Subgroups — #15750605

Order 3328.

202 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	C₂²×#88034	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₃×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
5	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
6	C₂²×#6090	C₄	1	1	yes	characteristic
7	C₂×C₁₃×C₃₂	—	2	1	no	—
8	C₂×C₁₃×C₃₂	—	2	1	no	—
9	C₂×C₁₃×C₃₂	C₄	1	2	yes	complemented
10	C₁₆×Dic₁₃	—	2	1	no	—
11	C₁₆×Dic₁₃	—	2	1	no	—
12	C₂×#88034	—	2	1	no	—
13	C₂×#88034	—	2	1	no	—
14	C₂×#88034	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
17	C₂²×#1055	C₈	1	1	yes	characteristic
18	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	Q₈	1	1	yes	characteristic
24	C₈×Dic₁₃	—	2	1	no	—
25	C₈×Dic₁₃	—	2	1	no	—
26	C₂×#6090	—	2	1	no	—
27	C₂×#6090	—	2	1	no	—
28	C₂×#6090	C₈	1	2	yes	—
29	C₁₃×C₃₂	—	2	2	no	—
30	#88034	—	2	2	no	—
31	#7465	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×Dic₁₃	C₁₆	1	2	yes	—
34	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
40	C₄×Dic₁₃	—	2	1	no	—
41	C₄×Dic₁₃	—	2	1	no	—
42	C₂×#1055	—	2	1	no	—
43	C₂×#1055	—	2	1	no	—
44	C₂×#1055	C₁₆	1	2	yes	—
45	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
48	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
49	#6090	—	2	2	no	—
50	C₂²×C₃₂	—	13	1	no	—
51	C₂²×C₃₂	D₁₃	1	1	yes	characteristic
52	C₂×C₄×C₁₆	—	13	1	no	—
53	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₃	M₃₂	1	1	yes	characteristic
59	C₂×C₄×C₁₃	M₃₂	1	1	yes	characteristic
60	C₂×Dic₁₃	—	2	2	no	—
61	C₂×Dic₁₃	—	2	2	no	—
62	C₂×Dic₁₃	C₃₂	1	4	yes	complemented
63	C₈×C₁₃	#95	1	1	yes	characteristic
64	C₈×C₁₃	#97	1	1	yes	characteristic
65	C₈×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
66	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
67	#1055	—	2	2	no	—
68	C₂²×C₁₆	—	13	1	no	—
69	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
70	C₂×C₄×C₈	—	13	1	no	—
71	C₂×C₃₂	—	2	1	no	—
72	C₂×C₃₂	—	2	1	no	—
73	C₂×C₃₂	—	13	2	no	—
74	C₂×C₃₂	—	26	1	no	—
75	C₂×C₃₂	—	26	1	no	—
76	C₂×C₃₂	Dic₁₃	1	2	yes	complemented
77	C₄×C₁₆	—	26	1	no	—
78	C₄×C₁₆	—	26	1	no	—
79	C₂²×C₁₃	C₄×C₁₆	1	1	yes	characteristic
80	C₂²×C₁₃	C₂²⋊C₁₆	1	1	yes	characteristic
81	C₂²×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
82	C₂²×C₁₃	C₂×C₃₂	1	2	yes	—
83	C₂²×C₁₃	M₆₄	1	2	yes	—
84	C₄×C₁₃	#336	1	1	yes	characteristic
85	C₄×C₁₃	#346	1	1	yes	characteristic
86	C₄×C₁₃	C₂²⋊C₁₆	1	1	yes	characteristic
87	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
88	Dic₁₃	—	2	4	no	—
89	C₂²×C₈	—	13	1	no	—
90	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
91	C₂×C₄²	—	13	1	no	—
92	C₂×C₁₆	—	13	2	no	—
93	C₂×C₁₆	—	26	1	no	—
94	C₂×C₁₆	—	26	1	no	—
95	C₂×C₁₆	Dic₂₆	1	1	yes	characteristic
96	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
97	C₂×C₁₆	D₅₂	1	1	yes	characteristic
98	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
99	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
100	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
101	C₄×C₈	—	26	1	no	—
102	C₄×C₈	—	26	1	no	—
103	C₃₂	—	2	2	no	—
104	C₃₂	—	26	2	no	—
105	C₂×C₁₃	#1374	1	1	yes	characteristic
106	C₂×C₁₃	C₄×C₃₂	1	1	yes	characteristic
107	C₂×C₁₃	#1397	1	1	yes	characteristic
108	C₂×C₁₃	C₂²⋊C₃₂	1	2	yes	—
109	C₂×C₁₃	C₄⋊C₃₂	1	2	yes	—
110	C₂²×C₄	—	13	2	no	—
111	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
112	C₂×C₈	—	13	2	no	—
113	C₂×C₈	—	26	1	no	—
114	C₂×C₈	—	26	1	no	—
115	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
116	C₂×C₈	#6094	1	1	yes	characteristic
117	C₂×C₈	#6094	1	1	yes	characteristic
118	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
119	C₂×C₈	#6101	1	1	yes	characteristic
120	C₂×C₈	#6103	1	1	yes	characteristic
121	C₄²	—	26	1	no	—
122	C₄²	—	26	1	no	—
123	C₁₆	—	26	2	no	—
124	C₁₆	#6101	1	1	yes	characteristic
125	C₁₆	C₄⋊Dic₁₃	1	1	yes	characteristic
126	C₁₆	#6103	1	1	yes	characteristic
127	C₁₆	C₂²⋊Dic₁₃	1	1	yes	characteristic
128	C₁₃	#7465	1	1	yes	characteristic, complemented, Sylow
129	C₂³	C₁₆×D₁₃	1	1	yes	characteristic
130	C₂×C₄	—	13	4	no	—
131	C₂×C₄	—	26	2	no	—
132	C₂×C₄	—	26	2	no	—
133	C₂×C₄	C₁₆×D₁₃	1	1	yes	characteristic
134	C₂×C₄	#88038	1	1	yes	characteristic
135	C₂×C₄	#88038	1	1	yes	characteristic
136	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
137	C₂×C₄	#88054	1	1	yes	characteristic
138	C₂×C₄	#88060	1	1	yes	characteristic
139	C₈	—	26	2	no	—
140	C₈	#88054	1	1	yes	characteristic
141	C₈	#88055	1	1	yes	characteristic
142	C₈	#88060	1	1	yes	characteristic
143	C₈	#88071	1	1	yes	characteristic
144	C₂²	D₁₃×C₃₂	1	2	yes	—
145	C₂²	#11718039	1	2	yes	—
146	C₂²	C₁₆×Dic₁₃	1	1	yes	characteristic
147	C₂²	#11718093	1	1	yes	characteristic
148	C₂²	#11718099	1	1	yes	characteristic
149	C₄	—	26	4	no	—
150	C₄	#11718093	1	1	yes	characteristic
151	C₄	#11718094	1	1	yes	characteristic
152	C₄	#11718099	1	1	yes	characteristic
153	C₄	#11718142	1	1	yes	characteristic
154	C₂	#13112913	1	1	yes	characteristic
155	C₂	C₃₂×Dic₁₃	1	1	yes	characteristic
156	C₂	#13112969	1	1	yes	characteristic
157	C₂	#13112973	1	2	yes	—
158	C₂	#13113031	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₁₆	C₂×D₁₃
commutator	C₂×C₁₃	C₄×C₃₂
Fitting	C₂²×C₁₃×C₃₂	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₁₆
cosocle	C₂²×C₁₃×C₁₆	C₂²
hypercenter	C₂²×C₃₂	D₁₃
hypocenter	C₁₃	#7465
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
9. C₂×C₁₃×C₃₂	C₄	semidirect
14. C₂×#88034	C₄	semidirect
62. C₂×Dic₁₃	C₃₂	semidirect
76. C₂×C₃₂	Dic₁₃	semidirect
128. C₁₃	#7465	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88034	2
C₂×C₁₆×Dic₁₃	2
C₂²×C₁₃×C₃₂	2
#7465	13