Subgroups — #15779356

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718097	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	#11718097	—	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	#12906	—	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×#367	—	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	SD₃₂	1	8	yes	—
46	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×Dic₁₃	—	8	4	no	—
53	C₂×Dic₁₃	—	8	8	no	—
54	C₈×C₁₃	#107	1	4	yes	—
55	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
57	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
58	C₂×#107	—	13	4	no	—
59	C₂×#107	—	26	2	no	—
60	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
61	#384	—	26	2	no	—
62	#367	—	26	8	no	—
63	C₄×C₁₆	Dic₁₃	1	4	yes	complemented
64	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
65	C₂²×C₁₃	#493	1	1	yes	characteristic
66	C₂²×C₁₃	#500	1	1	yes	characteristic
67	C₂²×C₁₃	C₂×SD₃₂	1	4	yes	—
68	C₄×C₁₃	#367	1	8	yes	complemented
69	C₄×C₁₃	#384	1	1	yes	characteristic
70	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
71	C₄×C₁₃	#494	1	1	yes	characteristic
72	C₄×C₁₃	#500	1	1	yes	characteristic
73	Dic₁₃	—	16	4	no	—
74	C₂²×C₈	D₅₂	1	1	yes	characteristic
75	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	26	4	no	—
77	C₂×C₄⋊C₄	—	52	4	no	—
78	C₂×C₄²	D₅₂	1	1	yes	characteristic
79	C₂×C₁₆	Dic₂₆	1	4	yes	—
80	C₂×C₁₆	D₅₂	1	4	yes	—
81	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
82	#107	—	26	8	no	—
83	#107	—	52	4	no	—
84	C₄×C₈	Dic₂₆	1	2	yes	—
85	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
86	C₂×C₁₃	#1845	1	1	yes	characteristic
87	C₂×C₁₃	C₂×#367	1	2	yes	—
88	C₂×C₁₃	#2248	1	2	yes	—
89	C₂×C₁₃	#2254	1	2	yes	—
90	C₂²×C₄	—	52	4	no	—
91	C₂²×C₄	D₁₀₄	1	2	yes	—
92	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
93	C₂×C₈	D₁₀₄	1	2	yes	—
94	C₂×C₈	Dic₅₂	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
96	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
97	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
98	C₂×C₈	C₂×D₅₂	1	2	yes	—
99	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
100	C₄⋊C₄	—	52	4	no	—
101	C₄⋊C₄	—	52	4	no	—
102	C₄⋊C₄	—	104	8	no	—
103	C₄²	Dic₅₂	1	2	yes	—
104	C₄²	C₄⋊Dic₁₃	1	2	yes	—
105	C₁₆	C₄⋊Dic₁₃	1	8	yes	complemented
106	C₁₃	#12906	1	1	yes	characteristic, complemented, Sylow
107	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
108	C₂×C₄	—	104	8	no	—
109	C₂×C₄	—	104	4	no	—
110	C₂×C₄	#88040	1	8	yes	—
111	C₂×C₄	#88058	1	4	yes	—
112	C₂×C₄	#88123	1	1	yes	characteristic
113	C₂×C₄	#88128	1	1	yes	characteristic
114	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
118	C₈	#88058	1	4	yes	—
119	C₈	#88123	1	2	yes	—
120	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
121	C₂²	#11718343	1	1	yes	characteristic
122	C₂²	#11718356	1	1	yes	characteristic
123	C₂²	C₂×#88040	1	4	yes	—
124	C₂²	C₂×#88058	1	1	yes	characteristic
125	C₄	—	208	4	no	—
126	C₄	#11718097	1	8	yes	complemented
127	C₄	#11718343	1	1	yes	characteristic
128	C₄	#11718360	1	1	yes	characteristic
129	C₄	#11718596	1	1	yes	characteristic
130	C₄	C₂×#88058	1	1	yes	characteristic
131	C₂	#13114996	1	1	yes	characteristic
132	C₂	C₂×#11718097	1	2	yes	—
133	C₂	#13117092	1	2	yes	—
134	C₂	#13117118	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	#12906
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
63. C₄×C₁₆	Dic₁₃	semidirect
68. C₄×C₁₃	#367	semidirect
105. C₁₆	C₄⋊Dic₁₃	semidirect
106. C₁₃	#12906	semidirect
126. C₄	#11718097	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718097	2
#13114996	2
C₂×C₄×C₁₃×C₁₆	2
#12906	13