Subgroups — #17940947

Order 4352.

10930 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×Dic₁₇	C₂	1	32	yes	maximal, complemented
2	#14541987	C₂	1	20	yes	maximal, complemented
3	C₂×#11763287	C₂	1	10	yes	maximal, complemented
4	C₂²×C₁₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
5	#11763292	C₂²	1	160	yes	complemented
6	#11763287	C₂²	1	40	yes	—
7	#11763287	C₂²	1	80	yes	complemented
8	#11763284	C₂²	1	40	yes	—
9	#41×Dic₁₇	C₂²	1	160	yes	complemented
10	C₂×Q₈×Dic₁₇	C₂²	1	80	yes	complemented
11	C₂×#10587612	C₂²	1	60	yes	—
12	C₂×C₁₇×2₋¹⁺⁴	C₄	1	32	yes	complemented
13	C₂×C₁₇×2₋¹⁺⁴	C₂²	1	16	yes	—
14	C₂²×#41×C₁₇	C₂²	1	10	yes	—
15	C₂³×Q₈×C₁₇	C₂²	1	5	yes	—
16	C₁₇×2₋¹⁺⁴	C₂×C₄	1	256	yes	complemented
17	C₂×#41×C₁₇	C₂×C₄	1	160	yes	complemented
18	C₂×#41×C₁₇	C₂³	1	80	yes	—
19	C₂²×Q₈×C₁₇	C₂×C₄	1	80	yes	complemented
20	C₂²×Q₈×C₁₇	C₂³	1	40	yes	—
21	C₂²×Q₈×C₁₇	C₂³	1	10	yes	—
22	C₂²×D₄×C₁₇	C₂³	1	10	yes	—
23	C₂³×C₄×C₁₇	C₂³	1	15	yes	—
24	C₂×C₂²⋊Dic₁₇	C₂³	1	40	yes	—
25	Q₈×Dic₁₇	C₂³	1	320	yes	—
26	D₄×Dic₁₇	C₂³	1	320	yes	—
27	#10587612	C₂³	1	80	yes	—
28	#10587612	C₂³	1	240	yes	—
29	C₂×C₄⋊Dic₁₇	C₂³	1	120	yes	—
30	C₂×C₄×Dic₁₇	C₂³	1	120	yes	—
31	C₂⁴×C₁₇	C₂⁴	1	5	yes	—
32	#41×C₁₇	C₂²×C₄	1	640	yes	—
33	C₂×Q₈×C₁₇	C₂²×C₄	1	80	yes	—
34	C₂×Q₈×C₁₇	C₂²×C₄	1	320	yes	—
35	C₂×Q₈×C₁₇	C₂⁴	1	40	yes	—
36	C₂×D₄×C₁₇	C₂²×C₄	1	80	yes	—
37	C₂×D₄×C₁₇	C₂⁴	1	40	yes	—
38	C₂²×C₄×C₁₇	C₂²×C₄	1	120	yes	—
39	C₂²×C₄×C₁₇	C₂⁴	1	10	yes	—
40	C₂²×C₄×C₁₇	C₂⁴	1	60	yes	—
41	C₂²×Dic₁₇	C₂⁴	1	80	yes	—
42	C₂²⋊Dic₁₇	C₂⁴	1	16	yes	—
43	C₂²⋊Dic₁₇	C₂⁴	1	80	yes	—
44	C₄⋊Dic₁₇	C₂⁴	1	160	yes	—
45	C₄×Dic₁₇	C₂⁴	1	160	yes	—
46	#62653	—	17	1	no	maximal, Sylow
47	C₂³×C₁₇	—	4	10	no	—
48	C₂³×C₁₇	C₂³×C₄	1	20	yes	—
49	C₂³×C₁₇	C₂⁵	1	1	yes	characteristic
50	C₂³×C₁₇	C₂⁵	1	10	yes	—
51	Q₈×C₁₇	C₂³×C₄	1	160	yes	—
52	D₄×C₁₇	C₂³×C₄	1	160	yes	—
53	C₂×C₄×C₁₇	C₂³×C₄	1	40	yes	—
54	C₂×C₄×C₁₇	C₂³×C₄	1	240	yes	—
55	C₂×C₄×C₁₇	C₂⁵	1	20	yes	—
56	C₂×Dic₁₇	—	4	80	no	—
57	C₂×Dic₁₇	C₂⁵	1	32	yes	—
58	C₂²×2₋¹⁺⁴	D₁₇	1	1	yes	characteristic
59	C₄×2₋¹⁺⁴	—	17	32	no	—
60	#3428	—	17	20	no	—
61	C₂×#519	—	17	10	no	—
62	C₂²×C₁₇	—	2	20	no	—
63	C₂²×C₁₇	—	2	2	no	—
64	C₂²×C₁₇	—	4	20	no	—
65	C₂²×C₁₇	C₂⁴×C₄	1	2	yes	—
66	C₂²×C₁₇	C₂⁴×C₄	1	20	yes	—
67	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
68	C₄×C₁₇	C₂⁴×C₄	1	40	yes	—
69	Dic₁₇	—	2	32	no	—
70	C₂×2₋¹⁺⁴	Dic₁₇	1	32	yes	complemented
71	C₂×2₋¹⁺⁴	C₂×D₁₇	1	16	yes	—
72	C₂²×#41	C₂×D₁₇	1	10	yes	—
73	C₂³×Q₈	C₂×D₁₇	1	5	yes	—
74	#520	—	17	160	no	—
75	#519	—	17	40	no	—
76	#519	—	17	80	no	—
77	#518	—	17	40	no	—
78	C₄×#41	—	17	160	no	—
79	C₂×C₄×Q₈	—	17	80	no	—
80	C₂×#117	—	17	60	no	—
81	C₂×C₁₇	—	2	2	no	—
82	C₂×C₁₇	—	2	20	no	—
83	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
84	C₂×C₁₇	2₋¹⁺⁶	1	2	yes	—
85	2₋¹⁺⁴	C₂×Dic₁₇	1	256	yes	complemented
86	C₂×#41	C₂×Dic₁₇	1	160	yes	complemented
87	C₂×#41	C₂²×D₁₇	1	80	yes	—
88	C₂²×Q₈	C₂×Dic₁₇	1	80	yes	complemented
89	C₂²×Q₈	C₂²×D₁₇	1	40	yes	—
90	C₂²×Q₈	C₂²×D₁₇	1	10	yes	—
91	C₂²×D₄	C₂²×D₁₇	1	10	yes	—
92	C₂³×C₄	C₂²×D₁₇	1	15	yes	—
93	C₄×Q₈	—	17	320	no	—
94	C₄×D₄	—	17	320	no	—
95	#117	—	17	80	no	—
96	#117	—	17	240	no	—
97	C₂×C₄⋊C₄	—	17	120	no	—
98	C₂×C₂²⋊C₄	—	17	40	no	—
99	C₂×C₄²	—	17	120	no	—
100	C₁₇	#62653	1	1	yes	characteristic, complemented, Sylow
101	C₂⁴	C₂³×D₁₇	1	5	yes	—
102	#41	C₂²×Dic₁₇	1	640	yes	—
103	C₂×Q₈	C₂²×Dic₁₇	1	80	yes	—
104	C₂×Q₈	C₂²×Dic₁₇	1	320	yes	—
105	C₂×Q₈	C₂³×D₁₇	1	40	yes	—
106	C₂×D₄	C₂²×Dic₁₇	1	80	yes	—
107	C₂×D₄	C₂³×D₁₇	1	40	yes	—
108	C₂²×C₄	—	17	80	no	—
109	C₂²×C₄	C₂²×Dic₁₇	1	120	yes	—
110	C₂²×C₄	C₂³×D₁₇	1	60	yes	—
111	C₂²×C₄	C₂³×D₁₇	1	10	yes	—
112	C₄⋊C₄	—	17	160	no	—
113	C₂²⋊C₄	—	17	16	no	—
114	C₂²⋊C₄	—	17	80	no	—
115	C₄²	—	17	160	no	—
116	C₂³	—	4	10	no	—
117	C₂³	C₂³×Dic₁₇	1	20	yes	—
118	C₂³	C₂⁴×D₁₇	1	1	yes	characteristic
119	C₂³	C₂⁴×D₁₇	1	10	yes	—
120	Q₈	C₂³×Dic₁₇	1	160	yes	—
121	D₄	C₂³×Dic₁₇	1	160	yes	—
122	C₂×C₄	—	17	32	no	—
123	C₂×C₄	—	68	80	no	—
124	C₂×C₄	C₂³×Dic₁₇	1	40	yes	—
125	C₂×C₄	C₂³×Dic₁₇	1	240	yes	—
126	C₂×C₄	C₂⁴×D₁₇	1	20	yes	—
127	C₂²	—	2	20	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	20	no	—
130	C₂²	C₂⁴×Dic₁₇	1	2	yes	—
131	C₂²	C₂⁴×Dic₁₇	1	20	yes	—
132	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
133	C₄	—	34	32	no	—
134	C₄	C₂⁴×Dic₁₇	1	40	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	2	20	no	—
137	C₂	C₂⁵×Dic₁₇	1	1	yes	characteristic
138	C₂	#14543596	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Fitting, prime radical	C₂²×C₁₇×2₋¹⁺⁴	C₂
socle, cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₁₇
hypocenter	C₁₇	#62653
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₁₇	C₂	semidirect
2. #14541987	C₂	semidirect
3. C₂×#11763287	C₂	semidirect
5. #11763292	C₂²	semidirect
7. #11763287	C₂²	semidirect
9. #41×Dic₁₇	C₂²	semidirect
10. C₂×Q₈×Dic₁₇	C₂²	semidirect
12. C₂×C₁₇×2₋¹⁺⁴	C₄	semidirect
16. C₁₇×2₋¹⁺⁴	C₂×C₄	semidirect
17. C₂×#41×C₁₇	C₂×C₄	semidirect
19. C₂²×Q₈×C₁₇	C₂×C₄	semidirect
70. C₂×2₋¹⁺⁴	Dic₁₇	semidirect
85. 2₋¹⁺⁴	C₂×Dic₁₇	semidirect
86. C₂×#41	C₂×Dic₁₇	semidirect
88. C₂²×Q₈	C₂×Dic₁₇	semidirect
100. C₁₇	#62653	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×Dic₁₇	2
#14541987	2
C₂×#11763287	2
C₂²×C₁₇×2₋¹⁺⁴	2
#62653	17