Subgroups — #15779353

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718096	C₂	1	4	yes	maximal
2	#13114996	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88058	—	2	2	no	—
7	C₂×#88058	C₂²	1	4	yes	—
8	#11718596	—	2	2	no	—
9	C₄×C₁₃×C₁₆	C₄	1	4	yes	complemented
10	#11718096	—	2	8	no	—
11	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃	—	2	4	no	—
15	C₂×C₄⋊Dic₁₃	—	4	4	no	—
16	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
17	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
18	C₂×C₁₃×C₁₆	Q₈	1	4	yes	—
19	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₃	Q₈	1	2	yes	—
21	#88058	—	2	8	no	—
22	#88058	—	4	4	no	—
23	#12905	—	13	1	no	maximal, Sylow
24	C₂²×C₄×C₁₃	D₈	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
26	C₂²×Dic₁₃	—	4	4	no	—
27	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₁₃	D₈	1	2	yes	—
29	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
30	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
32	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
33	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
35	C₄²×C₁₃	Q₁₆	1	2	yes	—
36	C₄⋊Dic₁₃	—	4	4	no	—
37	C₄⋊Dic₁₃	—	4	4	no	—
38	C₄⋊Dic₁₃	—	8	8	no	—
39	C₁₃×C₁₆	C₄⋊C₄	1	8	yes	complemented
40	C₂×#366	—	13	4	no	—
41	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
42	#1845	—	13	2	no	—
43	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃	#107	1	4	yes	—
45	C₂×C₄×C₁₃	D₁₆	1	4	yes	—
46	C₂×C₄×C₁₃	Q₃₂	1	4	yes	—
47	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×Dic₁₃	—	8	4	no	—
54	C₂×Dic₁₃	—	8	8	no	—
55	C₈×C₁₃	#107	1	4	yes	—
56	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
58	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
59	C₂×#107	—	13	4	no	—
60	C₂×#107	—	26	2	no	—
61	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
62	#384	—	26	2	no	—
63	#366	—	26	8	no	—
64	C₄×C₁₆	Dic₁₃	1	4	yes	complemented
65	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
66	C₂²×C₁₃	#493	1	1	yes	characteristic
67	C₂²×C₁₃	#500	1	1	yes	characteristic
68	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
69	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
70	C₄×C₁₃	#366	1	8	yes	complemented
71	C₄×C₁₃	#384	1	1	yes	characteristic
72	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
73	C₄×C₁₃	#494	1	1	yes	characteristic
74	C₄×C₁₃	#500	1	1	yes	characteristic
75	Dic₁₃	—	16	4	no	—
76	C₂²×C₈	D₅₂	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	—	26	4	no	—
79	C₂×C₄⋊C₄	—	52	4	no	—
80	C₂×C₄²	D₅₂	1	1	yes	characteristic
81	C₂×C₁₆	Dic₂₆	1	4	yes	—
82	C₂×C₁₆	D₅₂	1	4	yes	—
83	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
84	#107	—	26	8	no	—
85	#107	—	52	4	no	—
86	C₄×C₈	Dic₂₆	1	2	yes	—
87	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
88	C₂×C₁₃	#1845	1	1	yes	characteristic
89	C₂×C₁₃	C₂×#366	1	2	yes	—
90	C₂×C₁₃	#2246	1	1	yes	characteristic
91	C₂×C₁₃	#2247	1	1	yes	characteristic
92	C₂×C₁₃	#2252	1	2	yes	—
93	C₂²×C₄	—	52	4	no	—
94	C₂²×C₄	D₁₀₄	1	2	yes	—
95	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
96	C₂×C₈	D₁₀₄	1	2	yes	—
97	C₂×C₈	Dic₅₂	1	2	yes	—
98	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
99	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
100	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
101	C₂×C₈	C₂×D₅₂	1	2	yes	—
102	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
103	C₄⋊C₄	—	52	4	no	—
104	C₄⋊C₄	—	52	4	no	—
105	C₄⋊C₄	—	104	8	no	—
106	C₄²	Dic₅₂	1	2	yes	—
107	C₄²	C₄⋊Dic₁₃	1	2	yes	—
108	C₁₆	C₄⋊Dic₁₃	1	8	yes	complemented
109	C₁₃	#12905	1	1	yes	characteristic, complemented, Sylow
110	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
111	C₂×C₄	—	104	8	no	—
112	C₂×C₄	—	104	4	no	—
113	C₂×C₄	D₂₀₈	1	4	yes	—
114	C₂×C₄	Dic₁₀₄	1	4	yes	—
115	C₂×C₄	#88058	1	4	yes	—
116	C₂×C₄	#88123	1	1	yes	characteristic
117	C₂×C₄	#88128	1	1	yes	characteristic
118	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
122	C₈	#88058	1	4	yes	—
123	C₈	#88123	1	2	yes	—
124	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
125	C₂²	#11718343	1	1	yes	characteristic
126	C₂²	#11718356	1	1	yes	characteristic
127	C₂²	C₂×D₂₀₈	1	2	yes	—
128	C₂²	C₂×Dic₁₀₄	1	2	yes	—
129	C₂²	C₂×#88058	1	1	yes	characteristic
130	C₄	—	208	4	no	—
131	C₄	#11718096	1	8	yes	complemented
132	C₄	#11718343	1	1	yes	characteristic
133	C₄	#11718360	1	1	yes	characteristic
134	C₄	#11718596	1	1	yes	characteristic
135	C₄	C₂×#88058	1	1	yes	characteristic
136	C₂	#13114996	1	1	yes	characteristic
137	C₂	C₂×#11718096	1	2	yes	—
138	C₂	#13117086	1	1	yes	characteristic
139	C₂	#13117089	1	1	yes	characteristic
140	C₂	#13117111	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#12905
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
9. C₄×C₁₃×C₁₆	C₄	semidirect
39. C₁₃×C₁₆	C₄⋊C₄	semidirect
64. C₄×C₁₆	Dic₁₃	semidirect
70. C₄×C₁₃	#366	semidirect
108. C₁₆	C₄⋊Dic₁₃	semidirect
109. C₁₃	#12905	semidirect
131. C₄	#11718096	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718096	2
#13114996	2
C₂×C₄×C₁₃×C₁₆	2
#12905	13