Subgroups — #15774464

Order 3328.

270 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	#13116592	C₂	1	2	yes	maximal
2	#13112900	C₂	1	4	yes	maximal, complemented
3	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88052	—	2	2	no	—
8	C₈²×C₁₃	C₄	1	2	yes	—
9	C₈²×C₁₃	C₂²	1	2	yes	—
10	C₄⋊#6090	—	2	2	no	—
11	C₄⋊#6090	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₃	—	2	1	no	—
13	C₂²×C₈×C₁₃	C₈	1	2	yes	—
14	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₄×C₈×C₁₃	—	2	1	no	—
18	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
19	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
20	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	#88052	—	4	4	no	—
27	C₂×#6090	—	2	4	no	—
28	#11996	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₁₃	—	2	4	no	—
33	C₂×C₈×C₁₃	—	2	2	no	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	—	2	4	no	—
36	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
41	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
46	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	#6090	—	4	4	no	—
48	#2150	—	13	2	no	—
49	C₂×C₈²	D₁₃	1	1	yes	characteristic
50	#1367	—	13	4	no	—
51	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#106	1	2	yes	—
58	C₂×C₄×C₁₃	#108	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₁₃	—	2	4	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	—	2	1	no	—
65	C₈×C₁₃	—	2	2	no	—
66	C₈×C₁₃	#106	1	2	yes	—
67	C₈×C₁₃	M₃₂	1	4	yes	complemented
68	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
69	C₂×M₃₂	—	26	2	no	—
70	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
71	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
73	C₄⋊C₁₆	—	13	4	no	—
74	C₄⋊C₁₆	—	26	2	no	—
75	C₈²	Dic₁₃	1	2	yes	—
76	C₈²	C₂×D₁₃	1	2	yes	—
77	C₂²×C₁₃	—	2	1	no	—
78	C₂²×C₁₃	—	2	1	no	—
79	C₂²×C₁₃	#334	1	2	yes	—
80	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	—	2	1	no	—
83	C₄×C₁₃	#334	1	2	yes	—
84	C₄×C₁₃	#364	1	2	yes	—
85	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
86	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
87	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	#1055	1	2	yes	—
90	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
91	C₂²×C₈	D₅₂	1	1	yes	characteristic
92	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
93	M₃₂	—	52	4	no	—
94	C₂×C₁₆	—	26	4	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	#1055	1	1	yes	characteristic
97	C₄×C₈	#1055	1	1	yes	characteristic
98	C₄×C₈	Dic₂₆	1	1	yes	characteristic
99	C₄×C₈	D₅₂	1	1	yes	characteristic
100	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
101	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
102	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
103	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
104	C₂×C₁₃	—	2	2	no	—
105	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
106	C₂×C₁₃	#2150	1	1	yes	characteristic
107	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
108	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
109	C₂²×C₄	#6099	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	#6095	1	4	yes	—
116	C₂×C₈	C₂×#1055	1	2	yes	—
117	C₂×C₈	C₂×#1055	1	2	yes	—
118	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
120	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
121	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
122	C₄²	C₂×#1055	1	1	yes	characteristic
123	C₄²	#6099	1	1	yes	characteristic
124	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
125	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
126	C₁₆	—	52	4	no	—
127	C₁₃	#11996	1	1	yes	characteristic, complemented, Sylow
128	C₂³	C₄⋊#1055	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
132	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
133	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
134	C₂×C₄	#88057	1	2	yes	—
135	C₂×C₄	#88059	1	2	yes	—
136	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
137	C₂×C₄	C₂×#6099	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	2	2	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	#88052	1	4	yes	complemented
144	C₈	#88057	1	2	yes	—
145	C₈	C₂×#6095	1	2	yes	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	#11718049	1	2	yes	—
149	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#11718049	1	2	yes	—
153	C₄	#11718055	1	2	yes	—
154	C₄	C₂×#88052	1	2	yes	—
155	C₄	C₂×#88057	1	1	yes	characteristic
156	C₄	C₂×#88059	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#11718049	1	1	yes	characteristic
159	C₂	#13116592	1	1	yes	characteristic
160	C₂	C₂×#11718055	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#11996
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #13112900	C₂	semidirect
67. C₈×C₁₃	M₃₂	semidirect
127. C₁₃	#11996	semidirect
143. C₈	#88052	semidirect

Maximal subgroups

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

subgroup	index
#13116592	2
#13112900	2
C₂×C₈²×C₁₃	2
#11996	13