Subgroups — #15781524

Order 3328.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
2	#13115000	C₂	1	1	yes	maximal, characteristic
3	#13112928	C₂	1	4	yes	maximal
4	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	1	no	—
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	—	2	1	no	—
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88058	C₂²	1	2	yes	—
11	C₂×#88057	—	2	1	no	—
12	#11718597	—	2	1	no	—
13	#11718343	—	2	2	no	—
14	#11718343	C₄	1	4	yes	complemented
15	C₁₃×#346	—	2	2	no	—
16	C₂²×Dic₂₆	—	4	1	no	—
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×C₄⋊Dic₁₃	—	2	2	no	—
22	C₂×C₄⋊Dic₁₃	—	4	1	no	—
23	#88123	—	2	4	no	—
24	#88123	—	4	2	no	—
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	D₄	1	2	yes	—
30	#88071	—	4	1	no	—
31	#88071	—	4	1	no	—
32	#88058	—	2	4	no	—
33	#88058	—	2	2	no	—
34	#88058	—	2	4	no	—
35	#88058	C₂×C₄	1	4	yes	—
36	#88057	—	4	2	no	—
37	#13283	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	4	2	no	—
42	C₂²×Dic₁₃	—	4	2	no	—
43	C₂×Dic₂₆	—	4	4	no	—
44	C₂×Dic₂₆	—	8	4	no	—
45	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
47	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₃	#41	1	2	yes	—
52	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₁₃	SD₁₆	1	2	yes	—
54	C₄⋊Dic₁₃	—	2	4	no	—
55	C₄⋊Dic₁₃	—	4	4	no	—
56	C₄⋊Dic₁₃	—	4	2	no	—
57	C₄⋊Dic₁₃	—	4	2	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	C₁₃×C₁₆	—	2	4	no	—
60	C₂×#500	—	13	1	no	—
61	C₂×#346	D₁₃	1	1	yes	characteristic
62	#1846	—	13	1	no	—
63	#1380	—	13	4	no	—
64	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	#102	1	2	yes	—
68	C₂×C₄×C₁₃	#102	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₁₃	—	4	4	no	—
76	C₂×Dic₁₃	—	8	4	no	—
77	C₂×Dic₁₃	—	8	2	no	—
78	C₂×Dic₁₃	—	8	2	no	—
79	C₂×Dic₁₃	—	8	2	no	—
80	Dic₂₆	—	8	8	no	—
81	C₈×C₁₃	#103	1	4	yes	—
82	C₈×C₁₃	C₄×Q₈	1	2	yes	—
83	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄⋊Q₈	—	26	1	no	—
85	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
86	#500	—	13	4	no	—
87	#500	—	26	2	no	—
88	C₂×#107	—	13	2	no	—
89	C₂×#107	—	13	2	no	—
90	C₂×#107	—	26	1	no	—
91	C₂×#106	—	26	1	no	—
92	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
93	#382	—	26	1	no	—
94	#346	—	2	2	no	—
95	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
96	C₂²×C₁₃	#486	1	1	yes	characteristic
97	C₂²×C₁₃	#500	1	1	yes	characteristic
98	C₂²×C₁₃	#509	1	2	yes	—
99	C₂²×C₁₃	#510	1	2	yes	—
100	C₄×C₁₃	—	2	2	no	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	#391	1	1	yes	characteristic
103	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
104	C₄×C₁₃	#487	1	1	yes	characteristic
105	C₄×C₁₃	#498	1	1	yes	characteristic
106	Dic₁₃	—	8	4	no	—
107	Dic₁₃	—	16	2	no	—
108	C₂²×Q₈	—	52	1	no	—
109	C₂²×C₈	D₅₂	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
111	C₄⋊Q₈	—	26	4	no	—
112	C₄⋊Q₈	—	52	2	no	—
113	C₂×C₄⋊C₄	—	26	2	no	—
114	C₂×C₄⋊C₄	—	26	2	no	—
115	C₂×C₄⋊C₄	—	52	1	no	—
116	C₂×C₄²	D₅₂	1	1	yes	characteristic
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
120	#107	—	13	4	no	—
121	#107	—	26	4	no	—
122	#107	—	26	2	no	—
123	#107	—	26	4	no	—
124	#106	—	52	2	no	—
125	C₄×C₈	C₄×D₁₃	1	2	yes	—
126	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
127	#95	—	52	1	no	—
128	#95	—	52	1	no	—
129	C₂×C₁₃	#1990	1	1	yes	characteristic
130	C₂×C₁₃	#2140	1	1	yes	characteristic
131	C₂×C₁₃	#2141	1	1	yes	characteristic
132	C₂×C₁₃	#2243	1	2	yes	—
133	C₂×C₁₃	#2255	1	2	yes	—
134	C₂×Q₈	—	52	4	no	—
135	C₂×Q₈	—	104	4	no	—
136	C₂²×C₄	—	52	2	no	—
137	C₂²×C₄	—	52	2	no	—
138	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
139	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
141	C₂×C₈	#6095	1	2	yes	—
142	C₂×C₈	Dic₅₂	1	2	yes	—
143	C₂×C₈	#6103	1	2	yes	—
144	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
145	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
146	C₂×C₈	#6127	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
148	C₄⋊C₄	—	26	4	no	—
149	C₄⋊C₄	—	52	2	no	—
150	C₄⋊C₄	—	52	4	no	—
151	C₄⋊C₄	—	52	2	no	—
152	C₄⋊C₄	—	52	2	no	—
153	C₄²	#6095	1	2	yes	—
154	C₄²	#6103	1	2	yes	—
155	C₁₆	—	2	4	no	—
156	C₁₃	#13283	1	1	yes	characteristic, complemented, Sylow
157	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
158	Q₈	—	104	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	52	4	no	—
162	C₂×C₄	—	104	4	no	—
163	C₂×C₄	—	104	2	no	—
164	C₂×C₄	—	104	2	no	—
165	C₂×C₄	—	104	2	no	—
166	C₂×C₄	#88061	1	2	yes	—
167	C₂×C₄	#88061	1	2	yes	—
168	C₂×C₄	#88123	1	1	yes	characteristic
169	C₂×C₄	#88129	1	1	yes	characteristic
170	C₂×C₄	C₂×#6095	1	1	yes	characteristic
171	C₂×C₄	C₂×#6095	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
173	C₂×C₄	C₂×#6103	1	1	yes	characteristic
174	C₈	#88056	1	4	yes	—
175	C₈	C₄×Dic₂₆	1	2	yes	—
176	C₈	#88178	1	2	yes	—
177	C₂²	#11718343	1	1	yes	characteristic
178	C₂²	#11718355	1	1	yes	characteristic
179	C₂²	#11718569	1	2	yes	—
180	C₂²	#11718570	1	2	yes	—
181	C₂²	C₂×#88061	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	104	4	no	—
185	C₄	—	208	2	no	—
186	C₄	#11718341	1	1	yes	characteristic
187	C₄	#11718342	1	1	yes	characteristic
188	C₄	#11718590	1	1	yes	characteristic
189	C₄	C₂×#88056	1	1	yes	characteristic
190	C₂	#13115797	1	1	yes	characteristic
191	C₂	#13116553	1	1	yes	characteristic
192	C₂	#13116557	1	1	yes	characteristic
193	C₂	#13117073	1	2	yes	—
194	C₂	#13117122	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#13283
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
14. #11718343	C₄	semidirect
27. C₂×C₁₃×C₁₆	Q₈	semidirect
119. C₂×C₁₆	Dic₂₆	semidirect
156. C₁₃	#13283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718343	2
#13115000	2
#13112928	2
C₂×C₁₃×#346	2
#13283	13