Subgroups — C₁₇×#13283 #16871027

Order 4352.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1846	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1380	C₂	1	4	yes	maximal
5	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
6	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
7	C₁₇×#500	—	2	2	no	—
8	C₁₇×#500	C₄	1	4	yes	complemented
9	C₂×C₁₇×#107	—	2	1	no	—
10	C₂×C₁₇×#107	C₂²	1	2	yes	—
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×#346	—	2	2	no	—
16	C₂²×Q₈×C₁₇	—	4	1	no	—
17	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₁₇×C₄⋊Q₈	—	2	4	no	—
20	C₁₇×C₄⋊Q₈	—	4	2	no	—
21	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
24	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₂×C₁₆×C₁₇	Q₈	1	4	yes	complemented
28	C₁₇×#107	—	2	4	no	—
29	C₁₇×#107	—	2	2	no	—
30	C₁₇×#107	—	2	4	no	—
31	C₁₇×#107	C₂×C₄	1	4	yes	—
32	C₁₇×#106	—	4	2	no	—
33	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₇	D₄	1	2	yes	—
35	C₁₇×#95	—	4	1	no	—
36	C₁₇×#95	—	4	1	no	—
37	C₂×Q₈×C₁₇	—	4	4	no	—
38	C₂×Q₈×C₁₇	—	8	4	no	—
39	C₂²×C₄×C₁₇	—	4	2	no	—
40	C₂²×C₄×C₁₇	—	4	2	no	—
41	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
47	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₁₇	#41	1	2	yes	—
51	C₄⋊C₄×C₁₇	—	2	4	no	—
52	C₄⋊C₄×C₁₇	—	4	4	no	—
53	C₄⋊C₄×C₁₇	—	4	2	no	—
54	C₄⋊C₄×C₁₇	—	4	2	no	—
55	C₄⋊C₄×C₁₇	—	4	2	no	—
56	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₁₇	SD₁₆	1	2	yes	—
58	C₁₆×C₁₇	—	2	4	no	—
59	#13283	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
61	Q₈×C₁₇	—	8	8	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	4	4	no	—
65	C₂×C₄×C₁₇	—	8	4	no	—
66	C₂×C₄×C₁₇	—	8	2	no	—
67	C₂×C₄×C₁₇	—	8	2	no	—
68	C₂×C₄×C₁₇	—	8	2	no	—
69	C₂×C₄×C₁₇	#102	1	2	yes	—
70	C₂×C₄×C₁₇	#102	1	2	yes	—
71	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₈×C₁₇	#103	1	4	yes	—
78	C₈×C₁₇	C₄×Q₈	1	2	yes	—
79	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
80	C₂×#500	C₂×C₁₇	1	1	yes	characteristic
81	C₂×#346	C₂×C₁₇	1	1	yes	characteristic
82	#1846	C₂×C₁₇	1	1	yes	characteristic
83	#1380	C₂×C₁₇	1	4	yes	—
84	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
85	C₂²×C₁₇	#486	1	1	yes	characteristic
86	C₂²×C₁₇	#500	1	1	yes	characteristic
87	C₂²×C₁₇	#509	1	2	yes	—
88	C₂²×C₁₇	#510	1	2	yes	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	8	4	no	—
92	C₄×C₁₇	—	16	2	no	—
93	C₄×C₁₇	#391	1	1	yes	characteristic
94	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
95	C₄×C₁₇	#487	1	1	yes	characteristic
96	C₄×C₁₇	#498	1	1	yes	characteristic
97	C₂×C₄⋊Q₈	—	2	1	no	—
98	C₂²×C₁₆	C₂²×C₁₇	1	2	yes	—
99	#500	—	2	2	no	—
100	#500	C₄×C₁₇	1	4	yes	complemented
101	C₂×#107	—	2	1	no	—
102	C₂×#107	C₂²×C₁₇	1	2	yes	—
103	C₂×#107	C₂²×C₁₇	1	2	yes	—
104	C₂×#106	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
106	#382	—	2	1	no	—
107	#346	—	2	2	no	—
108	C₂×C₁₇	#1990	1	1	yes	characteristic
109	C₂×C₁₇	#2140	1	1	yes	characteristic
110	C₂×C₁₇	#2141	1	1	yes	characteristic
111	C₂×C₁₇	#2243	1	2	yes	—
112	C₂×C₁₇	#2255	1	2	yes	—
113	C₂²×Q₈	—	4	1	no	—
114	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	Q₈×C₁₇	1	4	yes	complemented
125	#107	—	2	4	no	—
126	#107	—	2	2	no	—
127	#107	—	2	4	no	—
128	#107	C₂×C₄×C₁₇	1	4	yes	—
129	#106	—	4	2	no	—
130	C₄×C₈	C₂×C₄×C₁₇	1	2	yes	—
131	C₄×C₈	D₄×C₁₇	1	2	yes	—
132	#95	—	4	1	no	—
133	#95	—	4	1	no	—
134	C₁₇	#13283	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	8	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
140	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
142	C₂×C₈	C₂²⋊C₄×C₁₇	1	2	yes	—
143	C₂×C₈	SD₁₆×C₁₇	1	2	yes	—
144	C₂×C₈	Q₁₆×C₁₇	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
147	C₂×C₈	C₂×Q₈×C₁₇	1	2	yes	—
148	C₂×C₈	#41×C₁₇	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	C₂²⋊C₄×C₁₇	1	2	yes	—
155	C₄²	SD₁₆×C₁₇	1	2	yes	—
156	C₁₆	—	2	4	no	—
157	C₂³	C₂×D₈×C₁₇	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₁₇×#102	1	2	yes	—
167	C₂×C₄	C₁₇×#102	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
169	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
170	C₂×C₄	C₁₇×C₄⋊Q₈	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
172	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₁₆×C₁₇	1	1	yes	characteristic
174	C₈	C₁₇×#103	1	4	yes	—
175	C₈	C₄×Q₈×C₁₇	1	2	yes	—
176	C₈	C₁₇×C₂²⋊Q₈	1	2	yes	—
177	C₂²	C₂×C₁₇×#102	1	1	yes	characteristic
178	C₂²	C₁₇×#486	1	1	yes	characteristic
179	C₂²	C₁₇×#500	1	1	yes	characteristic
180	C₂²	C₁₇×#509	1	2	yes	—
181	C₂²	C₁₇×#510	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	16	2	no	—
186	C₄	C₁₇×#391	1	1	yes	characteristic
187	C₄	C₂×C₁₇×#103	1	1	yes	characteristic
188	C₄	C₁₇×#487	1	1	yes	characteristic
189	C₄	C₁₇×#498	1	1	yes	characteristic
190	C₂	C₁₇×#1990	1	1	yes	characteristic
191	C₂	C₁₇×#2140	1	1	yes	characteristic
192	C₂	C₁₇×#2141	1	1	yes	characteristic
193	C₂	C₁₇×#2243	1	2	yes	—
194	C₂	C₁₇×#2255	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
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
8. C₁₇×#500	C₄	semidirect
27. C₂×C₁₆×C₁₇	Q₈	semidirect
59. #13283	C₁₇	direct (N × H)
100. #500	C₄×C₁₇	semidirect
124. C₂×C₁₆	Q₈×C₁₇	semidirect
134. C₁₇	#13283	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×#500	2
C₂×C₁₇×#346	2
C₁₇×#1846	2
C₁₇×#1380	2
#13283	17

Subgroups — C17×#13283 #16871027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13283 #16871027