Subgroups — #15771934

Order 3328.

306 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	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
2	C₈×#6099	C₂	1	1	yes	maximal, characteristic
3	#13112755	C₂	1	2	yes	maximal, complemented
4	C₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#6090	C₄	1	2	yes	—
8	#11718586	C₄	1	1	yes	characteristic
9	C₄×#6099	C₄	1	1	yes	characteristic
10	C₁₃×C₄⋊C₁₆	C₄	1	2	yes	—
11	C₁₃×C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
13	C₄×#6090	C₄	1	2	yes	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₄×#6090	C₂²	1	1	yes	characteristic
16	#11718046	C₄	1	2	yes	—
17	C₈×#1055	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#6099	C₈	1	2	yes	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²⋊#1055	C₈	1	2	yes	—
28	C₂×#6090	—	2	4	no	—
29	C₂×#6090	C₂×C₄	1	2	yes	—
30	C₂×#6090	C₂×C₄	1	2	yes	—
31	C₄⋊#1055	C₈	1	2	yes	—
32	#88043	C₈	1	2	yes	—
33	C₄×#1055	C₂×C₄	1	2	yes	—
34	C₄×#1055	C₂×C₄	1	2	yes	—
35	C₁₆×M₁₆	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₁₃	C₄²	1	1	yes	characteristic
46	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	#6099	C₁₆	1	8	yes	complemented
50	C₂×#1055	C₂×C₈	1	4	yes	—
51	C₂×#1055	C₂×C₈	1	4	yes	—
52	C₁₃×C₁₆	—	2	4	no	—
53	#6090	—	2	2	no	—
54	#6090	M₁₆	1	4	yes	complemented
55	#2162	D₁₃	1	1	yes	characteristic
56	C₂×C₄×C₁₆	—	13	1	no	—
57	C₈×M₁₆	—	13	1	no	—
58	#1312	—	13	2	no	—
59	C₈×C₁₆	—	13	2	no	—
60	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	1	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
68	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	C₈×C₁₃	—	2	1	no	—
72	C₈×C₁₃	—	2	1	no	—
73	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
74	C₈×C₁₃	#131	1	2	yes	—
75	#1055	—	2	4	no	—
76	#1055	C₂×C₁₆	1	8	yes	complemented
77	C₂²×C₁₆	—	13	2	no	—
78	#431	—	13	1	no	—
79	C₄×M₁₆	—	13	1	no	—
80	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
81	C₄⋊C₁₆	Dic₁₃	1	2	yes	—
82	C₂²⋊C₁₆	Dic₁₃	1	2	yes	—
83	C₄×C₁₆	—	13	2	no	—
84	C₄×C₁₆	—	13	1	no	—
85	C₄×C₁₆	—	13	1	no	—
86	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
87	#322	—	13	2	no	—
88	C₈²	—	13	2	no	—
89	C₂²×C₁₃	—	2	1	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	C₄×C₁₆	1	2	yes	—
92	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄×C₁₃	—	2	1	no	—
94	C₄×C₁₃	—	2	1	no	—
95	C₄×C₁₃	C₄×C₁₆	1	2	yes	—
96	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₁₃	#405	1	1	yes	characteristic
98	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
99	C₄×C₁₃	#504	1	2	yes	—
100	C₂×M₁₆	—	13	2	no	—
101	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
102	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
104	C₂×C₁₆	—	13	2	no	—
105	C₂×C₁₆	—	13	2	no	—
106	C₂×C₁₆	—	26	4	no	—
107	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
108	C₄⋊C₈	—	13	2	no	—
109	C₂²⋊C₈	—	13	2	no	—
110	#97	—	13	2	no	—
111	C₄×C₈	—	13	2	no	—
112	C₄×C₈	—	13	2	no	—
113	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
114	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
115	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
116	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
117	C₂×C₁₃	—	2	2	no	—
118	C₂×C₁₃	C₈×M₁₆	1	1	yes	characteristic
119	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
120	C₂×C₁₃	#2108	1	1	yes	characteristic
121	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
122	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
123	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
124	M₁₆	—	13	8	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	13	4	no	—
128	C₂×C₈	—	13	4	no	—
129	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
130	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
132	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
133	C₄²	C₈×D₁₃	1	1	yes	characteristic
134	C₄²	C₈×D₁₃	1	1	yes	characteristic
135	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
136	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
137	C₁₆	—	2	4	no	—
138	C₁₆	—	13	4	no	—
139	C₁₆	—	26	2	no	—
140	C₁₃	C₁₆×M₁₆	1	1	yes	characteristic, complemented, Sylow
141	C₂³	C₈×Dic₁₃	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₁₆×D₁₃	1	4	yes	—
146	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
147	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
148	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
149	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	13	8	no	—
155	C₈	—	26	4	no	—
156	C₈	M₁₆×D₁₃	1	2	yes	—
157	C₈	#88202	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₁₆×Dic₁₃	1	2	yes	—
161	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₁₆×Dic₁₃	1	2	yes	—
165	C₄	C₂×C₁₆×D₁₃	1	2	yes	—
166	C₄	#11718568	1	2	yes	—
167	C₄	#11718658	1	1	yes	characteristic
168	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#13113117	1	1	yes	characteristic
171	C₂	C₂×C₁₆×Dic₁₃	1	1	yes	characteristic
172	C₂	#13116435	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Fitting	C₁₃×#2162	C₂
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂³×C₁₃	C₄×C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#2162	D₁₃
hypocenter	C₁₃	C₁₆×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #13112755	C₂	semidirect
4. C₁₆×#1055	C₂	semidirect
49. #6099	C₁₆	semidirect
54. #6090	M₁₆	semidirect
76. #1055	C₂×C₁₆	semidirect
140. C₁₃	C₁₆×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#6090	2
C₈×#6099	2
#13112755	2
C₁₆×#1055	2
C₁₃×#2162	2
C₁₆×M₁₆	13