Subgroups — #15779305

Order 3328.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115000	C₂	1	1	yes	maximal, characteristic
2	#13114996	C₂	1	1	yes	maximal, characteristic
3	#13112928	C₂	1	4	yes	maximal
4	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88058	—	2	1	no	—
8	C₂×#88058	C₂²	1	2	yes	—
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88057	—	2	1	no	—
11	#11718597	—	2	1	no	—
12	#11718596	—	2	1	no	—
13	C₁₃×#346	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	—	2	2	no	—
18	C₂×C₄⋊Dic₁₃	—	2	2	no	—
19	C₂×C₄⋊Dic₁₃	—	4	2	no	—
20	C₂×C₁₃×C₁₆	—	2	2	no	—
21	C₂×C₁₃×C₁₆	—	2	2	no	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₃	Q₈	1	2	yes	—
25	#88071	—	4	2	no	—
26	#88058	—	2	4	no	—
27	#88058	—	2	4	no	—
28	#88058	—	4	2	no	—
29	#88057	—	4	2	no	—
30	#12890	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	D₈	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₁₃	—	4	2	no	—
34	C₂²×Dic₁₃	—	4	2	no	—
35	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	#41	1	2	yes	—
41	C₂×C₈×C₁₃	#41	1	2	yes	—
42	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₄²×C₁₃	Q₁₆	1	2	yes	—
44	C₄⋊Dic₁₃	—	4	2	no	—
45	C₄⋊Dic₁₃	—	4	2	no	—
46	C₄⋊Dic₁₃	—	4	2	no	—
47	C₄⋊Dic₁₃	—	4	2	no	—
48	C₄⋊Dic₁₃	—	8	4	no	—
49	C₁₃×C₁₆	—	2	4	no	—
50	C₂×#346	D₁₃	1	1	yes	characteristic
51	#1846	—	13	1	no	—
52	#1845	—	13	1	no	—
53	#1380	—	13	4	no	—
54	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	#107	1	4	yes	—
58	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₁₃	—	8	2	no	—
65	C₂×Dic₁₃	—	8	2	no	—
66	C₂×Dic₁₃	—	8	4	no	—
67	C₂×Dic₁₃	—	8	2	no	—
68	C₂×Dic₁₃	—	8	2	no	—
69	C₈×C₁₃	#106	1	4	yes	—
70	C₈×C₁₃	#117	1	2	yes	—
71	C₈×C₁₃	#125	1	2	yes	—
72	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
73	C₂×#107	—	13	2	no	—
74	C₂×#107	—	13	2	no	—
75	C₂×#107	—	26	1	no	—
76	C₂×#106	—	26	1	no	—
77	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
78	#384	—	26	1	no	—
79	#382	—	26	1	no	—
80	#346	Dic₁₃	1	4	yes	complemented
81	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
82	C₂²×C₁₃	#486	1	1	yes	characteristic
83	C₂²×C₁₃	#486	1	1	yes	characteristic
84	C₂²×C₁₃	#509	1	2	yes	—
85	C₂²×C₁₃	#510	1	2	yes	—
86	C₄×C₁₃	—	2	4	no	—
87	C₄×C₁₃	#382	1	1	yes	characteristic
88	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
89	C₄×C₁₃	#487	1	1	yes	characteristic
90	C₄×C₁₃	#487	1	1	yes	characteristic
91	Dic₁₃	—	16	2	no	—
92	Dic₁₃	—	16	2	no	—
93	C₂²×C₈	D₅₂	1	1	yes	characteristic
94	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	26	2	no	—
96	C₂×C₄⋊C₄	—	26	2	no	—
97	C₂×C₄⋊C₄	—	52	2	no	—
98	C₂×C₄²	D₅₂	1	1	yes	characteristic
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
102	#107	—	26	4	no	—
103	#107	—	26	4	no	—
104	#107	—	52	2	no	—
105	#106	—	52	2	no	—
106	C₄×C₈	Dic₂₆	1	2	yes	—
107	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
108	#95	—	52	2	no	—
109	C₂×C₁₃	#1834	1	1	yes	characteristic
110	C₂×C₁₃	#2159	1	2	yes	—
111	C₂×C₁₃	#2243	1	2	yes	—
112	C₂×C₁₃	#2244	1	1	yes	characteristic
113	C₂×C₁₃	#2245	1	1	yes	characteristic
114	C₂²×C₄	—	52	2	no	—
115	C₂²×C₄	—	52	2	no	—
116	C₂²×C₄	D₁₀₄	1	2	yes	—
117	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
118	C₂×C₈	#6095	1	2	yes	—
119	C₂×C₈	#6095	1	2	yes	—
120	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
121	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
122	C₂×C₈	#6127	1	2	yes	—
123	C₂×C₈	#6127	1	2	yes	—
124	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
125	C₄⋊C₄	—	52	2	no	—
126	C₄⋊C₄	—	52	2	no	—
127	C₄⋊C₄	—	52	2	no	—
128	C₄⋊C₄	—	52	2	no	—
129	C₄⋊C₄	—	104	4	no	—
130	C₄²	Dic₅₂	1	2	yes	—
131	C₄²	C₄⋊Dic₁₃	1	2	yes	—
132	C₁₆	—	2	4	no	—
133	C₁₃	#12890	1	1	yes	characteristic, complemented, Sylow
134	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	104	4	no	—
138	C₂×C₄	—	104	2	no	—
139	C₂×C₄	—	104	2	no	—
140	C₂×C₄	—	104	2	no	—
141	C₂×C₄	—	104	2	no	—
142	C₂×C₄	#88058	1	4	yes	—
143	C₂×C₄	#88129	1	1	yes	characteristic
144	C₂×C₄	#88129	1	1	yes	characteristic
145	C₂×C₄	C₂×#6095	1	1	yes	characteristic
146	C₂×C₄	C₂×#6095	1	1	yes	characteristic
147	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
149	C₈	#88057	1	4	yes	—
150	C₈	#88124	1	2	yes	—
151	C₈	#88180	1	2	yes	—
152	C₂²	#11718355	1	1	yes	characteristic
153	C₂²	#11718355	1	1	yes	characteristic
154	C₂²	#11718569	1	2	yes	—
155	C₂²	#11718570	1	2	yes	—
156	C₂²	C₂×#88058	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	208	2	no	—
159	C₄	—	208	2	no	—
160	C₄	#11718342	1	1	yes	characteristic
161	C₄	#11718342	1	1	yes	characteristic
162	C₄	#11718597	1	1	yes	characteristic
163	C₄	C₂×#88057	1	1	yes	characteristic
164	C₂	#13114964	1	1	yes	characteristic
165	C₂	#13116624	1	2	yes	—
166	C₂	#13117073	1	2	yes	—
167	C₂	#13117078	1	1	yes	characteristic
168	C₂	#13117082	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#12890
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. C₁₃×#346	C₄	semidirect
80. #346	Dic₁₃	semidirect
133. C₁₃	#12890	semidirect

Maximal subgroups

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

subgroup	index
#13115000	2
#13114996	2
#13112928	2
C₂×C₁₃×#346	2
#12890	13