Subgroups — #15754126

Order 3328.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114689	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718049	C₂	1	2	yes	maximal
4	#13112671	C₂	1	2	yes	maximal
5	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	C₂²	1	2	yes	—
10	C₂×C₄⋊#1055	C₂²	1	2	yes	—
11	C₂×#88043	—	2	1	no	—
12	C₂×C₄×#1055	—	2	1	no	—
13	C₁₃×#336	C₄	1	4	yes	—
14	#11718116	—	2	1	no	—
15	#11718116	—	2	1	no	—
16	#11718049	C₄	1	8	yes	complemented
17	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	—	2	2	no	—
25	C₂²×#1055	—	2	2	no	—
26	C₄×C₈×C₁₃	—	2	2	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄⋊#1055	C₂×C₄	1	4	yes	—
31	C₄⋊#1055	C₂×C₄	1	4	yes	—
32	#88043	—	4	2	no	—
33	C₄×#1055	—	4	2	no	—
34	#8277	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	—	2	4	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	2	4	no	—
43	C₂×C₈×C₁₃	C₄²	1	4	yes	—
44	C₄²×C₁₃	—	2	4	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	—	4	2	no	—
47	C₄²×C₁₃	SD₁₆	1	2	yes	—
48	C₄²×C₁₃	SD₁₆	1	2	yes	—
49	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
50	C₄²×C₁₃	#41	1	2	yes	—
51	C₂×#1055	—	2	4	no	—
52	C₂×#1055	—	2	4	no	—
53	C₂×#1055	—	4	2	no	—
54	C₂×#1055	—	4	2	no	—
55	C₂×#1055	—	4	2	no	—
56	C₂×#1055	—	4	2	no	—
57	#1767	—	13	1	no	—
58	C₄×C₄⋊C₈	—	13	1	no	—
59	#1744	D₁₃	1	1	yes	characteristic
60	C₂×#334	—	13	2	no	—
61	#1277	—	13	2	no	—
62	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	#106	1	4	yes	—
70	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₁₃	#131	1	4	yes	—
75	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#135	1	2	yes	—
78	C₈×C₁₃	—	2	4	no	—
79	C₈×C₁₃	—	4	2	no	—
80	#1055	—	4	4	no	—
81	#1055	—	4	4	no	—
82	C₂×C₄⋊C₈	—	13	2	no	—
83	C₂×C₄⋊C₈	—	13	2	no	—
84	C₂×#97	—	26	1	no	—
85	C₂×C₄×C₈	—	26	1	no	—
86	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
88	C₄³	C₂×D₁₃	1	1	yes	characteristic
89	#336	—	26	1	no	—
90	#336	—	26	1	no	—
91	#336	Dic₁₃	1	4	yes	—
92	#334	—	13	8	no	—
93	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₁₃	#424	1	2	yes	—
95	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
96	C₂²×C₁₃	#427	1	1	yes	characteristic
97	C₂²×C₁₃	#430	1	2	yes	—
98	C₄×C₁₃	—	2	4	no	—
99	C₄×C₁₃	—	2	2	no	—
100	C₄×C₁₃	—	4	2	no	—
101	C₄×C₁₃	#405	1	2	yes	—
102	C₄×C₁₃	#433	1	2	yes	—
103	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
104	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
105	C₂²×C₈	—	26	2	no	—
106	C₂²×C₈	—	26	2	no	—
107	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
108	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
109	C₂×C₄²	Dic₂₆	1	2	yes	—
110	C₂×C₄²	D₅₂	1	1	yes	characteristic
111	C₂×C₄²	D₅₂	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
113	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
114	C₄⋊C₈	—	13	4	no	—
115	C₄⋊C₈	—	13	4	no	—
116	C₄⋊C₈	—	26	2	no	—
117	C₄⋊C₈	—	26	2	no	—
118	#97	—	52	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	52	2	no	—
121	C₄×C₈	C₄×D₁₃	1	4	yes	—
122	C₂×C₁₃	#1570	1	2	yes	—
123	C₂×C₁₃	#1572	1	2	yes	—
124	C₂×C₁₃	#1768	1	1	yes	characteristic
125	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
126	C₂×C₁₃	#1778	1	1	yes	characteristic
127	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
128	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
129	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
131	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	26	4	no	—
136	C₂×C₈	—	26	4	no	—
137	C₂×C₈	—	52	2	no	—
138	C₂×C₈	—	52	2	no	—
139	C₂×C₈	—	52	2	no	—
140	C₂×C₈	—	52	2	no	—
141	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	2	no	—
145	C₄²	#6095	1	2	yes	—
146	C₄²	#6095	1	2	yes	—
147	C₄²	C₂×C₄×D₁₃	1	2	yes	—
148	C₄²	#6127	1	2	yes	—
149	C₁₃	#8277	1	1	yes	characteristic, complemented, Sylow
150	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#88057	1	4	yes	—
158	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
159	C₂×C₄	C₄×D₅₂	1	2	yes	—
160	C₂×C₄	C₂×#6095	1	1	yes	characteristic
161	C₂×C₄	C₂×#6095	1	1	yes	characteristic
162	C₂×C₄	#88158	1	2	yes	—
163	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
164	C₂×C₄	#88180	1	1	yes	characteristic
165	C₂×C₄	#88202	1	4	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₈	—	52	4	no	—
169	C₈	—	52	4	no	—
170	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
171	C₂²	#11718660	1	2	yes	—
172	C₂²	C₂×#88057	1	1	yes	characteristic
173	C₂²	#11718767	1	1	yes	characteristic
174	C₂²	#11718786	1	2	yes	—
175	C₄	—	2	4	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	C₄×#6095	1	2	yes	—
179	C₄	C₄×#6095	1	2	yes	—
180	C₄	#11718658	1	2	yes	—
181	C₄	#11718661	1	2	yes	—
182	C₂	#13113607	1	2	yes	—
183	C₂	#13113616	1	2	yes	—
184	C₂	#13114693	1	1	yes	characteristic
185	C₂	C₄×#88057	1	1	yes	characteristic
186	C₂	#13114742	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1744	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#8277
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
16. #11718049	C₄	semidirect
149. C₁₃	#8277	semidirect

Maximal subgroups

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

subgroup	index
#13114689	2
C₄×C₄⋊#1055	2
C₂×#11718049	2
#13112671	2
C₁₃×#1744	2
#8277	13