Subgroups — #16015963

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123436	C₂	1	2	yes	maximal
2	#13116624	C₂	1	6	yes	maximal, complemented
3	#13116620	C₂	1	3	yes	maximal, complemented
4	C₂×#11718096	C₂	1	3	yes	maximal, complemented
5	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
7	#11719532	—	2	2	no	—
8	C₁₃×#504	C₄	1	8	yes	complemented
9	C₂×C₁₃×M₃₂	C₂²	1	3	yes	—
10	C₂²×C₁₃×C₁₆	C₂²	1	3	yes	—
11	#11718782	—	2	6	no	—
12	#11718767	C₂²	1	6	yes	—
13	C₂×#88058	C₂²	1	6	yes	—
14	#11718097	C₂²	1	6	yes	—
15	#11718096	C₂²	1	3	yes	—
16	#11718096	C₂²	1	6	yes	—
17	#11718096	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
19	C₁₃×#131	C₂×C₄	1	4	yes	—
20	C₁₃×#131	Q₈	1	4	yes	—
21	C₂×C₁₃×M₁₆	D₄	1	3	yes	—
22	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
23	Q₈×Dic₁₃	—	4	2	no	—
24	D₄×Dic₁₃	—	4	6	no	—
25	#88180	—	2	6	no	—
26	C₂×C₄⋊Dic₁₃	—	2	6	no	—
27	C₁₃×M₃₂	C₂×C₄	1	12	yes	complemented
28	C₂×C₁₃×C₁₆	C₂×C₄	1	12	yes	complemented
29	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
30	C₂×C₁₃×C₁₆	C₂³	1	3	yes	—
31	#88058	—	2	6	no	—
32	#88058	C₂³	1	6	yes	—
33	#88058	C₂³	1	2	yes	—
34	#88057	—	2	6	no	—
35	#33410	—	13	1	no	maximal, Sylow
36	C₁₃×#41	C₄⋊C₄	1	4	yes	—
37	C₁₃×#41	Q₁₆	1	4	yes	—
38	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
39	C₂×D₄×C₁₃	D₈	1	3	yes	—
40	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
41	C₂²×Dic₁₃	—	4	6	no	—
42	C₁₃×M₁₆	C₄⋊C₄	1	12	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
44	C₂×C₈×C₁₃	C₂×D₄	1	3	yes	—
45	C₂×C₈×C₁₃	C₂×Q₈	1	6	yes	—
46	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
47	C₂²⋊Dic₁₃	—	4	6	no	—
48	C₄⋊Dic₁₃	—	2	6	no	—
49	C₄⋊Dic₁₃	—	2	2	no	—
50	C₄⋊Dic₁₃	—	4	6	no	—
51	C₄×Dic₁₃	—	4	6	no	—
52	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
53	C₁₃×C₁₆	C₂²×C₄	1	6	yes	—
54	C₂×#504	D₁₃	1	1	yes	characteristic
55	#2913	—	13	2	no	—
56	#2159	—	13	6	no	—
57	#2158	—	13	3	no	—
58	C₂×#366	—	13	3	no	—
59	C₂³×C₁₃	C₂×D₈	1	3	yes	—
60	Q₈×C₁₃	#107	1	4	yes	—
61	D₄×C₁₃	#107	1	12	yes	—
62	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
63	C₂×C₄×C₁₃	C₂×D₈	1	3	yes	—
64	C₂×C₄×C₁₃	C₂×Q₁₆	1	6	yes	—
65	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
66	C₂×Dic₁₃	—	4	2	no	—
67	C₂×Dic₁₃	—	4	6	no	—
68	C₂×Dic₁₃	—	16	6	no	—
69	C₈×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
70	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
71	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
73	#520	—	26	2	no	—
74	#504	Dic₁₃	1	8	yes	complemented
75	C₂×M₃₂	C₂×D₁₃	1	3	yes	—
76	C₂²×C₁₆	C₂×D₁₃	1	3	yes	—
77	#428	—	26	6	no	—
78	#427	—	13	6	no	—
79	C₂×#107	—	13	6	no	—
80	#367	—	13	6	no	—
81	#366	—	13	3	no	—
82	#366	—	13	6	no	—
83	#366	—	13	1	no	—
84	C₂²×C₁₃	—	2	3	no	—
85	C₂²×C₁₃	—	2	3	no	—
86	C₂²×C₁₃	C₂×#107	1	6	yes	—
87	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₁₃	C₂×#107	1	6	yes	—
89	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
91	Dic₁₃	—	8	6	no	—
92	Dic₁₃	—	8	2	no	—
93	C₂×#41	D₅₂	1	1	yes	characteristic
94	#131	Dic₂₆	1	4	yes	—
95	#131	C₂×Dic₁₃	1	4	yes	—
96	C₂×M₁₆	D₅₂	1	3	yes	—
97	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
98	C₄×Q₈	—	52	2	no	—
99	C₄×D₄	—	52	6	no	—
100	#117	—	26	6	no	—
101	C₂×C₄⋊C₄	—	26	6	no	—
102	M₃₂	C₂×Dic₁₃	1	12	yes	complemented
103	C₂×C₁₆	C₂×Dic₁₃	1	12	yes	complemented
104	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
105	C₂×C₁₆	C₂²×D₁₃	1	3	yes	—
106	#107	—	13	6	no	—
107	#107	—	13	2	no	—
108	#107	—	26	6	no	—
109	#106	—	26	6	no	—
110	C₂×C₁₃	—	2	6	no	—
111	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
112	C₂×C₁₃	#3415	1	1	yes	characteristic
113	C₂×C₁₃	#3417	1	1	yes	characteristic
114	#41	Dic₅₂	1	4	yes	—
115	#41	C₄⋊Dic₁₃	1	4	yes	—
116	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
117	C₂×D₄	D₁₀₄	1	3	yes	—
118	C₂²×C₄	—	52	6	no	—
119	C₂²×C₄	C₂×D₅₂	1	3	yes	—
120	M₁₆	C₄⋊Dic₁₃	1	12	yes	—
121	C₂×C₈	C₂×Dic₂₆	1	6	yes	—
122	C₂×C₈	C₂×D₅₂	1	3	yes	—
123	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
124	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
125	C₄⋊C₄	—	26	2	no	—
126	C₄⋊C₄	—	26	6	no	—
127	C₄⋊C₄	—	52	6	no	—
128	C₂²⋊C₄	—	52	6	no	—
129	C₄²	—	52	6	no	—
130	C₁₆	C₂²×Dic₁₃	1	2	yes	—
131	C₁₆	C₂²×Dic₁₃	1	6	yes	—
132	C₁₃	#33410	1	1	yes	characteristic, complemented, Sylow
133	C₂³	C₂×D₁₀₄	1	3	yes	—
134	Q₈	#88058	1	4	yes	—
135	D₄	#88058	1	12	yes	—
136	C₂×C₄	—	52	2	no	—
137	C₂×C₄	—	52	6	no	—
138	C₂×C₄	—	208	6	no	—
139	C₂×C₄	C₂×D₁₀₄	1	3	yes	—
140	C₂×C₄	C₂×Dic₅₂	1	6	yes	—
141	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
142	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
143	C₈	C₂×C₄⋊Dic₁₃	1	6	yes	—
144	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
145	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×#88058	1	6	yes	—
149	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
150	C₄	—	104	6	no	—
151	C₄	—	104	2	no	—
152	C₄	C₂×#88058	1	6	yes	—
153	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
154	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
155	C₂	—	2	6	no	—
156	C₂	C₂²×#88058	1	1	yes	characteristic
157	C₂	#13127977	1	1	yes	characteristic
158	C₂	#13127989	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#504	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#504	D₁₃
hypocenter	C₁₃	#33410
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #13116624	C₂	semidirect
3. #13116620	C₂	semidirect
4. C₂×#11718096	C₂	semidirect
8. C₁₃×#504	C₄	semidirect
27. C₁₃×M₃₂	C₂×C₄	semidirect
28. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
74. #504	Dic₁₃	semidirect
102. M₃₂	C₂×Dic₁₃	semidirect
103. C₂×C₁₆	C₂×Dic₁₃	semidirect
132. C₁₃	#33410	semidirect

Maximal subgroups

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

subgroup	index
#13123436	2
#13116624	2
#13116620	2
C₂×#11718096	2
C₂×C₁₃×#504	2
#33410	13