Subgroups — #15774504

Order 3328.

242 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113584	C₂	1	2	yes	maximal
2	#13112910	C₂	1	2	yes	maximal, complemented
3	#13112908	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718586	—	2	2	no	—
9	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
10	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	#11718050	C₂²	1	4	yes	—
14	#11718049	—	2	2	no	—
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₁₃×C₁₆	—	2	2	no	—
19	C₂×C₁₃×C₁₆	C₈	1	4	yes	complemented
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²⋊#1055	—	4	2	no	—
29	C₄⋊#1055	—	2	4	no	—
30	C₄×#1055	—	4	2	no	—
31	#12008	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
35	C₂×C₈×C₁₃	—	2	2	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	M₁₆	1	2	yes	—
40	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₃	D₈	1	1	yes	characteristic
44	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
45	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×#1055	—	4	4	no	—
47	C₁₃×C₁₆	—	2	1	no	—
48	C₁₃×C₁₆	—	2	1	no	—
49	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
50	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
51	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
52	#1564	—	13	2	no	—
53	#1372	—	13	2	no	—
54	#1371	—	13	2	no	—
55	C₂³×C₁₃	#107	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#108	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₁₃	—	2	1	no	—
67	C₈×C₁₃	—	2	1	no	—
68	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
69	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
70	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
71	#1055	—	8	4	no	—
72	C₂²×C₁₆	Dic₁₃	1	1	yes	characteristic
73	C₂²×C₁₆	Dic₁₃	1	1	yes	characteristic
74	#431	—	26	2	no	—
75	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
76	C₄×C₁₆	Dic₁₃	1	1	yes	characteristic
77	C₄×C₁₆	Dic₁₃	1	1	yes	characteristic
78	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
79	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
80	#335	—	13	4	no	—
81	#334	—	26	2	no	—
82	C₂²×C₁₃	—	2	1	no	—
83	C₂²×C₁₃	—	2	1	no	—
84	C₂²×C₁₃	#335	1	2	yes	—
85	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
86	C₄×C₁₃	—	2	1	no	—
87	C₄×C₁₃	—	2	1	no	—
88	C₄×C₁₃	#335	1	2	yes	—
89	C₄×C₁₃	#368	1	2	yes	—
90	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
92	C₄×C₁₃	#508	1	2	yes	—
93	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
95	C₂×C₄²	D₅₂	1	1	yes	characteristic
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	#1055	1	4	yes	complemented
98	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
99	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
100	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
101	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
102	C₄⋊C₈	—	26	4	no	—
103	C₂²⋊C₈	—	52	2	no	—
104	C₄×C₈	—	52	2	no	—
105	C₄×C₈	Dic₂₆	1	1	yes	characteristic
106	C₄×C₈	D₅₂	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
109	C₂×C₁₃	—	2	2	no	—
110	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
111	C₂×C₁₃	#2158	1	1	yes	characteristic
112	C₂×C₁₃	C₂×#368	1	1	yes	characteristic
113	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
114	C₂²×C₄	Dic₅₂	1	1	yes	characteristic
115	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	52	4	no	—
118	C₂×C₈	C₂×#1055	1	2	yes	—
119	C₂×C₈	#6099	1	2	yes	—
120	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
121	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
122	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
123	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
124	C₄²	D₁₀₄	1	1	yes	characteristic
125	C₄²	Dic₅₂	1	1	yes	characteristic
126	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
127	C₄²	C₂×D₅₂	1	1	yes	characteristic
128	C₁₆	—	2	1	no	—
129	C₁₆	—	2	1	no	—
130	C₁₆	C₂×#1055	1	2	yes	—
131	C₁₆	C₂×#1055	1	2	yes	—
132	C₁₃	#12008	1	1	yes	characteristic, complemented, Sylow
133	C₂³	#88058	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	C₄⋊#1055	1	2	yes	—
137	C₂×C₄	#88058	1	1	yes	characteristic
138	C₂×C₄	#88058	1	1	yes	characteristic
139	C₂×C₄	#88058	1	1	yes	characteristic
140	C₂×C₄	#88059	1	2	yes	—
141	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
142	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	104	4	no	—
147	C₈	C₄⋊#1055	1	2	yes	—
148	C₈	C₂²×#1055	1	1	yes	characteristic
149	C₈	C₂×#6099	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#11718050	1	2	yes	—
153	C₂²	C₂×#88058	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#11718050	1	2	yes	—
157	C₄	#11718098	1	2	yes	—
158	C₄	#11718565	1	2	yes	—
159	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
160	C₄	C₂×#88059	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#11718050	1	1	yes	characteristic
163	C₂	#13116620	1	1	yes	characteristic
164	C₂	C₂×#11718098	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₁₀₄
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	#107
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#12008
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13112910	C₂	semidirect
3. #13112908	C₂	semidirect
19. C₂×C₁₃×C₁₆	C₈	semidirect
97. C₂×C₁₆	#1055	semidirect
132. C₁₃	#12008	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
#13112910	2
#13112908	2
C₂×C₄×C₁₃×C₁₆	2
#12008	13