Subgroups — #15779291

Order 3328.

302 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115008	C₂	1	1	yes	maximal, characteristic
2	#13114993	C₂	1	1	yes	maximal, characteristic
3	#13112935	C₂	1	4	yes	maximal
4	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88059	—	2	1	no	—
8	C₂×#88059	C₂²	1	2	yes	—
9	C₂×#88057	—	2	1	no	—
10	C₂×#88057	C₂²	1	2	yes	—
11	#11718596	—	2	1	no	—
12	#11718585	—	2	1	no	—
13	C₁₃×#346	C₄	1	2	yes	complemented
14	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	—	2	2	no	—
19	C₂×C₄⋊Dic₁₃	—	4	2	no	—
20	C₂×#6099	—	2	2	no	—
21	C₁₃×M₃₂	—	2	2	no	—
22	C₁₃×M₃₂	—	2	2	no	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
28	#88059	—	2	4	no	—
29	#88059	—	4	2	no	—
30	#88057	—	2	4	no	—
31	#88057	—	4	2	no	—
32	C₄⋊#1055	—	4	2	no	—
33	#12886	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₁₃	—	4	2	no	—
37	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	D₈	1	2	yes	—
40	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
41	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₁₃	#41	1	2	yes	—
44	C₂×C₈×C₁₃	#41	1	2	yes	—
45	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₃	SD₁₆	1	2	yes	—
48	C₄⋊Dic₁₃	—	4	2	no	—
49	C₄⋊Dic₁₃	—	4	2	no	—
50	C₄⋊Dic₁₃	—	8	4	no	—
51	#6099	—	4	2	no	—
52	#6099	—	4	2	no	—
53	C₂×#1055	—	4	2	no	—
54	C₁₃×C₁₆	—	2	4	no	—
55	C₄×M₃₂	D₁₃	1	1	yes	characteristic
56	#1848	—	13	1	no	—
57	#1844	—	13	1	no	—
58	#1383	—	13	4	no	—
59	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	#106	1	2	yes	—
62	C₂×C₄×C₁₃	#106	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
65	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×Dic₁₃	—	8	4	no	—
70	C₂×Dic₁₃	—	8	2	no	—
71	C₈×C₁₃	#107	1	2	yes	—
72	C₈×C₁₃	#107	1	2	yes	—
73	C₈×C₁₃	#117	1	2	yes	—
74	C₈×C₁₃	#125	1	2	yes	—
75	#1055	—	8	2	no	—
76	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
77	C₂×#108	—	13	2	no	—
78	C₂×#108	—	26	1	no	—
79	C₂×#106	—	13	2	no	—
80	C₂×#106	—	26	1	no	—
81	#423	—	26	1	no	—
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	#384	—	26	1	no	—
84	#346	Dic₁₃	1	2	yes	complemented
85	C₄×C₁₆	Dic₁₃	1	2	yes	complemented
86	C₂²×C₁₃	—	2	2	no	—
87	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
88	C₂²×C₁₃	#486	1	1	yes	characteristic
89	C₂²×C₁₃	#487	1	1	yes	characteristic
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	#365	1	4	yes	—
92	C₄×C₁₃	#382	1	1	yes	characteristic
93	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
94	C₄×C₁₃	#486	1	1	yes	characteristic
95	C₄×C₁₃	#487	1	1	yes	characteristic
96	Dic₁₃	—	16	2	no	—
97	C₂×M₁₆	—	26	2	no	—
98	C₂²×C₈	D₅₂	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	26	2	no	—
101	C₂×C₄⋊C₄	—	52	2	no	—
102	C₂×C₄²	D₅₂	1	1	yes	characteristic
103	M₃₂	—	2	2	no	—
104	M₃₂	—	2	2	no	—
105	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
106	#108	—	26	4	no	—
107	#108	—	52	2	no	—
108	#106	—	26	4	no	—
109	#106	—	52	2	no	—
110	C₄⋊C₈	—	52	2	no	—
111	C₄×C₈	Dic₂₆	1	1	yes	characteristic
112	C₄×C₈	Dic₂₆	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
115	C₂×C₁₃	—	2	1	no	—
116	C₂×C₁₃	—	2	1	no	—
117	C₂×C₁₃	#1834	1	1	yes	characteristic
118	C₂×C₁₃	C₂×#365	1	2	yes	—
119	C₂²×C₄	—	52	2	no	—
120	C₂²×C₄	#6095	1	2	yes	—
121	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
122	M₁₆	—	52	2	no	—
123	M₁₆	—	52	2	no	—
124	C₂×C₈	—	52	2	no	—
125	C₂×C₈	D₁₀₄	1	2	yes	—
126	C₂×C₈	Dic₅₂	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
130	C₂×C₈	#6127	1	2	yes	—
131	C₂×C₈	#6127	1	2	yes	—
132	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
133	C₄⋊C₄	—	52	2	no	—
134	C₄⋊C₄	—	52	2	no	—
135	C₄⋊C₄	—	104	4	no	—
136	C₄²	#6095	1	2	yes	—
137	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
138	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₁₃	#12886	1	1	yes	characteristic, complemented, Sylow
141	C₂³	C₂×#6095	1	1	yes	characteristic
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	104	4	no	—
144	C₂×C₄	—	104	2	no	—
145	C₂×C₄	#88057	1	2	yes	—
146	C₂×C₄	#88057	1	2	yes	—
147	C₂×C₄	#88129	1	1	yes	characteristic
148	C₂×C₄	#88129	1	1	yes	characteristic
149	C₂×C₄	C₂×#6095	1	1	yes	characteristic
150	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
153	C₈	—	104	2	no	—
154	C₈	#88058	1	2	yes	—
155	C₈	#88058	1	2	yes	—
156	C₈	#88124	1	2	yes	—
157	C₈	#88180	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	#11718342	1	1	yes	characteristic
160	C₂²	#11718355	1	1	yes	characteristic
161	C₂²	C₂×#88057	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	208	2	no	—
164	C₄	#11718105	1	4	yes	—
165	C₄	#11718342	1	1	yes	characteristic
166	C₄	#11718355	1	1	yes	characteristic
167	C₄	#11718597	1	1	yes	characteristic
168	C₄	C₂×#88058	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	#13114964	1	1	yes	characteristic
172	C₂	C₂×#11718105	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88057
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#12886
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. C₁₃×#346	C₄	semidirect
14. C₄×C₁₃×C₁₆	C₄	semidirect
84. #346	Dic₁₃	semidirect
85. C₄×C₁₆	Dic₁₃	semidirect
140. C₁₃	#12886	semidirect

Maximal subgroups

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

subgroup	index
#13115008	2
#13114993	2
#13112935	2
C₄×C₁₃×M₃₂	2
#12886	13