Subgroups — #13123931

Order 1664.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 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	#11718685	C₂	1	2	yes	maximal, complemented
3	#11718685	C₂	1	4	yes	maximal, complemented
4	Q₈×#1055	C₂	1	2	yes	maximal, complemented
5	#11718661	C₂	1	4	yes	maximal
6	#11718587	C₂	1	1	yes	maximal, characteristic
7	#11718586	C₂	1	1	yes	maximal, characteristic
8	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₁₃×#125	C₄	1	2	yes	—
11	C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
12	C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#117	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂²⋊#1055	C₂²	1	2	yes	—
18	C₂²⋊#1055	C₂²	1	2	yes	—
19	C₄⋊#1055	C₂²	1	4	yes	—
20	C₄⋊#1055	C₂²	1	4	yes	—
21	C₄⋊#1055	C₂²	1	2	yes	—
22	C₄⋊#1055	C₂²	1	2	yes	—
23	#88043	C₂²	1	4	yes	—
24	#88043	C₂²	1	2	yes	—
25	C₄×#1055	C₂²	1	4	yes	—
26	C₄×#1055	C₂²	1	1	yes	characteristic
27	C₄×#1055	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×Q₈×C₁₃	C₂×C₄	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	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
40	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₄²×C₁₃	C₂³	1	4	yes	—
45	C₂×#1055	C₂³	1	2	yes	—
46	C₂×#1055	C₂³	1	2	yes	—
47	C₂×#1055	C₂³	1	4	yes	—
48	#2985	—	13	1	no	maximal, Sylow
49	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₁₃	—	2	4	no	—
51	Q₈×C₁₃	—	4	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	4	1	no	—
55	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	4	yes	—
60	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	#1055	—	2	2	no	—
64	#1055	—	2	2	no	—
65	#1055	—	2	4	no	—
66	#533	D₁₃	1	1	yes	characteristic
67	#446	—	13	2	no	—
68	#446	—	13	4	no	—
69	C₈×Q₈	—	13	2	no	—
70	#433	—	13	4	no	—
71	#432	—	13	1	no	—
72	#431	—	13	1	no	—
73	C₂²×C₁₃	—	2	1	no	—
74	C₂²×C₁₃	—	2	1	no	—
75	C₂²×C₁₃	—	2	1	no	—
76	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₁₃	—	2	1	no	—
78	C₄×C₁₃	—	2	1	no	—
79	C₄×C₁₃	—	2	4	no	—
80	C₄×C₁₃	—	2	1	no	—
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	—	2	4	no	—
83	C₄×C₁₃	#131	1	2	yes	—
84	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
85	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
86	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
87	#125	Dic₁₃	1	2	yes	—
88	C₂²⋊Q₈	Dic₁₃	1	2	yes	—
89	C₂²⋊Q₈	Dic₁₃	1	2	yes	—
90	C₄×Q₈	C₂×D₁₃	1	2	yes	—
91	C₄×Q₈	C₂×D₁₃	1	2	yes	—
92	#117	C₂×D₁₃	1	2	yes	—
93	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
94	C₄⋊C₈	—	13	2	no	—
95	C₄⋊C₈	—	13	2	no	—
96	C₄⋊C₈	—	13	4	no	—
97	C₄⋊C₈	—	13	4	no	—
98	C₂²⋊C₈	—	13	2	no	—
99	C₂²⋊C₈	—	13	2	no	—
100	#97	—	13	2	no	—
101	#97	—	13	4	no	—
102	C₄×C₈	—	13	1	no	—
103	C₄×C₈	—	13	1	no	—
104	C₄×C₈	—	13	4	no	—
105	C₂×C₁₃	—	4	1	no	—
106	C₂×C₁₃	#519	1	1	yes	characteristic
107	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
108	C₂×C₁₃	#568	1	1	yes	characteristic
109	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
110	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
111	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
114	C₂×C₈	—	13	2	no	—
115	C₂×C₈	—	13	2	no	—
116	C₂×C₈	—	13	4	no	—
117	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
118	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
119	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
120	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
121	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
122	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
123	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
124	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
125	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
126	C₄²	C₂²×D₁₃	1	1	yes	characteristic
127	C₄²	C₂²×D₁₃	1	4	yes	—
128	C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	C₁₃	#2985	1	1	yes	characteristic, complemented, Sylow
130	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
137	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
138	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
139	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
141	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
142	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
144	C₈	—	26	2	no	—
145	C₈	—	26	2	no	—
146	C₈	—	26	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	4	no	—
157	C₄	#88202	1	2	yes	—
158	C₄	#88254	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#11719517	1	1	yes	characteristic
161	C₂	C₂×#88202	1	1	yes	characteristic
162	C₂	#11719524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#533	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#533	D₁₃
hypocenter	C₁₃	#2985
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
2. #11718685	C₂	semidirect
3. #11718685	C₂	semidirect
4. Q₈×#1055	C₂	semidirect
129. C₁₃	#2985	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
#11718685	2
Q₈×#1055	2
#11718661	2
#11718587	2
#11718586	2
#2985	13