Subgroups — C₂×C₄²×#1057 #16796716

Order 3328.

2046 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#1057	C₂	1	6	yes	maximal
2	C₄²×#1057	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#1057	C₂²	1	4	yes	—
5	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₁₃	C₄	1	3	yes	—
7	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	—
8	C₂×C₄×#1057	C₄	1	48	yes	direct product factor
9	C₂×C₄×#1057	C₂²	1	24	yes	—
10	C₄²×Dic₁₃	C₄	1	4	yes	—
11	C₄²×Dic₁₃	C₂²	1	4	yes	—
12	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
13	C₂³×Dic₁₃	C₂×C₄	1	3	yes	—
14	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
15	C₂²×#1057	C₂×C₄	1	48	yes	—
16	C₂²×#1057	C₂³	1	8	yes	—
17	C₂×C₄²×C₁₃	C₈	1	8	yes	complemented
18	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₁₃	C₈	1	24	yes	complemented
20	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₁₃	C₂×C₄	1	24	yes	—
22	C₂×C₄×Dic₁₃	C₂×C₄	1	12	yes	—
23	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
24	C₄×#1057	C₂×C₄	1	96	yes	direct product factor
25	C₂×C₄²×C₈	—	13	1	no	maximal, Sylow
26	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	C₄²	1	12	yes	—
28	C₂²×C₄×C₁₃	C₂×C₈	1	12	yes	—
29	C₂²×C₄×C₁₃	C₂×C₈	1	12	yes	—
30	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
31	C₂²×Dic₁₃	C₄²	1	12	yes	—
32	C₂²×Dic₁₃	C₂×C₈	1	24	yes	—
33	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₃	C₂²×C₄	1	12	yes	—
35	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
36	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
37	C₂×#1057	C₄²	1	64	yes	direct product factor
38	C₂×#1057	C₂²×C₄	1	48	yes	—
39	C₄²×C₁₃	C₂×C₈	1	16	yes	complemented
40	C₄×Dic₁₃	C₄²	1	24	yes	—
41	C₄×Dic₁₃	C₂×C₈	1	48	yes	complemented
42	C₄×Dic₁₃	C₂²×C₄	1	24	yes	—
43	C₂²×C₄×C₈	—	13	6	no	—
44	C₂×C₄³	—	13	1	no	—
45	C₄²×C₈	—	13	8	no	—
46	C₂³×C₁₃	C₂×C₄²	1	6	yes	—
47	C₂³×C₁₃	C₂²×C₈	1	6	yes	—
48	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
49	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₄×C₈	1	48	yes	complemented
51	C₂×C₄×C₁₃	C₂×C₄²	1	12	yes	—
52	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
53	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
54	C₂×Dic₁₃	C₄×C₈	1	48	yes	complemented
55	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
56	C₂×Dic₁₃	C₂×C₄²	1	16	yes	—
57	C₂×Dic₁₃	C₂×C₄²	1	12	yes	—
58	C₂×Dic₁₃	C₂²×C₈	1	24	yes	—
59	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
60	#1057	C₂×C₄²	1	32	yes	direct product factor
61	C₂³×C₈	—	13	4	no	—
62	C₂²×C₄²	—	13	3	no	—
63	C₂²×C₄²	—	13	3	no	—
64	C₂²×C₄²	#261	1	1	yes	characteristic
65	C₂×C₄×C₈	—	13	24	no	—
66	C₂×C₄×C₈	—	13	48	no	—
67	C₄³	—	13	4	no	—
68	C₄³	—	13	4	no	—
69	C₂²×C₁₃	C₄³	1	4	yes	—
70	C₂²×C₁₃	C₂×C₄×C₈	1	12	yes	—
71	C₂²×C₁₃	C₂×C₄×C₈	1	12	yes	—
72	C₂²×C₁₃	C₂²×C₄²	1	3	yes	—
73	C₂²×C₁₃	C₂³×C₈	1	3	yes	—
74	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
75	C₄×C₁₃	C₂×C₄×C₈	1	24	yes	complemented
76	Dic₁₃	C₄³	1	4	yes	—
77	Dic₁₃	C₂×C₄×C₈	1	24	yes	complemented
78	Dic₁₃	C₂²×C₄²	1	4	yes	—
79	C₂³×C₄	—	13	1	no	—
80	C₂³×C₄	—	13	3	no	—
81	C₂³×C₄	C₂×#261	1	3	yes	—
82	C₂²×C₈	—	13	8	no	—
83	C₂²×C₈	—	13	48	no	—
84	C₂×C₄²	—	13	24	no	—
85	C₂×C₄²	—	13	24	no	—
86	C₂×C₄²	—	13	12	no	—
87	C₂×C₄²	—	13	6	no	—
88	C₂×C₄²	—	13	6	no	—
89	C₂×C₄²	#1057	1	8	yes	direct product factor
90	C₂×C₄²	C₂×#261	1	4	yes	—
91	C₄×C₈	—	13	96	no	—
92	C₂×C₁₃	C₄²×C₈	1	8	yes	complemented
93	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
94	C₂×C₁₃	C₂²×C₄×C₈	1	3	yes	—
95	C₂×C₁₃	C₂²×C₄×C₈	1	3	yes	—
96	C₂⁴	C₂²×#261	1	1	yes	characteristic
97	C₂²×C₄	—	13	24	no	—
98	C₂²×C₄	—	13	12	no	—
99	C₂²×C₄	—	13	12	no	—
100	C₂²×C₄	—	13	1	no	—
101	C₂²×C₄	—	13	6	no	—
102	C₂²×C₄	—	13	1	no	—
103	C₂²×C₄	D₁₃⋊C₈	1	12	yes	—
104	C₂²×C₄	C₄×#261	1	12	yes	—
105	C₂²×C₄	C₂×#1057	1	12	yes	—
106	C₂²×C₄	C₂²×#261	1	6	yes	—
107	C₂×C₈	—	13	48	no	—
108	C₂×C₈	—	13	64	no	—
109	C₄²	—	13	48	no	—
110	C₄²	—	13	24	no	—
111	C₄²	—	13	24	no	—
112	C₄²	C₂×#1057	1	16	yes	direct product factor
113	C₁₃	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
114	C₂³	C₂×D₁₃⋊C₈	1	6	yes	—
115	C₂³	C₂×C₄×#261	1	6	yes	—
116	C₂³	C₂²×#1057	1	2	yes	—
117	C₂³	C₂³×#261	1	1	yes	characteristic
118	C₂×C₄	—	13	48	no	—
119	C₂×C₄	—	13	16	no	—
120	C₂×C₄	—	13	24	no	—
121	C₂×C₄	—	13	6	no	—
122	C₂×C₄	—	13	12	no	—
123	C₂×C₄	—	13	6	no	—
124	C₂×C₄	C₄×#1057	1	48	yes	direct product factor
125	C₂×C₄	C₂×D₁₃⋊C₈	1	12	yes	—
126	C₂×C₄	C₂×C₄×#261	1	12	yes	—
127	C₂×C₄	C₂²×#1057	1	12	yes	—
128	C₈	—	13	32	no	—
129	C₂²	C₄×D₁₃⋊C₈	1	12	yes	—
130	C₂²	C₄²×#261	1	4	yes	—
131	C₂²	C₂×C₄×#1057	1	12	yes	—
132	C₂²	C₂²×D₁₃⋊C₈	1	3	yes	—
133	C₂²	C₂²×C₄×#261	1	3	yes	—
134	C₂²	C₂³×#1057	1	1	yes	characteristic
135	C₄	—	13	24	no	—
136	C₄	—	13	4	no	—
137	C₄	—	13	4	no	—
138	C₄	C₂×C₄×#1057	1	24	yes	direct product factor
139	C₂	C₄²×#1057	1	8	yes	direct product factor
140	C₂	C₂×C₄²×#261	1	1	yes	characteristic
141	C₂	C₂²×C₄×#1057	1	3	yes	—
142	C₂	C₂×C₄×D₁₃⋊C₈	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	#261
commutator, hypocenter	C₁₃	C₂×C₄²×C₈
Frattini	C₂³	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×Dic₁₃	C₂⁴
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×#1057	C₂	direct (N × H)
8. C₂×C₄×#1057	C₄	direct (N × H)
17. C₂×C₄²×C₁₃	C₈	semidirect
19. C₂×C₄×Dic₁₃	C₈	semidirect
24. C₄×#1057	C₂×C₄	direct (N × H)
37. C₂×#1057	C₄²	direct (N × H)
39. C₄²×C₁₃	C₂×C₈	semidirect
41. C₄×Dic₁₃	C₂×C₈	semidirect
50. C₂×C₄×C₁₃	C₄×C₈	semidirect
54. C₂×Dic₁₃	C₄×C₈	semidirect
60. #1057	C₂×C₄²	direct (N × H)
75. C₄×C₁₃	C₂×C₄×C₈	semidirect
77. Dic₁₃	C₂×C₄×C₈	semidirect
89. C₂×C₄²	#1057	direct (N × H)
92. C₂×C₁₃	C₄²×C₈	semidirect
112. C₄²	C₂×#1057	direct (N × H)
113. C₁₃	C₂×C₄²×C₈	semidirect
124. C₂×C₄	C₄×#1057	direct (N × H)
138. C₄	C₂×C₄×#1057	direct (N × H)
139. C₂	C₄²×#1057	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×C₄×#1057	2
C₄²×#1057	2
C₂×C₄²×Dic₁₃	2
C₂×C₄²×C₈	13

Subgroups — C2×C42×#1057 #16796716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×#1057 #16796716