Finite Group Databasebeta

Subgroups — #13127986

Order 1664.

350 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 113 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	#11719476	C₂	1	2	yes	maximal, complemented
2	C₁₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718856	C₂	1	2	yes	maximal, complemented
4	D₁₃×Q₃₂	C₂	1	2	yes	maximal, complemented
5	#11718576	C₂	1	4	yes	maximal, complemented
6	#11718573	C₂	1	2	yes	maximal, complemented
7	C₂×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718562	C₂	1	1	yes	maximal, characteristic, complemented
9	#88257	—	2	2	no	—
10	C₁₃×#135	C₂²	1	2	yes	—
11	#88205	—	2	2	no	—
12	Q₁₆×D₁₃	C₂²	1	4	yes	—
13	#88169	—	2	4	no	—
14	#88166	C₂²	1	4	yes	—
15	C₂×Dic₅₂	C₂²	1	2	yes	—
16	#88155	C₂²	1	1	yes	characteristic
17	C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
18	C₁₃×SD₃₂	C₂²	1	2	yes	—
19	C₁₃×D₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
21	#88069	C₂²	1	4	yes	—
22	#88067	C₂²	1	4	yes	—
23	#88052	C₂²	1	1	yes	characteristic
24	Dic₁₀₄	C₂²	1	2	yes	—
25	Dic₁₀₄	C₂²	1	2	yes	—
26	#88038	C₂²	1	2	yes	—
27	C₁₆×D₁₃	C₂²	1	2	yes	—
28	C₁₃×#41	—	2	2	no	—
29	Q₈×D₁₃	—	2	4	no	—
30	#6129	—	2	4	no	—
31	#6129	—	4	4	no	—
32	#6127	—	4	2	no	—
33	#6127	D₄	1	1	yes	characteristic
34	C₂×Dic₂₆	—	2	2	no	—
35	C₂×Dic₂₆	—	4	2	no	—
36	C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₁₃×SD₁₆	—	2	2	no	—
38	C₁₃×D₈	C₂³	1	2	yes	—
39	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	#6107	—	2	4	no	—
41	#6105	—	2	4	no	—
42	#6099	D₄	1	1	yes	characteristic
43	Dic₅₂	—	2	2	no	—
44	Dic₅₂	C₂³	1	4	yes	—
45	#6094	D₄	1	2	yes	—
46	C₈×D₁₃	C₂³	1	2	yes	—
47	C₁₃×C₁₆	C₂³	1	1	yes	characteristic
48	C₁₃×C₁₆	C₂³	1	1	yes	characteristic
49	#6090	C₂³	1	2	yes	—
50	#3417	—	13	1	no	maximal, Sylow
51	Q₈×C₁₃	—	2	2	no	—
52	D₄×C₁₃	—	2	2	no	—
53	D₄×C₁₃	—	4	2	no	—
54	C₂×C₄×C₁₃	—	4	2	no	—
55	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²⋊D₁₃	—	4	4	no	—
57	C₂²⋊D₁₃	D₈	1	2	yes	—
58	C₂×Dic₁₃	—	4	2	no	—
59	C₂×Dic₁₃	—	4	4	no	—
60	D₅₂	D₈	1	1	yes	characteristic
61	C₄×D₁₃	—	4	4	no	—
62	C₄×D₁₃	C₂×D₄	1	2	yes	—
63	Dic₂₆	—	2	4	no	—
64	Dic₂₆	—	4	2	no	—
65	Dic₂₆	—	4	4	no	—
66	Dic₂₆	D₈	1	1	yes	characteristic
67	C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₈×C₁₃	C₂⁴	1	1	yes	characteristic
69	#1055	C₂×D₄	1	2	yes	—
70	#578	—	13	2	no	—
71	#510	—	13	2	no	—
72	#510	—	13	4	no	—
73	#508	—	13	2	no	—
74	#508	D₁₃	1	1	yes	characteristic, complemented
75	C₂×Q₃₂	—	13	2	no	—
76	C₂×Q₃₂	—	13	1	no	—
77	#504	—	13	1	no	—
78	C₂²×C₁₃	—	4	2	no	—
79	C₂²×C₁₃	C₂×D₈	1	1	yes	characteristic
80	C₂×D₁₃	C₂×D₈	1	2	yes	—
81	C₄×C₁₃	—	4	2	no	—
82	C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
83	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	Dic₁₃	—	4	4	no	—
85	Dic₁₃	C₂×D₈	1	2	yes	—
86	2₋¹⁺⁴	—	26	2	no	—
87	#137	—	26	4	no	—
88	#137	—	26	2	no	—
89	#135	—	13	4	no	—
90	#135	C₂×D₁₃	1	2	yes	—
91	C₂×Q₁₆	—	13	4	no	—
92	C₂×Q₁₆	—	13	2	no	—
93	#131	—	13	1	no	—
94	Q₃₂	—	13	4	no	—
95	Q₃₂	—	13	2	no	—
96	Q₃₂	—	13	2	no	—
97	Q₃₂	C₂×D₁₃	1	1	yes	characteristic
98	SD₃₂	—	13	4	no	—
99	SD₃₂	C₂×D₁₃	1	2	yes	—
100	D₁₆	C₂×D₁₃	1	1	yes	characteristic
101	M₃₂	—	13	1	no	—
102	M₃₂	—	13	2	no	—
103	C₂×C₁₆	—	13	2	no	—
104	C₂×C₁₆	C₂×D₁₃	1	1	yes	characteristic
105	C₂×C₁₃	—	2	1	no	—
106	C₂×C₁₃	—	8	2	no	—
107	C₂×C₁₃	C₂²×D₈	1	1	yes	characteristic
108	D₁₃	—	2	2	no	—
109	#41	—	2	2	no	—
110	#41	—	13	1	no	—
111	#41	—	26	4	no	—
112	#41	—	52	2	no	—
113	#41	—	52	4	no	—
114	C₂×Q₈	—	26	4	no	—
115	C₂×Q₈	—	26	2	no	—
116	C₂×Q₈	—	52	2	no	—
117	Q₁₆	—	13	4	no	—
118	Q₁₆	—	26	4	no	—
119	Q₁₆	—	26	2	no	—
120	Q₁₆	C₂²×D₁₃	1	2	yes	—
121	SD₁₆	—	2	2	no	—
122	SD₁₆	—	26	4	no	—
123	D₈	C₂²×D₁₃	1	2	yes	—
124	M₁₆	—	13	1	no	—
125	M₁₆	—	13	2	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
128	C₁₆	—	13	2	no	—
129	C₁₆	C₂²×D₁₃	1	1	yes	characteristic
130	C₁₆	C₂²×D₁₃	1	1	yes	characteristic
131	C₁₃	#3417	1	1	yes	characteristic, complemented, Sylow
132	Q₈	—	2	2	no	—
133	Q₈	—	13	1	no	—
134	Q₈	—	26	4	no	—
135	Q₈	—	52	4	no	—
136	Q₈	—	52	2	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	13	2	no	—
140	D₄	—	13	1	no	—
141	D₄	—	52	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	13	2	no	—
144	C₂×C₄	—	52	4	no	—
145	C₂×C₄	—	52	4	no	—
146	C₂×C₄	—	52	2	no	—
147	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
148	C₈	—	13	2	no	—
149	C₈	D₄×D₁₃	1	1	yes	characteristic
150	C₈	C₂³×D₁₃	1	1	yes	characteristic
151	C₂²	—	4	2	no	—
152	C₂²	—	13	2	no	—
153	C₂²	D₈×D₁₃	1	1	yes	characteristic
154	C₄	—	4	2	no	—
155	C₄	—	13	2	no	—
156	C₄	—	52	4	no	—
157	C₄	D₈×D₁₃	1	1	yes	characteristic
158	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	8	2	no	—
161	C₂	—	26	2	no	—
162	C₂	C₂×D₈×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×D₁₃
commutator, cosocle	C₈×C₁₃	C₂⁴
Frattini	C₈	C₂³×D₁₃
Fitting	C₁₃×#508	C₂
socle	C₂×C₁₃	C₂²×D₈
hypercenter	#508	D₁₃
hypocenter	C₁₃	#3417
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #11719476	C₂	semidirect
2. C₁₃×#508	C₂	semidirect
3. #11718856	C₂	semidirect
4. D₁₃×Q₃₂	C₂	semidirect
5. #11718576	C₂	semidirect
6. #11718573	C₂	semidirect
7. C₂×Dic₁₀₄	C₂	semidirect
8. #11718562	C₂	semidirect
74. #508	D₁₃	semidirect
131. C₁₃	#3417	semidirect

Maximal subgroups

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

subgroup	index
#11719476	2
C₁₃×#508	2
#11718856	2
D₁₃×Q₃₂	2
#11718576	2
#11718573	2
C₂×Dic₁₀₄	2
#11718562	2
#3417	13