Finite Group Databasebeta

Subgroups — #16789035

Order 3328.

350 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131059	C₂	1	1	yes	maximal, characteristic
2	#13131057	C₂	1	2	yes	maximal
3	#13131047	C₂	1	1	yes	maximal, characteristic
4	#13131045	C₂	1	2	yes	maximal
5	#13114590	C₂	1	1	yes	maximal, characteristic
6	C₂×#88106	C₂²	1	2	yes	—
7	C₂×#88106	C₂²	1	1	yes	characteristic
8	C₂²×#6092	C₂²	1	1	yes	characteristic
9	#11718783	C₂²	1	1	yes	characteristic
10	#11718658	C₂²	1	2	yes	—
11	#11718262	C₄	1	4	yes	complemented
12	#11718258	C₄	1	2	yes	complemented
13	#11718257	C₄	1	2	yes	complemented
14	#11718232	C₄	1	8	yes	complemented
15	#11718231	C₄	1	4	yes	complemented
16	C₄×#6092	C₄	1	4	yes	complemented
17	C₁₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
18	#11718152	C₄	1	1	yes	characteristic
19	#11718123	C₄	1	2	yes	—
20	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
21	C₁₃×#117	C₂×C₄	1	2	yes	—
22	#88180	C₂×C₄	1	1	yes	characteristic
23	C₂×#6099	C₂×C₄	1	2	yes	—
24	C₂²×#1055	C₂³	1	1	yes	characteristic
25	#88106	C₂×C₄	1	4	yes	—
26	#88106	C₂×C₄	1	4	yes	—
27	#88106	C₂×C₄	1	4	yes	—
28	C₂×#6092	C₂×C₄	1	2	yes	—
29	C₂×#6092	C₂×C₄	1	4	yes	—
30	C₂×#6092	C₂×C₄	1	2	yes	—
31	C₂×#6092	C₂×C₄	1	1	yes	characteristic
32	C₂×#6092	C₂×C₄	1	2	yes	—
33	C₂×#6092	C₂×C₄	1	1	yes	characteristic
34	#88055	C₂×C₄	1	2	yes	—
35	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
36	#88043	C₂×C₄	1	4	yes	—
37	C₄×#1055	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	C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₁₃×C₂²⋊C₄	—	2	2	no	—
44	C₄²×C₁₃	C₄²	1	4	yes	—
45	C₂²⋊Dic₁₃	—	2	1	no	—
46	C₄⋊Dic₁₃	—	2	1	no	—
47	C₄×Dic₁₃	C₄²	1	2	yes	—
48	#6099	—	2	2	no	—
49	#6099	—	2	2	no	—
50	C₂×#1055	C₄²	1	4	yes	—
51	C₂×#1055	C₂²×C₄	1	2	yes	—
52	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
53	C₂×#1055	C₂²×C₄	1	2	yes	—
54	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
55	#6092	C₄²	1	1	yes	characteristic
56	#6092	C₄²	1	2	yes	—
57	#6092	C₄²	1	1	yes	characteristic
58	#6092	C₄²	1	8	yes	—
59	#6092	C₄²	1	4	yes	—
60	#2110	—	13	2	no	—
61	#2110	—	13	1	no	—
62	#2108	—	13	2	no	—
63	#2108	—	13	1	no	—
64	#1747	—	13	1	no	—
65	C₂³×C₁₃	#131	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	4	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	2	yes	—
70	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
71	C₂×Dic₁₃	—	2	2	no	—
72	C₈×C₁₃	—	2	2	no	—
73	#1055	—	2	4	no	—
74	#1055	C₂×C₄²	1	1	yes	characteristic
75	#1055	C₂×C₄²	1	2	yes	—
76	#1055	C₂×C₄²	1	1	yes	characteristic
77	C₂×M₃₂	—	13	2	no	—
78	C₂×M₃₂	—	13	1	no	—
79	C₂²×C₁₆	—	13	1	no	—
80	#405	—	13	1	no	—
81	#405	—	13	2	no	—
82	#347	—	13	8	no	—
83	#347	—	13	4	no	—
84	#346	—	13	4	no	—
85	#346	—	13	2	no	—
86	C₄×C₁₆	—	13	4	no	—
87	C₄×C₁₆	—	13	2	no	—
88	#338	—	13	1	no	—
89	H(3,ℤ/4)	—	13	2	no	—
90	H(3,ℤ/4)	#261	1	1	yes	characteristic
91	C₂²×C₁₃	—	4	1	no	—
92	C₂²×C₁₃	#405	1	2	yes	—
93	C₂²×C₁₃	#405	1	1	yes	characteristic
94	C₄×C₁₃	—	4	4	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	2	no	—
99	C₂×M₁₆	—	13	2	no	—
100	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
101	C₂²×C₈	—	13	1	no	—
102	#117	—	13	1	no	—
103	#117	C₂×#261	1	2	yes	—
104	M₃₂	—	13	4	no	—
105	M₃₂	—	13	4	no	—
106	M₃₂	—	13	4	no	—
107	C₂×C₁₆	—	13	2	no	—
108	C₂×C₁₆	—	13	1	no	—
109	C₂×C₁₆	—	13	4	no	—
110	C₂×C₁₆	—	13	2	no	—
111	C₂×C₁₆	—	13	2	no	—
112	C₂×C₁₆	—	13	1	no	—
113	#97	—	13	2	no	—
114	#97	—	13	4	no	—
115	C₄×C₈	—	13	2	no	—
116	C₄×C₈	—	13	4	no	—
117	C₂×C₁₃	—	2	1	no	—
118	C₂×C₁₃	—	2	2	no	—
119	C₂×C₁₃	#1726	1	1	yes	characteristic
120	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
121	M₁₆	—	2	2	no	—
122	M₁₆	—	26	2	no	—
123	M₁₆	—	26	2	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	13	2	no	—
126	C₂×C₈	—	13	1	no	—
127	C₂×C₈	—	13	2	no	—
128	C₂×C₈	—	13	4	no	—
129	C₂×C₈	C₄×#261	1	2	yes	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	26	1	no	—
132	C₂²⋊C₄	—	2	2	no	—
133	C₂²⋊C₄	—	26	1	no	—
134	C₄²	—	13	2	no	—
135	C₄²	C₄×#261	1	4	yes	—
136	C₁₆	—	13	1	no	—
137	C₁₆	—	13	2	no	—
138	C₁₆	—	13	1	no	—
139	C₁₆	—	13	8	no	—
140	C₁₆	—	13	4	no	—
141	C₁₃	#12542	1	1	yes	characteristic, complemented, Sylow
142	C₂³	#88238	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	26	2	no	—
145	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×#261	1	2	yes	—
147	C₂×C₄	#88238	1	2	yes	—
148	C₂×C₄	#88240	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	13	1	no	—
151	C₈	—	13	2	no	—
152	C₈	—	13	1	no	—
153	C₈	—	26	4	no	—
154	C₂²	—	4	1	no	—
155	C₂²	#11719062	1	2	yes	—
156	C₂²	#11719072	1	1	yes	characteristic
157	C₄	—	4	4	no	—
158	C₄	—	52	2	no	—
159	C₄	C₄²×#261	1	1	yes	characteristic
160	C₄	#11719062	1	1	yes	characteristic
161	C₄	#11719072	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	2	1	no	—
164	C₂	#13130493	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#261
commutator	C₄×C₁₃	C₄³
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×H(3,ℤ/4)	C₄
socle	C₂×C₁₃	#1726
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×#1055	C₂³
hypercenter	H(3,ℤ/4)	#261
hypocenter	C₁₃	#12542
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
11. #11718262	C₄	semidirect
12. #11718258	C₄	semidirect
13. #11718257	C₄	semidirect
14. #11718232	C₄	semidirect
15. #11718231	C₄	semidirect
16. C₄×#6092	C₄	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
#13131059	2
#13131057	2
#13131047	2
#13131045	2
#13114590	2
#12542	13