Subgroups — #13123923

Order 1664.

364 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	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#431	C₂	1	1	yes	maximal, characteristic
3	#11718587	C₂	1	1	yes	maximal, characteristic
4	#11718489	C₂	1	4	yes	maximal, complemented
5	#11718380	C₂	1	2	yes	maximal
6	#11718379	C₂	1	2	yes	maximal
7	#11718340	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
9	C₂²×Dic₂₆	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×#6101	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
14	C₄×Dic₂₆	C₄	1	1	yes	characteristic
15	C₄×Dic₂₆	C₄	1	2	yes	—
16	C₄×Dic₂₆	C₄	1	1	yes	characteristic
17	C₄×Dic₂₆	C₂²	1	2	yes	—
18	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
20	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
22	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₂²	1	2	yes	—
24	C₂²⋊#1055	C₂²	1	2	yes	—
25	#88055	C₂²	1	4	yes	—
26	#88054	C₂²	1	4	yes	—
27	#88054	C₂²	1	4	yes	—
28	C₈×Dic₁₃	C₂²	1	4	yes	—
29	C₄⋊#1055	C₂²	1	2	yes	—
30	#88043	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	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
35	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
36	C₂×Dic₂₆	—	2	4	no	—
37	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
39	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂×C₈×C₁₃	C₂³	1	2	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	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	#6101	C₂×C₄	1	2	yes	—
50	#6101	C₂×C₄	1	2	yes	—
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	C₄×Dic₁₃	C₂³	1	2	yes	—
57	C₂×#1055	C₂³	1	4	yes	—
58	#2984	—	13	1	no	maximal, Sylow
59	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
69	C₂×Dic₁₃	—	2	4	no	—
70	C₂×Dic₁₃	—	2	4	no	—
71	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
75	Dic₂₆	—	2	2	no	—
76	Dic₂₆	—	2	4	no	—
77	Dic₂₆	M₁₆	1	4	yes	—
78	C₈×C₁₃	—	2	2	no	—
79	C₈×C₁₃	—	2	2	no	—
80	#1055	—	2	4	no	—
81	C₂×C₄×Q₈	—	13	1	no	—
82	#446	—	13	4	no	—
83	#446	—	13	2	no	—
84	C₈×Q₈	—	13	2	no	—
85	#432	—	13	1	no	—
86	#432	—	13	2	no	—
87	#431	—	13	2	no	—
88	#431	D₁₃	1	1	yes	characteristic
89	C₂²×C₁₃	—	2	1	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	#131	1	2	yes	—
92	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₁₃	—	2	1	no	—
94	C₄×C₁₃	—	2	1	no	—
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	—	2	1	no	—
97	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
98	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
99	Dic₁₃	—	2	2	no	—
100	Dic₁₃	—	2	2	no	—
101	Dic₁₃	—	2	2	no	—
102	Dic₁₃	C₂×M₁₆	1	4	yes	—
103	C₂²×Q₈	—	13	1	no	—
104	C₄×Q₈	—	13	1	no	—
105	C₄×Q₈	—	13	2	no	—
106	C₄×Q₈	—	13	1	no	—
107	C₄×Q₈	—	13	1	no	—
108	C₄×Q₈	—	13	2	no	—
109	C₄×Q₈	—	13	1	no	—
110	C₂×C₄⋊C₄	—	13	2	no	—
111	C₂×C₄⋊C₄	—	13	1	no	—
112	C₂×C₄²	—	13	2	no	—
113	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
114	C₄⋊C₈	—	13	4	no	—
115	C₄⋊C₈	—	13	4	no	—
116	C₄⋊C₈	—	13	2	no	—
117	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
118	C₂²⋊C₈	—	13	2	no	—
119	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
120	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
121	#97	—	13	4	no	—
122	#97	—	13	2	no	—
123	C₄×C₈	—	13	4	no	—
124	C₄×C₈	C₂×D₁₃	1	2	yes	—
125	C₂×C₁₃	—	2	2	no	—
126	C₂×C₁₃	#519	1	1	yes	characteristic
127	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
128	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
129	C₂×Q₈	—	13	1	no	—
130	C₂×Q₈	—	13	2	no	—
131	C₂×Q₈	—	13	1	no	—
132	C₂×Q₈	—	26	4	no	—
133	C₂²×C₄	—	13	2	no	—
134	C₂²×C₄	—	13	2	no	—
135	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
138	C₂×C₈	—	13	4	no	—
139	C₂×C₈	C₂²×D₁₃	1	2	yes	—
140	C₂×C₈	C₂²×D₁₃	1	2	yes	—
141	C₄⋊C₄	—	13	2	no	—
142	C₄⋊C₄	—	13	2	no	—
143	C₄⋊C₄	—	13	2	no	—
144	C₄⋊C₄	—	13	2	no	—
145	C₄⋊C₄	—	13	2	no	—
146	C₄⋊C₄	—	13	1	no	—
147	C₄⋊C₄	—	13	1	no	—
148	C₄²	—	13	2	no	—
149	C₄²	—	13	2	no	—
150	C₄²	—	13	2	no	—
151	C₄²	—	13	2	no	—
152	C₄²	C₄×D₁₃	1	1	yes	characteristic
153	C₄²	C₄×D₁₃	1	1	yes	characteristic
154	C₄²	C₂²×D₁₃	1	1	yes	characteristic
155	C₄²	C₂²×D₁₃	1	1	yes	characteristic
156	C₁₃	#2984	1	1	yes	characteristic, complemented, Sylow
157	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
158	Q₈	—	13	4	no	—
159	Q₈	—	26	4	no	—
160	Q₈	—	26	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	13	2	no	—
165	C₂×C₄	—	13	2	no	—
166	C₂×C₄	—	13	2	no	—
167	C₂×C₄	—	13	2	no	—
168	C₂×C₄	—	26	4	no	—
169	C₂×C₄	—	26	4	no	—
170	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	26	4	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	#88155	1	2	yes	—
182	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	13	4	no	—
188	C₄	—	26	2	no	—
189	C₄	—	26	2	no	—
190	C₄	—	26	2	no	—
191	C₄	M₁₆×D₁₃	1	2	yes	—
192	C₄	#88257	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	#11719221	1	1	yes	characteristic
195	C₂	C₂×#88155	1	1	yes	characteristic
196	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#431	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#431	D₁₃
hypocenter	C₁₃	#2984
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
4. #11718489	C₂	semidirect
7. #11718340	C₂	semidirect
8. C₈×Dic₂₆	C₂	semidirect
156. C₁₃	#2984	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₆	2
C₁₃×#431	2
#11718587	2
#11718489	2
#11718380	2
#11718379	2
#11718340	2
C₈×Dic₂₆	2
#2984	13