Subgroups — #15774490

Order 3328.

270 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116592	C₂	1	2	yes	maximal
2	#13112906	C₂	1	4	yes	maximal, complemented
3	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88052	—	2	2	no	—
8	C₈²×C₁₃	C₄	1	2	yes	—
9	C₈²×C₁₃	C₂²	1	2	yes	—
10	C₄⋊#6090	—	2	2	no	—
11	C₄⋊#6090	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₃	—	2	1	no	—
13	C₂²×C₈×C₁₃	C₈	1	2	yes	—
14	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₄×C₈×C₁₃	—	2	1	no	—
18	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
19	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
20	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
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	#88052	—	4	4	no	—
27	C₂×#6090	—	2	4	no	—
28	#12004	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₁₃	—	2	4	no	—
33	C₂×C₈×C₁₃	—	2	2	no	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	—	2	4	no	—
36	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₁₃	D₈	1	2	yes	—
41	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
47	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	#6090	—	4	4	no	—
49	#2150	—	13	2	no	—
50	C₂×C₈²	D₁₃	1	1	yes	characteristic
51	#1370	—	13	4	no	—
52	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#107	1	2	yes	—
59	C₂×C₄×C₁₃	#108	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₁₃	—	2	4	no	—
64	C₈×C₁₃	—	2	1	no	—
65	C₈×C₁₃	—	2	1	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	#107	1	2	yes	—
68	C₈×C₁₃	M₃₂	1	4	yes	complemented
69	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
70	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×M₃₂	—	26	2	no	—
72	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
73	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
75	C₄⋊C₁₆	—	13	4	no	—
76	C₄⋊C₁₆	—	26	2	no	—
77	C₈²	Dic₁₃	1	2	yes	—
78	C₈²	C₂×D₁₃	1	2	yes	—
79	C₂²×C₁₃	—	2	1	no	—
80	C₂²×C₁₃	—	2	1	no	—
81	C₂²×C₁₃	#335	1	2	yes	—
82	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	#335	1	2	yes	—
86	C₄×C₁₃	#364	1	2	yes	—
87	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
88	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
89	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
90	C₂²×C₈	—	2	1	no	—
91	C₂²×C₈	#1055	1	2	yes	—
92	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
93	C₂²×C₈	D₅₂	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
95	M₃₂	—	52	4	no	—
96	C₂×C₁₆	—	26	4	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	#1055	1	1	yes	characteristic
99	C₄×C₈	#1055	1	1	yes	characteristic
100	C₄×C₈	Dic₂₆	1	1	yes	characteristic
101	C₄×C₈	D₅₂	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
103	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
104	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
105	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
106	C₂×C₁₃	—	2	2	no	—
107	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
108	C₂×C₁₃	#2150	1	1	yes	characteristic
109	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
110	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
111	C₂²×C₄	#6099	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	D₁₀₄	1	2	yes	—
118	C₂×C₈	Dic₅₂	1	2	yes	—
119	C₂×C₈	C₂×#1055	1	2	yes	—
120	C₂×C₈	C₂×#1055	1	2	yes	—
121	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
124	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
125	C₄²	C₂×#1055	1	1	yes	characteristic
126	C₄²	#6099	1	1	yes	characteristic
127	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
128	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
129	C₁₆	—	52	4	no	—
130	C₁₃	#12004	1	1	yes	characteristic, complemented, Sylow
131	C₂³	C₄⋊#1055	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
135	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
136	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
137	C₂×C₄	#88058	1	2	yes	—
138	C₂×C₄	#88059	1	2	yes	—
139	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
140	C₂×C₄	C₂×#6099	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	#88052	1	4	yes	complemented
147	C₈	#88058	1	2	yes	—
148	C₈	C₂×D₁₀₄	1	1	yes	characteristic
149	C₈	C₂×Dic₅₂	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#11718050	1	2	yes	—
153	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#11718050	1	2	yes	—
157	C₄	#11718055	1	2	yes	—
158	C₄	C₂×#88052	1	2	yes	—
159	C₄	C₂×#88058	1	1	yes	characteristic
160	C₄	C₂×#88059	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#11718050	1	1	yes	characteristic
163	C₂	#13116592	1	1	yes	characteristic
164	C₂	C₂×#11718055	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#12004
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #13112906	C₂	semidirect
68. C₈×C₁₃	M₃₂	semidirect
130. C₁₃	#12004	semidirect
146. C₈	#88052	semidirect

Maximal subgroups

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

subgroup	index
#13116592	2
#13112906	2
C₂×C₈²×C₁₃	2
#12004	13