Subgroups — #15783411

Order 3328.

198 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718091	C₂	1	1	yes	maximal, characteristic
2	#13116432	C₂	1	1	yes	maximal, characteristic
3	#13113033	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
6	#11718762	C₄	1	1	yes	characteristic
7	C₂×#88052	C₄	1	1	yes	characteristic
8	C₁₃×M₆₄	—	2	1	no	—
9	C₁₃×M₆₄	C₂²	1	2	yes	—
10	C₂×C₁₃×C₃₂	—	2	1	no	—
11	#11718094	C₄	1	2	yes	—
12	C₁₆×Dic₁₃	C₂²	1	2	yes	—
13	#11718091	—	2	1	no	—
14	#11718091	C₂²	1	2	yes	—
15	C₂×#88034	—	2	1	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	#88180	C₈	1	1	yes	characteristic
18	C₂×#6099	C₈	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	Q₈	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	Q₈	1	1	yes	characteristic
24	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
25	#88055	C₈	1	2	yes	—
26	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
27	#88052	—	2	1	no	—
28	#88052	—	2	1	no	—
29	C₂×#6090	C₂×C₄	1	2	yes	—
30	C₁₃×C₃₂	—	2	2	no	—
31	#88034	—	2	2	no	—
32	#13630	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²⋊Dic₁₃	—	2	1	no	—
41	C₄⋊Dic₁₃	—	2	1	no	—
42	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
43	#6099	C₁₆	1	2	yes	—
44	#6099	C₁₆	1	2	yes	—
45	C₂×#1055	C₂×C₈	1	2	yes	—
46	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
48	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
49	C₁₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
50	#6090	—	2	2	no	—
51	C₂×M₆₄	—	13	1	no	—
52	C₂×M₆₄	D₁₃	1	1	yes	characteristic
53	#2108	—	13	1	no	—
54	#1422	—	13	4	no	—
55	C₂³×C₁₃	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₃	M₃₂	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×Dic₁₃	—	2	2	no	—
63	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
64	C₈×C₁₃	C₄⋊C₈	1	1	yes	characteristic
65	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
67	#1055	C₂×C₁₆	1	4	yes	—
68	C₂×M₃₂	—	13	1	no	—
69	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
70	#405	—	13	1	no	—
71	M₆₄	—	2	1	no	—
72	M₆₄	—	13	2	no	—
73	M₆₄	—	26	1	no	—
74	M₆₄	C₂×D₁₃	1	2	yes	—
75	C₂×C₃₂	—	2	1	no	—
76	C₂×C₃₂	—	26	1	no	—
77	#346	—	13	2	no	—
78	C₄×C₁₆	—	13	2	no	—
79	C₂²×C₁₃	—	4	1	no	—
80	C₂²×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
81	C₂²×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
82	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
83	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
84	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
85	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
87	Dic₁₃	—	4	2	no	—
88	C₂×M₁₆	—	13	1	no	—
89	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
90	#117	—	13	1	no	—
91	M₃₂	—	26	1	no	—
92	M₃₂	—	26	1	no	—
93	C₂×C₁₆	—	13	2	no	—
94	C₂×C₁₆	Dic₂₆	1	1	yes	characteristic
95	C₂×C₁₆	Dic₂₆	1	1	yes	characteristic
96	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
98	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
99	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
100	#97	—	13	2	no	—
101	C₄×C₈	—	13	2	no	—
102	C₃₂	—	2	2	no	—
103	C₃₂	—	26	2	no	—
104	C₂×C₁₃	—	2	1	no	—
105	C₂×C₁₃	—	2	1	no	—
106	C₂×C₁₃	—	2	1	no	—
107	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
108	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
109	M₁₆	—	13	2	no	—
110	M₁₆	—	13	2	no	—
111	C₂×C₈	—	13	2	no	—
112	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
113	C₂×C₈	#6094	1	1	yes	characteristic
114	C₂×C₈	#6094	1	1	yes	characteristic
115	C₂×C₈	#6101	1	1	yes	characteristic
116	C₂×C₈	#6101	1	1	yes	characteristic
117	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
118	C₄⋊C₄	—	26	1	no	—
119	C₂²⋊C₄	—	26	1	no	—
120	C₄²	—	13	2	no	—
121	C₁₆	—	26	2	no	—
122	C₁₆	#6101	1	1	yes	characteristic
123	C₁₆	#6101	1	1	yes	characteristic
124	C₁₆	C₂×Dic₂₆	1	1	yes	characteristic
125	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
126	C₁₃	#13630	1	1	yes	characteristic, complemented, Sylow
127	C₂³	#88038	1	1	yes	characteristic
128	C₂×C₄	—	26	2	no	—
129	C₂×C₄	C₁₆×D₁₃	1	1	yes	characteristic
130	C₂×C₄	C₁₆×D₁₃	1	1	yes	characteristic
131	C₂×C₄	#88038	1	1	yes	characteristic
132	C₂×C₄	#88054	1	1	yes	characteristic
133	C₂×C₄	#88054	1	1	yes	characteristic
134	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
135	C₈	—	13	4	no	—
136	C₈	#88054	1	1	yes	characteristic
137	C₈	#88054	1	1	yes	characteristic
138	C₈	C₂×#6094	1	1	yes	characteristic
139	C₈	C₂×#6101	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	#11718093	1	1	yes	characteristic
142	C₂²	#11718093	1	1	yes	characteristic
143	C₂²	C₂×#88038	1	1	yes	characteristic
144	C₄	—	52	2	no	—
145	C₄	#11718093	1	1	yes	characteristic
146	C₄	#11718093	1	1	yes	characteristic
147	C₄	C₂×C₁₆×D₁₃	1	1	yes	characteristic
148	C₄	C₂×#88054	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	C₂×#11718093	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₂×C₁₃×M₆₄	C₂
socle	C₂×C₁₃	C₂×C₄⋊C₁₆
prime radical	C₂³×C₁₃	M₃₂
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	C₂×M₆₄	D₁₃
hypocenter	C₁₃	#13630
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #13113033	C₂	semidirect
126. C₁₃	#13630	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718091	2
#13116432	2
#13113033	2
C₂×C₁₃×M₆₄	2
#13630	13