Subgroups — #15781528

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718344	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	C₂×#88124	—	2	1	no	—
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	—	2	1	no	—
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88058	C₂²	1	2	yes	—
11	C₂×#88057	—	2	1	no	—
12	#11718596	—	2	1	no	—
13	#11718344	—	2	2	no	—
14	#11718344	C₄	1	4	yes	complemented
15	C₁₃×#346	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	—	2	2	no	—
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×C₄⋊Dic₁₃	—	4	2	no	—
22	C₂×#6101	—	4	2	no	—
23	#88124	—	2	4	no	—
24	#88124	—	4	2	no	—
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₂×C₁₃×C₁₆	Q₈	1	4	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	D₄	1	2	yes	—
30	#88058	—	2	2	no	—
31	#88058	—	2	4	no	—
32	#88058	—	4	2	no	—
33	#88058	C₂×C₄	1	4	yes	—
34	#88057	—	2	4	no	—
35	#13284	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	D₈	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₃	—	4	2	no	—
39	C₂²×Dic₁₃	—	4	2	no	—
40	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
42	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₁₃	#41	1	2	yes	—
47	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₄²×C₁₃	SD₁₆	1	2	yes	—
49	C₄⋊Dic₁₃	—	2	4	no	—
50	C₄⋊Dic₁₃	—	4	2	no	—
51	C₄⋊Dic₁₃	—	4	2	no	—
52	C₄⋊Dic₁₃	—	4	2	no	—
53	C₄⋊Dic₁₃	—	8	4	no	—
54	#6101	—	4	8	no	—
55	C₁₃×C₁₆	—	2	4	no	—
56	C₂×#499	—	13	1	no	—
57	C₂×#346	D₁₃	1	1	yes	characteristic
58	#1845	—	13	1	no	—
59	#1380	—	13	4	no	—
60	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	#102	1	4	yes	—
64	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×Dic₁₃	—	4	4	no	—
71	C₂×Dic₁₃	—	8	2	no	—
72	C₂×Dic₁₃	—	8	2	no	—
73	C₂×Dic₁₃	—	8	2	no	—
74	C₂×Dic₁₃	—	8	4	no	—
75	C₈×C₁₃	#103	1	4	yes	—
76	C₈×C₁₃	C₄×Q₈	1	2	yes	—
77	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×#125	—	26	1	no	—
79	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
80	#499	—	13	4	no	—
81	#499	—	26	2	no	—
82	C₂×#107	—	13	2	no	—
83	C₂×#107	—	13	2	no	—
84	C₂×#107	—	26	1	no	—
85	C₂×#106	—	26	1	no	—
86	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
87	#384	—	26	1	no	—
88	#346	—	2	2	no	—
89	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
90	C₂²×C₁₃	#486	1	1	yes	characteristic
91	C₂²×C₁₃	#500	1	1	yes	characteristic
92	C₂²×C₁₃	#509	1	2	yes	—
93	C₂²×C₁₃	#510	1	2	yes	—
94	C₄×C₁₃	—	2	4	no	—
95	C₄×C₁₃	#391	1	1	yes	characteristic
96	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
97	C₄×C₁₃	#487	1	1	yes	characteristic
98	C₄×C₁₃	#498	1	1	yes	characteristic
99	Dic₁₃	—	8	4	no	—
100	Dic₁₃	—	16	2	no	—
101	C₂²×C₈	D₅₂	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
103	#125	—	26	4	no	—
104	#125	—	52	2	no	—
105	C₂×C₄⋊C₄	—	26	2	no	—
106	C₂×C₄⋊C₄	—	26	2	no	—
107	C₂×C₄⋊C₄	—	52	2	no	—
108	C₂×C₄⋊C₄	—	52	2	no	—
109	C₂×C₄²	D₅₂	1	1	yes	characteristic
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	Dic₂₆	1	4	yes	—
113	#107	—	13	4	no	—
114	#107	—	26	2	no	—
115	#107	—	26	4	no	—
116	#107	—	52	2	no	—
117	#106	—	26	4	no	—
118	C₄×C₈	C₄×D₁₃	1	2	yes	—
119	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
120	C₂×C₁₃	#1990	1	1	yes	characteristic
121	C₂×C₁₃	#2139	1	2	yes	—
122	C₂×C₁₃	#2244	1	1	yes	characteristic
123	C₂×C₁₃	#2245	1	1	yes	characteristic
124	C₂×C₁₃	#2255	1	2	yes	—
125	C₂²×C₄	—	52	2	no	—
126	C₂²×C₄	—	52	2	no	—
127	C₂²×C₄	D₁₀₄	1	2	yes	—
128	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
129	C₂×C₈	#6095	1	2	yes	—
130	C₂×C₈	Dic₅₂	1	2	yes	—
131	C₂×C₈	#6103	1	2	yes	—
132	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
134	C₂×C₈	#6127	1	2	yes	—
135	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
136	C₄⋊C₄	—	26	4	no	—
137	C₄⋊C₄	—	52	2	no	—
138	C₄⋊C₄	—	52	8	no	—
139	C₄⋊C₄	—	52	2	no	—
140	C₄⋊C₄	—	52	2	no	—
141	C₄⋊C₄	—	104	4	no	—
142	C₄²	#6095	1	2	yes	—
143	C₄²	#6103	1	2	yes	—
144	C₁₆	—	2	4	no	—
145	C₁₃	#13284	1	1	yes	characteristic, complemented, Sylow
146	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	52	4	no	—
150	C₂×C₄	—	104	2	no	—
151	C₂×C₄	—	104	2	no	—
152	C₂×C₄	—	104	4	no	—
153	C₂×C₄	—	104	2	no	—
154	C₂×C₄	#88061	1	4	yes	—
155	C₂×C₄	#88123	1	1	yes	characteristic
156	C₂×C₄	#88129	1	1	yes	characteristic
157	C₂×C₄	C₂×#6095	1	1	yes	characteristic
158	C₂×C₄	C₂×#6095	1	1	yes	characteristic
159	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
160	C₂×C₄	C₂×#6103	1	1	yes	characteristic
161	C₈	#88056	1	4	yes	—
162	C₈	C₄×Dic₂₆	1	2	yes	—
163	C₈	#88178	1	2	yes	—
164	C₂²	#11718343	1	1	yes	characteristic
165	C₂²	#11718355	1	1	yes	characteristic
166	C₂²	#11718569	1	2	yes	—
167	C₂²	#11718570	1	2	yes	—
168	C₂²	C₂×#88061	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	104	4	no	—
171	C₄	—	208	2	no	—
172	C₄	#11718341	1	1	yes	characteristic
173	C₄	#11718342	1	1	yes	characteristic
174	C₄	#11718590	1	1	yes	characteristic
175	C₄	C₂×#88056	1	1	yes	characteristic
176	C₂	#13115797	1	1	yes	characteristic
177	C₂	#13116549	1	2	yes	—
178	C₂	#13117078	1	1	yes	characteristic
179	C₂	#13117082	1	1	yes	characteristic
180	C₂	#13117122	1	2	yes	—

	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	#13284
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
14. #11718344	C₄	semidirect
145. C₁₃	#13284	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718344	2
#13114996	2
#13112928	2
C₂×C₁₃×#346	2
#13284	13