Subgroups — C₁₇×#13206 #16870950

Order 4352.

590 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₇×#357	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₁₇	—	2	1	no	—
6	C₂²×D₈×C₁₇	C₂²	1	2	yes	—
7	C₂×C₁₇×#127	—	2	1	no	—
8	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
9	C₁₇×#493	—	2	2	no	—
10	C₁₇×#493	C₄	1	4	yes	complemented
11	C₂×C₁₇×#107	C₂²	1	2	yes	—
12	C₂×C₁₇×#106	—	2	1	no	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#382	—	2	1	no	—
15	C₁₇×#357	—	2	8	no	—
16	C₁₇×#346	—	2	2	no	—
17	C₂²×D₄×C₁₇	—	2	2	no	—
18	C₂²×D₄×C₁₇	—	4	2	no	—
19	C₂×D₈×C₁₇	—	2	4	no	—
20	C₂×D₈×C₁₇	—	2	4	no	—
21	C₂×D₈×C₁₇	—	2	2	no	—
22	C₂×D₈×C₁₇	—	2	4	no	—
23	C₂×D₈×C₁₇	—	4	2	no	—
24	C₂×D₈×C₁₇	—	4	2	no	—
25	C₂×D₈×C₁₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×#127	—	2	4	no	—
29	C₁₇×#127	—	4	2	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
31	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₇	—	2	2	no	—
33	C₂×C₁₆×C₁₇	—	2	2	no	—
34	C₂×C₁₆×C₁₇	D₄	1	4	yes	complemented
35	C₁₇×#107	—	2	4	no	—
36	C₁₇×#106	—	4	2	no	—
37	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₇	D₄	1	2	yes	—
39	C₁₇×#95	—	4	2	no	—
40	C₂⁴×C₁₇	—	4	2	no	—
41	C₂×D₄×C₁₇	—	2	4	no	—
42	C₂×D₄×C₁₇	—	4	8	no	—
43	C₂×D₄×C₁₇	—	4	2	no	—
44	C₂×D₄×C₁₇	—	4	2	no	—
45	C₂×D₄×C₁₇	—	4	2	no	—
46	C₂×D₄×C₁₇	—	4	2	no	—
47	C₂×D₄×C₁₇	—	4	2	no	—
48	C₂×D₄×C₁₇	—	8	8	no	—
49	C₂²×C₄×C₁₇	—	4	2	no	—
50	C₂²×C₄×C₁₇	D₈	1	2	yes	—
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	D₈×C₁₇	—	2	8	no	—
53	D₈×C₁₇	—	4	8	no	—
54	D₈×C₁₇	—	4	4	no	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₇	D₈	1	2	yes	—
57	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
58	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	#41	1	2	yes	—
62	C₄⋊C₄×C₁₇	—	4	2	no	—
63	C₄⋊C₄×C₁₇	—	4	2	no	—
64	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₁₇	SD₁₆	1	2	yes	—
66	C₁₆×C₁₇	—	2	4	no	—
67	#13206	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₁₇	—	4	4	no	—
69	C₂³×C₁₇	—	8	2	no	—
70	C₂³×C₁₇	—	8	2	no	—
71	C₂³×C₁₇	—	8	8	no	—
72	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
73	D₄×C₁₇	—	4	4	no	—
74	D₄×C₁₇	—	4	4	no	—
75	D₄×C₁₇	—	8	16	no	—
76	D₄×C₁₇	—	8	2	no	—
77	D₄×C₁₇	—	8	2	no	—
78	C₂×C₄×C₁₇	—	2	4	no	—
79	C₂×C₄×C₁₇	—	8	2	no	—
80	C₂×C₄×C₁₇	—	8	4	no	—
81	C₂×C₄×C₁₇	#102	1	4	yes	—
82	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₈×C₁₇	#102	1	4	yes	—
89	C₈×C₁₇	C₄×D₄	1	2	yes	—
90	C₈×C₁₇	#121	1	2	yes	—
91	C₂×#493	C₂×C₁₇	1	1	yes	characteristic
92	C₂×#357	C₂×C₁₇	1	4	yes	complemented
93	C₂×#346	C₂×C₁₇	1	1	yes	characteristic, complemented
94	#1846	C₂×C₁₇	1	1	yes	characteristic, complemented
95	C₂²×C₁₇	—	8	8	no	—
96	C₂²×C₁₇	—	8	4	no	—
97	C₂²×C₁₇	—	16	4	no	—
98	C₂²×C₁₇	—	16	4	no	—
99	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
100	C₂²×C₁₇	#486	1	1	yes	characteristic
101	C₂²×C₁₇	#493	1	1	yes	characteristic
102	C₂²×C₁₇	#509	1	4	yes	—
103	C₄×C₁₇	—	2	4	no	—
104	C₄×C₁₇	—	16	2	no	—
105	C₄×C₁₇	#390	1	1	yes	characteristic
106	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
107	C₄×C₁₇	#486	1	1	yes	characteristic
108	C₄×C₁₇	#492	1	1	yes	characteristic
109	C₂²×D₈	—	2	1	no	—
110	C₂²×D₈	C₂²×C₁₇	1	2	yes	—
111	C₂×#127	—	2	1	no	—
112	C₂²×C₁₆	C₂²×C₁₇	1	2	yes	—
113	#493	—	2	2	no	—
114	#493	C₄×C₁₇	1	4	yes	complemented
115	C₂×#107	C₂²×C₁₇	1	2	yes	—
116	C₂×#106	—	2	1	no	—
117	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
118	#382	—	2	1	no	—
119	#357	—	2	8	no	—
120	#346	—	2	2	no	—
121	C₂×C₁₇	—	16	4	no	—
122	C₂×C₁₇	#1957	1	1	yes	characteristic
123	C₂×C₁₇	#2140	1	2	yes	—
124	C₂×C₁₇	#2244	1	2	yes	—
125	C₂×C₁₇	#2250	1	2	yes	—
126	C₂²×D₄	—	2	2	no	—
127	C₂²×D₄	—	4	2	no	—
128	C₂×D₈	—	2	4	no	—
129	C₂×D₈	—	2	4	no	—
130	C₂×D₈	—	2	2	no	—
131	C₂×D₈	—	2	4	no	—
132	C₂×D₈	—	4	2	no	—
133	C₂×D₈	—	4	2	no	—
134	C₂×D₈	C₂×C₄×C₁₇	1	4	yes	—
135	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
137	#127	—	2	4	no	—
138	#127	—	4	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	D₄×C₁₇	1	4	yes	complemented
144	#107	—	2	4	no	—
145	#106	—	4	2	no	—
146	C₄×C₈	C₂×C₄×C₁₇	1	2	yes	—
147	C₄×C₈	D₄×C₁₇	1	2	yes	—
148	#95	—	4	2	no	—
149	C₁₇	#13206	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁴	—	4	2	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	2	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	4	8	no	—
158	C₂×D₄	—	8	8	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	D₈×C₁₇	1	2	yes	—
161	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
162	D₈	—	2	8	no	—
163	D₈	—	4	8	no	—
164	D₈	—	4	4	no	—
165	C₂×C₈	C₂²⋊C₄×C₁₇	1	2	yes	—
166	C₂×C₈	D₈×C₁₇	1	2	yes	—
167	C₂×C₈	SD₁₆×C₁₇	1	2	yes	—
168	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
170	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
171	C₂×C₈	#41×C₁₇	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	C₂²⋊C₄×C₁₇	1	2	yes	—
175	C₄²	SD₁₆×C₁₇	1	2	yes	—
176	C₁₆	—	2	4	no	—
177	C₂³	—	4	4	no	—
178	C₂³	—	8	2	no	—
179	C₂³	—	8	8	no	—
180	C₂³	—	8	2	no	—
181	C₂³	C₂×D₈×C₁₇	1	1	yes	characteristic
182	D₄	—	4	4	no	—
183	D₄	—	4	4	no	—
184	D₄	—	8	16	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	C₁₇×#102	1	4	yes	—
191	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
192	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
193	C₂×C₄	C₁₇×#127	1	1	yes	characteristic
194	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
195	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
196	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
197	C₈	C₁₇×#102	1	4	yes	—
198	C₈	C₄×D₄×C₁₇	1	2	yes	—
199	C₈	C₁₇×#121	1	2	yes	—
200	C₂²	—	8	4	no	—
201	C₂²	—	8	8	no	—
202	C₂²	—	16	4	no	—
203	C₂²	—	16	4	no	—
204	C₂²	C₂×C₁₇×#102	1	1	yes	characteristic
205	C₂²	C₁₇×#486	1	1	yes	characteristic
206	C₂²	C₁₇×#493	1	1	yes	characteristic
207	C₂²	C₁₇×#509	1	4	yes	—
208	C₄	—	2	4	no	—
209	C₄	—	16	2	no	—
210	C₄	C₁₇×#390	1	1	yes	characteristic
211	C₄	C₂×C₁₇×#102	1	1	yes	characteristic
212	C₄	C₁₇×#486	1	1	yes	characteristic
213	C₄	C₁₇×#492	1	1	yes	characteristic
214	C₂	—	16	4	no	—
215	C₂	C₁₇×#1957	1	1	yes	characteristic
216	C₂	C₁₇×#2140	1	2	yes	—
217	C₂	C₁₇×#2244	1	2	yes	—
218	C₂	C₁₇×#2250	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
prime radical	C₂×C₁₇×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ C₂×C₄×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₇×#357	C₂	semidirect
3. C₂×C₁₇×#346	C₂	semidirect
4. C₁₇×#1846	C₂	semidirect
10. C₁₇×#493	C₄	semidirect
34. C₂×C₁₆×C₁₇	D₄	semidirect
67. #13206	C₁₇	direct (N × H)
92. C₂×#357	C₂×C₁₇	semidirect
93. C₂×#346	C₂×C₁₇	semidirect
94. #1846	C₂×C₁₇	semidirect
114. #493	C₄×C₁₇	semidirect
143. C₂×C₁₆	D₄×C₁₇	semidirect
149. C₁₇	#13206	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×#493	2
C₂×C₁₇×#357	2
C₂×C₁₇×#346	2
C₁₇×#1846	2
#13206	17

Subgroups — C17×#13206 #16870950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13206 #16870950