Subgroups — #16786652

Order 3328.

306 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131193	C₂	1	2	yes	maximal
2	#13129749	C₂	1	2	yes	maximal, complemented
3	C₈×#6092	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×#1055	C₄	1	1	yes	characteristic
7	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
8	C₂²⋊#6092	C₄	1	4	yes	—
9	C₄⋊#6092	C₄	1	4	yes	—
10	C₄×#6092	C₂²	1	4	yes	—
11	C₈×#1055	C₄	1	1	yes	characteristic
12	C₈×#1055	C₄	1	1	yes	characteristic
13	C₈×#1055	C₂²	1	1	yes	characteristic
14	C₈×#1055	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₈	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×#1055	C₈	1	2	yes	—
18	C₂²×#1055	C₂×C₄	1	2	yes	—
19	C₂×#6092	C₂×C₄	1	8	yes	—
20	C₄×C₈×C₁₃	C₈	1	2	yes	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×#1055	C₈	1	2	yes	—
24	C₄×#1055	C₂×C₄	1	1	yes	characteristic
25	C₄×#1055	C₂×C₄	1	2	yes	—
26	C₄×#1055	C₂×C₄	1	1	yes	characteristic
27	C₄×#1055	C₂×C₄	1	1	yes	characteristic
28	C₄×#1055	C₂³	1	1	yes	characteristic
29	#11410	—	13	1	no	maximal, Sylow
30	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₁₆	1	8	yes	complemented
34	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
36	C₄²×C₁₃	C₄²	1	1	yes	characteristic
37	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×#1055	—	2	4	no	—
41	C₂×#1055	—	2	4	no	—
42	C₂×#1055	C₄²	1	2	yes	—
43	C₂×#1055	C₂×C₈	1	2	yes	—
44	C₂×#1055	C₂×C₈	1	2	yes	—
45	C₂×#1055	C₂²×C₄	1	2	yes	—
46	#6092	—	2	8	no	—
47	#2162	—	13	2	no	—
48	C₂×C₈²	—	13	1	no	—
49	#1312	—	13	2	no	—
50	C₈×C₁₆	—	13	2	no	—
51	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
58	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₈×C₁₃	C₂×C₁₆	1	4	yes	—
62	C₈×C₁₃	C₂×C₁₆	1	4	yes	—
63	#1055	—	2	2	no	—
64	#1055	—	2	2	no	—
65	#1055	M₃₂	1	4	yes	—
66	#1055	#131	1	4	yes	—
67	C₂×C₄×C₈	—	13	1	no	—
68	C₂×C₄×C₈	—	13	1	no	—
69	C₂×C₄×C₈	#261	1	1	yes	characteristic
70	C₄⋊C₁₆	—	13	4	no	—
71	C₂²⋊C₁₆	—	13	4	no	—
72	C₄×C₁₆	—	13	4	no	—
73	C₈²	—	13	1	no	—
74	C₈²	—	13	1	no	—
75	C₈²	—	13	1	no	—
76	C₈²	—	13	1	no	—
77	C₂²×C₁₃	—	2	1	no	—
78	C₂²×C₁₃	—	2	1	no	—
79	C₂²×C₁₃	C₄×C₁₆	1	2	yes	—
80	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	—	2	1	no	—
83	C₄×C₁₃	C₄×C₁₆	1	2	yes	—
84	C₄×C₁₃	#405	1	2	yes	—
85	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
86	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
87	C₂²×C₈	—	13	2	no	—
88	C₂²×C₈	—	13	2	no	—
89	C₂²×C₈	#1057	1	2	yes	—
90	C₂×C₄²	C₂×#261	1	1	yes	characteristic
91	C₂×C₁₆	—	13	8	no	—
92	C₄×C₈	—	13	2	no	—
93	C₄×C₈	—	13	2	no	—
94	C₄×C₈	—	13	1	no	—
95	C₄×C₈	—	13	1	no	—
96	C₄×C₈	—	13	1	no	—
97	C₄×C₈	—	13	1	no	—
98	C₄×C₈	#1057	1	2	yes	—
99	C₄×C₈	C₂×#261	1	1	yes	characteristic
100	C₄×C₈	C₂×#261	1	1	yes	characteristic
101	C₂×C₁₃	—	2	2	no	—
102	C₂×C₁₃	#1450	1	1	yes	characteristic
103	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
104	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
105	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
106	C₂²×C₄	C₄×#261	1	1	yes	characteristic
107	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
108	C₂×C₈	—	13	2	no	—
109	C₂×C₈	—	13	2	no	—
110	C₂×C₈	—	13	2	no	—
111	C₂×C₈	—	13	2	no	—
112	C₂×C₈	—	26	4	no	—
113	C₂×C₈	—	26	4	no	—
114	C₂×C₈	#6092	1	8	yes	complemented
115	C₂×C₈	C₂×#1057	1	2	yes	—
116	C₂×C₈	C₂×#1057	1	2	yes	—
117	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
118	C₄²	C₄×#261	1	1	yes	characteristic
119	C₄²	C₂×#1057	1	1	yes	characteristic
120	C₄²	C₂²×#261	1	1	yes	characteristic
121	C₁₆	—	26	8	no	—
122	C₁₃	#11410	1	1	yes	characteristic, complemented, Sylow
123	C₂³	C₄×#1057	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₂×#6092	1	4	yes	—
127	C₂×C₄	C₄×#1057	1	1	yes	characteristic
128	C₂×C₄	C₄×#1057	1	1	yes	characteristic
129	C₂×C₄	C₄×#1057	1	1	yes	characteristic
130	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
132	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
133	C₈	—	13	4	no	—
134	C₈	—	13	4	no	—
135	C₈	—	26	2	no	—
136	C₈	—	26	2	no	—
137	C₈	C₂×#6092	1	4	yes	—
138	C₈	C₂×#6092	1	4	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₄×#6092	1	2	yes	—
142	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	C₄×#6092	1	2	yes	—
146	C₄	D₁₃⋊M₃₂	1	2	yes	—
147	C₄	#11719089	1	2	yes	—
148	C₄	C₂²×#6092	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#13130039	1	1	yes	characteristic
151	C₂	C₂×C₄×#6092	1	1	yes	characteristic
152	C₂	#13131042	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₄²	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂³×C₁₃	C₄×C₈
cosocle	C₄×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#11410
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13129749	C₂	semidirect
3. C₈×#6092	C₂	semidirect
33. C₂×C₈×C₁₃	C₁₆	semidirect
114. C₂×C₈	#6092	semidirect
122. C₁₃	#11410	semidirect

Maximal subgroups

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

subgroup	index
#13131193	2
#13129749	2
C₈×#6092	2
C₂×C₈×#1055	2
#11410	13