₂×D₅₅₆	direct (N × H)
163. C₄	C₂×C₈×D₁₃₉	semidirect
167. C₂	C₈×D₅₅₆	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×C₈×D₁₃₉	2
C₂×C₄×D₅₅₆	2
C₈×D₅₅₆	2
C₂×C₄⋊(C₁₃₉⋊C₈)	2
C₂×(C₂×D₁₃₉)⋊C₈	2
C₂×C₄×C₈×C₁₃₉	2
C₂×C₈×D₄	139

Subgroups — C2×C8×D556 #31815078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×D₅₅₆ #31815078