Subgroups — #13123937

Order 1664.

330 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719175	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#431	C₂	1	1	yes	maximal, characteristic
3	#11718586	C₂	1	1	yes	maximal, characteristic
4	#11718492	C₂	1	2	yes	maximal, complemented
5	#11718491	C₂	1	2	yes	maximal, complemented
6	#11718381	C₂	1	4	yes	maximal
7	#11718340	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	#88180	C₂²	1	2	yes	—
11	#88178	C₄	1	2	yes	—
12	#88178	C₄	1	2	yes	—
13	#88124	C₄	1	2	yes	—
14	#88123	C₄	1	1	yes	characteristic
15	#88123	C₄	1	1	yes	characteristic
16	C₄×Dic₂₆	C₂²	1	2	yes	—
17	C₄×Dic₂₆	C₂²	1	2	yes	—
18	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	C₂²⋊#1055	C₂²	1	2	yes	—
23	#88055	C₂²	1	4	yes	—
24	#88054	C₂²	1	4	yes	—
25	#88054	C₂²	1	4	yes	—
26	C₈×Dic₁₃	C₂²	1	4	yes	—
27	C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₄⋊#1055	C₂²	1	1	yes	characteristic
29	C₄×#1055	C₂²	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₈×C₁₃	C₂³	1	2	yes	—
37	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
44	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
46	#6101	C₂×C₄	1	4	yes	—
47	#6101	C₂×C₄	1	4	yes	—
48	C₄×Dic₁₃	C₂³	1	4	yes	—
49	C₂×#1055	C₂³	1	2	yes	—
50	C₂×#1055	C₂³	1	2	yes	—
51	#2986	—	13	1	no	maximal, Sylow
52	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
64	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
65	Dic₂₆	—	2	4	no	—
66	Dic₂₆	—	2	4	no	—
67	C₈×C₁₃	—	2	2	no	—
68	C₈×C₁₃	—	2	2	no	—
69	#1055	—	2	2	no	—
70	#1055	—	2	2	no	—
71	#533	—	13	1	no	—
72	#446	—	13	2	no	—
73	#446	—	13	2	no	—
74	C₈×Q₈	—	13	2	no	—
75	C₈×Q₈	—	13	2	no	—
76	#433	—	13	4	no	—
77	#431	—	13	1	no	—
78	#431	D₁₃	1	1	yes	characteristic
79	C₂²×C₁₃	—	2	1	no	—
80	C₂²×C₁₃	—	2	1	no	—
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	—	2	1	no	—
86	C₄×C₁₃	#131	1	2	yes	—
87	C₄×C₁₃	#131	1	2	yes	—
88	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
89	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
90	Dic₁₃	—	2	4	no	—
91	Dic₁₃	—	2	4	no	—
92	C₄⋊Q₈	—	13	1	no	—
93	C₄⋊Q₈	—	13	1	no	—
94	#125	—	13	2	no	—
95	C₂²⋊Q₈	—	13	2	no	—
96	C₂²⋊Q₈	—	13	2	no	—
97	C₄×Q₈	—	13	2	no	—
98	C₄×Q₈	—	13	2	no	—
99	#117	—	13	2	no	—
100	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
101	C₄⋊C₈	—	13	4	no	—
102	C₄⋊C₈	—	13	4	no	—
103	C₄⋊C₈	—	13	1	no	—
104	C₄⋊C₈	—	13	1	no	—
105	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
106	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
107	C₂²⋊C₈	—	13	2	no	—
108	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
109	#97	—	13	4	no	—
110	C₄×C₈	—	13	4	no	—
111	C₄×C₈	—	13	2	no	—
112	C₄×C₈	C₂×D₁₃	1	2	yes	—
113	C₂×C₁₃	—	4	1	no	—
114	C₂×C₁₃	#519	1	1	yes	characteristic
115	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
116	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
117	C₂×Q₈	—	13	2	no	—
118	C₂×Q₈	—	13	2	no	—
119	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
122	C₂×C₈	—	13	2	no	—
123	C₂×C₈	—	13	2	no	—
124	C₂×C₈	C₂²×D₁₃	1	2	yes	—
125	C₂×C₈	C₂²×D₁₃	1	2	yes	—
126	C₄⋊C₄	—	13	2	no	—
127	C₄⋊C₄	—	13	2	no	—
128	C₄⋊C₄	—	13	4	no	—
129	C₄⋊C₄	—	13	4	no	—
130	C₄⋊C₄	—	13	2	no	—
131	C₄⋊C₄	—	13	2	no	—
132	C₂²⋊C₄	—	13	4	no	—
133	C₄²	—	13	4	no	—
134	C₄²	C₄×D₁₃	1	1	yes	characteristic
135	C₄²	C₄×D₁₃	1	1	yes	characteristic
136	C₄²	C₂²×D₁₃	1	1	yes	characteristic
137	C₄²	C₂²×D₁₃	1	1	yes	characteristic
138	C₁₃	#2986	1	1	yes	characteristic, complemented, Sylow
139	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
140	Q₈	—	26	4	no	—
141	Q₈	—	26	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	13	4	no	—
147	C₂×C₄	—	13	4	no	—
148	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	26	2	no	—
157	C₈	—	26	2	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	26	4	no	—
166	C₄	—	26	4	no	—
167	C₄	#88155	1	2	yes	—
168	C₄	#88161	1	2	yes	—
169	C₄	#88257	1	1	yes	characteristic
170	C₄	#88257	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#11719221	1	1	yes	characteristic
173	C₂	C₂×#88155	1	1	yes	characteristic
174	C₂	C₂×#88161	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	#2986
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. #11718492	C₂	semidirect
5. #11718491	C₂	semidirect
7. #11718340	C₂	semidirect
8. C₈×Dic₂₆	C₂	semidirect
138. C₁₃	#2986	semidirect

Maximal subgroups

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

subgroup	index
#11719175	2
C₁₃×#431	2
#11718586	2
#11718492	2
#11718491	2
#11718381	2
#11718340	2
C₈×Dic₂₆	2
#2986	13