Subgroups — C₂³×C₂²⋊C₄×D₁₇ #17940874

Order 4352.

43678 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	C₂⁶×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₁₇	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#63227	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊C₄×D₁₇	C₂	1	56	yes	maximal, direct product factor
6	C₂³×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₇	—	2	32	no	—
8	C₂⁵×D₁₇	C₄	1	32	yes	complemented
9	C₂⁵×D₁₇	C₂²	1	28	yes	—
10	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
11	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₁₇	C₂²	1	2	yes	complemented
13	C₂³×C₄×D₁₇	—	2	32	no	—
14	C₂³×C₄×D₁₇	C₂²	1	56	yes	complemented
15	C₂²×C₂²⋊Dic₁₇	C₂²	1	28	yes	complemented
16	C₂²×#63227	C₂²	1	56	yes	complemented
17	C₂×C₂²⋊C₄×D₁₇	C₂²	1	448	yes	direct product factor
18	C₂⁶×C₁₇	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	complemented
20	C₂²×C₂²⋊C₄×C₁₇	C₂²	1	28	yes	complemented
21	C₂⁵×C₁₇	—	2	16	no	—
22	C₂⁵×C₁₇	C₂×C₄	1	16	yes	complemented
23	C₂⁵×C₁₇	C₂³	1	14	yes	—
24	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
25	C₂⁴×D₁₇	—	2	16	no	—
26	C₂⁴×D₁₇	—	2	512	no	—
27	C₂⁴×D₁₇	—	2	448	no	—
28	C₂⁴×D₁₇	C₂×C₄	1	16	yes	—
29	C₂⁴×D₁₇	C₂×C₄	1	448	yes	complemented
30	C₂⁴×D₁₇	C₂×C₄	1	16	yes	complemented
31	C₂⁴×D₁₇	D₄	1	32	yes	—
32	C₂⁴×D₁₇	C₂³	1	112	yes	—
33	C₂⁴×D₁₇	C₂³	1	14	yes	—
34	C₂⁴×D₁₇	C₂³	1	14	yes	—
35	C₂³×C₄×C₁₇	—	2	16	no	—
36	C₂³×C₄×C₁₇	C₂³	1	28	yes	complemented
37	C₂³×Dic₁₇	—	2	16	no	—
38	C₂³×Dic₁₇	C₂³	1	28	yes	complemented
39	C₂²×C₄×D₁₇	—	2	448	no	—
40	C₂²×C₄×D₁₇	C₂³	1	224	yes	complemented
41	C₂×C₂²⋊C₄×C₁₇	C₂³	1	112	yes	complemented
42	C₂×C₂²⋊Dic₁₇	C₂³	1	112	yes	complemented
43	C₂×#63227	C₂³	1	224	yes	complemented
44	C₂²⋊C₄×D₁₇	C₂³	1	512	yes	direct product factor
45	C₂⁴×C₁₇	—	2	128	no	—
46	C₂⁴×C₁₇	—	2	112	no	—
47	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
48	C₂⁴×C₁₇	C₂²×C₄	1	112	yes	complemented
49	C₂⁴×C₁₇	C₂×D₄	1	16	yes	—
50	C₂⁴×C₁₇	C₂⁴	1	28	yes	—
51	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
52	C₂³×D₁₇	—	2	128	no	—
53	C₂³×D₁₇	—	2	112	no	—
54	C₂³×D₁₇	—	2	3584	no	—
55	C₂³×D₁₇	—	2	896	no	—
56	C₂³×D₁₇	C₂²⋊C₄	1	256	yes	direct product factor
57	C₂³×D₁₇	C₂²×C₄	1	112	yes	—
58	C₂³×D₁₇	C₂²×C₄	1	896	yes	complemented
59	C₂³×D₁₇	C₂²×C₄	1	112	yes	complemented
60	C₂³×D₁₇	C₂×D₄	1	224	yes	—
61	C₂³×D₁₇	C₂⁴	1	64	yes	—
62	C₂³×D₁₇	C₂⁴	1	28	yes	—
63	C₂³×D₁₇	C₂⁴	1	28	yes	—
64	C₂²×C₄×C₁₇	—	2	112	no	—
65	C₂²×C₄×C₁₇	C₂⁴	1	56	yes	complemented
66	C₂²×Dic₁₇	—	2	112	no	—
67	C₂²×Dic₁₇	C₂⁴	1	56	yes	complemented
68	C₂×C₄×D₁₇	—	2	896	no	—
69	C₂×C₄×D₁₇	C₂⁴	1	128	yes	complemented
70	C₂²⋊C₄×C₁₇	C₂⁴	1	64	yes	complemented
71	C₂²⋊Dic₁₇	C₂⁴	1	64	yes	complemented
72	#63227	C₂⁴	1	128	yes	complemented
73	C₂⁴×C₂²⋊C₄	—	17	1	no	maximal, Sylow
74	C₂³×C₁₇	—	2	448	no	—
75	C₂³×C₁₇	—	2	112	no	—
76	C₂³×C₁₇	C₂×C₂²⋊C₄	1	64	yes	complemented
77	C₂³×C₁₇	C₂³×C₄	1	28	yes	—
78	C₂³×C₁₇	C₂³×C₄	1	112	yes	complemented
79	C₂³×C₁₇	C₂²×D₄	1	56	yes	—
80	C₂³×C₁₇	C₂⁵	1	8	yes	—
81	C₂³×C₁₇	C₂⁵	1	7	yes	—
82	C₂²×D₁₇	—	2	448	no	—
83	C₂²×D₁₇	—	2	112	no	—
84	C₂²×D₁₇	—	2	3584	no	—
85	C₂²×D₁₇	—	2	256	no	—
86	C₂²×D₁₇	C₂×C₂²⋊C₄	1	896	yes	direct product factor
87	C₂²×D₁₇	C₂³×C₄	1	112	yes	—
88	C₂²×D₁₇	C₂³×C₄	1	256	yes	complemented
89	C₂²×D₁₇	C₂³×C₄	1	112	yes	complemented
90	C₂²×D₁₇	C₂²×D₄	1	224	yes	—
91	C₂²×D₁₇	C₂⁵	1	8	yes	—
92	C₂²×D₁₇	C₂⁵	1	8	yes	—
93	C₂×C₄×C₁₇	—	2	112	no	—
94	C₂×C₄×C₁₇	C₂⁵	1	16	yes	complemented
95	C₂×Dic₁₇	—	2	112	no	—
96	C₂×Dic₁₇	C₂⁵	1	16	yes	complemented
97	C₄×D₁₇	—	2	256	no	—
98	C₂⁷	—	17	1	no	—
99	C₂⁵×C₄	—	17	2	no	—
100	C₂³×C₂²⋊C₄	—	17	1	no	—
101	C₂³×C₂²⋊C₄	—	17	2	no	—
102	C₂³×C₂²⋊C₄	—	17	56	no	—
103	C₂³×C₂²⋊C₄	D₁₇	1	1	yes	characteristic, direct product factor
104	C₂²×C₁₇	—	2	16	no	—
105	C₂²×C₁₇	—	2	224	no	—
106	C₂²×C₁₇	C₂²×C₂²⋊C₄	1	112	yes	complemented
107	C₂²×C₁₇	C₂⁴×C₄	1	14	yes	—
108	C₂²×C₁₇	C₂⁴×C₄	1	16	yes	complemented
109	C₂²×C₁₇	C₂³×D₄	1	28	yes	—
110	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
111	C₂×D₁₇	—	2	16	no	—
112	C₂×D₁₇	—	2	224	no	—
113	C₂×D₁₇	—	2	512	no	—
114	C₂×D₁₇	C₂²×C₂²⋊C₄	1	448	yes	direct product factor
115	C₂×D₁₇	C₂⁴×C₄	1	16	yes	—
116	C₂×D₁₇	C₂⁴×C₄	1	16	yes	complemented
117	C₂×D₁₇	C₂³×D₄	1	32	yes	—
118	C₄×C₁₇	—	2	16	no	—
119	Dic₁₇	—	2	16	no	—
120	C₂⁶	—	17	1	no	—
121	C₂⁶	—	17	32	no	—
122	C₂⁶	—	17	28	no	—
123	C₂⁶	—	17	1	no	—
124	C₂⁶	—	34	32	no	—
125	C₂⁶	C₂×D₁₇	1	1	yes	characteristic
126	C₂⁴×C₄	—	17	2	no	—
127	C₂⁴×C₄	—	17	56	no	—
128	C₂⁴×C₄	—	34	32	no	—
129	C₂⁴×C₄	C₂×D₁₇	1	2	yes	complemented
130	C₂²×C₂²⋊C₄	—	17	28	no	—
131	C₂²×C₂²⋊C₄	—	17	56	no	—
132	C₂²×C₂²⋊C₄	—	17	448	no	—
133	C₂²×C₂²⋊C₄	C₂×D₁₇	1	28	yes	direct product factor
134	C₂×C₁₇	—	2	16	no	—
135	C₂×C₁₇	C₂³×C₂²⋊C₄	1	28	yes	complemented
136	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
137	C₂×C₁₇	C₂⁴×D₄	1	2	yes	—
138	D₁₇	—	2	16	no	—
139	D₁₇	C₂³×C₂²⋊C₄	1	32	yes	direct product factor
140	C₂⁵	—	2	16	no	—
141	C₂⁵	—	17	16	no	—
142	C₂⁵	—	17	32	no	—
143	C₂⁵	—	17	14	no	—
144	C₂⁵	—	17	16	no	—
145	C₂⁵	—	17	448	no	—
146	C₂⁵	—	17	112	no	—
147	C₂⁵	—	17	14	no	—
148	C₂⁵	—	34	448	no	—
149	C₂⁵	—	34	512	no	—
150	C₂⁵	—	34	16	no	—
151	C₂⁵	C₄×D₁₇	1	16	yes	complemented
152	C₂⁵	C₂²×D₁₇	1	14	yes	—
153	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
154	C₂³×C₄	—	2	16	no	—
155	C₂³×C₄	—	17	28	no	—
156	C₂³×C₄	—	17	224	no	—
157	C₂³×C₄	—	34	448	no	—
158	C₂³×C₄	—	34	16	no	—
159	C₂³×C₄	C₂²×D₁₇	1	28	yes	complemented
160	C₂×C₂²⋊C₄	—	17	112	no	—
161	C₂×C₂²⋊C₄	—	17	224	no	—
162	C₂×C₂²⋊C₄	—	17	512	no	—
163	C₂×C₂²⋊C₄	C₂²×D₁₇	1	112	yes	direct product factor
164	C₁₇	C₂⁴×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
165	C₂⁴	—	2	112	no	—
166	C₂⁴	—	2	128	no	—
167	C₂⁴	—	17	112	no	—
168	C₂⁴	—	17	224	no	—
169	C₂⁴	—	17	28	no	—
170	C₂⁴	—	17	256	no	—
171	C₂⁴	—	17	112	no	—
172	C₂⁴	—	17	896	no	—
173	C₂⁴	—	17	64	no	—
174	C₂⁴	—	17	28	no	—
175	C₂⁴	—	34	3584	no	—
176	C₂⁴	—	34	896	no	—
177	C₂⁴	—	34	112	no	—
178	C₂⁴	—	34	128	no	—
179	C₂⁴	C₂×C₄×D₁₇	1	8	yes	—
180	C₂⁴	C₂×C₄×D₁₇	1	112	yes	complemented
181	C₂⁴	D₄×D₁₇	1	16	yes	—
182	C₂⁴	C₂³×D₁₇	1	28	yes	—
183	C₂⁴	C₂³×D₁₇	1	7	yes	—
184	C₂²×C₄	—	2	112	no	—
185	C₂²×C₄	—	17	56	no	—
186	C₂²×C₄	—	17	128	no	—
187	C₂²×C₄	—	34	896	no	—
188	C₂²×C₄	—	34	112	no	—
189	C₂²×C₄	C₂³×D₁₇	1	56	yes	complemented
190	C₂²⋊C₄	—	17	64	no	—
191	C₂²⋊C₄	—	17	128	no	—
192	C₂²⋊C₄	C₂³×D₁₇	1	64	yes	direct product factor
193	C₂³	—	2	448	no	—
194	C₂³	—	2	112	no	—
195	C₂³	—	17	896	no	—
196	C₂³	—	17	112	no	—
197	C₂³	—	17	224	no	—
198	C₂³	—	17	8	no	—
199	C₂³	—	17	112	no	—
200	C₂³	—	17	256	no	—
201	C₂³	—	17	8	no	—
202	C₂³	—	34	3584	no	—
203	C₂³	—	34	256	no	—
204	C₂³	—	34	448	no	—
205	C₂³	—	34	112	no	—
206	C₂³	C₂²⋊C₄×D₁₇	1	64	yes	direct product factor
207	C₂³	C₂²×C₄×D₁₇	1	28	yes	—
208	C₂³	C₂²×C₄×D₁₇	1	112	yes	complemented
209	C₂³	C₂×D₄×D₁₇	1	56	yes	—
210	C₂³	C₂⁴×D₁₇	1	8	yes	—
211	C₂³	C₂⁴×D₁₇	1	7	yes	—
212	C₂×C₄	—	2	112	no	—
213	C₂×C₄	—	17	16	no	—
214	C₂×C₄	—	34	256	no	—
215	C₂×C₄	—	34	112	no	—
216	C₂×C₄	C₂⁴×D₁₇	1	16	yes	complemented
217	C₂²	—	2	224	no	—
218	C₂²	—	2	16	no	—
219	C₂²	—	17	448	no	—
220	C₂²	—	17	16	no	—
221	C₂²	—	17	32	no	—
222	C₂²	—	17	16	no	—
223	C₂²	—	34	512	no	—
224	C₂²	—	34	224	no	—
225	C₂²	—	34	16	no	—
226	C₂²	C₂×C₂²⋊C₄×D₁₇	1	112	yes	direct product factor
227	C₂²	C₂³×C₄×D₁₇	1	14	yes	—
228	C₂²	C₂³×C₄×D₁₇	1	16	yes	complemented
229	C₂²	C₂²×D₄×D₁₇	1	28	yes	—
230	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
231	C₄	—	2	16	no	—
232	C₄	—	34	16	no	—
233	C₂	—	2	16	no	—
234	C₂	—	17	32	no	—
235	C₂	—	34	16	no	—
236	C₂	C₂²×C₂²⋊C₄×D₁₇	1	28	yes	direct product factor
237	C₂	C₂⁴×C₄×D₁₇	1	1	yes	characteristic
238	C₂	C₂³×D₄×D₁₇	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₁₇
commutator	C₂×C₁₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₇
Fitting	C₂³×C₂²⋊C₄×C₁₇	C₂
socle	C₂⁵×C₁₇	C₂³
prime radical	C₂⁶×D₁₇	C₂
cosocle	C₂²×C₁₇	C₂⁶
hypercenter	C₂³×C₂²⋊C₄	D₁₇
hypocenter	C₁₇	C₂⁴×C₂²⋊C₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₁₇	C₂	semidirect
3. C₂³×C₂²⋊Dic₁₇	C₂	semidirect
4. C₂³×#63227	C₂	semidirect
5. C₂²×C₂²⋊C₄×D₁₇	C₂	direct (N × H)
6. C₂³×C₂²⋊C₄×C₁₇	C₂	semidirect
8. C₂⁵×D₁₇	C₄	semidirect
12. C₂⁴×Dic₁₇	C₂²	semidirect
14. C₂³×C₄×D₁₇	C₂²	semidirect
15. C₂²×C₂²⋊Dic₁₇	C₂²	semidirect
16. C₂²×#63227	C₂²	semidirect
17. C₂×C₂²⋊C₄×D₁₇	C₂²	direct (N × H)
19. C₂⁴×C₄×C₁₇	C₂²	semidirect
20. C₂²×C₂²⋊C₄×C₁₇	C₂²	semidirect
22. C₂⁵×C₁₇	C₂×C₄	semidirect
29. C₂⁴×D₁₇	C₂×C₄	semidirect
30. C₂⁴×D₁₇	C₂×C₄	semidirect
36. C₂³×C₄×C₁₇	C₂³	semidirect
38. C₂³×Dic₁₇	C₂³	semidirect
40. C₂²×C₄×D₁₇	C₂³	semidirect
41. C₂×C₂²⋊C₄×C₁₇	C₂³	semidirect
42. C₂×C₂²⋊Dic₁₇	C₂³	semidirect
43. C₂×#63227	C₂³	semidirect
44. C₂²⋊C₄×D₁₇	C₂³	direct (N × H)
48. C₂⁴×C₁₇	C₂²×C₄	semidirect
56. C₂³×D₁₇	C₂²⋊C₄	direct (N × H)
58. C₂³×D₁₇	C₂²×C₄	semidirect
59. C₂³×D₁₇	C₂²×C₄	semidirect
65. C₂²×C₄×C₁₇	C₂⁴	semidirect
67. C₂²×Dic₁₇	C₂⁴	semidirect
69. C₂×C₄×D₁₇	C₂⁴	semidirect
70. C₂²⋊C₄×C₁₇	C₂⁴	semidirect
71. C₂²⋊Dic₁₇	C₂⁴	semidirect
72. #63227	C₂⁴	semidirect
76. C₂³×C₁₇	C₂×C₂²⋊C₄	semidirect
78. C₂³×C₁₇	C₂³×C₄	semidirect
86. C₂²×D₁₇	C₂×C₂²⋊C₄	direct (N × H)
88. C₂²×D₁₇	C₂³×C₄	semidirect
89. C₂²×D₁₇	C₂³×C₄	semidirect
94. C₂×C₄×C₁₇	C₂⁵	semidirect
96. C₂×Dic₁₇	C₂⁵	semidirect
103. C₂³×C₂²⋊C₄	D₁₇	direct (N × H)
106. C₂²×C₁₇	C₂²×C₂²⋊C₄	semidirect
108. C₂²×C₁₇	C₂⁴×C₄	semidirect
114. C₂×D₁₇	C₂²×C₂²⋊C₄	direct (N × H)
116. C₂×D₁₇	C₂⁴×C₄	semidirect
129. C₂⁴×C₄	C₂×D₁₇	semidirect
133. C₂²×C₂²⋊C₄	C₂×D₁₇	direct (N × H)
135. C₂×C₁₇	C₂³×C₂²⋊C₄	semidirect
139. D₁₇	C₂³×C₂²⋊C₄	direct (N × H)
151. C₂⁵	C₄×D₁₇	semidirect
159. C₂³×C₄	C₂²×D₁₇	semidirect
163. C₂×C₂²⋊C₄	C₂²×D₁₇	direct (N × H)
164. C₁₇	C₂⁴×C₂²⋊C₄	semidirect
180. C₂⁴	C₂×C₄×D₁₇	semidirect
189. C₂²×C₄	C₂³×D₁₇	semidirect
192. C₂²⋊C₄	C₂³×D₁₇	direct (N × H)
206. C₂³	C₂²⋊C₄×D₁₇	direct (N × H)
208. C₂³	C₂²×C₄×D₁₇	semidirect
216. C₂×C₄	C₂⁴×D₁₇	semidirect
226. C₂²	C₂×C₂²⋊C₄×D₁₇	direct (N × H)
228. C₂²	C₂³×C₄×D₁₇	semidirect
236. C₂	C₂²×C₂²⋊C₄×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
C₂⁶×D₁₇	2
C₂⁴×C₄×D₁₇	2
C₂³×C₂²⋊Dic₁₇	2
C₂³×#63227	2
C₂²×C₂²⋊C₄×D₁₇	2
C₂³×C₂²⋊C₄×C₁₇	2
C₂⁴×C₂²⋊C₄	17

Subgroups — C23×C22⋊C4×D17 #17940874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊C₄×D₁₇ #17940874