Subgroups — #15774476

Order 3328.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88052	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13112901	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88052	C₄	1	1	yes	characteristic
8	C₂×#88052	C₄	1	1	yes	characteristic
9	#11718586	—	2	1	no	—
10	C₁₃×#334	—	2	1	no	—
11	C₁₃×#334	C₂²	1	2	yes	—
12	#11718053	C₂²	1	2	yes	—
13	C₄×#6090	C₄	1	2	yes	—
14	#11718050	—	2	1	no	—
15	#11718049	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₃	—	4	1	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₁₃×C₄⋊C₈	—	2	2	no	—
21	C₁₃×C₄⋊C₈	—	4	1	no	—
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²⋊#1055	—	4	1	no	—
27	#88052	—	2	2	no	—
28	#88052	C₈	1	4	yes	complemented
29	C₂×#6090	C₂×C₄	1	2	yes	—
30	C₂×#6090	C₂×C₄	1	2	yes	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄×#1055	—	4	1	no	—
33	#12000	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	4	2	no	—
39	C₂×C₈×C₁₃	—	8	2	no	—
40	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
48	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×#1055	—	4	2	no	—
51	#6090	—	2	2	no	—
52	#6090	C₂×C₈	1	4	yes	complemented
53	C₄×M₃₂	—	13	1	no	—
54	#1564	—	13	1	no	—
55	C₂×#334	D₁₃	1	1	yes	characteristic
56	#1368	—	13	4	no	—
57	C₂³×C₁₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#108	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	—	2	1	no	—
70	C₈×C₁₃	—	8	2	no	—
71	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
72	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
74	#1055	—	8	2	no	—
75	C₂×M₃₂	—	13	1	no	—
76	C₂×M₃₂	—	13	1	no	—
77	#431	—	26	1	no	—
78	C₂×C₄⋊C₈	—	2	1	no	—
79	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
80	#346	—	13	2	no	—
81	C₄×C₁₆	—	13	2	no	—
82	#335	—	26	1	no	—
83	#334	—	2	1	no	—
84	#334	—	13	2	no	—
85	#334	C₂×D₁₃	1	2	yes	—
86	C₂²×C₁₃	—	2	1	no	—
87	C₂²×C₁₃	—	2	1	no	—
88	C₂²×C₁₃	#334	1	2	yes	—
89	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	—	2	1	no	—
93	C₄×C₁₃	#334	1	2	yes	—
94	C₄×C₁₃	#365	1	2	yes	—
95	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
97	C₂²×C₈	—	4	1	no	—
98	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
99	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
100	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
101	M₃₂	—	13	4	no	—
102	M₃₂	—	26	2	no	—
103	C₂×C₁₆	—	13	2	no	—
104	C₂×C₁₆	—	13	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄⋊C₈	—	26	2	no	—
108	C₂²⋊C₈	—	52	1	no	—
109	C₄×C₈	—	52	1	no	—
110	C₄×C₈	Dic₂₆	1	1	yes	characteristic
111	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₂×C₁₃	—	2	2	no	—
115	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
116	C₂×C₁₃	C₂×#365	1	1	yes	characteristic
117	C₂×C₁₃	#2155	1	1	yes	characteristic
118	C₂²×C₄	#6101	1	1	yes	characteristic
119	C₂²×C₄	#6105	1	1	yes	characteristic
120	C₂²×C₄	#6106	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	8	2	no	—
124	C₂×C₈	—	52	2	no	—
125	C₂×C₈	C₈×D₁₃	1	2	yes	—
126	C₂×C₈	#6094	1	2	yes	—
127	C₂×C₈	#6101	1	1	yes	characteristic
128	C₂×C₈	#6101	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
131	C₄²	#6101	1	1	yes	characteristic
132	C₄²	#6105	1	1	yes	characteristic
133	C₄²	#6106	1	1	yes	characteristic
134	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
135	C₁₆	—	13	4	no	—
136	C₁₆	—	26	2	no	—
137	C₁₃	#12000	1	1	yes	characteristic, complemented, Sylow
138	C₂³	#88048	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	#88048	1	1	yes	characteristic
142	C₂×C₄	#88048	1	1	yes	characteristic
143	C₂×C₄	#88048	1	1	yes	characteristic
144	C₂×C₄	#88054	1	2	yes	—
145	C₂×C₄	#88062	1	2	yes	—
146	C₂×C₄	C₂×#6101	1	1	yes	characteristic
147	C₂×C₄	C₂×#6105	1	1	yes	characteristic
148	C₂×C₄	C₂×#6106	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	104	2	no	—
153	C₈	#88054	1	2	yes	—
154	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
155	C₈	C₂×#6094	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#11718072	1	2	yes	—
159	C₂²	C₂×#88048	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#11718072	1	2	yes	—
164	C₄	#11718079	1	2	yes	—
165	C₄	C₂×#88054	1	1	yes	characteristic
166	C₄	C₂×#88062	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#11718072	1	1	yes	characteristic
169	C₂	C₂×#11718079	1	1	yes	characteristic
170	C₂	#13116610	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88048
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#334	C₂
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂³×C₁₃	#106
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×#334	D₁₃
hypocenter	C₁₃	#12000
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #13112901	C₂	semidirect
28. #88052	C₈	semidirect
52. #6090	C₂×C₈	semidirect
137. C₁₃	#12000	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88052	2
#13113584	2
#13112901	2
C₂×C₁₃×#334	2
#12000	13