Subgroups — C₂×#13113034 #15783409

Order 3328.

214 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×#11718091	C₂	1	2	yes	maximal
2	#13113034	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₄×C₁₃×C₁₆	C₄	1	2	yes	—
8	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
9	#11718091	—	2	2	no	—
10	#11718091	C₂²	1	4	yes	—
11	C₂×#88034	—	2	2	no	—
12	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
15	C₂×C₁₃×C₁₆	—	2	1	no	—
16	C₂×C₁₃×C₁₆	—	2	1	no	—
17	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	Q₈	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	Q₈	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₈	1	2	yes	—
25	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	#88034	—	2	4	no	—
27	C₂×#1422	—	13	1	no	maximal, Sylow
28	C₂²×C₄×C₁₃	—	2	1	no	—
29	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₁₃	C₁₆	1	2	yes	—
31	C₂×C₈×C₁₃	C₁₆	1	2	yes	—
32	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₁₃	—	2	1	no	—
40	C₄²×C₁₃	C₂×C₈	1	2	yes	—
41	C₁₃×C₁₆	—	2	2	no	—
42	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
43	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
44	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
45	C₁₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
46	C₂×M₆₄	—	13	2	no	—
47	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
48	#1422	—	13	4	no	—
49	C₂³×C₁₃	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₁₃	M₃₂	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
58	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₁₃	C₂×C₁₆	1	4	yes	—
61	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
63	C₂²×C₁₆	Dic₁₃	1	1	yes	characteristic
64	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
65	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
66	M₆₄	—	13	4	no	—
67	M₆₄	—	26	2	no	—
68	C₂×C₃₂	—	26	2	no	—
69	C₄×C₁₆	Dic₁₃	1	2	yes	—
70	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
71	C₂²×C₁₃	—	2	2	no	—
72	C₂²×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
73	C₂²×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
74	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
75	C₄×C₁₃	—	2	4	no	—
76	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
77	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
78	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
80	C₂²×C₈	#1055	1	1	yes	characteristic
81	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
82	C₂×C₄²	#1055	1	1	yes	characteristic
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	Dic₂₆	1	1	yes	characteristic
86	C₂×C₁₆	Dic₂₆	1	1	yes	characteristic
87	C₂×C₁₆	D₅₂	1	1	yes	characteristic
88	C₂×C₁₆	D₅₂	1	1	yes	characteristic
89	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
90	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
91	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
92	C₄×C₈	#1055	1	2	yes	—
93	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
94	C₃₂	—	26	4	no	—
95	C₂×C₁₃	—	2	1	no	—
96	C₂×C₁₃	—	2	1	no	—
97	C₂×C₁₃	#1422	1	2	yes	complemented
98	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
101	C₂×C₈	#6090	1	2	yes	—
102	C₂×C₈	#6090	1	2	yes	—
103	C₂×C₈	C₂×#1055	1	1	yes	characteristic
104	C₂×C₈	C₂×#1055	1	2	yes	—
105	C₂×C₈	#6099	1	1	yes	characteristic
106	C₂×C₈	#6099	1	1	yes	characteristic
107	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
108	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
109	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
110	C₄²	—	2	1	no	—
111	C₄²	C₂×#1055	1	2	yes	—
112	C₁₆	—	2	2	no	—
113	C₁₆	C₄⋊Dic₁₃	1	1	yes	characteristic
114	C₁₆	C₄⋊Dic₁₃	1	1	yes	characteristic
115	C₁₆	C₂×Dic₂₆	1	1	yes	characteristic
116	C₁₆	C₂×D₅₂	1	1	yes	characteristic
117	C₁₃	C₂×#1422	1	1	yes	characteristic, complemented, Sylow
118	C₂³	#88052	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
122	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
123	C₂×C₄	C₂×#6090	1	1	yes	characteristic
124	C₂×C₄	C₂×#6090	1	1	yes	characteristic
125	C₂×C₄	#88052	1	1	yes	characteristic
126	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
127	C₈	C₄⋊#1055	1	1	yes	characteristic
128	C₈	C₄⋊#1055	1	1	yes	characteristic
129	C₈	C₂×#6090	1	4	yes	—
130	C₈	C₂×#6099	1	1	yes	characteristic
131	C₈	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
132	C₂²	—	2	2	no	—
133	C₂²	C₄⋊#6090	1	1	yes	characteristic
134	C₂²	C₄⋊#6090	1	1	yes	characteristic
135	C₂²	C₂×#88052	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	C₄⋊#6090	1	1	yes	characteristic
138	C₄	C₄⋊#6090	1	1	yes	characteristic
139	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
140	C₄	C₂²×#6090	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#13113034	1	2	yes	direct product factor
144	C₂	C₂×C₄⋊#6090	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₅₂
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₄⋊C₁₆
prime radical	C₂³×C₁₃	M₃₂
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₂×#1422
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13113034	C₂	direct (N × H)
97. C₂×C₁₃	#1422	semidirect
117. C₁₃	C₂×#1422	semidirect
143. C₂	#13113034	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×#11718091	2
#13113034	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#1422	13

Subgroups — C2×#13113034 #15783409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113034 #15783409