Subgroups — C₈×#41×C₁₇ #14525970

Order 2176.

456 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×C₁₇	C₂	1	2	yes	maximal, complemented
3	C₈×D₄×C₁₇	C₂	1	6	yes	maximal, complemented
4	C₁₇×#431	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₈×C₁₇	C₂	1	3	yes	maximal, complemented
6	C₂×#41×C₁₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₇	C₂²	1	6	yes	—
8	C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₄×D₄×C₁₇	C₄	1	3	yes	—
11	C₄×D₄×C₁₇	C₂²	1	3	yes	—
12	C₁₇×#117	C₄	1	3	yes	—
13	C₂×C₄²×C₁₇	C₂²	1	3	yes	—
14	C₁₇×C₄⋊C₈	C₂²	1	6	yes	—
15	C₁₇×C₂²⋊C₈	C₂²	1	6	yes	—
16	C₄×C₈×C₁₇	C₂²	1	3	yes	—
17	C₄×C₈×C₁₇	C₂²	1	6	yes	—
18	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	#41×C₁₇	C₈	1	8	yes	direct product factor
20	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₁₇	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
25	C₂×C₈×C₁₇	—	2	12	no	—
26	C₂×C₈×C₁₇	C₂³	1	6	yes	—
27	C₂×C₈×C₁₇	C₂³	1	2	yes	—
28	C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄×C₁₇	C₂×C₄	1	6	yes	—
31	C₄²×C₁₇	C₂×C₄	1	3	yes	—
32	C₄²×C₁₇	C₂×C₄	1	3	yes	—
33	C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₄²×C₁₇	C₂³	1	3	yes	—
35	C₂³×C₁₇	C₂²×C₄	1	3	yes	—
36	Q₈×C₁₇	C₂×C₈	1	4	yes	complemented
37	D₄×C₁₇	C₂×C₈	1	12	yes	complemented
38	C₂×C₄×C₁₇	—	2	6	no	—
39	C₂×C₄×C₁₇	—	2	6	no	—
40	C₂×C₄×C₁₇	C₂×C₈	1	12	yes	complemented
41	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
47	C₈×C₁₇	—	2	6	no	—
48	C₈×C₁₇	#41	1	4	yes	direct product factor
49	C₈×#41	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₁₇	—	2	3	no	—
51	C₂²×C₁₇	—	2	3	no	—
52	C₂²×C₁₇	C₂²×C₈	1	6	yes	—
53	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₇	—	2	3	no	—
55	C₄×C₁₇	—	2	3	no	—
56	C₄×C₁₇	C₂²×C₈	1	2	yes	—
57	C₄×C₁₇	C₂²×C₈	1	6	yes	—
58	C₄×C₁₇	#131	1	2	yes	—
59	C₄×C₁₇	C₂×#41	1	2	yes	—
60	C₄×#41	C₂×C₁₇	1	1	yes	characteristic
61	C₈×Q₈	C₂×C₁₇	1	2	yes	complemented
62	C₈×D₄	C₂×C₁₇	1	6	yes	complemented
63	#431	C₂×C₁₇	1	3	yes	complemented
64	C₂×C₄×C₈	C₂×C₁₇	1	3	yes	complemented
65	C₂×C₁₇	—	2	6	no	—
66	C₂×C₁₇	C₄×#41	1	1	yes	characteristic
67	C₂×C₁₇	C₂³×C₈	1	1	yes	characteristic
68	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
69	C₂×#41	C₄×C₁₇	1	1	yes	characteristic
70	C₂²×C₈	C₂²×C₁₇	1	6	yes	—
71	C₄×Q₈	C₄×C₁₇	1	1	yes	characteristic
72	C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
73	C₄×D₄	C₄×C₁₇	1	3	yes	—
74	C₄×D₄	C₂²×C₁₇	1	3	yes	—
75	#117	C₄×C₁₇	1	3	yes	—
76	C₂×C₄²	C₂²×C₁₇	1	3	yes	—
77	C₄⋊C₈	C₂²×C₁₇	1	6	yes	—
78	C₂²⋊C₈	C₂²×C₁₇	1	6	yes	—
79	C₄×C₈	C₂²×C₁₇	1	3	yes	—
80	C₄×C₈	C₂²×C₁₇	1	6	yes	—
81	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
82	C₁₇	C₈×#41	1	1	yes	characteristic, direct product factor, Sylow
83	#41	C₈×C₁₇	1	8	yes	direct product factor
84	C₂×Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
85	C₂×D₄	C₂×C₄×C₁₇	1	3	yes	—
86	C₂²×C₄	C₂×C₄×C₁₇	1	3	yes	—
87	C₂²×C₄	C₂×C₄×C₁₇	1	3	yes	—
88	C₂²×C₄	C₂³×C₁₇	1	3	yes	—
89	C₂×C₈	—	2	12	no	—
90	C₂×C₈	C₂³×C₁₇	1	6	yes	—
91	C₂×C₈	C₂³×C₁₇	1	2	yes	—
92	C₄⋊C₄	C₂×C₄×C₁₇	1	3	yes	—
93	C₄⋊C₄	C₂×C₄×C₁₇	1	3	yes	—
94	C₂²⋊C₄	C₂×C₄×C₁₇	1	6	yes	—
95	C₄²	C₂×C₄×C₁₇	1	3	yes	—
96	C₄²	C₂×C₄×C₁₇	1	3	yes	—
97	C₄²	C₂³×C₁₇	1	1	yes	characteristic
98	C₄²	C₂³×C₁₇	1	3	yes	—
99	C₂³	C₂²×C₄×C₁₇	1	3	yes	—
100	Q₈	C₂×C₈×C₁₇	1	4	yes	complemented
101	D₄	C₂×C₈×C₁₇	1	12	yes	complemented
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	C₂×C₈×C₁₇	1	12	yes	complemented
105	C₂×C₄	C₂²×C₄×C₁₇	1	3	yes	—
106	C₂×C₄	C₂²×C₄×C₁₇	1	3	yes	—
107	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
108	C₂×C₄	C₂²×C₄×C₁₇	1	3	yes	—
109	C₂×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
110	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
111	C₈	—	2	6	no	—
112	C₈	#41×C₁₇	1	4	yes	direct product factor
113	C₂²	—	2	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	C₂²×C₈×C₁₇	1	6	yes	—
116	C₂²	C₂³×C₄×C₁₇	1	1	yes	characteristic
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂²×C₈×C₁₇	1	2	yes	—
120	C₄	C₂²×C₈×C₁₇	1	6	yes	—
121	C₄	C₁₇×#131	1	2	yes	—
122	C₄	C₂×#41×C₁₇	1	2	yes	—
123	C₂	—	2	6	no	—
124	C₂	C₄×#41×C₁₇	1	1	yes	characteristic
125	C₂	C₂³×C₈×C₁₇	1	1	yes	characteristic
126	C₂	C₂×C₁₇×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈×C₁₇	C₂²
commutator	C₂	C₂³×C₈×C₁₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
socle	C₂²×C₁₇	C₂³×C₄
prime radical	C₂×#41×C₁₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×C₁₇	C₂	semidirect
3. C₈×D₄×C₁₇	C₂	semidirect
4. C₁₇×#431	C₂	semidirect
5. C₂×C₄×C₈×C₁₇	C₂	semidirect
19. #41×C₁₇	C₈	direct (N × H)
36. Q₈×C₁₇	C₂×C₈	semidirect
37. D₄×C₁₇	C₂×C₈	semidirect
40. C₂×C₄×C₁₇	C₂×C₈	semidirect
48. C₈×C₁₇	#41	direct (N × H)
49. C₈×#41	C₁₇	direct (N × H)
61. C₈×Q₈	C₂×C₁₇	semidirect
62. C₈×D₄	C₂×C₁₇	semidirect
63. #431	C₂×C₁₇	semidirect
64. C₂×C₄×C₈	C₂×C₁₇	semidirect
82. C₁₇	C₈×#41	direct (N × H)
83. #41	C₈×C₁₇	direct (N × H)
100. Q₈	C₂×C₈×C₁₇	semidirect
101. D₄	C₂×C₈×C₁₇	semidirect
104. C₂×C₄	C₂×C₈×C₁₇	semidirect
112. C₈	#41×C₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#41×C₁₇	2
C₈×Q₈×C₁₇	2
C₈×D₄×C₁₇	2
C₁₇×#431	2
C₂×C₄×C₈×C₁₇	2
C₈×#41	17

Subgroups — C8×#41×C17 #14525970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#41×C₁₇ #14525970