Subgroups — #13123912

Order 1664.

402 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
2	#11718895	C₂	1	4	yes	maximal, complemented
3	#11718673	C₂	1	2	yes	maximal, complemented
4	#11718672	C₂	1	4	yes	maximal, complemented
5	D₄×#1055	C₂	1	2	yes	maximal, complemented
6	#11718587	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#41	C₂²	1	2	yes	—
9	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₁₃×#127	C₄	1	1	yes	characteristic
11	C₁₃×#124	C₄	1	2	yes	—
12	C₁₃×#121	C₄	1	2	yes	—
13	C₁₃×#121	C₄	1	2	yes	—
14	C₄×D₄×C₁₃	C₂²	1	2	yes	—
15	C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#6099	C₂²	1	4	yes	—
18	C₂²×#1055	C₂²	1	4	yes	—
19	C₂²⋊#1055	C₂²	1	4	yes	—
20	C₂²⋊#1055	C₂²	1	4	yes	—
21	C₂²⋊#1055	C₂²	1	2	yes	—
22	C₂²⋊#1055	C₂²	1	2	yes	—
23	C₄⋊#1055	C₂²	1	2	yes	—
24	C₄⋊#1055	C₂²	1	2	yes	—
25	#88043	C₂²	1	2	yes	—
26	C₄×#1055	C₂²	1	1	yes	characteristic
27	C₄×#1055	C₂²	1	1	yes	characteristic
28	C₁₃×#41	—	4	4	no	—
29	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
38	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	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	#6099	—	4	4	no	—
47	C₂×#1055	—	4	4	no	—
48	C₂×#1055	C₂³	1	2	yes	—
49	C₂×#1055	C₂³	1	4	yes	—
50	C₂×#1055	C₂³	1	2	yes	—
51	#2981	—	13	1	no	maximal, Sylow
52	C₂³×C₁₃	C₂²×C₄	1	4	yes	—
53	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	Q₈×C₁₃	—	4	2	no	—
55	D₄×C₁₃	—	2	4	no	—
56	D₄×C₁₃	—	4	2	no	—
57	D₄×C₁₃	—	4	2	no	—
58	D₄×C₁₃	—	4	4	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	—	2	4	no	—
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	4	yes	—
69	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	#1055	—	2	2	no	—
72	#1055	—	2	4	no	—
73	#1055	—	2	2	no	—
74	#532	D₁₃	1	1	yes	characteristic
75	#436	—	13	2	no	—
76	#435	—	13	4	no	—
77	C₈×D₄	—	13	2	no	—
78	#432	—	13	1	no	—
79	#431	—	13	1	no	—
80	#408	—	13	4	no	—
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	—	2	4	no	—
83	C₂²×C₁₃	—	2	1	no	—
84	C₂²×C₁₃	—	2	4	no	—
85	C₂²×C₁₃	—	2	1	no	—
86	C₂²×C₁₃	—	2	4	no	—
87	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	—	2	4	no	—
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	—	2	1	no	—
93	C₄×C₁₃	#131	1	2	yes	—
94	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
95	C₂×#41	C₂×D₁₃	1	2	yes	—
96	C₂×M₁₆	—	13	4	no	—
97	C₂²×C₈	—	13	4	no	—
98	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
99	#127	Dic₁₃	1	1	yes	characteristic
100	#124	Dic₁₃	1	2	yes	—
101	#121	Dic₁₃	1	2	yes	—
102	#121	Dic₁₃	1	2	yes	—
103	C₄×D₄	C₂×D₁₃	1	2	yes	—
104	C₄×D₄	C₂×D₁₃	1	2	yes	—
105	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
106	C₄⋊C₈	—	13	2	no	—
107	C₄⋊C₈	—	13	2	no	—
108	C₂²⋊C₈	—	13	2	no	—
109	C₂²⋊C₈	—	13	2	no	—
110	C₂²⋊C₈	—	13	4	no	—
111	C₂²⋊C₈	—	13	4	no	—
112	#97	—	13	2	no	—
113	C₄×C₈	—	13	1	no	—
114	C₄×C₈	—	13	1	no	—
115	C₂×C₁₃	—	4	1	no	—
116	C₂×C₁₃	—	4	4	no	—
117	C₂×C₁₃	#518	1	1	yes	characteristic
118	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₃	#568	1	1	yes	characteristic
120	#41	—	4	4	no	—
121	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
122	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
123	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
124	C₂×D₄	C₂×Dic₁₃	1	4	yes	—
125	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
126	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
127	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
130	M₁₆	—	52	4	no	—
131	C₂×C₈	—	13	2	no	—
132	C₂×C₈	—	13	2	no	—
133	C₂×C₈	—	13	4	no	—
134	C₂×C₈	—	52	4	no	—
135	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
136	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
137	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
138	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
139	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
140	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
141	C₄²	C₂²×D₁₃	1	1	yes	characteristic
142	C₄²	C₂²×D₁₃	1	1	yes	characteristic
143	C₁₃	#2981	1	1	yes	characteristic, complemented, Sylow
144	C₂³	C₂²×Dic₁₃	1	4	yes	—
145	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
146	Q₈	—	4	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
161	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
163	C₈	—	26	2	no	—
164	C₈	—	26	4	no	—
165	C₈	—	26	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#88202	1	2	yes	—
179	C₄	#88251	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	4	4	no	—
182	C₂	#11719503	1	1	yes	characteristic
183	C₂	C₂×#88202	1	1	yes	characteristic
184	C₂	#11719524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting, prime radical	C₁₃×#532	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#532	D₁₃
hypocenter	C₁₃	#2981
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
2. #11718895	C₂	semidirect
3. #11718673	C₂	semidirect
4. #11718672	C₂	semidirect
5. D₄×#1055	C₂	semidirect
6. #11718587	C₂	semidirect
7. #11718586	C₂	semidirect
143. C₁₃	#2981	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#532	2
#11718895	2
#11718673	2
#11718672	2
D₄×#1055	2
#11718587	2
#11718586	2
#2981	13