Subgroups — #15781532

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic
2	#13115008	C₂	1	1	yes	maximal, characteristic
3	#13112932	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#11719175	—	2	1	no	—
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88059	—	2	1	no	—
9	C₂×#88059	C₂²	1	2	yes	—
10	#11718767	—	2	1	no	—
11	#11718767	C₂²	1	2	yes	—
12	#11718585	—	2	1	no	—
13	#11718344	—	2	1	no	—
14	#11718344	C₄	1	2	yes	—
15	#11718343	C₄	1	2	yes	—
16	#11718341	—	2	1	no	—
17	C₄×C₁₃×C₁₆	—	2	1	no	—
18	C₄×C₁₃×C₁₆	—	2	1	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	#88180	—	2	2	no	—
23	#88178	—	4	2	no	—
24	C₂×#6099	—	2	2	no	—
25	#88124	—	2	2	no	—
26	#88123	—	2	2	no	—
27	C₄×Dic₂₆	—	4	2	no	—
28	C₂×C₁₃×C₁₆	—	2	2	no	—
29	C₂×C₁₃×C₁₆	—	2	2	no	—
30	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
35	#88059	—	2	4	no	—
36	#88059	—	4	2	no	—
37	#88058	—	2	2	no	—
38	#88058	C₂×C₄	1	4	yes	—
39	#88057	—	2	2	no	—
40	#88057	—	2	2	no	—
41	C₄⋊#1055	—	4	2	no	—
42	#13285	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₂₆	—	4	2	no	—
46	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
49	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₃	#41	1	2	yes	—
54	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₁₃	D₈	1	2	yes	—
57	C₂²⋊Dic₁₃	—	4	2	no	—
58	C₄⋊Dic₁₃	—	2	4	no	—
59	C₄⋊Dic₁₃	—	4	2	no	—
60	#6101	—	4	4	no	—
61	C₄×Dic₁₃	—	4	2	no	—
62	#6099	—	4	2	no	—
63	#6099	—	4	2	no	—
64	C₂×#1055	—	4	2	no	—
65	C₁₃×C₁₆	—	2	4	no	—
66	#3160	—	13	1	no	—
67	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
68	#1848	—	13	1	no	—
69	#1382	—	13	4	no	—
70	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	#102	1	2	yes	—
73	C₂×C₄×C₁₃	#102	1	2	yes	—
74	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×Dic₁₃	—	4	4	no	—
81	Dic₂₆	—	8	4	no	—
82	C₈×C₁₃	#103	1	2	yes	—
83	C₈×C₁₃	#103	1	2	yes	—
84	C₈×C₁₃	C₄×Q₈	1	2	yes	—
85	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
86	#1055	—	8	2	no	—
87	#533	—	26	1	no	—
88	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
89	#500	—	13	2	no	—
90	#499	—	13	2	no	—
91	#499	—	26	1	no	—
92	#498	—	26	1	no	—
93	C₂×#108	—	13	2	no	—
94	C₂×#108	—	26	1	no	—
95	#427	—	13	2	no	—
96	#427	—	26	1	no	—
97	#423	—	26	1	no	—
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	C₄×C₁₆	—	2	1	no	—
100	C₄×C₁₆	—	2	1	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
103	C₂²×C₁₃	#487	1	1	yes	characteristic
104	C₂²×C₁₃	#498	1	1	yes	characteristic
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	#359	1	4	yes	—
107	C₄×C₁₃	#391	1	1	yes	characteristic
108	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
109	C₄×C₁₃	#486	1	1	yes	characteristic
110	C₄×C₁₃	#500	1	1	yes	characteristic
111	Dic₁₃	—	8	4	no	—
112	C₂×M₁₆	—	26	2	no	—
113	C₂²×C₈	D₅₂	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
115	C₄⋊Q₈	—	26	2	no	—
116	#125	—	26	2	no	—
117	C₂²⋊Q₈	—	52	2	no	—
118	C₄×Q₈	—	52	2	no	—
119	#117	—	26	2	no	—
120	C₂×C₄²	D₅₂	1	1	yes	characteristic
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
124	#108	—	26	4	no	—
125	#108	—	52	2	no	—
126	#107	—	13	4	no	—
127	#107	—	26	2	no	—
128	#106	—	26	2	no	—
129	#106	—	26	2	no	—
130	C₄⋊C₈	—	52	2	no	—
131	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
132	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
135	C₂×C₁₃	—	2	1	no	—
136	C₂×C₁₃	—	2	1	no	—
137	C₂×C₁₃	#1990	1	1	yes	characteristic
138	C₂×C₁₃	C₂×#359	1	2	yes	—
139	C₂×Q₈	—	52	2	no	—
140	C₂²×C₄	#6095	1	2	yes	—
141	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
142	M₁₆	—	52	2	no	—
143	M₁₆	—	52	2	no	—
144	C₂×C₈	—	52	2	no	—
145	C₂×C₈	#6095	1	2	yes	—
146	C₂×C₈	Dic₅₂	1	2	yes	—
147	C₂×C₈	#6103	1	1	yes	characteristic
148	C₂×C₈	#6103	1	1	yes	characteristic
149	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
151	C₂×C₈	#6127	1	2	yes	—
152	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	26	4	no	—
154	C₄⋊C₄	—	52	4	no	—
155	C₄⋊C₄	—	52	2	no	—
156	C₂²⋊C₄	—	52	2	no	—
157	C₄²	—	52	2	no	—
158	C₄²	D₁₀₄	1	2	yes	—
159	C₄²	#6103	1	1	yes	characteristic
160	C₄²	#6103	1	1	yes	characteristic
161	C₁₆	—	2	4	no	—
162	C₁₃	#13285	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂×#6095	1	1	yes	characteristic
164	Q₈	—	104	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	52	4	no	—
167	C₂×C₄	#88061	1	2	yes	—
168	C₂×C₄	#88061	1	2	yes	—
169	C₂×C₄	#88123	1	1	yes	characteristic
170	C₂×C₄	#88129	1	1	yes	characteristic
171	C₂×C₄	C₂×#6095	1	1	yes	characteristic
172	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
174	C₂×C₄	C₂×#6103	1	1	yes	characteristic
175	C₈	—	104	2	no	—
176	C₈	#88056	1	2	yes	—
177	C₈	#88056	1	2	yes	—
178	C₈	C₄×Dic₂₆	1	2	yes	—
179	C₈	#88178	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	#11718341	1	1	yes	characteristic
182	C₂²	#11718342	1	1	yes	characteristic
183	C₂²	C₂×#88061	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	104	4	no	—
186	C₄	#11718102	1	4	yes	—
187	C₄	#11718343	1	1	yes	characteristic
188	C₄	#11718355	1	1	yes	characteristic
189	C₄	#11718590	1	1	yes	characteristic
190	C₄	C₂×#88056	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#13115797	1	1	yes	characteristic
194	C₂	C₂×#11718102	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#13285
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
30. C₂×C₁₃×C₁₆	Q₈	semidirect
123. C₂×C₁₆	Dic₂₆	semidirect
162. C₁₃	#13285	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13115008	2
#13112932	2
C₂×C₄×C₁₃×C₁₆	2
#13285	13