Subgroups — #15754163

Order 3328.

430 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88058	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
3	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
4	#13112676	C₂	1	2	yes	maximal
5	C₂×C₁₃×#322	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	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₃	—	2	2	no	—
11	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₁₃×#322	C₄	1	4	yes	complemented
14	#11718143	C₄	1	2	yes	—
15	#11718142	—	2	2	no	—
16	#11718049	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	—	2	1	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
23	C₂×C₄×Dic₁₃	—	2	2	no	—
24	C₂²×#1055	—	2	2	no	—
25	C₄×C₈×C₁₃	—	2	1	no	—
26	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	Q₈	1	2	yes	—
31	#88071	—	2	2	no	—
32	#88058	C₈	1	8	yes	complemented
33	C₈×Dic₁₃	—	4	4	no	—
34	C₄⋊#1055	—	2	2	no	—
35	C₄⋊#1055	C₂×C₄	1	4	yes	—
36	#8284	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
40	C₂²×Dic₁₃	—	2	4	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	C₄²	1	2	yes	—
47	C₂×C₈×C₁₃	C₄²	1	2	yes	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
50	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₁₃	#41	1	1	yes	characteristic
54	C₄⋊Dic₁₃	—	2	4	no	—
55	C₄⋊Dic₁₃	C₂×C₈	1	8	yes	—
56	C₄×Dic₁₃	—	4	2	no	—
57	C₄×Dic₁₃	—	4	2	no	—
58	C₂×#1055	—	2	4	no	—
59	C₂×#1055	—	4	2	no	—
60	C₂×#1055	—	4	2	no	—
61	C₄×#107	—	13	1	no	—
62	C₈×C₄⋊C₄	—	13	2	no	—
63	C₂×#334	—	13	1	no	—
64	C₂×#322	D₁₃	1	1	yes	characteristic
65	#1279	—	13	2	no	—
66	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₁₃	#131	1	2	yes	—
76	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₁₃	#137	1	2	yes	—
78	C₂×Dic₁₃	—	2	8	no	—
79	C₂×Dic₁₃	—	4	4	no	—
80	C₂×Dic₁₃	—	4	4	no	—
81	C₈×C₁₃	—	2	2	no	—
82	C₈×C₁₃	—	2	4	no	—
83	C₈×C₁₃	—	4	2	no	—
84	C₈×C₁₃	C₄×C₈	1	4	yes	complemented
85	#1055	—	4	4	no	—
86	C₂×#107	—	13	2	no	—
87	C₂×C₄⋊C₈	—	13	2	no	—
88	C₂×C₄×C₈	—	26	2	no	—
89	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
92	C₄×C₄⋊C₄	—	13	2	no	—
93	#340	—	13	2	no	—
94	#336	—	26	2	no	—
95	#334	—	13	4	no	—
96	#322	Dic₁₃	1	4	yes	complemented
97	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₁₃	C₈×D₄	1	2	yes	—
99	C₂²×C₁₃	#441	1	1	yes	characteristic
100	C₂²×C₁₃	#442	1	1	yes	characteristic
101	C₂²×C₁₃	#444	1	2	yes	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
105	C₄×C₁₃	#405	1	1	yes	characteristic
106	C₄×C₁₃	#428	1	2	yes	—
107	C₄×C₁₃	#440	1	2	yes	—
108	C₄×C₁₃	C₈×Q₈	1	2	yes	—
109	Dic₁₃	—	4	8	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	26	2	no	—
112	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
113	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
114	C₂²×C₈	D₅₂	1	2	yes	—
115	C₂×C₄⋊C₄	—	13	4	no	—
116	C₂×C₄²	—	26	2	no	—
117	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
118	#107	—	13	8	no	—
119	C₄⋊C₈	—	13	4	no	—
120	C₄⋊C₈	—	26	2	no	—
121	C₄×C₈	—	2	1	no	—
122	C₄×C₈	—	52	4	no	—
123	C₄×C₈	Dic₂₆	1	2	yes	—
124	C₄×C₈	C₄×D₁₃	1	2	yes	—
125	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
126	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
127	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
128	#95	—	26	2	no	—
129	C₂×C₁₃	#1579	1	2	yes	—
130	C₂×C₁₃	#1580	1	2	yes	—
131	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₁₃	#1773	1	1	yes	characteristic
133	C₂×C₁₃	#1776	1	1	yes	characteristic
134	C₂²×C₄	—	26	4	no	—
135	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
137	C₂²×C₄	#6127	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	26	4	no	—
144	C₂×C₈	—	52	2	no	—
145	C₂×C₈	—	52	2	no	—
146	C₂×C₈	C₈×D₁₃	1	4	yes	—
147	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
148	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
149	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
150	C₄⋊C₄	—	13	8	no	—
151	C₄⋊C₄	—	26	4	no	—
152	C₄²	—	52	2	no	—
153	C₄²	—	52	2	no	—
154	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
156	C₄²	#6127	1	1	yes	characteristic
157	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
158	C₁₃	#8284	1	1	yes	characteristic, complemented, Sylow
159	C₂³	C₄×D₅₂	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	26	8	no	—
162	C₂×C₄	—	52	4	no	—
163	C₂×C₄	—	52	4	no	—
164	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
165	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
166	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
167	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
168	C₂×C₄	#88155	1	2	yes	—
169	C₂×C₄	#88162	1	2	yes	—
170	C₂×C₄	#88163	1	2	yes	—
171	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
173	C₂×C₄	#88180	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	4	no	—
176	C₈	—	4	2	no	—
177	C₈	—	52	4	no	—
178	C₈	C₈×Dic₁₃	1	4	yes	complemented
179	C₂²	C₈×D₅₂	1	2	yes	—
180	C₂²	#11718372	1	1	yes	characteristic
181	C₂²	#11718377	1	1	yes	characteristic
182	C₂²	#11718378	1	2	yes	—
183	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	52	8	no	—
187	C₄	C₈×Dic₂₆	1	2	yes	—
188	C₄	#11718371	1	2	yes	—
189	C₄	C₂×C₈×Dic₁₃	1	1	yes	characteristic
190	C₄	#11718762	1	1	yes	characteristic
191	C₄	#11718782	1	2	yes	—
192	C₂	#13113656	1	2	yes	—
193	C₂	#13113661	1	2	yes	—
194	C₂	C₈×C₄⋊Dic₁₃	1	1	yes	characteristic
195	C₂	#13114720	1	1	yes	characteristic
196	C₂	#13114733	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#8284
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
13. C₁₃×#322	C₄	semidirect
16. #11718049	C₄	semidirect
32. #88058	C₈	semidirect
84. C₈×C₁₃	C₄×C₈	semidirect
96. #322	Dic₁₃	semidirect
158. C₁₃	#8284	semidirect
178. C₈	C₈×Dic₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88058	2
C₈×C₄⋊Dic₁₃	2
C₂×#11718049	2
#13112676	2
C₂×C₁₃×#322	2
#8284	13