Subgroups — C₂³×C₄⋊C₄×D₁₇ #17940877

Order 4352.

19742 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₁₇	C₂	1	2	yes	maximal
2	C₂⁴×C₄×D₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#63225	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄⋊C₄×D₁₇	C₂	1	56	yes	maximal, direct product factor
6	C₂³×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₇	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₁₇	C₂²	1	2	yes	—
9	C₂⁴×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×D₁₇	—	2	32	no	—
11	C₂³×C₄×D₁₇	C₄	1	32	yes	complemented
12	C₂³×C₄×D₁₇	C₂²	1	56	yes	—
13	C₂³×C₄×D₁₇	C₂²	1	28	yes	—
14	C₂²×C₄⋊Dic₁₇	C₂²	1	28	yes	complemented
15	C₂²×#63225	C₂²	1	56	yes	complemented
16	C₂×C₄⋊C₄×D₁₇	C₂²	1	448	yes	direct product factor
17	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₁₇	C₂²	1	28	yes	complemented
20	C₂⁵×C₁₇	C₂³	1	1	yes	characteristic
21	C₂⁴×D₁₇	C₂×C₄	1	16	yes	—
22	C₂⁴×D₁₇	D₄	1	16	yes	—
23	C₂⁴×D₁₇	Q₈	1	16	yes	—
24	C₂⁴×D₁₇	C₂³	1	14	yes	—
25	C₂³×C₄×C₁₇	—	2	16	no	—
26	C₂³×C₄×C₁₇	C₂×C₄	1	16	yes	complemented
27	C₂³×C₄×C₁₇	C₂³	1	28	yes	—
28	C₂³×C₄×C₁₇	C₂³	1	14	yes	—
29	C₂³×Dic₁₇	—	2	16	no	—
30	C₂³×Dic₁₇	C₂×C₄	1	16	yes	complemented
31	C₂³×Dic₁₇	C₂³	1	28	yes	—
32	C₂³×Dic₁₇	C₂³	1	14	yes	—
33	C₂²×C₄×D₁₇	—	2	448	no	—
34	C₂²×C₄×D₁₇	C₂×C₄	1	448	yes	complemented
35	C₂²×C₄×D₁₇	C₂³	1	224	yes	—
36	C₂²×C₄×D₁₇	C₂³	1	112	yes	—
37	C₂×C₄⋊C₄×C₁₇	C₂³	1	112	yes	complemented
38	C₂×C₄⋊Dic₁₇	C₂³	1	112	yes	complemented
39	C₂×#63225	C₂³	1	224	yes	complemented
40	C₄⋊C₄×D₁₇	C₂³	1	512	yes	direct product factor
41	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
42	C₂⁴×C₁₇	C₂×D₄	1	8	yes	—
43	C₂⁴×C₁₇	C₂×Q₈	1	8	yes	—
44	C₂⁴×C₁₇	C₂⁴	1	7	yes	—
45	C₂³×D₁₇	C₄⋊C₄	1	256	yes	direct product factor
46	C₂³×D₁₇	C₂²×C₄	1	112	yes	—
47	C₂³×D₁₇	C₂×D₄	1	112	yes	—
48	C₂³×D₁₇	C₂×Q₈	1	112	yes	—
49	C₂³×D₁₇	C₂⁴	1	28	yes	—
50	C₂²×C₄×C₁₇	—	2	112	no	—
51	C₂²×C₄×C₁₇	C₂²×C₄	1	112	yes	complemented
52	C₂²×C₄×C₁₇	C₂⁴	1	56	yes	—
53	C₂²×C₄×C₁₇	C₂⁴	1	28	yes	—
54	C₂²×Dic₁₇	—	2	112	no	—
55	C₂²×Dic₁₇	C₂²×C₄	1	112	yes	complemented
56	C₂²×Dic₁₇	C₂⁴	1	56	yes	—
57	C₂²×Dic₁₇	C₂⁴	1	28	yes	—
58	C₂×C₄×D₁₇	—	2	896	no	—
59	C₂×C₄×D₁₇	C₂²×C₄	1	896	yes	complemented
60	C₂×C₄×D₁₇	C₂⁴	1	128	yes	—
61	C₂×C₄×D₁₇	C₂⁴	1	64	yes	—
62	C₄⋊C₄×C₁₇	C₂⁴	1	64	yes	complemented
63	C₄⋊Dic₁₇	C₂⁴	1	64	yes	complemented
64	#63225	C₂⁴	1	128	yes	complemented
65	C₂⁴×C₄⋊C₄	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×C₄⋊C₄	1	64	yes	complemented
67	C₂³×C₁₇	C₂³×C₄	1	28	yes	—
68	C₂³×C₁₇	C₂²×D₄	1	28	yes	—
69	C₂³×C₁₇	C₂²×Q₈	1	28	yes	—
70	C₂³×C₁₇	C₂⁵	1	7	yes	—
71	C₂²×D₁₇	C₂×C₄⋊C₄	1	896	yes	direct product factor
72	C₂²×D₁₇	C₂³×C₄	1	112	yes	—
73	C₂²×D₁₇	C₂²×D₄	1	112	yes	—
74	C₂²×D₁₇	C₂²×Q₈	1	112	yes	—
75	C₂²×D₁₇	C₂⁵	1	8	yes	—
76	C₂×C₄×C₁₇	—	2	112	no	—
77	C₂×C₄×C₁₇	C₂³×C₄	1	112	yes	complemented
78	C₂×C₄×C₁₇	C₂⁵	1	16	yes	—
79	C₂×C₄×C₁₇	C₂⁵	1	8	yes	—
80	C₂×Dic₁₇	—	2	112	no	—
81	C₂×Dic₁₇	C₂³×C₄	1	112	yes	complemented
82	C₂×Dic₁₇	C₂⁵	1	16	yes	—
83	C₂×Dic₁₇	C₂⁵	1	8	yes	—
84	C₄×D₁₇	—	2	256	no	—
85	C₄×D₁₇	C₂³×C₄	1	256	yes	complemented
86	C₂⁵×C₄	—	17	1	no	—
87	C₂⁵×C₄	—	17	2	no	—
88	C₂³×C₄⋊C₄	—	17	2	no	—
89	C₂³×C₄⋊C₄	—	17	1	no	—
90	C₂³×C₄⋊C₄	—	17	56	no	—
91	C₂³×C₄⋊C₄	D₁₇	1	1	yes	characteristic, direct product factor
92	C₂²×C₁₇	C₂²×C₄⋊C₄	1	112	yes	complemented
93	C₂²×C₁₇	C₂⁴×C₄	1	14	yes	—
94	C₂²×C₁₇	C₂³×D₄	1	14	yes	—
95	C₂²×C₁₇	C₂³×Q₈	1	14	yes	—
96	C₂²×C₁₇	C₂⁶	1	1	yes	characteristic
97	C₂×D₁₇	C₂²×C₄⋊C₄	1	448	yes	direct product factor
98	C₂×D₁₇	C₂⁴×C₄	1	16	yes	—
99	C₂×D₁₇	C₂³×D₄	1	16	yes	—
100	C₂×D₁₇	C₂³×Q₈	1	16	yes	—
101	C₄×C₁₇	—	2	16	no	—
102	C₄×C₁₇	C₂⁴×C₄	1	16	yes	complemented
103	Dic₁₇	—	2	16	no	—
104	Dic₁₇	C₂⁴×C₄	1	16	yes	complemented
105	C₂⁶	—	17	1	no	—
106	C₂⁴×C₄	—	17	1	no	—
107	C₂⁴×C₄	—	17	32	no	—
108	C₂⁴×C₄	—	17	28	no	—
109	C₂⁴×C₄	—	17	2	no	—
110	C₂⁴×C₄	—	17	56	no	—
111	C₂⁴×C₄	—	34	32	no	—
112	C₂⁴×C₄	C₂×D₁₇	1	2	yes	—
113	C₂⁴×C₄	C₂×D₁₇	1	1	yes	characteristic
114	C₂²×C₄⋊C₄	—	17	56	no	—
115	C₂²×C₄⋊C₄	—	17	28	no	—
116	C₂²×C₄⋊C₄	—	17	448	no	—
117	C₂²×C₄⋊C₄	C₂×D₁₇	1	28	yes	direct product factor
118	C₂×C₁₇	C₂³×C₄⋊C₄	1	28	yes	complemented
119	C₂×C₁₇	C₂⁵×C₄	1	1	yes	characteristic
120	C₂×C₁₇	C₂⁴×D₄	1	1	yes	characteristic
121	C₂×C₁₇	C₂⁴×Q₈	1	1	yes	characteristic
122	D₁₇	C₂³×C₄⋊C₄	1	32	yes	direct product factor
123	C₂⁵	—	17	16	no	—
124	C₂⁵	—	17	16	no	—
125	C₂⁵	—	17	16	no	—
126	C₂⁵	—	17	14	no	—
127	C₂⁵	C₂²×D₁₇	1	1	yes	characteristic
128	C₂³×C₄	—	2	16	no	—
129	C₂³×C₄	—	17	16	no	—
130	C₂³×C₄	—	17	14	no	—
131	C₂³×C₄	—	17	448	no	—
132	C₂³×C₄	—	17	112	no	—
133	C₂³×C₄	—	17	28	no	—
134	C₂³×C₄	—	17	224	no	—
135	C₂³×C₄	—	34	448	no	—
136	C₂³×C₄	—	34	16	no	—
137	C₂³×C₄	C₄×D₁₇	1	16	yes	complemented
138	C₂³×C₄	C₂²×D₁₇	1	28	yes	—
139	C₂³×C₄	C₂²×D₁₇	1	14	yes	—
140	C₂×C₄⋊C₄	—	17	224	no	—
141	C₂×C₄⋊C₄	—	17	112	no	—
142	C₂×C₄⋊C₄	—	17	512	no	—
143	C₂×C₄⋊C₄	C₂²×D₁₇	1	112	yes	direct product factor
144	C₁₇	C₂⁴×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
145	C₂⁴	—	17	112	no	—
146	C₂⁴	—	17	112	no	—
147	C₂⁴	—	17	112	no	—
148	C₂⁴	—	17	28	no	—
149	C₂⁴	—	17	256	no	—
150	C₂⁴	C₂×C₄×D₁₇	1	8	yes	—
151	C₂⁴	D₄×D₁₇	1	8	yes	—
152	C₂⁴	Q₈×D₁₇	1	8	yes	—
153	C₂⁴	C₂³×D₁₇	1	7	yes	—
154	C₂²×C₄	—	2	112	no	—
155	C₂²×C₄	—	17	112	no	—
156	C₂²×C₄	—	17	28	no	—
157	C₂²×C₄	—	17	896	no	—
158	C₂²×C₄	—	17	64	no	—
159	C₂²×C₄	—	17	56	no	—
160	C₂²×C₄	—	17	128	no	—
161	C₂²×C₄	—	34	896	no	—
162	C₂²×C₄	—	34	112	no	—
163	C₂²×C₄	C₂×C₄×D₁₇	1	112	yes	complemented
164	C₂²×C₄	C₂³×D₁₇	1	56	yes	—
165	C₂²×C₄	C₂³×D₁₇	1	28	yes	—
166	C₄⋊C₄	—	17	128	no	—
167	C₄⋊C₄	—	17	64	no	—
168	C₄⋊C₄	C₂³×D₁₇	1	64	yes	direct product factor
169	C₂³	—	17	896	no	—
170	C₂³	—	17	112	no	—
171	C₂³	—	17	112	no	—
172	C₂³	—	17	112	no	—
173	C₂³	—	17	8	no	—
174	C₂³	C₄⋊C₄×D₁₇	1	64	yes	direct product factor
175	C₂³	C₂²×C₄×D₁₇	1	28	yes	—
176	C₂³	C₂×D₄×D₁₇	1	28	yes	—
177	C₂³	C₂×Q₈×D₁₇	1	28	yes	—
178	C₂³	C₂⁴×D₁₇	1	7	yes	—
179	C₂×C₄	—	2	112	no	—
180	C₂×C₄	—	17	112	no	—
181	C₂×C₄	—	17	8	no	—
182	C₂×C₄	—	17	256	no	—
183	C₂×C₄	—	17	16	no	—
184	C₂×C₄	—	34	256	no	—
185	C₂×C₄	—	34	112	no	—
186	C₂×C₄	C₂²×C₄×D₁₇	1	112	yes	complemented
187	C₂×C₄	C₂⁴×D₁₇	1	16	yes	—
188	C₂×C₄	C₂⁴×D₁₇	1	8	yes	—
189	C₂²	—	17	448	no	—
190	C₂²	—	17	16	no	—
191	C₂²	—	17	16	no	—
192	C₂²	—	17	16	no	—
193	C₂²	C₂×C₄⋊C₄×D₁₇	1	112	yes	direct product factor
194	C₂²	C₂³×C₄×D₁₇	1	14	yes	—
195	C₂²	C₂²×D₄×D₁₇	1	14	yes	—
196	C₂²	C₂²×Q₈×D₁₇	1	14	yes	—
197	C₂²	C₂⁵×D₁₇	1	1	yes	characteristic
198	C₄	—	2	16	no	—
199	C₄	—	17	16	no	—
200	C₄	—	34	16	no	—
201	C₄	C₂³×C₄×D₁₇	1	16	yes	complemented
202	C₂	—	17	32	no	—
203	C₂	C₂²×C₄⋊C₄×D₁₇	1	28	yes	direct product factor
204	C₂	C₂⁴×C₄×D₁₇	1	1	yes	characteristic
205	C₂	C₂³×D₄×D₁₇	1	1	yes	characteristic
206	C₂	C₂³×Q₈×D₁₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂³×C₄⋊Dic₁₇	C₂	semidirect
4. C₂³×#63225	C₂	semidirect
5. C₂²×C₄⋊C₄×D₁₇	C₂	direct (N × H)
6. C₂³×C₄⋊C₄×C₁₇	C₂	semidirect
11. C₂³×C₄×D₁₇	C₄	semidirect
14. C₂²×C₄⋊Dic₁₇	C₂²	semidirect
15. C₂²×#63225	C₂²	semidirect
16. C₂×C₄⋊C₄×D₁₇	C₂²	direct (N × H)
19. C₂²×C₄⋊C₄×C₁₇	C₂²	semidirect
26. C₂³×C₄×C₁₇	C₂×C₄	semidirect
30. C₂³×Dic₁₇	C₂×C₄	semidirect
34. C₂²×C₄×D₁₇	C₂×C₄	semidirect
37. C₂×C₄⋊C₄×C₁₇	C₂³	semidirect
38. C₂×C₄⋊Dic₁₇	C₂³	semidirect
39. C₂×#63225	C₂³	semidirect
40. C₄⋊C₄×D₁₇	C₂³	direct (N × H)
45. C₂³×D₁₇	C₄⋊C₄	direct (N × H)
51. C₂²×C₄×C₁₇	C₂²×C₄	semidirect
55. C₂²×Dic₁₇	C₂²×C₄	semidirect
59. C₂×C₄×D₁₇	C₂²×C₄	semidirect
62. C₄⋊C₄×C₁₇	C₂⁴	semidirect
63. C₄⋊Dic₁₇	C₂⁴	semidirect
64. #63225	C₂⁴	semidirect
66. C₂³×C₁₇	C₂×C₄⋊C₄	semidirect
71. C₂²×D₁₇	C₂×C₄⋊C₄	direct (N × H)
77. C₂×C₄×C₁₇	C₂³×C₄	semidirect
81. C₂×Dic₁₇	C₂³×C₄	semidirect
85. C₄×D₁₇	C₂³×C₄	semidirect
91. C₂³×C₄⋊C₄	D₁₇	direct (N × H)
92. C₂²×C₁₇	C₂²×C₄⋊C₄	semidirect
97. C₂×D₁₇	C₂²×C₄⋊C₄	direct (N × H)
102. C₄×C₁₇	C₂⁴×C₄	semidirect
104. Dic₁₇	C₂⁴×C₄	semidirect
117. C₂²×C₄⋊C₄	C₂×D₁₇	direct (N × H)
118. C₂×C₁₇	C₂³×C₄⋊C₄	semidirect
122. D₁₇	C₂³×C₄⋊C₄	direct (N × H)
137. C₂³×C₄	C₄×D₁₇	semidirect
143. C₂×C₄⋊C₄	C₂²×D₁₇	direct (N × H)
144. C₁₇	C₂⁴×C₄⋊C₄	semidirect
163. C₂²×C₄	C₂×C₄×D₁₇	semidirect
168. C₄⋊C₄	C₂³×D₁₇	direct (N × H)
174. C₂³	C₄⋊C₄×D₁₇	direct (N × H)
186. C₂×C₄	C₂²×C₄×D₁₇	semidirect
193. C₂²	C₂×C₄⋊C₄×D₁₇	direct (N × H)
201. C₄	C₂³×C₄×D₁₇	semidirect
203. C₂	C₂²×C₄⋊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
C₂⁴×C₄×D₁₇	2
C₂³×C₄⋊Dic₁₇	2
C₂³×#63225	2
C₂²×C₄⋊C₄×D₁₇	2
C₂³×C₄⋊C₄×C₁₇	2
C₂⁴×C₄⋊C₄	17

Subgroups — C23×C4⋊C4×D17 #17940877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊C₄×D₁₇ #17940877