Subgroups — #16015581

Order 3328.

716 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127939	C₂	1	2	yes	maximal
2	#13116441	C₂	1	6	yes	maximal, complemented
3	C₂×#11718058	C₂	1	6	yes	maximal, complemented
4	C₁₃×#2875	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
6	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
7	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
8	C₁₃×#405	C₄	1	3	yes	—
9	C₁₃×#405	C₂²	1	3	yes	—
10	C₄×C₁₃×M₁₆	C₄	1	3	yes	—
11	C₂×C₁₃×#97	C₂²	1	3	yes	—
12	#11718801	C₄	1	16	yes	complemented
13	C₂×#88052	C₂²	1	12	yes	—
14	#11718058	C₂²	1	4	yes	—
15	#11718058	C₂²	1	12	yes	—
16	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
17	C₁₃×#131	—	2	4	no	—
18	C₁₃×#131	C₂×C₄	1	8	yes	—
19	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
20	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
22	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
23	C₄×Q₈×C₁₃	—	2	1	no	—
24	C₄×D₄×C₁₃	—	2	3	no	—
25	C₁₃×#117	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
27	C₁₃×#97	C₂×C₄	1	3	yes	—
28	C₁₃×#97	C₂×C₄	1	3	yes	—
29	C₁₃×#97	C₂³	1	1	yes	characteristic
30	C₁₃×#97	C₂³	1	3	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	6	yes	—
32	#88052	C₂×C₄	1	24	yes	complemented
33	#88052	C₂³	1	8	yes	—
34	C₂×#6090	C₂×C₄	1	24	yes	complemented
35	#33329	—	13	1	no	maximal, Sylow
36	C₁₃×#41	C₄²	1	4	yes	—
37	C₁₃×#41	M₁₆	1	4	yes	—
38	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
39	C₂×D₄×C₁₃	M₁₆	1	3	yes	—
40	C₂²×C₄×C₁₃	—	2	3	no	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
42	C₁₃×M₁₆	—	2	6	no	—
43	C₁₃×M₁₆	C₄²	1	12	yes	—
44	C₂×C₈×C₁₃	—	2	6	no	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	12	yes	—
49	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₁₃×C₄⋊C₄	—	2	3	no	—
51	C₁₃×C₂²⋊C₄	—	2	3	no	—
52	C₄²×C₁₃	—	2	3	no	—
53	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
54	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	#6090	C₂²×C₄	1	16	yes	—
56	#3407	—	13	2	no	—
57	#2875	D₁₃	1	1	yes	characteristic
58	#2110	—	13	6	no	—
59	C₂×#347	—	13	6	no	—
60	C₂³×C₁₃	C₂×M₁₆	1	3	yes	—
61	Q₈×C₁₃	#97	1	4	yes	—
62	D₄×C₁₃	#97	1	12	yes	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	3	no	—
65	C₂×C₄×C₁₃	—	8	3	no	—
66	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
67	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
68	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
69	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
70	C₈×C₁₃	—	2	1	no	—
71	C₈×C₁₃	—	2	3	no	—
72	C₈×C₁₃	C₂×C₄²	1	6	yes	—
73	C₈×C₁₃	C₂³×C₄	1	2	yes	—
74	C₂×#131	Dic₁₃	1	1	yes	characteristic
75	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
76	C₄×#41	Dic₁₃	1	1	yes	characteristic
77	#504	—	13	16	no	—
78	C₂×M₃₂	—	13	12	no	—
79	#405	Dic₁₃	1	3	yes	—
80	#405	C₂×D₁₃	1	3	yes	—
81	C₄×M₁₆	Dic₁₃	1	3	yes	—
82	C₂×#97	C₂×D₁₃	1	3	yes	—
83	#347	—	13	4	no	—
84	#347	—	13	12	no	—
85	C₂²×C₁₃	—	4	3	no	—
86	C₂²×C₁₃	C₂×#97	1	6	yes	—
87	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
88	C₄×C₁₃	—	4	3	no	—
89	C₄×C₁₃	—	4	1	no	—
90	C₄×C₁₃	C₂×#97	1	6	yes	—
91	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
92	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
93	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
94	#131	—	2	4	no	—
95	#131	C₄×D₁₃	1	8	yes	—
96	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
97	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
98	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
99	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
100	C₄×Q₈	—	2	1	no	—
101	C₄×D₄	—	2	3	no	—
102	#117	C₂×Dic₁₃	1	3	yes	—
103	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
104	M₃₂	—	13	24	no	—
105	M₃₂	—	13	8	no	—
106	C₂×C₁₆	—	13	24	no	—
107	#97	C₂×Dic₁₃	1	3	yes	—
108	#97	C₂×Dic₁₃	1	3	yes	—
109	#97	C₂²×D₁₃	1	1	yes	characteristic
110	#97	C₂²×D₁₃	1	3	yes	—
111	C₄×C₈	C₂×Dic₁₃	1	6	yes	—
112	C₂×C₁₃	—	2	1	no	—
113	C₂×C₁₃	—	2	6	no	—
114	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
115	#41	#6099	1	4	yes	—
116	#41	C₄×Dic₁₃	1	4	yes	—
117	C₂×Q₈	#6099	1	1	yes	characteristic
118	C₂×D₄	#6099	1	3	yes	—
119	C₂²×C₄	—	2	3	no	—
120	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
121	M₁₆	—	2	6	no	—
122	M₁₆	C₄×Dic₁₃	1	12	yes	—
123	C₂×C₈	—	2	6	no	—
124	C₂×C₈	C₂×C₄×D₁₃	1	12	yes	—
125	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
126	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
127	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
128	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
129	C₄⋊C₄	—	2	3	no	—
130	C₂²⋊C₄	—	2	3	no	—
131	C₄²	—	2	3	no	—
132	C₄²	C₂²×Dic₁₃	1	3	yes	—
133	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
134	C₁₆	—	13	16	no	—
135	C₁₃	#33329	1	1	yes	characteristic, complemented, Sylow
136	C₂³	C₂×#6099	1	3	yes	—
137	Q₈	#88043	1	4	yes	—
138	D₄	#88043	1	12	yes	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	—	8	3	no	—
142	C₂×C₄	C₂×#6099	1	6	yes	—
143	C₂×C₄	C₂×#6099	1	3	yes	—
144	C₂×C₄	C₂×C₄×Dic₁₃	1	6	yes	—
145	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	3	no	—
148	C₈	C₂×C₄×Dic₁₃	1	6	yes	—
149	C₈	C₂²×C₄×D₁₃	1	2	yes	—
150	C₂²	—	4	3	no	—
151	C₂²	C₂×#88043	1	6	yes	—
152	C₂²	C₂²×#6099	1	1	yes	characteristic
153	C₄	—	4	3	no	—
154	C₄	—	4	1	no	—
155	C₄	C₂×#88043	1	6	yes	—
156	C₄	C₂²×#6099	1	1	yes	characteristic
157	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×#88043	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2875	C₂
socle	C₂×C₁₃	C₂²×#97
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2875	D₁₃
hypocenter	C₁₃	#33329
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #2875

Complements

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

N	H ≅ G/N	type
2. #13116441	C₂	semidirect
3. C₂×#11718058	C₂	semidirect
12. #11718801	C₄	semidirect
32. #88052	C₂×C₄	semidirect
34. C₂×#6090	C₂×C₄	semidirect
135. C₁₃	#33329	semidirect

Maximal subgroups

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

subgroup	index
#13127939	2
#13116441	2
C₂×#11718058	2
C₁₃×#2875	2
#33329	13