Subgroups — D₁₇×#2253 #17189802

Order 4352.

494 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	D₁₇×#499	C₂	1	2	yes	maximal
2	#14531063	C₂	1	1	yes	maximal, characteristic, complemented
3	#14531062	C₂	1	2	yes	maximal, complemented
4	#14531061	C₂	1	2	yes	maximal, complemented
5	#14531060	C₂	1	2	yes	maximal, complemented
6	D₁₇×#367	C₂	1	2	yes	maximal
7	D₁₇×#366	C₂	1	2	yes	maximal
8	C₄×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
10	#125×D₁₇	—	2	2	no	—
11	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
13	#11763211	C₂²	1	2	yes	—
14	#11763210	C₂²	1	4	yes	—
15	#11763209	—	2	2	no	—
16	#11763208	—	2	2	no	—
17	#107×D₁₇	C₂²	1	4	yes	—
18	#106×D₁₇	—	2	2	no	—
19	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
20	#11762637	C₂²	1	2	yes	—
21	#11762636	C₂²	1	4	yes	—
22	#11762635	C₂²	1	2	yes	—
23	#11762634	C₂²	1	4	yes	—
24	C₄×#63214	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
27	C₁₇×#499	C₂²	1	2	yes	—
28	C₁₇×#367	C₂²	1	2	yes	—
29	C₁₇×#366	C₂²	1	2	yes	—
30	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
31	C₁₇×#125	—	2	2	no	—
32	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
33	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×D₁₇	—	2	4	no	—
35	C₄⋊C₄×D₁₇	—	4	4	no	—
36	#10587576	—	2	4	no	—
37	#10587575	—	4	4	no	—
38	C₄²×D₁₇	D₄	1	1	yes	characteristic
39	#10587556	—	2	2	no	—
40	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
41	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₁₇×#107	C₂³	1	4	yes	—
43	C₁₇×#106	—	2	2	no	—
44	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
45	#10587489	C₂³	1	4	yes	—
46	#10587488	—	2	2	no	—
47	C₈×Dic₁₇	D₄	1	1	yes	characteristic
48	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
49	C₂×#63214	C₂³	1	1	yes	characteristic
50	C₂×#63214	C₂³	1	1	yes	characteristic
51	#10587479	—	2	4	no	—
52	#10587478	—	2	4	no	—
53	C₄×#3622	D₄	1	1	yes	characteristic
54	C₁₆×D₁₇	Q₈	1	4	yes	—
55	C₁₆×D₁₇	Q₈	1	4	yes	—
56	C₂×C₄×D₁₇	—	4	4	no	—
57	C₂×C₄×D₁₇	D₈	1	2	yes	—
58	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₁₇	—	2	4	no	—
62	C₄⋊C₄×C₁₇	—	4	4	no	—
63	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Dic₁₇	—	2	4	no	—
65	C₄⋊Dic₁₇	—	4	4	no	—
66	#63225	—	4	4	no	—
67	#63225	—	4	4	no	—
68	#63225	—	4	4	no	—
69	C₄×Dic₁₇	D₈	1	2	yes	—
70	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
71	C₂×#3622	C₂×D₄	1	1	yes	characteristic
72	C₂×#3622	C₂×D₄	1	1	yes	characteristic
73	C₈×D₁₇	Q₁₆	1	4	yes	—
74	C₈×D₁₇	C₂×Q₈	1	2	yes	—
75	C₈×D₁₇	C₂×Q₈	1	2	yes	—
76	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
77	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
78	#63214	C₂×Q₈	1	2	yes	—
79	#63214	C₂×Q₈	1	2	yes	—
80	C₂×#2253	—	17	1	no	maximal, Sylow
81	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	—	4	4	no	—
83	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₁₇	—	4	4	no	—
86	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
87	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
88	C₄×D₁₇	—	2	4	no	—
89	C₄×D₁₇	—	8	8	no	—
90	C₄×D₁₇	C₄⋊Q₈	1	2	yes	—
91	C₄×D₁₇	C₂×Q₁₆	1	2	yes	—
92	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
93	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
94	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
95	#3622	C₄⋊Q₈	1	2	yes	—
96	#3622	C₂×Q₁₆	1	2	yes	—
97	C₂×#499	—	17	2	no	—
98	#2253	—	17	2	no	—
99	#2253	—	17	2	no	—
100	#2253	—	17	2	no	—
101	#2253	—	17	1	no	—
102	#2253	D₁₇	1	1	yes	characteristic, direct product factor
103	C₂×#367	—	17	2	no	—
104	C₂×#366	—	17	2	no	—
105	C₂×C₄×C₁₆	—	17	1	no	—
106	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
107	C₂×D₁₇	#500	1	2	yes	—
108	C₂×D₁₇	#508	1	4	yes	—
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	—	8	4	no	—
111	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
112	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
113	Dic₁₇	—	2	2	no	—
114	Dic₁₇	—	8	4	no	—
115	Dic₁₇	#500	1	2	yes	—
116	C₂×#125	—	34	2	no	—
117	C₂²×C₁₆	—	17	1	no	—
118	C₂²×C₁₆	—	17	1	no	—
119	#499	—	17	4	no	—
120	#499	—	17	2	no	—
121	#499	—	34	2	no	—
122	#499	—	34	2	no	—
123	#499	C₂×D₁₇	1	2	yes	—
124	C₂×#107	—	17	4	no	—
125	C₂×#106	—	34	2	no	—
126	C₂×C₄×C₈	—	17	1	no	—
127	#367	—	17	2	no	—
128	#367	—	17	4	no	—
129	#367	C₂×D₁₇	1	2	yes	—
130	#366	—	17	2	no	—
131	#366	—	17	4	no	—
132	#366	C₂×D₁₇	1	2	yes	—
133	C₄×C₁₆	—	17	1	no	—
134	C₄×C₁₆	—	17	1	no	—
135	C₄×C₁₆	—	17	1	no	—
136	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
137	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
138	C₂×C₁₇	C₂×#508	1	2	yes	—
139	D₁₇	#2253	1	4	yes	direct product factor
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	17	1	no	—
142	#125	—	2	2	no	—
143	#125	—	34	4	no	—
144	#125	—	34	2	no	—
145	#125	—	68	4	no	—
146	C₂×C₄⋊C₄	—	34	4	no	—
147	C₂×C₄⋊C₄	—	68	4	no	—
148	C₂×C₄²	—	17	1	no	—
149	C₂×C₁₆	—	17	1	no	—
150	C₂×C₁₆	—	17	1	no	—
151	C₂×C₁₆	—	17	4	no	—
152	C₂×C₁₆	—	17	4	no	—
153	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
154	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
155	#107	—	17	4	no	—
156	#107	—	34	4	no	—
157	#107	C₂²×D₁₇	1	4	yes	—
158	#106	—	2	2	no	—
159	#106	—	34	4	no	—
160	#106	—	34	2	no	—
161	C₄×C₈	—	17	1	no	—
162	C₄×C₈	—	17	1	no	—
163	C₄×C₈	—	17	1	no	—
164	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
165	C₁₇	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
166	C₂²×C₄	—	17	1	no	—
167	C₂²×C₄	—	17	2	no	—
168	C₂²×C₄	—	68	4	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	1	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	17	2	no	—
173	C₂×C₈	—	17	4	no	—
174	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	34	4	no	—
179	C₄⋊C₄	—	68	4	no	—
180	C₄⋊C₄	—	68	4	no	—
181	C₄⋊C₄	—	68	4	no	—
182	C₄⋊C₄	—	68	4	no	—
183	C₄²	—	17	1	no	—
184	C₄²	—	17	2	no	—
185	C₄²	D₄×D₁₇	1	1	yes	characteristic
186	C₁₆	—	17	2	no	—
187	C₁₆	—	17	2	no	—
188	C₁₆	Q₈×D₁₇	1	2	yes	—
189	C₁₆	Q₈×D₁₇	1	2	yes	—
190	C₂³	—	17	1	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	2	no	—
194	C₂×C₄	—	17	2	no	—
195	C₂×C₄	—	17	2	no	—
196	C₂×C₄	—	34	4	no	—
197	C₂×C₄	—	68	4	no	—
198	C₂×C₄	—	136	8	no	—
199	C₂×C₄	D₈×D₁₇	1	2	yes	—
200	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
201	C₈	—	17	2	no	—
202	C₈	—	17	2	no	—
203	C₈	Q₁₆×D₁₇	1	2	yes	—
204	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
205	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
206	C₂²	—	17	2	no	—
207	C₂²	—	17	4	no	—
208	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	8	4	no	—
211	C₄	—	17	2	no	—
212	C₄	—	34	2	no	—
213	C₄	—	136	4	no	—
214	C₄	C₄⋊Q₈×D₁₇	1	1	yes	characteristic
215	C₄	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
216	C₂	—	17	4	no	—
217	C₂	D₁₇×#500	1	1	yes	characteristic
218	C₂	D₁₇×#508	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2253	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂²×D₁₇	C₂×D₈
hypercenter	#2253	D₁₇
hypocenter	C₁₇	C₂×#2253
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #14531063	C₂	semidirect
3. #14531062	C₂	semidirect
4. #14531061	C₂	semidirect
5. #14531060	C₂	semidirect
9. C₁₇×#2253	C₂	semidirect
102. #2253	D₁₇	direct (N × H)
139. D₁₇	#2253	direct (N × H)
165. C₁₇	C₂×#2253	semidirect

Maximal subgroups

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

subgroup	index
D₁₇×#499	2
#14531063	2
#14531062	2
#14531061	2
#14531060	2
D₁₇×#367	2
D₁₇×#366	2
C₄×C₁₆×D₁₇	2
C₁₇×#2253	2
C₂×#2253	17

Subgroups — D17×#2253 #17189802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₇×#2253 #17189802