Subgroups — #16015923

Order 3328.

684 conjugacy classes of subgroups.

The subgroup lattice has 168 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	#13116602	C₂	1	6	yes	maximal, complemented
3	C₂×#11718055	C₂	1	6	yes	maximal, complemented
4	C₄×C₁₃×#131	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×C₄×C₈×C₁₃	C₂²	1	3	yes	—
12	#11718801	—	2	8	no	—
13	C₂×#88052	C₂²	1	12	yes	—
14	#11718055	C₂²	1	4	yes	—
15	#11718055	C₂²	1	12	yes	—
16	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
17	C₁₃×#131	C₈	1	8	yes	—
18	C₁₃×#131	D₄	1	4	yes	—
19	C₁₃×#131	Q₈	1	4	yes	—
20	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
24	C₄×Q₈×C₁₃	—	2	1	no	—
25	C₄×D₄×C₁₃	—	2	3	no	—
26	C₁₃×#117	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
28	C₁₃×#97	C₂×C₄	1	6	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
31	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₁₃	C₂³	1	3	yes	—
33	#88052	—	2	12	no	—
34	#88052	C₂³	1	8	yes	—
35	C₂×#6090	—	2	12	no	—
36	#33401	—	13	1	no	maximal, Sylow
37	C₁₃×#41	C₄⋊C₄	1	4	yes	—
38	C₁₃×#41	C₂×C₈	1	4	yes	—
39	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
40	C₂×D₄×C₁₃	M₁₆	1	3	yes	—
41	C₂²×C₄×C₁₃	—	2	3	no	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
43	C₁₃×M₁₆	C₄⋊C₄	1	12	yes	—
44	C₁₃×M₁₆	C₂×C₈	1	12	yes	—
45	C₂×C₈×C₁₃	C₂×C₈	1	12	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
50	C₂×C₈×C₁₃	C₂×Q₈	1	6	yes	—
51	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	3	no	—
53	C₁₃×C₂²⋊C₄	—	2	3	no	—
54	C₄²×C₁₃	—	2	3	no	—
55	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
57	#6090	—	2	8	no	—
58	#3407	—	13	2	no	—
59	C₄×#131	D₁₃	1	1	yes	characteristic
60	#2153	—	13	6	no	—
61	C₂×#364	—	13	6	no	—
62	C₂³×C₁₃	C₂×M₁₆	1	3	yes	—
63	Q₈×C₁₃	C₄⋊C₈	1	4	yes	—
64	D₄×C₁₃	C₄⋊C₈	1	12	yes	—
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	3	no	—
67	C₂×C₄×C₁₃	—	4	6	no	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
69	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
70	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
71	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
73	C₈×C₁₃	C₂²×C₈	1	2	yes	—
74	C₈×C₁₃	C₂²×C₈	1	6	yes	—
75	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
76	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂×#131	Dic₁₃	1	1	yes	characteristic
78	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
79	C₄×#41	Dic₁₃	1	1	yes	characteristic
80	#504	—	26	8	no	—
81	C₂×M₃₂	—	13	12	no	—
82	#405	Dic₁₃	1	3	yes	—
83	#405	C₂×D₁₃	1	3	yes	—
84	C₄×M₁₆	Dic₁₃	1	3	yes	—
85	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
86	#364	—	13	4	no	—
87	#364	—	13	12	no	—
88	C₂²×C₁₃	—	4	3	no	—
89	C₂²×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
90	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	4	3	no	—
93	C₄×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
94	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
96	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
97	#131	#1055	1	8	yes	—
98	#131	Dic₂₆	1	4	yes	—
99	#131	D₅₂	1	4	yes	—
100	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
101	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
102	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
103	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
104	C₄×Q₈	—	2	1	no	—
105	C₄×D₄	—	2	3	no	—
106	#117	C₂×Dic₁₃	1	3	yes	—
107	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
108	M₃₂	—	13	8	no	—
109	M₃₂	—	26	12	no	—
110	C₂×C₁₆	—	26	12	no	—
111	#97	C₂×Dic₁₃	1	6	yes	—
112	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
113	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
114	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
115	C₄×C₈	C₂²×D₁₃	1	3	yes	—
116	C₂×C₁₃	—	2	1	no	—
117	C₂×C₁₃	—	2	6	no	—
118	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
119	#41	C₂×#1055	1	4	yes	—
120	#41	C₄⋊Dic₁₃	1	4	yes	—
121	C₂×Q₈	#6099	1	1	yes	characteristic
122	C₂×D₄	#6099	1	3	yes	—
123	C₂²×C₄	—	2	3	no	—
124	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
125	M₁₆	C₂×#1055	1	12	yes	—
126	M₁₆	C₄⋊Dic₁₃	1	12	yes	—
127	C₂×C₈	C₂×#1055	1	12	yes	—
128	C₂×C₈	C₂×Dic₂₆	1	6	yes	—
129	C₂×C₈	C₂×D₅₂	1	6	yes	—
130	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
131	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
132	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
133	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
134	C₄⋊C₄	—	2	3	no	—
135	C₂²⋊C₄	—	2	3	no	—
136	C₄²	—	2	3	no	—
137	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
138	C₄²	C₂²×Dic₁₃	1	3	yes	—
139	C₁₆	—	26	8	no	—
140	C₁₃	#33401	1	1	yes	characteristic, complemented, Sylow
141	C₂³	C₂×#6099	1	3	yes	—
142	Q₈	C₄⋊#1055	1	4	yes	—
143	D₄	C₄⋊#1055	1	12	yes	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	3	no	—
146	C₂×C₄	—	4	6	no	—
147	C₂×C₄	C₂²×#1055	1	6	yes	—
148	C₂×C₄	C₂×#6099	1	3	yes	—
149	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
150	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
151	C₈	C₂²×#1055	1	2	yes	—
152	C₈	C₂²×#1055	1	6	yes	—
153	C₈	C₂×C₄⋊Dic₁₃	1	6	yes	—
154	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
155	C₈	C₂²×D₅₂	1	1	yes	characteristic
156	C₂²	—	4	3	no	—
157	C₂²	C₂×C₄⋊#1055	1	6	yes	—
158	C₂²	C₂²×#6099	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	3	no	—
161	C₄	C₂×C₄⋊#1055	1	6	yes	—
162	C₄	C₂³×#1055	1	1	yes	characteristic
163	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	6	no	—
166	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₅₂
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂×C₁₃	C₂²×C₄⋊C₈
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#33401
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13116602	C₂	semidirect
3. C₂×#11718055	C₂	semidirect
140. C₁₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#13127939	2
#13116602	2
C₂×#11718055	2
C₄×C₁₃×#131	2
#33401	13