Subgroups — #15771913

Order 3328.

322 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
3	#13112755	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×M₃₂	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88052	C₄	1	2	yes	—
8	C₂×C₄×#1055	C₄	1	2	yes	—
9	C₁₃×#346	C₄	1	2	yes	—
10	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
11	#11718053	C₄	1	2	yes	—
12	C₄×#6090	C₂²	1	2	yes	—
13	C₈×#1055	C₄	1	1	yes	characteristic
14	C₈×#1055	C₄	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×#1055	C₈	1	4	yes	—
20	C₁₃×M₃₂	—	2	4	no	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	#88052	—	2	4	no	—
27	C₂×#6090	C₂×C₄	1	4	yes	—
28	C₄×#1055	C₈	1	4	yes	—
29	C₄×#1055	C₂×C₄	1	2	yes	—
30	C₄×#1055	C₂×C₄	1	2	yes	—
31	#11402	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
37	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
38	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₁₃	C₄²	1	1	yes	characteristic
41	C₄²×C₁₃	C₂×C₈	1	2	yes	—
42	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×#1055	—	2	8	no	—
44	C₂×#1055	C₂×C₈	1	4	yes	—
45	C₂×#1055	C₂×C₈	1	4	yes	—
46	C₁₃×C₁₆	—	2	4	no	—
47	#6090	—	2	4	no	—
48	C₄×M₃₂	—	13	1	no	—
49	C₄×M₃₂	D₁₃	1	1	yes	characteristic
50	C₂×C₈²	—	13	1	no	—
51	#1312	—	13	4	no	—
52	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
55	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#97	1	2	yes	—
57	C₂×C₄×C₁₃	#97	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
60	C₈×C₁₃	#97	1	2	yes	—
61	C₈×C₁₃	#97	1	2	yes	—
62	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
63	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
64	#1055	—	2	4	no	—
65	#1055	M₃₂	1	8	yes	complemented
66	C₂×M₃₂	—	13	2	no	—
67	C₂×M₃₂	Dic₁₃	1	2	yes	—
68	C₂×C₄×C₈	—	13	2	no	—
69	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
70	#346	—	13	2	no	—
71	#346	Dic₁₃	1	2	yes	—
72	C₄×C₁₆	—	13	2	no	—
73	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
74	C₈²	—	13	1	no	—
75	C₈²	—	13	1	no	—
76	C₈²	—	13	2	no	—
77	C₂²×C₁₃	—	2	2	no	—
78	C₂²×C₁₃	#322	1	2	yes	—
79	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	#322	1	2	yes	—
82	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
83	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
84	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
85	C₂²×C₈	—	13	4	no	—
86	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
87	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
88	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
89	M₃₂	—	2	4	no	—
90	M₃₂	—	26	4	no	—
91	C₂×C₁₆	—	13	4	no	—
92	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
93	C₄×C₈	—	13	4	no	—
94	C₄×C₈	—	13	2	no	—
95	C₄×C₈	—	13	2	no	—
96	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
97	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
99	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
100	C₂×C₁₃	—	2	2	no	—
101	C₂×C₁₃	C₂×#322	1	1	yes	characteristic
102	C₂×C₁₃	C₄×M₃₂	1	2	yes	—
103	C₂²×C₄	C₈×D₁₃	1	2	yes	—
104	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
105	C₂×C₈	—	13	4	no	—
106	C₂×C₈	—	13	4	no	—
107	C₂×C₈	—	26	8	no	—
108	C₂×C₈	#6094	1	4	yes	—
109	C₂×C₈	#6099	1	4	yes	—
110	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
111	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
112	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
113	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
114	C₄²	C₈×D₁₃	1	2	yes	—
115	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
116	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
117	C₁₆	—	2	4	no	—
118	C₁₆	—	26	4	no	—
119	C₁₃	#11402	1	1	yes	characteristic, complemented, Sylow
120	C₂³	C₈×Dic₁₃	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	#88043	1	2	yes	—
123	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
124	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
125	C₂×C₄	#88055	1	2	yes	—
126	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
127	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
128	C₈	—	13	8	no	—
129	C₈	—	26	4	no	—
130	C₈	#88043	1	2	yes	—
131	C₈	#88055	1	2	yes	—
132	C₈	C₂×#6094	1	2	yes	—
133	C₈	C₂×#6099	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#11718046	1	2	yes	—
136	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	#11718046	1	2	yes	—
139	C₄	D₁₃×M₃₂	1	4	yes	—
140	C₄	C₂×#88043	1	1	yes	characteristic
141	C₄	C₂×#88055	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#11718046	1	1	yes	characteristic
144	C₂	M₃₂×Dic₁₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₄×C₁₃×M₃₂	C₂
prime radical	C₂³×C₁₃	C₄×C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#11402
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13112755	C₂	semidirect
65. #1055	M₃₂	semidirect
119. C₁₃	#11402	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88052	2
C₂×C₈×#1055	2
#13112755	2
C₄×C₁₃×M₃₂	2
#11402	13