Subgroups — #15777751

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116441	C₂	1	1	yes	maximal, characteristic
2	#13116440	C₂	1	2	yes	maximal
3	#13116432	C₂	1	2	yes	maximal
4	#13114591	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₁₃×#405	C₂²	1	1	yes	characteristic
9	#11718762	C₂²	1	2	yes	—
10	C₂×#88052	C₂²	1	2	yes	—
11	C₁₃×#346	C₄	1	2	yes	complemented
12	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
13	#11718152	C₄	1	1	yes	characteristic
14	#11718149	C₄	1	1	yes	characteristic
15	#11718123	C₄	1	2	yes	—
16	#11718104	C₄	1	8	yes	complemented
17	#11718094	C₄	1	4	yes	complemented
18	C₁₆×Dic₁₃	C₄	1	4	yes	complemented
19	#11718058	C₄	1	4	yes	complemented
20	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
23	#88180	C₂×C₄	1	2	yes	—
24	C₂×#6099	C₂×C₄	1	2	yes	—
25	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₁₃×#97	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	#88055	C₂×C₄	1	4	yes	—
34	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
35	#88052	C₂×C₄	1	4	yes	—
36	#88052	C₂×C₄	1	4	yes	—
37	C₂×#6090	C₂×C₄	1	4	yes	—
38	#12542	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₁₃×M₁₆	—	2	2	no	—
41	C₂×C₈×C₁₃	C₄²	1	2	yes	—
42	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
44	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₃×C₄⋊C₄	—	2	1	no	—
47	C₁₃×C₂²⋊C₄	—	2	1	no	—
48	C₄²×C₁₃	C₄²	1	2	yes	—
49	C₂²⋊Dic₁₃	—	2	2	no	—
50	C₄⋊Dic₁₃	—	2	2	no	—
51	C₄×Dic₁₃	C₄²	1	4	yes	—
52	#6099	—	2	2	no	—
53	#6099	—	2	2	no	—
54	C₂×#1055	C₄²	1	4	yes	—
55	C₁₃×C₁₆	C₄²	1	2	yes	—
56	C₁₃×C₁₆	C₄²	1	1	yes	characteristic
57	C₁₃×C₁₆	C₄²	1	4	yes	—
58	C₁₃×C₁₆	C₄²	1	1	yes	characteristic
59	#6090	C₄²	1	8	yes	—
60	#2110	—	13	2	no	—
61	#2110	—	13	1	no	—
62	#2108	—	13	2	no	—
63	#2108	D₁₃	1	1	yes	characteristic
64	#1747	—	13	1	no	—
65	C₂³×C₁₃	#131	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#131	1	2	yes	—
71	C₂×Dic₁₃	—	2	4	no	—
72	C₈×C₁₃	—	2	2	no	—
73	C₈×C₁₃	C₂×C₄²	1	2	yes	—
74	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
76	#1055	—	2	4	no	—
77	C₂×M₃₂	—	13	2	no	—
78	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
79	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
80	#405	—	13	2	no	—
81	#405	C₂×D₁₃	1	1	yes	characteristic
82	#347	—	13	8	no	—
83	#347	—	13	4	no	—
84	#346	—	13	4	no	—
85	#346	Dic₁₃	1	2	yes	complemented
86	C₄×C₁₆	—	13	4	no	—
87	C₄×C₁₆	Dic₁₃	1	2	yes	complemented
88	#338	—	13	1	no	—
89	H(3,ℤ/4)	—	13	1	no	—
90	H(3,ℤ/4)	—	13	2	no	—
91	C₂²×C₁₃	—	4	1	no	—
92	C₂²×C₁₃	#405	1	1	yes	characteristic
93	C₂²×C₁₃	#405	1	2	yes	—
94	C₄×C₁₃	—	4	2	no	—
95	C₄×C₁₃	C₄³	1	1	yes	characteristic
96	C₄×C₁₃	#405	1	1	yes	characteristic
97	C₄×C₁₃	#405	1	2	yes	—
98	Dic₁₃	—	4	4	no	—
99	C₂×M₁₆	—	13	2	no	—
100	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
102	#117	—	13	2	no	—
103	#117	C₄×D₁₃	1	1	yes	characteristic
104	M₃₂	—	13	4	no	—
105	M₃₂	—	13	4	no	—
106	M₃₂	C₄×D₁₃	1	4	yes	—
107	C₂×C₁₆	—	13	4	no	—
108	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
109	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
110	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
111	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
112	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
113	#97	—	13	4	no	—
114	#97	C₄×D₁₃	1	2	yes	—
115	C₄×C₈	—	13	4	no	—
116	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
117	C₂×C₁₃	—	2	2	no	—
118	C₂×C₁₃	—	2	1	no	—
119	C₂×C₁₃	#1726	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
121	M₁₆	—	2	2	no	—
122	M₁₆	—	26	2	no	—
123	M₁₆	—	26	2	no	—
124	C₂×C₈	—	13	4	no	—
125	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
126	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
127	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
129	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	26	2	no	—
132	C₂²⋊C₄	—	2	1	no	—
133	C₂²⋊C₄	—	26	2	no	—
134	C₄²	—	13	4	no	—
135	C₄²	C₄×Dic₁₃	1	2	yes	—
136	C₁₆	—	13	8	no	—
137	C₁₆	C₄×Dic₁₃	1	1	yes	characteristic
138	C₁₆	C₄×Dic₁₃	1	2	yes	—
139	C₁₆	C₄×Dic₁₃	1	4	yes	—
140	C₁₆	C₄×Dic₁₃	1	1	yes	characteristic
141	C₁₃	#12542	1	1	yes	characteristic, complemented, Sylow
142	C₂³	#88155	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	26	4	no	—
145	C₂×C₄	C₄²×D₁₃	1	2	yes	—
146	C₂×C₄	#88155	1	2	yes	—
147	C₂×C₄	#88155	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	26	4	no	—
151	C₈	C₄²×D₁₃	1	2	yes	—
152	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
153	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	#11718350	1	2	yes	—
156	C₂²	#11718762	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	—	52	4	no	—
159	C₄	#11718350	1	2	yes	—
160	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
161	C₄	#11718762	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	2	no	—
164	C₂	#13114522	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₁₃
commutator	C₄×C₁₃	C₄³
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2108	C₂
socle	C₂×C₁₃	#1726
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2108	D₁₃
hypocenter	C₁₃	#12542
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
11. C₁₃×#346	C₄	semidirect
12. C₄×C₁₃×C₁₆	C₄	semidirect
16. #11718104	C₄	semidirect
17. #11718094	C₄	semidirect
18. C₁₆×Dic₁₃	C₄	semidirect
19. #11718058	C₄	semidirect
85. #346	Dic₁₃	semidirect
87. C₄×C₁₆	Dic₁₃	semidirect
141. C₁₃	#12542	semidirect

Maximal subgroups

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

subgroup	index
#13116441	2
#13116440	2
#13116432	2
#13114591	2
C₁₃×#2108	2
#12542	13