Subgroups — #16954807

Order 4352.

218 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762643	C₂	1	2	yes	maximal
2	C₄⋊#10587465	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	#11762643	—	2	4	no	—
5	C₂×#10587465	C₂²	1	4	yes	—
6	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₁₇	C₄	1	2	yes	—
10	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₇	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₇	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₁₇	—	2	2	no	—
15	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
16	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
18	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
19	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₁₇	C₈	1	2	yes	—
21	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
22	#10587465	—	2	4	no	—
23	C₂²×C₄×C₁₇	C₁₆	1	2	yes	—
24	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₁₇	—	2	2	no	—
26	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
30	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₁₇	C₁₆	1	1	yes	characteristic
32	C₄²×C₁₇	C₁₆	1	1	yes	characteristic
33	C₄²×C₁₇	C₂×C₈	1	2	yes	—
34	C₁₆×C₁₇	—	2	2	no	—
35	C₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₁₇	C₂×Q₈	1	1	yes	characteristic
38	#13627	—	17	1	no	maximal, Sylow
39	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	2	4	no	—
41	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
42	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
45	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
46	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₁₇	—	2	2	no	—
48	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
49	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
51	C₂×M₆₄	—	17	2	no	—
52	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
53	C₄⋊C₃₂	—	17	4	no	—
54	C₂²×C₁₇	—	2	2	no	—
55	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
56	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₁₇	—	2	2	no	—
58	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
59	C₄×C₁₇	M₆₄	1	4	yes	complemented
60	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
61	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
62	C₂²×C₁₆	Dic₁₇	1	1	yes	characteristic
63	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
64	C₂×C₄×C₈	Dic₁₇	1	1	yes	characteristic
65	M₆₄	—	34	4	no	—
66	C₂×C₃₂	—	17	4	no	—
67	C₄×C₁₆	Dic₁₇	1	2	yes	—
68	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
69	C₂×C₁₇	—	2	2	no	—
70	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
71	C₂×C₁₇	C₂×M₆₄	1	2	yes	—
72	C₂²×C₈	#3622	1	1	yes	characteristic
73	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
74	C₂×C₄²	#3622	1	1	yes	characteristic
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	Dic₃₄	1	2	yes	—
77	C₂×C₁₆	D₆₈	1	2	yes	—
78	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
79	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
80	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
81	C₄×C₈	#3622	1	2	yes	—
82	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
83	C₃₂	—	34	4	no	—
84	C₁₇	#13627	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₄	#63214	1	2	yes	—
86	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	C₂×#3622	1	2	yes	—
89	C₂×C₈	C₂×#3622	1	1	yes	characteristic
90	C₂×C₈	#63223	1	2	yes	—
91	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
92	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
93	C₄²	#63214	1	1	yes	characteristic
94	C₄²	#63214	1	1	yes	characteristic
95	C₄²	C₂×#3622	1	2	yes	—
96	C₁₆	—	2	2	no	—
97	C₁₆	C₄⋊Dic₁₇	1	2	yes	—
98	C₁₆	C₂×Dic₃₄	1	1	yes	characteristic
99	C₁₆	C₂×D₆₈	1	1	yes	characteristic
100	C₂³	C₂×#63214	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₄⋊#3622	1	2	yes	—
103	C₂×C₄	C₂×#63214	1	2	yes	—
104	C₂×C₄	C₂×#63214	1	1	yes	characteristic
105	C₂×C₄	C₂×#63214	1	2	yes	—
106	C₂×C₄	#10587483	1	2	yes	—
107	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	C₄⋊#3622	1	2	yes	—
110	C₈	C₂×#63223	1	1	yes	characteristic
111	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	C₄⋊#63214	1	2	yes	—
114	C₂²	C₂²×#63214	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	C₄⋊#63214	1	2	yes	—
117	C₄	#11762643	1	4	yes	complemented
118	C₄	C₂×C₄⋊#3622	1	1	yes	characteristic
119	C₄	C₂×#10587483	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊#63214	1	1	yes	characteristic
122	C₂	C₂×#11762643	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₇
commutator, socle	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
prime radical	C₂³×C₁₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#13627
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄⋊#10587465	C₂	semidirect
59. C₄×C₁₇	M₆₄	semidirect
84. C₁₇	#13627	semidirect
117. C₄	#11762643	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762643	2
C₄⋊#10587465	2
C₂×C₄×C₁₆×C₁₇	2
#13627	17