Subgroups — #15774561

Order 3328.

242 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	#13113595	C₂	1	2	yes	maximal
2	#13112908	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₁₆	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	#11718585	—	2	2	no	—
7	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
8	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
10	#11718050	—	2	2	no	—
11	#11718050	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
15	C₂×#6099	—	4	2	no	—
16	C₂×C₁₃×C₁₆	—	2	4	no	—
17	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₄⋊#1055	—	2	4	no	—
24	C₄⋊#1055	—	4	2	no	—
25	#12022	—	13	1	no	maximal, Sylow
26	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
29	C₂×C₈×C₁₃	—	2	2	no	—
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₁₃	D₈	1	1	yes	characteristic
37	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	#6099	—	8	4	no	—
40	C₂×#1055	—	4	4	no	—
41	C₁₃×C₁₆	—	2	2	no	—
42	C₁₃×C₁₆	M₁₆	1	4	yes	complemented
43	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
44	#1567	—	13	2	no	—
45	#1371	—	13	4	no	—
46	C₂³×C₁₃	#107	1	1	yes	characteristic
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	1	no	—
49	C₂×C₄×C₁₃	—	2	1	no	—
50	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
51	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
53	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
54	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₁₃	—	2	1	no	—
59	C₈×C₁₃	—	2	1	no	—
60	C₈×C₁₃	#108	1	2	yes	—
61	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
62	#1055	—	8	4	no	—
63	C₂²×C₁₆	Dic₁₃	1	2	yes	—
64	#423	—	26	2	no	—
65	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
66	C₄×C₁₆	Dic₁₃	1	1	yes	characteristic
67	C₄×C₁₆	Dic₁₃	1	1	yes	characteristic
68	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
69	#335	—	13	4	no	—
70	#335	—	26	2	no	—
71	C₂²×C₁₃	—	2	1	no	—
72	C₂²×C₁₃	—	2	1	no	—
73	C₂²×C₁₃	#366	1	2	yes	—
74	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
75	C₄×C₁₃	—	2	1	no	—
76	C₄×C₁₃	—	2	1	no	—
77	C₄×C₁₃	#366	1	2	yes	—
78	C₄×C₁₃	#368	1	2	yes	—
79	C₄×C₁₃	#423	1	1	yes	characteristic
80	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
81	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
82	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
83	C₂×M₁₆	—	52	2	no	—
84	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
85	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
86	C₂×C₄²	D₅₂	1	1	yes	characteristic
87	C₂×C₁₆	—	2	4	no	—
88	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
89	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
90	C₄⋊C₈	—	26	4	no	—
91	C₄⋊C₈	—	52	2	no	—
92	C₄×C₈	Dic₂₆	1	1	yes	characteristic
93	C₄×C₈	D₅₂	1	1	yes	characteristic
94	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
95	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
96	C₂×C₁₃	—	2	2	no	—
97	C₂×C₁₃	#1567	1	1	yes	characteristic
98	C₂×C₁₃	C₂×#366	1	1	yes	characteristic
99	C₂×C₁₃	C₂×#368	1	1	yes	characteristic
100	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
101	C₂²×C₄	Dic₅₂	1	1	yes	characteristic
102	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
103	M₁₆	—	104	4	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	52	4	no	—
107	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
108	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
109	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
110	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
111	C₄²	D₁₀₄	1	1	yes	characteristic
112	C₄²	Dic₅₂	1	1	yes	characteristic
113	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
114	C₄²	C₂×D₅₂	1	1	yes	characteristic
115	C₁₆	—	2	2	no	—
116	C₁₆	#6099	1	4	yes	complemented
117	C₁₃	#12022	1	1	yes	characteristic, complemented, Sylow
118	C₂³	#88058	1	1	yes	characteristic
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	D₂₀₈	1	2	yes	—
123	C₂×C₄	Dic₁₀₄	1	2	yes	—
124	C₂×C₄	#88058	1	1	yes	characteristic
125	C₂×C₄	#88058	1	1	yes	characteristic
126	C₂×C₄	#88058	1	1	yes	characteristic
127	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
128	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	104	4	no	—
133	C₈	#88059	1	2	yes	—
134	C₈	C₂×#6099	1	2	yes	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	#11718096	1	2	yes	—
138	C₂²	C₂×#88058	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	#11718096	1	2	yes	—
142	C₄	#11718098	1	2	yes	—
143	C₄	C₂×D₂₀₈	1	1	yes	characteristic
144	C₄	C₂×Dic₁₀₄	1	1	yes	characteristic
145	C₄	#11718585	1	1	yes	characteristic
146	C₄	C₂×#88059	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#13113595	1	1	yes	characteristic
149	C₂	C₂×#11718096	1	1	yes	characteristic
150	C₂	C₂×#11718098	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	#107
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#12022
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ 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. #13112908	C₂	semidirect
42. C₁₃×C₁₆	M₁₆	semidirect
116. C₁₆	#6099	semidirect
117. C₁₃	#12022	semidirect

Maximal subgroups

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

subgroup	index
#13113595	2
#13112908	2
C₂×C₄×C₁₃×C₁₆	2
#12022	13