Subgroups — #17940952

Order 4352.

11222 conjugacy classes of subgroups.

The subgroup lattice has 174 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	12	yes	maximal, complemented
2	H(5,𝔽₂)×Dic₁₇	C₂	1	20	yes	maximal, complemented
3	#14541987	C₂	1	15	yes	maximal, complemented
4	C₂×#11763292	C₂	1	15	yes	maximal, complemented
5	C₂×C₁₇×#585	C₂	1	1	yes	maximal, characteristic
6	#11763292	C₂²	1	60	yes	complemented
7	#11763292	C₂²	1	120	yes	complemented
8	#11763292	C₂²	1	20	yes	complemented
9	#11763287	C₂²	1	30	yes	complemented
10	#11763284	C₂²	1	90	yes	complemented
11	#41×Dic₁₇	C₂²	1	120	yes	complemented
12	C₂×Q₈×Dic₁₇	C₂²	1	30	yes	complemented
13	C₂×D₄×Dic₁₇	C₂²	1	90	yes	complemented
14	C₂×#10587612	C₂²	1	45	yes	complemented
15	C₂²×C₄⋊Dic₁₇	C₂²	1	15	yes	complemented
16	C₁₇×#585	C₄	1	32	yes	complemented
17	C₂×C₁₇×2₋¹⁺⁴	C₂²	1	6	yes	—
18	C₂×C₁₇×H(5,𝔽₂)	C₂²	1	10	yes	—
19	C₂²×#41×C₁₇	C₂²	1	15	yes	—
20	C₁₇×2₋¹⁺⁴	C₂×C₄	1	96	yes	complemented
21	C₁₇×H(5,𝔽₂)	C₂×C₄	1	160	yes	complemented
22	C₂×#41×C₁₇	C₂×C₄	1	240	yes	complemented
23	C₂×#41×C₁₇	C₂³	1	60	yes	—
24	C₂×#41×C₁₇	C₂³	1	20	yes	—
25	C₂²×Q₈×C₁₇	C₂³	1	15	yes	—
26	C₂²×D₄×C₁₇	C₂³	1	45	yes	—
27	C₂³×C₄×C₁₇	C₂³	1	15	yes	—
28	C₂³×Dic₁₇	C₂³	1	30	yes	—
29	C₂×C₂²⋊Dic₁₇	C₂³	1	90	yes	—
30	Q₈×Dic₁₇	C₂³	1	120	yes	—
31	Q₈×Dic₁₇	C₂³	1	40	yes	—
32	D₄×Dic₁₇	C₂³	1	360	yes	—
33	D₄×Dic₁₇	C₂³	1	120	yes	—
34	#10587612	C₂³	1	60	yes	—
35	#10587612	C₂³	1	180	yes	—
36	C₂×C₄⋊Dic₁₇	C₂³	1	45	yes	—
37	C₂×C₄⋊Dic₁₇	C₂³	1	90	yes	—
38	C₂×C₄⋊Dic₁₇	C₂³	1	15	yes	—
39	C₂×C₄×Dic₁₇	C₂³	1	90	yes	—
40	C₂⁴×C₁₇	C₂⁴	1	15	yes	—
41	#41×C₁₇	C₂²×C₄	1	160	yes	complemented
42	#41×C₁₇	C₂²×C₄	1	480	yes	complemented
43	C₂×Q₈×C₁₇	C₂²×C₄	1	120	yes	complemented
44	C₂×Q₈×C₁₇	C₂⁴	1	20	yes	—
45	C₂×D₄×C₁₇	C₂²×C₄	1	360	yes	complemented
46	C₂×D₄×C₁₇	C₂⁴	1	60	yes	—
47	C₂²×C₄×C₁₇	C₂²×C₄	1	120	yes	complemented
48	C₂²×C₄×C₁₇	C₂⁴	1	15	yes	—
49	C₂²×C₄×C₁₇	C₂⁴	1	45	yes	—
50	C₂²×Dic₁₇	—	8	30	no	—
51	C₂²×Dic₁₇	C₂⁴	1	90	yes	—
52	C₂²×Dic₁₇	C₂⁴	1	30	yes	—
53	C₂²⋊Dic₁₇	C₂⁴	1	120	yes	—
54	C₄⋊Dic₁₇	C₂⁴	1	10	yes	—
55	C₄⋊Dic₁₇	C₂⁴	1	6	yes	—
56	C₄⋊Dic₁₇	C₂⁴	1	90	yes	—
57	C₄⋊Dic₁₇	C₂⁴	1	30	yes	—
58	C₄×Dic₁₇	C₂⁴	1	120	yes	—
59	#62654	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	—	4	15	no	—
61	C₂³×C₁₇	—	4	15	no	—
62	C₂³×C₁₇	C₂³×C₄	1	60	yes	—
63	C₂³×C₁₇	C₂⁵	1	15	yes	—
64	Q₈×C₁₇	C₂³×C₄	1	80	yes	—
65	D₄×C₁₇	C₂³×C₄	1	240	yes	—
66	C₂×C₄×C₁₇	C₂³×C₄	1	60	yes	—
67	C₂×C₄×C₁₇	C₂³×C₄	1	180	yes	—
68	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂⁵	1	15	yes	—
70	C₂×Dic₁₇	—	4	90	no	—
71	C₂×Dic₁₇	—	4	30	no	—
72	C₂×Dic₁₇	C₂⁵	1	12	yes	—
73	C₂×Dic₁₇	C₂⁵	1	20	yes	—
74	C₂×#585	D₁₇	1	1	yes	characteristic
75	C₄×2₋¹⁺⁴	—	17	12	no	—
76	C₄×H(5,𝔽₂)	—	17	20	no	—
77	#3428	—	17	15	no	—
78	C₂×#520	—	17	15	no	—
79	C₂²×C₁₇	—	2	15	no	—
80	C₂²×C₁₇	—	2	15	no	—
81	C₂²×C₁₇	—	4	60	no	—
82	C₂²×C₁₇	C₂⁴×C₄	1	30	yes	—
83	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
84	C₄×C₁₇	C₂⁴×C₄	1	2	yes	—
85	C₄×C₁₇	C₂⁴×C₄	1	30	yes	—
86	Dic₁₇	—	2	20	no	—
87	Dic₁₇	—	2	12	no	—
88	#585	Dic₁₇	1	32	yes	complemented
89	C₂×2₋¹⁺⁴	C₂×D₁₇	1	6	yes	—
90	C₂×H(5,𝔽₂)	C₂×D₁₇	1	10	yes	—
91	C₂²×#41	C₂×D₁₇	1	15	yes	—
92	#520	—	17	60	no	—
93	#520	—	17	120	no	—
94	#520	—	17	20	no	—
95	#519	—	17	30	no	—
96	#518	—	17	90	no	—
97	C₄×#41	—	17	120	no	—
98	C₂×C₄×Q₈	—	17	30	no	—
99	C₂×C₄×D₄	—	17	90	no	—
100	C₂×#117	—	17	45	no	—
101	C₂²×C₄⋊C₄	—	17	15	no	—
102	C₂×C₁₇	—	2	30	no	—
103	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
104	C₂×C₁₇	H(7,𝔽₂)	1	1	yes	characteristic
105	C₂×C₁₇	2₋¹⁺⁶	1	1	yes	characteristic
106	2₋¹⁺⁴	C₂×Dic₁₇	1	96	yes	complemented
107	H(5,𝔽₂)	C₂×Dic₁₇	1	160	yes	complemented
108	C₂×#41	C₂×Dic₁₇	1	240	yes	complemented
109	C₂×#41	C₂²×D₁₇	1	60	yes	—
110	C₂×#41	C₂²×D₁₇	1	20	yes	—
111	C₂²×Q₈	C₂²×D₁₇	1	15	yes	—
112	C₂²×D₄	C₂²×D₁₇	1	45	yes	—
113	C₂³×C₄	—	17	30	no	—
114	C₂³×C₄	C₂²×D₁₇	1	15	yes	—
115	C₄×Q₈	—	17	120	no	—
116	C₄×Q₈	—	17	40	no	—
117	C₄×D₄	—	17	360	no	—
118	C₄×D₄	—	17	120	no	—
119	#117	—	17	60	no	—
120	#117	—	17	180	no	—
121	C₂×C₄⋊C₄	—	17	45	no	—
122	C₂×C₄⋊C₄	—	17	90	no	—
123	C₂×C₄⋊C₄	—	17	15	no	—
124	C₂×C₂²⋊C₄	—	17	90	no	—
125	C₂×C₄²	—	17	90	no	—
126	C₁₇	#62654	1	1	yes	characteristic, complemented, Sylow
127	C₂⁴	C₂³×D₁₇	1	15	yes	—
128	#41	C₂²×Dic₁₇	1	160	yes	complemented
129	#41	C₂²×Dic₁₇	1	480	yes	complemented
130	C₂×Q₈	C₂²×Dic₁₇	1	120	yes	complemented
131	C₂×Q₈	C₂³×D₁₇	1	20	yes	—
132	C₂×D₄	C₂²×Dic₁₇	1	360	yes	complemented
133	C₂×D₄	C₂³×D₁₇	1	60	yes	—
134	C₂²×C₄	—	17	30	no	—
135	C₂²×C₄	—	17	90	no	—
136	C₂²×C₄	—	136	30	no	—
137	C₂²×C₄	C₂²×Dic₁₇	1	120	yes	complemented
138	C₂²×C₄	C₂³×D₁₇	1	45	yes	—
139	C₂²×C₄	C₂³×D₁₇	1	15	yes	—
140	C₄⋊C₄	—	17	10	no	—
141	C₄⋊C₄	—	17	6	no	—
142	C₄⋊C₄	—	17	90	no	—
143	C₄⋊C₄	—	17	30	no	—
144	C₂²⋊C₄	—	17	120	no	—
145	C₄²	—	17	120	no	—
146	C₂³	—	4	15	no	—
147	C₂³	—	4	15	no	—
148	C₂³	C₂³×Dic₁₇	1	60	yes	—
149	C₂³	C₂⁴×D₁₇	1	15	yes	—
150	Q₈	C₂³×Dic₁₇	1	80	yes	—
151	D₄	C₂³×Dic₁₇	1	240	yes	—
152	C₂×C₄	—	17	12	no	—
153	C₂×C₄	—	17	20	no	—
154	C₂×C₄	—	68	90	no	—
155	C₂×C₄	—	68	30	no	—
156	C₂×C₄	C₂³×Dic₁₇	1	60	yes	—
157	C₂×C₄	C₂³×Dic₁₇	1	180	yes	—
158	C₂×C₄	C₂⁴×D₁₇	1	15	yes	—
159	C₂×C₄	C₂⁴×D₁₇	1	1	yes	characteristic
160	C₂²	—	2	15	no	—
161	C₂²	—	2	15	no	—
162	C₂²	—	4	60	no	—
163	C₂²	C₂⁴×Dic₁₇	1	30	yes	—
164	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
165	C₄	—	34	20	no	—
166	C₄	—	34	12	no	—
167	C₄	C₂⁴×Dic₁₇	1	2	yes	—
168	C₄	C₂⁴×Dic₁₇	1	30	yes	—
169	C₂	—	2	30	no	—
170	C₂	C₂⁵×Dic₁₇	1	1	yes	characteristic
171	C₂	#14543595	1	1	yes	characteristic
172	C₂	#14543597	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Fitting, prime radical	C₂×C₁₇×#585	C₂
socle, cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂×#585	D₁₇
hypocenter	C₁₇	#62654
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₁₇	C₂	semidirect
2. H(5,𝔽₂)×Dic₁₇	C₂	semidirect
3. #14541987	C₂	semidirect
4. C₂×#11763292	C₂	semidirect
6. #11763292	C₂²	semidirect
7. #11763292	C₂²	semidirect
8. #11763292	C₂²	semidirect
9. #11763287	C₂²	semidirect
10. #11763284	C₂²	semidirect
11. #41×Dic₁₇	C₂²	semidirect
12. C₂×Q₈×Dic₁₇	C₂²	semidirect
13. C₂×D₄×Dic₁₇	C₂²	semidirect
14. C₂×#10587612	C₂²	semidirect
15. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
16. C₁₇×#585	C₄	semidirect
20. C₁₇×2₋¹⁺⁴	C₂×C₄	semidirect
21. C₁₇×H(5,𝔽₂)	C₂×C₄	semidirect
22. C₂×#41×C₁₇	C₂×C₄	semidirect
41. #41×C₁₇	C₂²×C₄	semidirect
42. #41×C₁₇	C₂²×C₄	semidirect
43. C₂×Q₈×C₁₇	C₂²×C₄	semidirect
45. C₂×D₄×C₁₇	C₂²×C₄	semidirect
47. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
88. #585	Dic₁₇	semidirect
106. 2₋¹⁺⁴	C₂×Dic₁₇	semidirect
107. H(5,𝔽₂)	C₂×Dic₁₇	semidirect
108. C₂×#41	C₂×Dic₁₇	semidirect
126. C₁₇	#62654	semidirect
128. #41	C₂²×Dic₁₇	semidirect
129. #41	C₂²×Dic₁₇	semidirect
130. C₂×Q₈	C₂²×Dic₁₇	semidirect
132. C₂×D₄	C₂²×Dic₁₇	semidirect
137. C₂²×C₄	C₂²×Dic₁₇	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
H(5,𝔽₂)×Dic₁₇	2
#14541987	2
C₂×#11763292	2
C₂×C₁₇×#585	2
#62654	17