Subgroups — C₂×C₁₇×#2263 #16891723

Order 4352.

502 conjugacy classes of subgroups.

The subgroup lattice has 230 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	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₇×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#2263	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₇×SD₆₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×D₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#135	—	2	1	no	—
8	C₂²×D₈×C₁₇	—	2	1	no	—
9	C₁₇×#508	—	2	2	no	—
10	C₁₇×#508	C₂²	1	4	yes	complemented
11	C₂×C₁₇×Q₃₂	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₁₇×SD₃₂	—	2	1	no	—
13	C₂×C₁₇×D₁₆	—	2	1	no	—
14	C₂×C₁₇×D₁₆	—	2	2	no	—
15	C₂×C₁₇×D₁₆	C₂²	1	2	yes	—
16	C₂×C₁₇×D₁₆	C₂²	1	4	yes	complemented
17	C₂×C₁₇×D₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×SD₆₄	C₂²	1	8	yes	complemented
20	C₁₇×D₃₂	C₂²	1	8	yes	complemented
21	C₁₇×M₆₄	C₂²	1	4	yes	complemented
22	C₂×C₁₇×C₃₂	C₂²	1	2	yes	complemented
23	C₂×#41×C₁₇	—	4	1	no	—
24	C₂²×D₄×C₁₇	—	4	1	no	—
25	C₁₇×#135	—	2	4	no	—
26	C₁₇×#135	—	4	2	no	—
27	C₂×Q₁₆×C₁₇	—	2	1	no	—
28	C₂×SD₁₆×C₁₇	—	4	1	no	—
29	C₂×D₈×C₁₇	—	2	2	no	—
30	C₂×D₈×C₁₇	—	2	4	no	—
31	C₂×D₈×C₁₇	—	2	1	no	—
32	C₂×D₈×C₁₇	—	4	1	no	—
33	C₂×D₈×C₁₇	—	4	2	no	—
34	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
35	C₁₇×Q₃₂	—	2	1	no	—
36	C₁₇×Q₃₂	C₂³	1	2	yes	complemented
37	C₁₇×SD₃₂	—	2	4	no	—
38	C₁₇×D₁₆	—	2	2	no	—
39	C₁₇×D₁₆	—	2	4	no	—
40	C₁₇×D₁₆	—	2	1	no	—
41	C₁₇×D₁₆	C₂³	1	4	yes	—
42	C₁₇×D₁₆	C₂³	1	2	yes	complemented
43	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
45	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
47	C₁₇×C₃₂	C₂³	1	4	yes	complemented
48	C₂⁴×C₁₇	—	8	1	no	—
49	#41×C₁₇	—	4	4	no	—
50	#41×C₁₇	—	8	2	no	—
51	C₂×Q₈×C₁₇	—	4	1	no	—
52	C₂×D₄×C₁₇	—	4	4	no	—
53	C₂×D₄×C₁₇	—	4	2	no	—
54	C₂×D₄×C₁₇	—	4	1	no	—
55	C₂×D₄×C₁₇	—	8	1	no	—
56	C₂×D₄×C₁₇	—	8	2	no	—
57	C₂×D₄×C₁₇	—	8	1	no	—
58	C₂²×C₄×C₁₇	—	8	1	no	—
59	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
60	Q₁₆×C₁₇	—	2	2	no	—
61	Q₁₆×C₁₇	—	4	1	no	—
62	SD₁₆×C₁₇	—	4	4	no	—
63	D₈×C₁₇	—	2	4	no	—
64	D₈×C₁₇	—	2	2	no	—
65	D₈×C₁₇	—	4	2	no	—
66	D₈×C₁₇	—	4	4	no	—
67	D₈×C₁₇	—	4	1	no	—
68	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
69	C₂×C₈×C₁₇	D₈	1	2	yes	—
70	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
72	C₁₆×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₁₇	C₂×D₄	1	2	yes	—
74	C₁₆×C₁₇	C₂⁴	1	1	yes	characteristic
75	C₂×#2263	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₁₇	—	8	4	no	—
77	C₂³×C₁₇	—	8	1	no	—
78	C₂³×C₁₇	—	8	2	no	—
79	C₂³×C₁₇	—	16	4	no	—
80	C₂³×C₁₇	D₁₆	1	1	yes	characteristic
81	Q₈×C₁₇	—	4	2	no	—
82	Q₈×C₁₇	—	8	1	no	—
83	D₄×C₁₇	—	4	4	no	—
84	D₄×C₁₇	—	4	2	no	—
85	D₄×C₁₇	—	8	4	no	—
86	D₄×C₁₇	—	8	4	no	—
87	D₄×C₁₇	—	8	2	no	—
88	D₄×C₁₇	—	8	1	no	—
89	C₂×C₄×C₁₇	—	8	1	no	—
90	C₂×C₄×C₁₇	—	8	4	no	—
91	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
92	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
95	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
96	C₈×C₁₇	C₂×D₈	1	2	yes	—
97	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×#508	C₂×C₁₇	1	1	yes	characteristic, complemented
99	C₂²×D₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
100	#2263	C₂×C₁₇	1	8	yes	direct product factor
101	C₂×SD₆₄	C₂×C₁₇	1	2	yes	complemented
102	C₂×D₃₂	C₂×C₁₇	1	2	yes	complemented
103	C₂×M₆₄	C₂×C₁₇	1	1	yes	characteristic, complemented
104	C₂²×C₁₇	—	2	2	no	—
105	C₂²×C₁₇	—	8	2	no	—
106	C₂²×C₁₇	—	8	4	no	—
107	C₂²×C₁₇	—	16	8	no	—
108	C₂²×C₁₇	—	16	2	no	—
109	C₂²×C₁₇	—	16	4	no	—
110	C₂²×C₁₇	C₂×D₁₆	1	2	yes	—
111	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
112	C₄×C₁₇	—	8	2	no	—
113	C₄×C₁₇	C₂×D₁₆	1	2	yes	—
114	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
115	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
116	C₂×#135	—	2	1	no	—
117	C₂²×D₈	—	2	1	no	—
118	#508	—	2	2	no	—
119	#508	C₂²×C₁₇	1	4	yes	complemented
120	C₂×Q₃₂	C₂²×C₁₇	1	1	yes	characteristic, complemented
121	C₂×SD₃₂	—	2	1	no	—
122	C₂×D₁₆	—	2	1	no	—
123	C₂×D₁₆	—	2	2	no	—
124	C₂×D₁₆	C₂²×C₁₇	1	2	yes	—
125	C₂×D₁₆	C₂²×C₁₇	1	4	yes	complemented
126	C₂×D₁₆	C₂²×C₁₇	1	1	yes	characteristic, complemented
127	C₂²×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
128	SD₆₄	C₂²×C₁₇	1	8	yes	complemented
129	D₃₂	C₂²×C₁₇	1	8	yes	complemented
130	M₆₄	C₂²×C₁₇	1	4	yes	complemented
131	C₂×C₃₂	C₂²×C₁₇	1	2	yes	complemented
132	C₂×C₁₇	—	2	2	no	—
133	C₂×C₁₇	—	16	2	no	—
134	C₂×C₁₇	—	16	4	no	—
135	C₂×C₁₇	#2263	1	2	yes	direct product factor
136	C₂×C₁₇	C₂²×D₁₆	1	1	yes	characteristic
137	C₂×#41	—	4	1	no	—
138	C₂²×D₄	—	4	1	no	—
139	#135	—	2	4	no	—
140	#135	—	4	2	no	—
141	C₂×Q₁₆	—	2	1	no	—
142	C₂×SD₁₆	—	4	1	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×D₈	—	2	4	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	—	4	1	no	—
147	C₂×D₈	—	4	2	no	—
148	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
149	Q₃₂	—	2	1	no	—
150	Q₃₂	C₂³×C₁₇	1	2	yes	complemented
151	SD₃₂	—	2	4	no	—
152	D₁₆	—	2	2	no	—
153	D₁₆	—	2	4	no	—
154	D₁₆	—	2	1	no	—
155	D₁₆	C₂³×C₁₇	1	4	yes	—
156	D₁₆	C₂³×C₁₇	1	2	yes	complemented
157	C₂×C₁₆	D₄×C₁₇	1	1	yes	characteristic
158	C₂×C₁₆	D₄×C₁₇	1	2	yes	—
159	C₂×C₁₆	C₂³×C₁₇	1	1	yes	characteristic
160	C₂×C₁₆	C₂³×C₁₇	1	2	yes	—
161	C₃₂	C₂³×C₁₇	1	4	yes	complemented
162	C₁₇	C₂×#2263	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	—	8	1	no	—
164	#41	—	4	4	no	—
165	#41	—	8	2	no	—
166	C₂×Q₈	—	4	1	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	1	no	—
170	C₂×D₄	—	8	1	no	—
171	C₂×D₄	—	8	1	no	—
172	C₂×D₄	—	8	2	no	—
173	C₂²×C₄	—	8	1	no	—
174	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
175	Q₁₆	—	2	2	no	—
176	Q₁₆	—	4	1	no	—
177	SD₁₆	—	4	4	no	—
178	D₈	—	2	4	no	—
179	D₈	—	2	2	no	—
180	D₈	—	4	2	no	—
181	D₈	—	4	4	no	—
182	D₈	—	4	1	no	—
183	C₂×C₈	D₈×C₁₇	1	1	yes	characteristic
184	C₂×C₈	D₈×C₁₇	1	2	yes	—
185	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
187	C₁₆	C₂×D₄×C₁₇	1	1	yes	characteristic
188	C₁₆	C₂×D₄×C₁₇	1	2	yes	—
189	C₁₆	C₂⁴×C₁₇	1	1	yes	characteristic
190	C₂³	—	8	1	no	—
191	C₂³	—	8	4	no	—
192	C₂³	—	8	2	no	—
193	C₂³	—	16	4	no	—
194	C₂³	C₁₇×D₁₆	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	8	1	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	2	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	2	no	—
202	D₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	C₁₇×D₁₆	1	1	yes	characteristic
206	C₂×C₄	C₁₇×D₁₆	1	2	yes	—
207	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
208	C₂×C₄	C₂×D₈×C₁₇	1	2	yes	—
209	C₈	C₂×D₈×C₁₇	1	1	yes	characteristic
210	C₈	C₂×D₈×C₁₇	1	2	yes	—
211	C₈	C₂²×D₄×C₁₇	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	4	no	—
215	C₂²	—	16	8	no	—
216	C₂²	—	16	4	no	—
217	C₂²	—	16	2	no	—
218	C₂²	C₂×C₁₇×D₁₆	1	2	yes	—
219	C₂²	C₂×C₁₇×D₁₆	1	1	yes	characteristic
220	C₄	—	8	2	no	—
221	C₄	C₂×C₁₇×D₁₆	1	2	yes	—
222	C₄	C₂×C₁₇×D₁₆	1	1	yes	characteristic
223	C₄	C₂²×D₈×C₁₇	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	16	4	no	—
226	C₂	—	16	2	no	—
227	C₂	C₁₇×#2263	1	2	yes	direct product factor
228	C₂	C₂²×C₁₇×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₇
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂²×C₁₇×D₁₆	C₂	semidirect
3. C₁₇×#2263	C₂	direct (N × H)
4. C₂×C₁₇×SD₆₄	C₂	semidirect
5. C₂×C₁₇×D₃₂	C₂	semidirect
6. C₂×C₁₇×M₆₄	C₂	semidirect
10. C₁₇×#508	C₂²	semidirect
11. C₂×C₁₇×Q₃₂	C₂²	semidirect
16. C₂×C₁₇×D₁₆	C₂²	semidirect
17. C₂×C₁₇×D₁₆	C₂²	semidirect
19. C₁₇×SD₆₄	C₂²	semidirect
20. C₁₇×D₃₂	C₂²	semidirect
21. C₁₇×M₆₄	C₂²	semidirect
22. C₂×C₁₇×C₃₂	C₂²	semidirect
36. C₁₇×Q₃₂	C₂³	semidirect
42. C₁₇×D₁₆	C₂³	semidirect
47. C₁₇×C₃₂	C₂³	semidirect
75. C₂×#2263	C₁₇	direct (N × H)
98. C₂×#508	C₂×C₁₇	semidirect
99. C₂²×D₁₆	C₂×C₁₇	semidirect
100. #2263	C₂×C₁₇	direct (N × H)
101. C₂×SD₆₄	C₂×C₁₇	semidirect
102. C₂×D₃₂	C₂×C₁₇	semidirect
103. C₂×M₆₄	C₂×C₁₇	semidirect
119. #508	C₂²×C₁₇	semidirect
120. C₂×Q₃₂	C₂²×C₁₇	semidirect
125. C₂×D₁₆	C₂²×C₁₇	semidirect
126. C₂×D₁₆	C₂²×C₁₇	semidirect
128. SD₆₄	C₂²×C₁₇	semidirect
129. D₃₂	C₂²×C₁₇	semidirect
130. M₆₄	C₂²×C₁₇	semidirect
131. C₂×C₃₂	C₂²×C₁₇	semidirect
135. C₂×C₁₇	#2263	direct (N × H)
150. Q₃₂	C₂³×C₁₇	semidirect
156. D₁₆	C₂³×C₁₇	semidirect
161. C₃₂	C₂³×C₁₇	semidirect
162. C₁₇	C₂×#2263	direct (N × H)
227. C₂	C₁₇×#2263	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
C₂²×C₁₇×D₁₆	2
C₁₇×#2263	2
C₂×C₁₇×SD₆₄	2
C₂×C₁₇×D₃₂	2
C₂×C₁₇×M₆₄	2
C₂×#2263	17

Subgroups — C2×C17×#2263 #16891723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#2263 #16891723