Subgroups — C₂×#13117111 #16018396

Order 3328.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	2	yes	maximal
2	#13117111	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718096	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	2	no	—
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	—	2	2	no	—
9	C₂×#88058	C₂²	1	4	yes	—
10	#11718343	—	2	4	no	—
11	#11718343	C₂²	1	8	yes	—
12	C₄×C₁₃×C₁₆	C₂²	1	4	yes	—
13	#11718096	C₂²	1	16	yes	—
14	C₂²×Dic₂₆	—	4	2	no	—
15	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	—	2	4	no	—
19	C₂×C₄⋊Dic₁₃	—	4	2	no	—
20	#88123	—	2	8	no	—
21	#88123	—	4	4	no	—
22	C₂×C₁₃×C₁₆	Q₈	1	8	yes	complemented
23	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
24	C₄×C₈×C₁₃	D₄	1	2	yes	—
25	C₄×C₈×C₁₃	C₂³	1	2	yes	—
26	#88058	—	2	8	no	—
27	#88058	—	2	4	no	—
28	#88058	C₂³	1	8	yes	—
29	C₂×#2252	—	13	1	no	maximal, Sylow
30	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₁₃	—	4	4	no	—
34	C₂×Dic₂₆	—	4	8	no	—
35	C₂×Dic₂₆	—	8	8	no	—
36	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
37	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
40	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₄²×C₁₃	D₈	1	2	yes	—
42	C₄²×C₁₃	C₂×D₄	1	2	yes	—
43	C₄⋊Dic₁₃	—	2	8	no	—
44	C₄⋊Dic₁₃	—	4	8	no	—
45	C₄⋊Dic₁₃	—	4	4	no	—
46	C₁₃×C₁₆	C₂×Q₈	1	8	yes	complemented
47	C₂×#500	—	13	2	no	—
48	#2252	—	13	8	no	—
49	C₂×#366	—	13	4	no	—
50	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
51	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	D₁₆	1	4	yes	—
53	C₂×C₄×C₁₃	Q₃₂	1	4	yes	—
54	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
57	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
59	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
60	C₂×Dic₁₃	—	4	8	no	—
61	C₂×Dic₁₃	—	8	8	no	—
62	Dic₂₆	—	8	16	no	—
63	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
64	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
65	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
66	C₂×C₄⋊Q₈	—	26	2	no	—
67	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
68	#500	—	13	8	no	—
69	#500	—	26	4	no	—
70	C₂×#107	—	13	4	no	—
71	C₂×#107	—	26	2	no	—
72	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
73	#366	—	13	16	no	—
74	C₄×C₁₆	C₂×D₁₃	1	4	yes	—
75	C₂²×C₁₃	#500	1	2	yes	—
76	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
77	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
78	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
79	C₄×C₁₃	#500	1	2	yes	—
80	C₄×C₁₃	C₂×D₁₆	1	4	yes	—
81	C₄×C₁₃	C₂×Q₃₂	1	4	yes	—
82	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
84	Dic₁₃	—	8	8	no	—
85	C₂²×Q₈	—	52	2	no	—
86	C₂²×C₈	D₅₂	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
88	C₄⋊Q₈	—	26	8	no	—
89	C₄⋊Q₈	—	52	4	no	—
90	C₂×C₄⋊C₄	—	26	4	no	—
91	C₂×C₄⋊C₄	—	52	2	no	—
92	C₂×C₄²	D₅₂	1	1	yes	characteristic
93	C₂×C₁₆	Dic₂₆	1	8	yes	complemented
94	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
95	#107	—	13	8	no	—
96	#107	—	26	4	no	—
97	#107	—	26	8	no	—
98	C₄×C₈	D₅₂	1	2	yes	—
99	C₄×C₈	C₂²×D₁₃	1	2	yes	—
100	C₂×C₁₃	#2252	1	4	yes	complemented
101	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
102	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
103	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
104	C₂×Q₈	—	52	8	no	—
105	C₂×Q₈	—	104	8	no	—
106	C₂²×C₄	—	52	4	no	—
107	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
108	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
109	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
110	C₂×C₈	Dic₅₂	1	4	yes	—
111	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
112	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
113	C₂×C₈	C₂×D₅₂	1	2	yes	—
114	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
115	C₄⋊C₄	—	26	8	no	—
116	C₄⋊C₄	—	52	4	no	—
117	C₄⋊C₄	—	52	8	no	—
118	C₄²	D₁₀₄	1	2	yes	—
119	C₄²	C₂×D₅₂	1	2	yes	—
120	C₁₆	C₂×Dic₂₆	1	8	yes	complemented
121	C₁₃	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
122	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
123	Q₈	—	104	16	no	—
124	C₂×C₄	—	52	8	no	—
125	C₂×C₄	—	104	8	no	—
126	C₂×C₄	D₂₀₈	1	4	yes	—
127	C₂×C₄	Dic₁₀₄	1	4	yes	—
128	C₂×C₄	#88123	1	2	yes	—
129	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
130	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
131	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
132	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
133	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
134	C₈	#88123	1	2	yes	—
135	C₈	C₂×Dic₅₂	1	4	yes	—
136	C₈	C₂²×Dic₂₆	1	2	yes	—
137	C₂²	#11718343	1	2	yes	—
138	C₂²	C₂×D₂₀₈	1	2	yes	—
139	C₂²	C₂×Dic₁₀₄	1	2	yes	—
140	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
141	C₄	—	104	8	no	—
142	C₄	#11718343	1	2	yes	—
143	C₄	C₂×D₂₀₈	1	4	yes	—
144	C₄	C₂×Dic₁₀₄	1	4	yes	—
145	C₄	C₂×#88123	1	1	yes	characteristic
146	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
147	C₂	#13117111	1	4	yes	direct product factor
148	C₂	C₂×#11718343	1	1	yes	characteristic
149	C₂	C₂²×D₂₀₈	1	1	yes	characteristic
150	C₂	C₂²×Dic₁₀₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2252
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13117111	C₂	direct (N × H)
22. C₂×C₁₃×C₁₆	Q₈	semidirect
46. C₁₃×C₁₆	C₂×Q₈	semidirect
93. C₂×C₁₆	Dic₂₆	semidirect
100. C₂×C₁₃	#2252	semidirect
120. C₁₆	C₂×Dic₂₆	semidirect
121. C₁₃	C₂×#2252	semidirect
147. C₂	#13117111	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×#11718343	2
#13117111	2
C₂×#11718096	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2252	13

Subgroups — C2×#13117111 #16018396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117111 #16018396