Subgroups — #15774530

Order 3328.

246 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116635	C₂	1	1	yes	maximal, characteristic
2	C₈×#6099	C₂	1	1	yes	maximal, characteristic
3	#13112905	C₂	1	2	yes	maximal, complemented
4	#13112899	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	—	2	1	no	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718586	C₄	1	1	yes	characteristic
9	C₄×#6099	C₄	1	1	yes	characteristic
10	C₁₃×C₄⋊C₁₆	—	2	1	no	—
11	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
12	C₂²⋊#6090	—	2	1	no	—
13	C₄⋊#6090	C₂²	1	1	yes	characteristic
14	C₄⋊#6090	C₂²	1	1	yes	characteristic
15	C₄×#6090	—	2	1	no	—
16	#11718046	C₄	1	2	yes	—
17	C₈×#1055	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#6099	C₈	1	1	yes	characteristic
22	C₂×#6099	C₈	1	1	yes	characteristic
23	C₂×C₁₃×C₁₆	—	2	2	no	—
24	C₂×C₁₃×C₁₆	—	4	2	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²⋊#1055	—	2	1	no	—
30	C₂×#6090	—	2	1	no	—
31	C₂×#6090	—	2	1	no	—
32	C₄⋊#1055	—	2	1	no	—
33	#88043	C₈	1	2	yes	—
34	C₄×#1055	C₂×C₄	1	2	yes	—
35	C₄×#1055	C₂×C₄	1	2	yes	—
36	#12015	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₃	—	2	1	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	4	1	no	—
43	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
50	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	#6099	—	2	2	no	—
52	#6099	C₁₆	1	4	yes	complemented
53	C₂×#1055	—	2	2	no	—
54	C₂×#1055	C₂×C₈	1	2	yes	—
55	C₂×#1055	C₂×C₈	1	2	yes	—
56	C₁₃×C₁₆	—	4	2	no	—
57	#6090	—	4	1	no	—
58	#6090	—	4	1	no	—
59	#2162	—	13	1	no	—
60	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic
61	C₈×M₁₆	—	13	1	no	—
62	#1370	—	13	2	no	—
63	#1367	—	13	2	no	—
64	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	4	1	no	—
68	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
72	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
73	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
76	C₈×C₁₃	—	2	1	no	—
77	C₈×C₁₃	—	2	1	no	—
78	C₈×C₁₃	—	2	1	no	—
79	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₈×C₁₃	#137	1	1	yes	characteristic
81	#1055	—	2	2	no	—
82	#1055	—	4	2	no	—
83	#1055	C₂×C₁₆	1	4	yes	complemented
84	C₂²×C₁₆	—	2	1	no	—
85	#431	—	13	1	no	—
86	C₄×M₁₆	—	13	1	no	—
87	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
88	C₄⋊C₁₆	—	2	1	no	—
89	C₄⋊C₁₆	—	13	1	no	—
90	C₄⋊C₁₆	—	13	1	no	—
91	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
92	C₂²⋊C₁₆	—	26	1	no	—
93	C₄×C₁₆	—	26	1	no	—
94	#322	—	13	2	no	—
95	C₈²	—	13	2	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	—	2	1	no	—
98	C₂²×C₁₃	C₄⋊C₁₆	1	2	yes	—
99	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	C₄⋊C₁₆	1	2	yes	—
104	C₄×C₁₃	#428	1	1	yes	characteristic
105	C₄×C₁₃	#430	1	1	yes	characteristic
106	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
107	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
108	C₂×M₁₆	—	13	1	no	—
109	C₂×M₁₆	—	13	1	no	—
110	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
111	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
112	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	4	2	no	—
115	C₂×C₁₆	—	26	1	no	—
116	C₂×C₁₆	—	26	1	no	—
117	C₄⋊C₈	—	26	1	no	—
118	C₂²⋊C₈	—	26	1	no	—
119	#97	—	13	2	no	—
120	C₄×C₈	—	13	2	no	—
121	C₄×C₈	—	13	2	no	—
122	C₄×C₈	Dic₂₆	1	1	yes	characteristic
123	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
126	C₂×C₁₃	—	2	2	no	—
127	C₂×C₁₃	#1565	1	1	yes	characteristic
128	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
129	C₂×C₁₃	#2153	1	1	yes	characteristic
130	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
131	C₂²×C₄	#6094	1	1	yes	characteristic
132	C₂²×C₄	#6101	1	1	yes	characteristic
133	M₁₆	—	13	4	no	—
134	M₁₆	—	26	2	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	—	13	2	no	—
139	C₂×C₈	—	13	2	no	—
140	C₂×C₈	—	26	2	no	—
141	C₂×C₈	#6101	1	1	yes	characteristic
142	C₂×C₈	#6101	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
145	C₄²	C₈×D₁₃	1	1	yes	characteristic
146	C₄²	#6094	1	1	yes	characteristic
147	C₄²	#6101	1	1	yes	characteristic
148	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
149	C₁₆	—	4	2	no	—
150	C₁₆	—	52	1	no	—
151	C₁₆	—	52	1	no	—
152	C₁₃	#12015	1	1	yes	characteristic, complemented, Sylow
153	C₂³	#88054	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₁₆×D₁₃	1	2	yes	—
158	C₂×C₄	#88038	1	2	yes	—
159	C₂×C₄	#88054	1	1	yes	characteristic
160	C₂×C₄	#88054	1	1	yes	characteristic
161	C₂×C₄	#88054	1	1	yes	characteristic
162	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
163	C₂×C₄	C₂×#6094	1	1	yes	characteristic
164	C₂×C₄	C₂×#6101	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	13	4	no	—
169	C₈	—	26	2	no	—
170	C₈	—	52	2	no	—
171	C₈	#88186	1	1	yes	characteristic
172	C₈	#88196	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#11718093	1	2	yes	—
176	C₂²	C₂×#88054	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#11718093	1	2	yes	—
181	C₄	C₂×C₁₆×D₁₃	1	1	yes	characteristic
182	C₄	C₂×#88038	1	1	yes	characteristic
183	C₄	#11718625	1	1	yes	characteristic
184	C₄	#11718785	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#13113586	1	1	yes	characteristic
187	C₂	C₂×#11718093	1	1	yes	characteristic
188	C₂	#13116601	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	#12015
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #13112905	C₂	semidirect
4. #13112899	C₂	semidirect
52. #6099	C₁₆	semidirect
83. #1055	C₂×C₁₆	semidirect
152. C₁₃	#12015	semidirect

Maximal subgroups

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

subgroup	index
#13116635	2
C₈×#6099	2
#13112905	2
#13112899	2
C₂×C₁₃×C₄⋊C₁₆	2
#12015	13