Subgroups — #13123579

Order 1664.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88155	C₂	1	2	yes	maximal, complemented
2	#11719182	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718771	C₂	1	2	yes	maximal, complemented
5	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718499	C₂	1	2	yes	maximal, complemented
7	#11718497	C₂	1	4	yes	maximal, complemented
8	#11718495	C₂	1	2	yes	maximal, complemented
9	C₂×#6127	C₂²	1	2	yes	—
10	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	#88184	C₄	1	2	yes	—
13	#88184	C₄	1	2	yes	—
14	C₂×#6099	C₂²	1	1	yes	characteristic
15	C₂×#6099	C₂²	1	1	yes	characteristic
16	#88155	—	2	8	no	—
17	C₂×#6094	C₂²	1	4	yes	—
18	C₂×C₈×D₁₃	C₂²	1	4	yes	—
19	#88129	C₄	1	2	yes	—
20	#88128	C₄	1	1	yes	characteristic
21	C₄×D₅₂	C₂²	1	2	yes	—
22	C₄×D₅₂	C₂²	1	2	yes	—
23	#88123	C₄	1	1	yes	characteristic
24	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
26	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	#88060	C₂²	1	4	yes	—
28	#88060	C₂²	1	4	yes	—
29	#88043	C₂²	1	2	yes	—
30	C₄×#1055	C₂²	1	2	yes	—
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	4	yes	—
35	#6127	—	2	4	no	—
36	#6127	—	2	4	no	—
37	C₂×D₅₂	C₂×C₄	1	2	yes	—
38	C₂×D₅₂	C₂×C₄	1	2	yes	—
39	C₂×D₅₂	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₁₃	C₂³	1	4	yes	—
41	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₁₃	—	2	4	no	—
43	C₂×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂×C₈×C₁₃	C₂³	1	2	yes	—
45	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	#6103	C₂×C₄	1	4	yes	—
50	#6103	C₂×C₄	1	4	yes	—
51	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
53	#6099	—	2	2	no	—
54	#6099	D₄	1	4	yes	complemented
55	C₂×#1055	C₂³	1	2	yes	—
56	C₂×#1055	C₂³	1	2	yes	—
57	#6094	—	2	8	no	—
58	C₈×D₁₃	—	2	8	no	—
59	#2935	—	13	1	no	maximal, Sylow
60	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	4	1	no	—
64	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₃	#41	1	2	yes	—
71	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₁₃	—	2	8	no	—
73	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
74	D₅₂	—	2	4	no	—
75	D₅₂	—	2	4	no	—
76	D₅₂	—	4	2	no	—
77	C₄×D₁₃	—	2	4	no	—
78	C₄×D₁₃	—	2	4	no	—
79	Dic₂₆	—	2	4	no	—
80	C₈×C₁₃	—	2	2	no	—
81	C₈×C₁₃	—	2	2	no	—
82	#1055	—	2	2	no	—
83	#1055	C₂×D₄	1	4	yes	—
84	C₂×#131	—	13	2	no	—
85	#532	—	13	1	no	—
86	#436	—	13	2	no	—
87	#435	—	13	4	no	—
88	C₈×D₄	—	13	2	no	—
89	C₂×C₄⋊C₈	D₁₃	1	1	yes	characteristic, complemented
90	#408	—	13	2	no	—
91	C₄×M₁₆	—	13	1	no	—
92	C₂²×C₁₃	—	2	1	no	—
93	C₂²×C₁₃	—	2	1	no	—
94	C₂²×C₁₃	C₄×D₄	1	2	yes	—
95	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₃	—	2	4	no	—
97	C₂×D₁₃	—	2	4	no	—
98	C₂×D₁₃	—	2	4	no	—
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	C₄×D₄	1	2	yes	—
103	C₄×C₁₃	#131	1	2	yes	complemented
104	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
106	Dic₁₃	—	2	4	no	—
107	C₂×#41	—	13	2	no	—
108	#131	—	26	8	no	—
109	C₂×M₁₆	—	13	1	no	—
110	C₂×M₁₆	—	13	1	no	—
111	C₂×M₁₆	—	13	4	no	—
112	C₂²×C₈	—	13	4	no	—
113	C₂²×C₈	C₂×D₁₃	1	2	yes	—
114	C₄⋊Q₈	—	13	1	no	—
115	#127	—	13	1	no	—
116	#124	—	13	2	no	—
117	#121	—	13	2	no	—
118	#121	—	13	2	no	—
119	C₄×D₄	—	13	2	no	—
120	C₄×D₄	—	13	2	no	—
121	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
122	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
123	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
124	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
125	C₂²⋊C₈	—	13	4	no	—
126	C₂²⋊C₈	—	13	4	no	—
127	#97	—	13	2	no	—
128	C₄×C₈	—	13	2	no	—
129	C₂×C₁₃	—	2	2	no	—
130	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
132	C₂×C₁₃	#568	1	1	yes	characteristic
133	D₁₃	—	4	4	no	—
134	#41	—	26	4	no	—
135	#41	—	26	4	no	—
136	C₂×Q₈	—	13	2	no	—
137	C₂×D₄	—	13	4	no	—
138	C₂×D₄	—	13	2	no	—
139	C₂×D₄	—	13	2	no	—
140	C₂×D₄	—	13	2	no	—
141	C₂²×C₄	—	13	4	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	M₁₆	—	13	4	no	—
146	M₁₆	—	26	8	no	—
147	M₁₆	—	26	2	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	13	2	no	—
150	C₂×C₈	—	13	2	no	—
151	C₂×C₈	—	26	8	no	—
152	C₂×C₈	C₂²×D₁₃	1	2	yes	—
153	C₂×C₈	C₂²×D₁₃	1	2	yes	—
154	C₄⋊C₄	—	13	2	no	—
155	C₄⋊C₄	—	13	2	no	—
156	C₂²⋊C₄	—	13	4	no	—
157	C₂²⋊C₄	—	13	4	no	—
158	C₄²	C₄×D₁₃	1	1	yes	characteristic
159	C₄²	C₄×D₁₃	1	1	yes	characteristic
160	C₄²	C₂²×D₁₃	1	1	yes	characteristic
161	C₄²	C₂²×D₁₃	1	1	yes	characteristic
162	C₁₃	#2935	1	1	yes	characteristic, complemented, Sylow
163	C₂³	—	13	4	no	—
164	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
165	Q₈	—	26	4	no	—
166	D₄	—	26	8	no	—
167	D₄	—	26	4	no	—
168	D₄	—	26	4	no	—
169	D₄	—	52	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	13	4	no	—
173	C₂×C₄	—	26	4	no	—
174	C₂×C₄	—	26	4	no	—
175	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
180	C₂×C₄	D₄×D₁₃	1	2	yes	—
181	C₂×C₄	#6131	1	2	yes	—
182	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	—	13	4	no	—
186	C₈	—	26	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	26	4	no	—
190	C₂²	—	26	4	no	—
191	C₂²	—	26	4	no	—
192	C₂²	#88147	1	2	yes	—
193	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	26	4	no	—
198	C₄	#88147	1	2	yes	—
199	C₄	#88155	1	2	yes	complemented
200	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
201	C₄	C₂×#6131	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	52	4	no	—
204	C₂	C₂×#88147	1	1	yes	characteristic
205	C₂	C₂×#88155	1	1	yes	characteristic
206	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×C₄⋊C₈	C₂
prime radical	#11719182	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#2935
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#88155	C₂	semidirect
3. C₂×C₁₃×C₄⋊C₈	C₂	semidirect
4. #11718771	C₂	semidirect
5. C₄×#6099	C₂	semidirect
6. #11718499	C₂	semidirect
7. #11718497	C₂	semidirect
8. #11718495	C₂	semidirect
54. #6099	D₄	semidirect
89. C₂×C₄⋊C₈	D₁₃	semidirect
103. C₄×C₁₃	#131	semidirect
162. C₁₃	#2935	semidirect
199. C₄	#88155	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88155	2
#11719182	2
C₂×C₁₃×C₄⋊C₈	2
#11718771	2
C₄×#6099	2
#11718499	2
#11718497	2
#11718495	2
#2935	13