Subgroups — C₂×C₁₇×#2262 #16891722

Order 4352.

470 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	C₂×C₁₇×#508	C₂	1	2	yes	maximal, complemented
2	C₁₇×#2262	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₁₇×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×SD₆₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×D₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#135	—	2	2	no	—
8	C₁₇×#508	—	2	4	no	—
9	C₁₇×#508	C₂²	1	8	yes	complemented
10	C₂×C₁₇×Q₃₂	C₂²	1	2	yes	—
11	C₂×C₁₇×SD₃₂	—	2	2	no	—
12	C₂×C₁₇×D₁₆	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×Q₆₄	C₂²	1	4	yes	complemented
15	C₁₇×SD₆₄	C₂²	1	8	yes	complemented
16	C₁₇×D₃₂	C₂²	1	4	yes	complemented
17	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×C₃₂	C₂²	1	4	yes	complemented
20	C₂×#41×C₁₇	—	4	2	no	—
21	C₁₇×#135	—	2	8	no	—
22	C₁₇×#135	—	4	4	no	—
23	C₂×Q₁₆×C₁₇	—	2	2	no	—
24	C₂×SD₁₆×C₁₇	—	4	2	no	—
25	C₂×D₈×C₁₇	—	2	2	no	—
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₁₇×Q₃₂	—	2	2	no	—
28	C₁₇×Q₃₂	C₂³	1	4	yes	—
29	C₁₇×SD₃₂	—	2	8	no	—
30	C₁₇×D₁₆	—	2	2	no	—
31	C₁₇×D₁₆	C₂³	1	4	yes	—
32	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
34	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
35	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
36	C₁₇×C₃₂	C₂³	1	2	yes	—
37	C₁₇×C₃₂	C₂³	1	2	yes	—
38	#41×C₁₇	—	4	8	no	—
39	#41×C₁₇	—	8	4	no	—
40	C₂×Q₈×C₁₇	—	4	2	no	—
41	C₂×D₄×C₁₇	—	4	2	no	—
42	C₂×D₄×C₁₇	—	8	2	no	—
43	C₂²×C₄×C₁₇	—	8	2	no	—
44	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
45	Q₁₆×C₁₇	—	2	4	no	—
46	Q₁₆×C₁₇	—	4	2	no	—
47	SD₁₆×C₁₇	—	4	8	no	—
48	D₈×C₁₇	—	2	4	no	—
49	D₈×C₁₇	—	4	2	no	—
50	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
51	C₂×C₈×C₁₇	D₈	1	2	yes	—
52	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
54	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₁₆×C₁₇	C₂×D₄	1	2	yes	—
56	C₁₆×C₁₇	C₂⁴	1	1	yes	characteristic
57	C₂×#2262	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₁₇	—	8	2	no	—
59	C₂³×C₁₇	D₁₆	1	1	yes	characteristic
60	Q₈×C₁₇	—	4	4	no	—
61	Q₈×C₁₇	—	8	2	no	—
62	D₄×C₁₇	—	4	4	no	—
63	D₄×C₁₇	—	8	8	no	—
64	D₄×C₁₇	—	8	2	no	—
65	C₂×C₄×C₁₇	—	8	2	no	—
66	C₂×C₄×C₁₇	—	8	8	no	—
67	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
68	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
71	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
72	C₈×C₁₇	C₂×D₈	1	2	yes	—
73	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
74	C₂×#508	C₂×C₁₇	1	2	yes	complemented
75	#2262	C₂×C₁₇	1	8	yes	direct product factor
76	C₂×Q₆₄	C₂×C₁₇	1	1	yes	characteristic, complemented
77	C₂×SD₆₄	C₂×C₁₇	1	2	yes	complemented
78	C₂×D₃₂	C₂×C₁₇	1	1	yes	characteristic, complemented
79	C₂²×C₃₂	C₂×C₁₇	1	1	yes	characteristic, complemented
80	C₂²×C₁₇	—	2	2	no	—
81	C₂²×C₁₇	—	8	4	no	—
82	C₂²×C₁₇	—	16	4	no	—
83	C₂²×C₁₇	C₂×D₁₆	1	2	yes	—
84	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
85	C₄×C₁₇	—	8	4	no	—
86	C₄×C₁₇	C₂×D₁₆	1	2	yes	—
87	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
88	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
89	C₂×#135	—	2	2	no	—
90	#508	—	2	4	no	—
91	#508	C₂²×C₁₇	1	8	yes	complemented
92	C₂×Q₃₂	C₂²×C₁₇	1	2	yes	—
93	C₂×SD₃₂	—	2	2	no	—
94	C₂×D₁₆	C₂²×C₁₇	1	2	yes	—
95	C₂²×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
96	Q₆₄	C₂²×C₁₇	1	4	yes	complemented
97	SD₆₄	C₂²×C₁₇	1	8	yes	complemented
98	D₃₂	C₂²×C₁₇	1	4	yes	complemented
99	C₂×C₃₂	C₂²×C₁₇	1	1	yes	characteristic
100	C₂×C₃₂	C₂²×C₁₇	1	1	yes	characteristic
101	C₂×C₃₂	C₂²×C₁₇	1	4	yes	complemented
102	C₂×C₁₇	—	2	2	no	—
103	C₂×C₁₇	—	16	4	no	—
104	C₂×C₁₇	#2262	1	2	yes	direct product factor
105	C₂×C₁₇	C₂²×D₁₆	1	1	yes	characteristic
106	C₂×#41	—	4	2	no	—
107	#135	—	2	8	no	—
108	#135	—	4	4	no	—
109	C₂×Q₁₆	—	2	2	no	—
110	C₂×SD₁₆	—	4	2	no	—
111	C₂×D₈	—	2	2	no	—
112	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
113	Q₃₂	—	2	2	no	—
114	Q₃₂	C₂³×C₁₇	1	4	yes	—
115	SD₃₂	—	2	8	no	—
116	D₁₆	—	2	2	no	—
117	D₁₆	C₂³×C₁₇	1	4	yes	—
118	C₂×C₁₆	D₄×C₁₇	1	1	yes	characteristic
119	C₂×C₁₆	D₄×C₁₇	1	2	yes	—
120	C₂×C₁₆	C₂³×C₁₇	1	1	yes	characteristic
121	C₂×C₁₆	C₂³×C₁₇	1	2	yes	—
122	C₃₂	C₂³×C₁₇	1	2	yes	—
123	C₃₂	C₂³×C₁₇	1	2	yes	—
124	C₁₇	C₂×#2262	1	1	yes	characteristic, direct product factor, Sylow
125	#41	—	4	8	no	—
126	#41	—	8	4	no	—
127	C₂×Q₈	—	4	2	no	—
128	C₂×D₄	—	4	2	no	—
129	C₂×D₄	—	8	2	no	—
130	C₂²×C₄	—	8	2	no	—
131	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
132	Q₁₆	—	2	4	no	—
133	Q₁₆	—	4	2	no	—
134	SD₁₆	—	4	8	no	—
135	D₈	—	2	4	no	—
136	D₈	—	4	2	no	—
137	C₂×C₈	D₈×C₁₇	1	1	yes	characteristic
138	C₂×C₈	D₈×C₁₇	1	2	yes	—
139	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
140	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
141	C₁₆	C₂×D₄×C₁₇	1	1	yes	characteristic
142	C₁₆	C₂×D₄×C₁₇	1	2	yes	—
143	C₁₆	C₂⁴×C₁₇	1	1	yes	characteristic
144	C₂³	—	8	2	no	—
145	C₂³	C₁₇×D₁₆	1	1	yes	characteristic
146	Q₈	—	4	4	no	—
147	Q₈	—	8	2	no	—
148	D₄	—	4	4	no	—
149	D₄	—	8	8	no	—
150	D₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	8	no	—
153	C₂×C₄	C₁₇×D₁₆	1	1	yes	characteristic
154	C₂×C₄	C₁₇×D₁₆	1	2	yes	—
155	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
156	C₂×C₄	C₂×D₈×C₁₇	1	2	yes	—
157	C₈	C₂×D₈×C₁₇	1	1	yes	characteristic
158	C₈	C₂×D₈×C₁₇	1	2	yes	—
159	C₈	C₂²×D₄×C₁₇	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	—	8	4	no	—
162	C₂²	—	16	4	no	—
163	C₂²	C₂×C₁₇×D₁₆	1	2	yes	—
164	C₂²	C₂×C₁₇×D₁₆	1	1	yes	characteristic
165	C₄	—	8	4	no	—
166	C₄	C₂×C₁₇×D₁₆	1	2	yes	—
167	C₄	C₂×C₁₇×D₁₆	1	1	yes	characteristic
168	C₄	C₂²×D₈×C₁₇	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	16	4	no	—
171	C₂	C₁₇×#2262	1	2	yes	direct product factor
172	C₂	C₂²×C₁₇×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₇
socle	C₂²×C₁₇	C₂×D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ C₂²×C₁₆×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×#508	C₂	semidirect
2. C₁₇×#2262	C₂	direct (N × H)
3. C₂×C₁₇×Q₆₄	C₂	semidirect
4. C₂×C₁₇×SD₆₄	C₂	semidirect
5. C₂×C₁₇×D₃₂	C₂	semidirect
6. C₂²×C₁₇×C₃₂	C₂	semidirect
9. C₁₇×#508	C₂²	semidirect
14. C₁₇×Q₆₄	C₂²	semidirect
15. C₁₇×SD₆₄	C₂²	semidirect
16. C₁₇×D₃₂	C₂²	semidirect
19. C₂×C₁₇×C₃₂	C₂²	semidirect
57. C₂×#2262	C₁₇	direct (N × H)
74. C₂×#508	C₂×C₁₇	semidirect
75. #2262	C₂×C₁₇	direct (N × H)
76. C₂×Q₆₄	C₂×C₁₇	semidirect
77. C₂×SD₆₄	C₂×C₁₇	semidirect
78. C₂×D₃₂	C₂×C₁₇	semidirect
79. C₂²×C₃₂	C₂×C₁₇	semidirect
91. #508	C₂²×C₁₇	semidirect
96. Q₆₄	C₂²×C₁₇	semidirect
97. SD₆₄	C₂²×C₁₇	semidirect
98. D₃₂	C₂²×C₁₇	semidirect
101. C₂×C₃₂	C₂²×C₁₇	semidirect
104. C₂×C₁₇	#2262	direct (N × H)
124. C₁₇	C₂×#2262	direct (N × H)
171. C₂	C₁₇×#2262	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#508	2
C₁₇×#2262	2
C₂×C₁₇×Q₆₄	2
C₂×C₁₇×SD₆₄	2
C₂×C₁₇×D₃₂	2
C₂²×C₁₇×C₃₂	2
C₂×#2262	17

Subgroups — C2×C17×#2262 #16891722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#2262 #16891722