Subgroups — #15777744

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718094	C₂	1	2	yes	maximal
2	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₁₃	C₂	1	2	yes	maximal
4	#13114522	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88055	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
10	C₂²×#6090	C₂²	1	1	yes	characteristic
11	C₂²×#6090	C₂²	1	1	yes	characteristic
12	#11718593	C₄	1	1	yes	characteristic
13	C₂×#88043	C₄	1	1	yes	characteristic
14	C₁₃×#346	C₄	1	4	yes	complemented
15	#11718094	C₄	1	8	yes	complemented
16	C₁₆×Dic₁₃	C₄	1	8	yes	complemented
17	C₄×#6090	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₂²×#1055	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
25	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
26	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	#88071	C₈	1	4	yes	—
29	#88055	C₈	1	8	yes	—
30	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
31	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
32	C₂×#6090	C₂×C₄	1	2	yes	—
33	C₂×#6090	C₂×C₄	1	2	yes	—
34	C₂×#6090	C₂×C₄	1	4	yes	—
35	C₂×#6090	C₂×C₄	1	4	yes	—
36	#88043	C₈	1	4	yes	—
37	#12539	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₁₃	C₄²	1	2	yes	—
42	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₄²	1	2	yes	—
48	C₄²×C₁₃	C₂×C₈	1	2	yes	—
49	C₄×Dic₁₃	C₄²	1	4	yes	—
50	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
51	C₂×#1055	C₄²	1	4	yes	—
52	C₂×#1055	C₂×C₈	1	4	yes	—
53	C₂×#1055	C₂×C₈	1	4	yes	—
54	C₁₃×C₁₆	C₄²	1	8	yes	—
55	#6090	C₄²	1	4	yes	—
56	#6090	C₄²	1	4	yes	—
57	C₂×#346	—	13	2	no	—
58	C₂×#346	D₁₃	1	1	yes	characteristic
59	C₂×C₄×C₁₆	—	13	2	no	—
60	C₂×C₄×C₁₆	—	13	1	no	—
61	#1726	—	13	1	no	—
62	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
65	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
66	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
69	C₂×Dic₁₃	—	2	8	no	—
70	C₂×Dic₁₃	C₄×C₈	1	8	yes	—
71	C₈×C₁₃	C₄×C₈	1	4	yes	—
72	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
73	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
74	C₈×C₁₃	C₂×C₄²	1	2	yes	—
75	#1055	C₄×C₈	1	8	yes	—
76	C₂²×C₁₆	—	13	1	no	—
77	C₂²×C₁₆	—	13	1	no	—
78	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
79	C₂×#97	—	13	2	no	—
80	C₂×#97	—	13	1	no	—
81	C₂×C₄×C₈	—	13	2	no	—
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	#376	—	13	1	no	—
84	#346	—	13	8	no	—
85	#346	Dic₁₃	1	4	yes	complemented
86	C₄×C₁₆	—	13	8	no	—
87	C₄×C₁₆	—	13	4	no	—
88	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
90	C₂²×C₁₃	#504	1	4	yes	—
91	C₄×C₁₃	—	2	4	no	—
92	C₄×C₁₃	C₄³	1	1	yes	characteristic
93	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
94	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
95	Dic₁₃	—	2	8	no	—
96	C₂²×C₈	—	13	2	no	—
97	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
99	C₂×C₄²	—	13	2	no	—
100	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
101	C₂×C₁₆	—	13	2	no	—
102	C₂×C₁₆	—	13	2	no	—
103	C₂×C₁₆	—	13	4	no	—
104	C₂×C₁₆	—	13	4	no	—
105	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
106	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
107	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
108	#97	—	13	8	no	—
109	#97	—	13	4	no	—
110	C₄×C₈	—	13	4	no	—
111	C₄×C₈	—	13	4	no	—
112	C₄×C₈	C₄×D₁₃	1	2	yes	—
113	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
114	#95	—	13	4	no	—
115	C₂×C₁₃	C₄²×C₈	1	1	yes	characteristic
116	C₂×C₁₃	#2108	1	4	yes	—
117	C₂×C₁₃	#2108	1	2	yes	—
118	C₂²×C₄	—	13	4	no	—
119	C₂²×C₄	C₈×D₁₃	1	2	yes	—
120	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
121	C₂×C₈	—	13	4	no	—
122	C₂×C₈	—	13	4	no	—
123	C₂×C₈	—	13	4	no	—
124	C₂×C₈	C₈×D₁₃	1	4	yes	—
125	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
126	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
127	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
129	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
130	C₄²	—	13	4	no	—
131	C₄²	—	13	4	no	—
132	C₄²	C₈×D₁₃	1	2	yes	—
133	C₄²	C₄×Dic₁₃	1	2	yes	—
134	C₁₆	—	13	4	no	—
135	C₁₆	—	13	4	no	—
136	C₁₆	C₄×Dic₁₃	1	8	yes	—
137	C₁₃	#12539	1	1	yes	characteristic, complemented, Sylow
138	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	13	8	no	—
141	C₂×C₄	—	26	8	no	—
142	C₂×C₄	C₈×Dic₁₃	1	4	yes	—
143	C₂×C₄	C₄²×D₁₃	1	2	yes	—
144	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
145	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
147	C₈	—	13	8	no	—
148	C₈	C₈×Dic₁₃	1	4	yes	—
149	C₈	C₄²×D₁₃	1	1	yes	characteristic
150	C₈	C₄²×D₁₃	1	1	yes	characteristic
151	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
152	C₂²	C₄×C₈×D₁₃	1	2	yes	—
153	C₂²	#11718568	1	4	yes	—
154	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
155	C₄	—	2	4	no	—
156	C₄	—	26	8	no	—
157	C₄	C₄×C₈×D₁₃	1	2	yes	—
158	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
159	C₄	C₂×C₈×Dic₁₃	1	1	yes	characteristic
160	C₂	C₄×C₈×Dic₁₃	1	1	yes	characteristic
161	C₂	#13116434	1	4	yes	—
162	C₂	#13116435	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₄²×C₈
Fitting	C₂×C₁₃×#346	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#12539
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
14. C₁₃×#346	C₄	semidirect
15. #11718094	C₄	semidirect
16. C₁₆×Dic₁₃	C₄	semidirect
17. C₄×#6090	C₄	semidirect
85. #346	Dic₁₃	semidirect
137. C₁₃	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718094	2
C₂×C₄×#6090	2
C₂×C₁₆×Dic₁₃	2
#13114522	2
C₂×C₁₃×#346	2
#12539	13