Subgroups — C₂×C₄×C₁₆×D₁₇ #17186945

Order 4352.

1022 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#63214	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×Dic₁₇	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×D₁₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₁₇	C₄	1	1	yes	characteristic
10	C₂²×#63214	C₂²	1	2	yes	—
11	C₂×C₁₆×D₁₇	C₄	1	16	yes	direct product factor
12	C₂×C₁₆×D₁₇	C₂²	1	8	yes	—
13	C₂×C₄×#3622	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₁₇	C₄	1	4	yes	—
17	C₄×C₈×D₁₇	C₂²	1	4	yes	—
18	C₄×#63214	C₂²	1	4	yes	complemented
19	C₁₆×Dic₁₇	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₁₇	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₁₇	C₈	1	2	yes	—
24	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	C₈	1	2	yes	—
29	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
31	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₁₇	C₈	1	8	yes	—
33	C₂×C₈×D₁₇	C₂×C₄	1	8	yes	—
34	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₁₇	C₂³	1	2	yes	—
37	C₄²×D₁₇	C₈	1	4	yes	—
38	C₄²×D₁₇	C₂×C₄	1	4	yes	—
39	C₂×C₁₆×C₁₇	C₂×C₄	1	8	yes	complemented
40	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
41	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₇	C₂³	1	2	yes	—
43	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
44	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
45	C₈×Dic₁₇	C₂³	1	2	yes	—
46	C₂×#63214	C₂×C₄	1	8	yes	complemented
47	C₂×#63214	C₂³	1	4	yes	—
48	C₄×#3622	C₂×C₄	1	4	yes	—
49	C₁₆×D₁₇	C₂×C₄	1	32	yes	direct product factor
50	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₁₇	C₁₆	1	16	yes	direct product factor
56	C₂×C₄×D₁₇	C₄²	1	4	yes	—
57	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
58	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
59	C₂×C₄×D₁₇	C₂×C₈	1	8	yes	—
60	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
66	C₄²×C₁₇	C₂×C₈	1	2	yes	—
67	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
68	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
69	C₄×Dic₁₇	C₂×C₈	1	8	yes	—
70	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
71	C₂×#3622	C₄²	1	4	yes	—
72	C₂×#3622	C₂×C₈	1	4	yes	—
73	C₂×#3622	C₂²×C₄	1	2	yes	—
74	C₂×#3622	C₂²×C₄	1	2	yes	—
75	C₈×D₁₇	C₄²	1	8	yes	—
76	C₈×D₁₇	C₂×C₈	1	16	yes	—
77	C₈×D₁₇	C₂²×C₄	1	8	yes	—
78	C₁₆×C₁₇	C₂²×C₄	1	8	yes	complemented
79	#63214	C₂²×C₄	1	8	yes	complemented
80	C₂²×C₄×C₁₆	—	17	1	no	maximal, Sylow
81	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₁₇	C₄×C₈	1	4	yes	—
83	C₂²×D₁₇	C₂×C₁₆	1	8	yes	—
84	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₁₇	C₂×C₁₆	1	8	yes	complemented
86	C₂×C₄×C₁₇	C₂×C₄²	1	2	yes	—
87	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×Dic₁₇	C₄×C₈	1	4	yes	—
92	C₂×Dic₁₇	C₂×C₁₆	1	8	yes	complemented
93	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
94	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
95	C₄×D₁₇	C₄×C₈	1	8	yes	—
96	C₄×D₁₇	C₂×C₁₆	1	32	yes	direct product factor
97	C₄×D₁₇	C₂×C₄²	1	4	yes	—
98	C₄×D₁₇	C₂²×C₈	1	4	yes	—
99	C₈×C₁₇	C₂×C₄²	1	2	yes	—
100	C₈×C₁₇	C₂²×C₈	1	4	yes	—
101	C₈×C₁₇	C₂³×C₄	1	2	yes	—
102	#3622	C₂×C₄²	1	4	yes	—
103	#3622	C₂²×C₈	1	4	yes	—
104	C₂³×C₁₆	—	17	2	no	—
105	C₂²×C₄×C₈	—	17	1	no	—
106	C₂×C₄×C₁₆	—	17	2	no	—
107	C₂×C₄×C₁₆	—	17	1	no	—
108	C₂×C₄×C₁₆	—	17	8	no	—
109	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic, direct product factor
110	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
111	C₂²×C₁₇	C₂²×C₁₆	1	4	yes	—
112	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
113	C₂×D₁₇	C₄×C₁₆	1	16	yes	direct product factor
114	C₂×D₁₇	C₂×C₄×C₈	1	4	yes	—
115	C₂×D₁₇	C₂²×C₁₆	1	8	yes	—
116	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
117	C₄×C₁₇	C₂²×C₁₆	1	8	yes	complemented
118	C₄×C₁₇	C₂²×C₄²	1	1	yes	characteristic
119	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
120	Dic₁₇	C₂×C₄×C₈	1	4	yes	—
121	Dic₁₇	C₂²×C₁₆	1	8	yes	complemented
122	C₂³×C₈	—	17	1	no	—
123	C₂³×C₈	—	17	1	no	—
124	C₂²×C₄²	—	17	1	no	—
125	C₂²×C₁₆	—	17	2	no	—
126	C₂²×C₁₆	—	17	16	no	—
127	C₂²×C₁₆	—	17	8	no	—
128	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
129	C₂×C₄×C₈	—	17	1	no	—
130	C₂×C₄×C₈	—	17	1	no	—
131	C₂×C₄×C₈	—	17	1	no	—
132	C₂×C₄×C₈	—	17	4	no	—
133	C₂×C₄×C₈	—	17	4	no	—
134	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
135	C₄×C₁₆	—	17	8	no	—
136	C₄×C₁₆	—	17	4	no	—
137	C₄×C₁₆	C₂×D₁₇	1	4	yes	direct product factor
138	C₂×C₁₇	C₂×C₄×C₁₆	1	4	yes	complemented
139	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
140	C₂×C₁₇	C₂³×C₁₆	1	2	yes	—
141	D₁₇	C₂×C₄×C₁₆	1	8	yes	direct product factor
142	C₂³×C₄	—	17	1	no	—
143	C₂³×C₄	—	17	2	no	—
144	C₂²×C₈	—	17	1	no	—
145	C₂²×C₈	—	17	1	no	—
146	C₂²×C₈	—	17	2	no	—
147	C₂²×C₈	—	17	4	no	—
148	C₂²×C₈	—	17	8	no	—
149	C₂²×C₈	—	17	2	no	—
150	C₂²×C₈	—	17	8	no	—
151	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
153	C₂×C₄²	—	17	1	no	—
154	C₂×C₄²	—	17	2	no	—
155	C₂×C₄²	—	17	4	no	—
156	C₂×C₄²	—	17	4	no	—
157	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
158	C₂×C₁₆	—	17	8	no	—
159	C₂×C₁₆	—	17	4	no	—
160	C₂×C₁₆	—	17	32	no	—
161	C₂×C₁₆	C₄×D₁₇	1	8	yes	direct product factor
162	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
163	C₄×C₈	—	17	2	no	—
164	C₄×C₈	—	17	4	no	—
165	C₄×C₈	—	17	2	no	—
166	C₄×C₈	—	17	4	no	—
167	C₄×C₈	C₄×D₁₇	1	2	yes	—
168	C₄×C₈	C₂²×D₁₇	1	2	yes	—
169	C₁₇	C₂²×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
170	C₂⁴	—	17	1	no	—
171	C₂²×C₄	—	17	2	no	—
172	C₂²×C₄	—	17	1	no	—
173	C₂²×C₄	—	17	2	no	—
174	C₂²×C₄	—	17	16	no	—
175	C₂²×C₄	—	17	4	no	—
176	C₂²×C₄	—	17	8	no	—
177	C₂²×C₄	—	17	2	no	—
178	C₂²×C₄	—	17	4	no	—
179	C₂²×C₄	C₈×D₁₇	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
181	C₂×C₈	—	17	2	no	—
182	C₂×C₈	—	17	2	no	—
183	C₂×C₈	—	17	4	no	—
184	C₂×C₈	—	17	4	no	—
185	C₂×C₈	—	17	8	no	—
186	C₂×C₈	—	17	8	no	—
187	C₂×C₈	—	17	16	no	—
188	C₂×C₈	C₈×D₁₇	1	4	yes	—
189	C₂×C₈	C₂×C₄×D₁₇	1	4	yes	—
190	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
192	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
193	C₄²	—	17	2	no	—
194	C₄²	—	17	8	no	—
195	C₄²	—	17	2	no	—
196	C₄²	C₈×D₁₇	1	2	yes	—
197	C₄²	C₂×C₄×D₁₇	1	2	yes	—
198	C₁₆	—	17	8	no	—
199	C₁₆	C₂×C₄×D₁₇	1	8	yes	direct product factor
200	C₂³	—	17	8	no	—
201	C₂³	—	17	4	no	—
202	C₂³	—	17	2	no	—
203	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
204	C₂×C₄	—	17	8	no	—
205	C₂×C₄	—	17	4	no	—
206	C₂×C₄	—	17	4	no	—
207	C₂×C₄	—	17	2	no	—
208	C₂×C₄	—	17	32	no	—
209	C₂×C₄	—	17	4	no	—
210	C₂×C₄	—	17	8	no	—
211	C₂×C₄	—	17	4	no	—
212	C₂×C₄	C₁₆×D₁₇	1	8	yes	direct product factor
213	C₂×C₄	C₄²×D₁₇	1	2	yes	—
214	C₂×C₄	C₂×C₈×D₁₇	1	4	yes	—
215	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
216	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
217	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
218	C₈	—	17	4	no	—
219	C₈	—	17	4	no	—
220	C₈	C₄²×D₁₇	1	2	yes	—
221	C₈	C₂×C₈×D₁₇	1	4	yes	—
222	C₈	C₂²×C₄×D₁₇	1	2	yes	—
223	C₂²	—	17	8	no	—
224	C₂²	—	17	16	no	—
225	C₂²	—	17	4	no	—
226	C₂²	C₄×C₈×D₁₇	1	2	yes	—
227	C₂²	C₂×C₁₆×D₁₇	1	4	yes	—
228	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
229	C₄	—	17	8	no	—
230	C₄	—	17	4	no	—
231	C₄	C₄×C₈×D₁₇	1	2	yes	—
232	C₄	C₂×C₁₆×D₁₇	1	8	yes	direct product factor
233	C₄	C₂×C₄²×D₁₇	1	1	yes	characteristic
234	C₄	C₂²×C₈×D₁₇	1	1	yes	characteristic
235	C₂	—	17	8	no	—
236	C₂	C₄×C₁₆×D₁₇	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₈×D₁₇	1	1	yes	characteristic
238	C₂	C₂²×C₁₆×D₁₇	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₁₆	D₁₇
commutator, hypocenter	C₁₇	C₂²×C₄×C₁₆
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#63214	C₂	semidirect
4. C₂×C₁₆×Dic₁₇	C₂	semidirect
5. C₄×C₁₆×D₁₇	C₂	direct (N × H)
6. C₂×C₄×C₁₆×C₁₇	C₂	semidirect
11. C₂×C₁₆×D₁₇	C₄	direct (N × H)
18. C₄×#63214	C₂²	semidirect
19. C₁₆×Dic₁₇	C₂²	semidirect
22. C₄×C₁₆×C₁₇	C₂²	semidirect
39. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
46. C₂×#63214	C₂×C₄	semidirect
49. C₁₆×D₁₇	C₂×C₄	direct (N × H)
55. C₂×C₄×D₁₇	C₁₆	direct (N × H)
78. C₁₆×C₁₇	C₂²×C₄	semidirect
79. #63214	C₂²×C₄	semidirect
85. C₂×C₄×C₁₇	C₂×C₁₆	semidirect
92. C₂×Dic₁₇	C₂×C₁₆	semidirect
96. C₄×D₁₇	C₂×C₁₆	direct (N × H)
109. C₂×C₄×C₁₆	D₁₇	direct (N × H)
113. C₂×D₁₇	C₄×C₁₆	direct (N × H)
117. C₄×C₁₇	C₂²×C₁₆	semidirect
121. Dic₁₇	C₂²×C₁₆	semidirect
137. C₄×C₁₆	C₂×D₁₇	direct (N × H)
138. C₂×C₁₇	C₂×C₄×C₁₆	semidirect
141. D₁₇	C₂×C₄×C₁₆	direct (N × H)
161. C₂×C₁₆	C₄×D₁₇	direct (N × H)
169. C₁₇	C₂²×C₄×C₁₆	semidirect
199. C₁₆	C₂×C₄×D₁₇	direct (N × H)
212. C₂×C₄	C₁₆×D₁₇	direct (N × H)
232. C₄	C₂×C₁₆×D₁₇	direct (N × H)
236. C₂	C₄×C₁₆×D₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₁₇	2
C₂×C₄×C₈×D₁₇	2
C₂×C₄×#63214	2
C₂×C₁₆×Dic₁₇	2
C₄×C₁₆×D₁₇	2
C₂×C₄×C₁₆×C₁₇	2
C₂²×C₄×C₁₆	17

Subgroups — C2×C4×C16×D17 #17186945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₆×D₁₇ #17186945