Subgroups — #13123792

Order 1664.

456 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6127	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#1055	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718495	C₂	1	2	yes	maximal, complemented
5	#11718491	C₂	1	2	yes	maximal, complemented
6	#11718386	C₂	1	4	yes	maximal, complemented
7	#11718346	C₂	1	2	yes	maximal, complemented
8	C₄×C₈×D₁₃	C₂	1	2	yes	maximal, complemented
9	C₂×#6127	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₂²⋊D₁₃	C₄	1	2	yes	—
12	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
13	#88180	C₄	1	1	yes	characteristic
14	C₂²×#1055	C₂²	1	2	yes	—
15	C₂×C₈×D₁₃	C₂²	1	4	yes	—
16	C₄×D₅₂	C₄	1	1	yes	characteristic
17	C₄×D₅₂	C₂²	1	1	yes	characteristic
18	#88126	C₄	1	2	yes	—
19	C₄²×D₁₃	C₂²	1	2	yes	—
20	C₄×Dic₂₆	C₄	1	1	yes	characteristic
21	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₄×C₈×C₁₃	C₂²	1	2	yes	—
25	#88060	C₂²	1	4	yes	—
26	#88054	C₂²	1	4	yes	—
27	C₈×Dic₁₃	C₂²	1	4	yes	—
28	C₄×#1055	C₂²	1	2	yes	—
29	C₄×#1055	C₂²	1	1	yes	characteristic
30	C₄×#1055	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
35	#6127	C₈	1	8	yes	complemented
36	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₃	C₂³	1	2	yes	—
40	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂³	1	4	yes	—
42	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
47	#6103	C₂×C₄	1	2	yes	—
48	#6103	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
51	#6101	C₂×C₄	1	2	yes	—
52	#6101	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₃	C₂³	1	2	yes	—
56	C₂×#1055	—	2	4	no	—
57	C₂×#1055	C₂³	1	2	yes	—
58	C₂×#1055	C₂³	1	2	yes	—
59	C₈×D₁₃	—	2	8	no	—
60	C₈×#41	—	13	1	no	maximal, Sylow
61	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	C₂×C₈	1	4	yes	complemented
66	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₁₃	C₂×C₈	1	8	yes	complemented
73	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
76	D₅₂	C₂×C₈	1	4	yes	complemented
77	C₄×D₁₃	—	2	4	no	—
78	C₄×D₁₃	—	2	4	no	—
79	C₄×D₁₃	C₂×C₈	1	8	yes	complemented
80	Dic₂₆	C₂×C₈	1	4	yes	complemented
81	C₈×C₁₃	—	2	4	no	—
82	#1055	—	2	2	no	—
83	#1055	#41	1	4	yes	complemented
84	C₄×#41	—	13	1	no	—
85	C₈×Q₈	—	13	2	no	—
86	C₈×D₄	—	13	2	no	—
87	C₈×D₄	—	13	4	no	—
88	#431	—	13	2	no	—
89	#431	D₁₃	1	1	yes	characteristic, complemented
90	C₂×C₄×C₈	—	13	1	no	—
91	C₂×C₄×C₈	—	13	2	no	—
92	C₂²×C₁₃	—	2	1	no	—
93	C₂²×C₁₃	—	2	1	no	—
94	C₂²×C₁₃	C₂²×C₈	1	2	yes	—
95	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₃	—	2	2	no	—
97	C₂×D₁₃	—	2	2	no	—
98	C₂×D₁₃	C₂²×C₈	1	4	yes	—
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	C₂²×C₈	1	2	yes	—
102	C₄×C₁₃	C₂²×C₈	1	2	yes	—
103	C₄×C₁₃	#131	1	2	yes	—
104	C₄×C₁₃	C₂×#41	1	2	yes	—
105	Dic₁₃	—	2	2	no	—
106	Dic₁₃	—	2	2	no	—
107	Dic₁₃	C₂²×C₈	1	4	yes	—
108	C₂×#41	—	13	1	no	—
109	C₂²×C₈	—	13	2	no	—
110	C₂²×C₈	—	13	4	no	—
111	C₄×Q₈	—	13	1	no	—
112	C₄×Q₈	—	13	1	no	—
113	C₄×D₄	—	13	2	no	—
114	C₄×D₄	—	13	2	no	—
115	C₄×D₄	—	13	1	no	—
116	C₄×D₄	—	13	1	no	—
117	#117	—	13	1	no	—
118	#117	—	13	2	no	—
119	C₂×C₄²	—	13	2	no	—
120	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
121	C₄⋊C₈	—	13	4	no	—
122	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
123	C₂²⋊C₈	—	13	4	no	—
124	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
125	C₄×C₈	—	13	4	no	—
126	C₄×C₈	—	13	2	no	—
127	C₄×C₈	—	13	1	no	—
128	C₄×C₈	—	13	1	no	—
129	C₄×C₈	C₂×D₁₃	1	2	yes	—
130	C₂×C₁₃	—	2	2	no	—
131	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
132	C₂×C₁₃	C₂³×C₈	1	1	yes	characteristic
133	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
134	D₁₃	—	2	4	no	—
135	#41	—	13	8	no	—
136	C₂×Q₈	—	13	1	no	—
137	C₂×D₄	—	13	2	no	—
138	C₂×D₄	—	13	1	no	—
139	C₂²×C₄	—	13	2	no	—
140	C₂²×C₄	—	13	2	no	—
141	C₂²×C₄	—	13	2	no	—
142	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
143	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
145	C₂×C₈	—	13	2	no	—
146	C₂×C₈	—	13	2	no	—
147	C₂×C₈	—	26	8	no	—
148	C₂×C₈	—	26	4	no	—
149	C₂×C₈	C₂²×D₁₃	1	4	yes	—
150	C₄⋊C₄	—	13	2	no	—
151	C₄⋊C₄	—	13	2	no	—
152	C₄⋊C₄	—	13	1	no	—
153	C₄⋊C₄	—	13	1	no	—
154	C₂²⋊C₄	—	13	2	no	—
155	C₂²⋊C₄	—	13	2	no	—
156	C₂²⋊C₄	—	13	2	no	—
157	C₄²	—	13	2	no	—
158	C₄²	—	13	2	no	—
159	C₄²	—	13	2	no	—
160	C₄²	C₄×D₁₃	1	1	yes	characteristic
161	C₄²	C₄×D₁₃	1	1	yes	characteristic
162	C₄²	C₂²×D₁₃	1	1	yes	characteristic
163	C₄²	C₂²×D₁₃	1	1	yes	characteristic
164	C₁₃	C₈×#41	1	1	yes	characteristic, complemented, Sylow
165	C₂³	—	13	2	no	—
166	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
167	Q₈	—	13	4	no	—
168	D₄	—	13	8	no	—
169	D₄	—	13	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	13	2	no	—
173	C₂×C₄	—	13	8	no	—
174	C₂×C₄	—	13	2	no	—
175	C₂×C₄	—	13	2	no	—
176	C₂×C₄	—	26	4	no	—
177	C₂×C₄	—	26	4	no	—
178	C₂×C₄	C₈×D₁₃	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	13	4	no	—
187	C₈	—	26	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	13	4	no	—
191	C₂²	—	26	2	no	—
192	C₂²	—	26	2	no	—
193	C₂²	C₂×C₈×D₁₃	1	2	yes	—
194	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	13	4	no	—
198	C₄	—	26	2	no	—
199	C₄	—	26	2	no	—
200	C₄	C₂×C₈×D₁₃	1	2	yes	—
201	C₄	C₂×C₈×D₁₃	1	2	yes	—
202	C₄	#88161	1	2	yes	—
203	C₄	#41×D₁₃	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	26	4	no	—
206	C₂	#11719227	1	1	yes	characteristic
207	C₂	C₂²×C₈×D₁₃	1	1	yes	characteristic
208	C₂	C₂×#88161	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#431	C₂
socle	C₂²×C₁₃	C₂³×C₄
prime radical	C₂×#6127	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#431	D₁₃
hypocenter	C₁₃	C₈×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. C₁₃×#431	C₂	semidirect
3. C₂×C₄×#1055	C₂	semidirect
4. #11718495	C₂	semidirect
5. #11718491	C₂	semidirect
6. #11718386	C₂	semidirect
7. #11718346	C₂	semidirect
8. C₄×C₈×D₁₃	C₂	semidirect
35. #6127	C₈	semidirect
65. C₂×C₄×C₁₃	C₂×C₈	semidirect
72. C₂²⋊D₁₃	C₂×C₈	semidirect
76. D₅₂	C₂×C₈	semidirect
79. C₄×D₁₃	C₂×C₈	semidirect
80. Dic₂₆	C₂×C₈	semidirect
83. #1055	#41	semidirect
89. #431	D₁₃	semidirect
164. C₁₃	C₈×#41	semidirect
178. C₂×C₄	C₈×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6127	2
C₁₃×#431	2
C₂×C₄×#1055	2
#11718495	2
#11718491	2
#11718386	2
#11718346	2
C₄×C₈×D₁₃	2
C₈×#41	13