Subgroups — #13129331

Order 1664.

890 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic
2	#11719426	C₂	1	4	yes	maximal, complemented
3	C₁₃⋊#564	C₂	1	4	yes	maximal, complemented
4	#11719330	C₂	1	8	yes	maximal
5	#11719275	C₂	1	4	yes	maximal, complemented
6	#11719272	C₂	1	4	yes	maximal, complemented
7	#11719223	C₂	1	4	yes	maximal
8	#11719175	C₂	1	2	yes	maximal
9	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₁₃×#125	C₂²	1	2	yes	—
11	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
12	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
13	C₁₃×#117	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	Q₈×Dic₁₃	C₂²	1	8	yes	—
16	#88198	C₂²	1	8	yes	—
17	#88180	C₂²	1	4	yes	—
18	#88178	C₂²	1	4	yes	—
19	#88178	C₂²	1	4	yes	—
20	#88144	C₂²	1	8	yes	—
21	#88144	C₂²	1	16	yes	—
22	#88143	C₂²	1	16	yes	—
23	#88143	C₂²	1	16	yes	—
24	#88141	C₂²	1	8	yes	—
25	#88133	C₂²	1	16	yes	—
26	#88124	C₂²	1	4	yes	—
27	#88124	C₂²	1	4	yes	—
28	#88123	C₂²	1	2	yes	—
29	#88123	C₂²	1	2	yes	—
30	#88123	C₂²	1	4	yes	—
31	C₄×Dic₂₆	C₂²	1	4	yes	—
32	C₄×Dic₂₆	C₂²	1	4	yes	—
33	C₄×Dic₂₆	C₂²	1	8	yes	—
34	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂×Dic₂₆	C₂³	1	4	yes	—
38	C₂×Dic₂₆	C₂³	1	4	yes	—
39	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
41	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
42	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
43	C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₄²×C₁₃	C₂³	1	2	yes	—
45	C₄²×C₁₃	C₂³	1	4	yes	—
46	C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
48	C₄⋊Dic₁₃	C₂³	1	4	yes	—
49	C₄⋊Dic₁₃	C₂³	1	4	yes	—
50	C₄⋊Dic₁₃	C₂³	1	4	yes	—
51	C₄⋊Dic₁₃	C₂³	1	4	yes	—
52	C₄⋊Dic₁₃	C₂³	1	16	yes	—
53	#6101	C₂³	1	8	yes	—
54	#6101	C₂³	1	8	yes	—
55	#6101	C₂³	1	16	yes	—
56	#6101	C₂³	1	16	yes	—
57	C₄×Dic₁₃	C₂³	1	8	yes	—
58	C₄×Dic₁₃	C₂³	1	16	yes	—
59	#3556	—	13	1	no	maximal, Sylow
60	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
61	Q₈×C₁₃	—	2	8	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
65	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
67	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
68	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×Dic₁₃	C₂⁴	1	8	yes	—
70	C₂×Dic₁₃	C₂⁴	1	8	yes	—
71	Dic₂₆	—	2	8	no	—
72	Dic₂₆	—	2	8	no	—
73	#564	—	13	4	no	—
74	#557	—	13	4	no	—
75	#557	—	13	4	no	—
76	#557	—	13	4	no	—
77	#541	—	13	8	no	—
78	#541	—	13	4	no	—
79	#533	—	13	2	no	—
80	#533	D₁₃	1	1	yes	characteristic
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	—	2	2	no	—
83	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	—	2	2	no	—
86	C₄×C₁₃	—	2	8	no	—
87	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
88	C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
89	Dic₁₃	—	2	8	no	—
90	Dic₁₃	—	2	8	no	—
91	C₄⋊Q₈	—	13	8	no	—
92	C₄⋊Q₈	—	13	2	no	—
93	C₄⋊Q₈	—	13	2	no	—
94	C₄⋊Q₈	—	13	4	no	—
95	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
96	#126	—	13	16	no	—
97	#125	—	13	8	no	—
98	#125	—	13	16	no	—
99	#125	—	13	16	no	—
100	#125	—	13	16	no	—
101	#125	—	13	4	no	—
102	#125	—	13	4	no	—
103	#125	C₂×D₁₃	1	2	yes	—
104	C₂²⋊Q₈	—	13	4	no	—
105	C₂²⋊Q₈	—	13	4	no	—
106	C₂²⋊Q₈	C₂×D₁₃	1	4	yes	—
107	C₄×Q₈	—	13	8	no	—
108	C₄×Q₈	—	13	8	no	—
109	C₄×Q₈	—	13	4	no	—
110	C₄×Q₈	—	13	4	no	—
111	C₄×Q₈	—	13	8	no	—
112	C₄×Q₈	C₂×D₁₃	1	4	yes	—
113	#117	—	13	4	no	—
114	#117	C₂×D₁₃	1	2	yes	—
115	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
116	C₂×C₁₃	—	4	1	no	—
117	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
119	C₂×Q₈	—	13	4	no	—
120	C₂×Q₈	—	13	4	no	—
121	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
122	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
123	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
124	C₄⋊C₄	—	13	4	no	—
125	C₄⋊C₄	—	13	4	no	—
126	C₄⋊C₄	—	13	4	no	—
127	C₄⋊C₄	—	13	4	no	—
128	C₄⋊C₄	—	13	8	no	—
129	C₄⋊C₄	—	13	8	no	—
130	C₄⋊C₄	—	13	16	no	—
131	C₄⋊C₄	—	13	16	no	—
132	C₄⋊C₄	—	13	16	no	—
133	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
134	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
135	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
136	C₂²⋊C₄	—	13	8	no	—
137	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
138	C₄²	—	13	8	no	—
139	C₄²	—	13	16	no	—
140	C₄²	C₂²×D₁₃	1	1	yes	characteristic
141	C₄²	C₂²×D₁₃	1	2	yes	—
142	C₄²	C₂²×D₁₃	1	1	yes	characteristic
143	C₄²	C₂²×D₁₃	1	4	yes	—
144	C₁₃	#3556	1	1	yes	characteristic, complemented, Sylow
145	C₂³	C₂³×D₁₃	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	Q₈	—	26	8	no	—
148	Q₈	—	26	8	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	13	8	no	—
152	C₂×C₄	—	13	8	no	—
153	C₂×C₄	C₂³×D₁₃	1	8	yes	—
154	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
155	C₂×C₄	C₂³×D₁₃	1	2	yes	—
156	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₁₃	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	8	no	—
164	C₄	—	26	8	no	—
165	C₄	—	26	8	no	—
166	C₄	#88254	1	2	yes	—
167	C₄	#88257	1	4	yes	—
168	C₂	—	4	1	no	—
169	C₂	C₂×#88254	1	1	yes	characteristic
170	C₂	C₂×#88257	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#533	C₂
prime radical	C₂³×C₁₃	C₂⁴
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3556
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #11719426	C₂	semidirect
3. C₁₃⋊#564	C₂	semidirect
5. #11719275	C₂	semidirect
6. #11719272	C₂	semidirect
144. C₁₃	#3556	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#533	2
#11719426	2
C₁₃⋊#564	2
#11719330	2
#11719275	2
#11719272	2
#11719223	2
#11719175	2
#3556	13