Subgroups — C₄×#11718094 #15777740

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718094	C₂	1	4	yes	maximal
2	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
9	C₂²×#6090	C₂²	1	2	yes	—
10	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₄×C₁₃×C₁₆	C₄	1	4	yes	complemented
13	#11718094	C₄	1	16	yes	direct product factor
14	C₄×#6090	C₄	1	4	yes	complemented
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₈	1	4	yes	—
19	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
20	C₂²×#1055	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	8	yes	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₈×Dic₁₃	C₈	1	8	yes	—
26	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
27	C₂×#6090	C₂×C₄	1	4	yes	—
28	C₂×#6090	C₂×C₄	1	8	yes	—
29	C₄×#1055	C₈	1	4	yes	—
30	C₄×#346	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₁₃	C₄²	1	2	yes	—
35	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
38	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₄²	1	2	yes	—
40	C₄²×C₁₃	C₂×C₈	1	2	yes	—
41	C₄×Dic₁₃	—	2	8	no	—
42	C₄×Dic₁₃	C₄²	1	4	yes	—
43	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
44	C₂×#1055	C₄²	1	4	yes	—
45	C₂×#1055	C₂×C₈	1	4	yes	—
46	C₂×#1055	C₂×C₈	1	4	yes	—
47	C₁₃×C₁₆	C₄²	1	8	yes	complemented
48	#6090	C₄²	1	8	yes	complemented
49	C₂×#346	—	13	4	no	—
50	C₂×C₄×C₁₆	—	13	1	no	—
51	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
52	C₄²×C₈	—	13	1	no	—
53	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
55	C₂×C₄×C₁₃	M₃₂	1	8	yes	—
56	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
60	C₂×Dic₁₃	—	2	8	no	—
61	C₂×Dic₁₃	C₄×C₈	1	8	yes	—
62	C₈×C₁₃	C₄×C₈	1	4	yes	—
63	C₈×C₁₃	C₂×C₄²	1	2	yes	—
64	C₈×C₁₃	C₂×C₄²	1	2	yes	—
65	#1055	C₄×C₈	1	8	yes	—
66	C₂²×C₁₆	—	13	2	no	—
67	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
68	C₂×C₄×C₈	—	13	2	no	—
69	C₂×C₄×C₈	—	13	2	no	—
70	C₂×C₄×C₈	—	13	1	no	—
71	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
72	C₄³	—	13	1	no	—
73	#346	—	13	16	no	—
74	C₄×C₁₆	—	13	4	no	—
75	C₄×C₁₆	Dic₁₃	1	4	yes	complemented
76	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
77	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
78	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
79	C₄×C₁₃	#346	1	8	yes	complemented
80	C₄×C₁₃	C₄³	1	1	yes	characteristic
81	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
82	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
83	Dic₁₃	—	2	8	no	—
84	C₂²×C₈	—	13	2	no	—
85	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
86	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
87	C₂×C₄²	—	13	2	no	—
88	C₂×C₄²	—	13	4	no	—
89	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
90	C₂×C₁₆	—	13	4	no	—
91	C₂×C₁₆	—	13	8	no	—
92	C₂×C₁₆	C₄×D₁₃	1	8	yes	—
93	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
94	C₄×C₈	—	13	8	no	—
95	C₄×C₈	—	13	8	no	—
96	C₄×C₈	—	13	4	no	—
97	C₄×C₈	C₄×D₁₃	1	2	yes	—
98	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
99	C₂×C₁₃	C₄²×C₈	1	1	yes	characteristic
100	C₂×C₁₃	C₂×#346	1	2	yes	—
101	C₂×C₁₃	C₄×M₃₂	1	4	yes	—
102	C₂²×C₄	—	13	4	no	—
103	C₂²×C₄	C₈×D₁₃	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
105	C₂×C₈	—	13	4	no	—
106	C₂×C₈	—	13	4	no	—
107	C₂×C₈	—	13	4	no	—
108	C₂×C₈	C₈×D₁₃	1	4	yes	—
109	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
110	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
111	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
112	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
113	C₄²	—	13	4	no	—
114	C₄²	—	13	4	no	—
115	C₄²	—	26	8	no	—
116	C₄²	C₈×D₁₃	1	2	yes	—
117	C₄²	C₄×Dic₁₃	1	2	yes	—
118	C₁₆	—	13	8	no	—
119	C₁₆	C₄×Dic₁₃	1	8	yes	complemented
120	C₁₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
121	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
122	C₂×C₄	—	13	8	no	—
123	C₂×C₄	—	26	8	no	—
124	C₂×C₄	#88038	1	8	yes	—
125	C₂×C₄	C₈×Dic₁₃	1	4	yes	—
126	C₂×C₄	C₄²×D₁₃	1	2	yes	—
127	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
128	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
130	C₈	—	13	8	no	—
131	C₈	C₈×Dic₁₃	1	4	yes	—
132	C₈	C₄²×D₁₃	1	2	yes	—
133	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
134	C₂²	C₄×C₈×D₁₃	1	2	yes	—
135	C₂²	C₂×#88038	1	4	yes	—
136	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
137	C₄	—	26	8	no	—
138	C₄	#11718094	1	8	yes	direct product factor
139	C₄	C₄×C₈×D₁₃	1	2	yes	—
140	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
141	C₄	C₂×C₈×Dic₁₃	1	1	yes	characteristic
142	C₂	C₄×C₈×Dic₁₃	1	1	yes	characteristic
143	C₂	C₂×#11718094	1	2	yes	—
144	C₂	C₄×#88038	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₄²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₄×#346
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₄×C₁₃×C₁₆	C₄	semidirect
13. #11718094	C₄	direct (N × H)
14. C₄×#6090	C₄	semidirect
47. C₁₃×C₁₆	C₄²	semidirect
48. #6090	C₄²	semidirect
75. C₄×C₁₆	Dic₁₃	semidirect
79. C₄×C₁₃	#346	semidirect
119. C₁₆	C₄×Dic₁₃	semidirect
120. C₁₃	C₄×#346	semidirect
138. C₄	#11718094	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718094	2
C₂×C₄×#6090	2
C₄×C₈×Dic₁₃	2
C₂×C₄×C₁₃×C₁₆	2
C₄×#346	13

Subgroups — C4×#11718094 #15777740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718094 #15777740