Subgroups — #17187361

Order 4352.

674 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587469	C₂	1	1	yes	maximal, characteristic
2	D₁₇×#427	C₂	1	2	yes	maximal
3	#14530571	C₂	1	4	yes	maximal, complemented
4	#14530569	C₂	1	4	yes	maximal, complemented
5	#14530568	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530566	C₂	1	2	yes	maximal, complemented
7	C₁₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	#117×D₁₇	—	2	2	no	—
10	C₂×#10587469	C₄	1	8	yes	complemented
11	C₂×#10587469	C₂²	1	2	yes	—
12	C₂×#10587469	C₂²	1	2	yes	—
13	C₂²×#63214	C₂²	1	1	yes	characteristic
14	#11762847	C₂²	1	2	yes	—
15	#11762846	C₂²	1	4	yes	—
16	#11762845	—	2	2	no	—
17	#11762844	—	2	2	no	—
18	#107×D₁₇	C₂²	1	4	yes	—
19	#106×D₁₇	—	2	2	no	—
20	#11762637	C₂²	1	2	yes	complemented
21	#11762636	C₂²	1	4	yes	complemented
22	#11762635	C₂²	1	2	yes	complemented
23	#11762634	C₂²	1	4	yes	complemented
24	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×#427	C₂²	1	2	yes	—
26	C₁₇×#367	C₂²	1	2	yes	complemented
27	C₁₇×#366	C₂²	1	2	yes	complemented
28	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₁₇×#117	—	2	2	no	—
31	#10587612	—	2	2	no	—
32	C₂²×#3622	D₄	1	1	yes	characteristic
33	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₁₇	D₄	1	2	yes	—
35	C₂×C₈×D₁₇	Q₈	1	4	yes	—
36	C₂×C₈×D₁₇	C₂³	1	2	yes	—
37	#10587578	—	2	4	no	—
38	C₄⋊C₄×D₁₇	—	2	4	no	—
39	C₂²⋊C₄×D₁₇	—	4	2	no	—
40	#10587563	—	4	2	no	—
41	#10587558	—	4	2	no	—
42	C₄²×D₁₇	—	4	2	no	—
43	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
44	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
46	C₁₇×#107	C₂³	1	4	yes	—
47	C₁₇×#106	—	2	2	no	—
48	#10587489	C₂³	1	4	yes	—
49	#10587488	—	2	2	no	—
50	C₂×#63214	C₂×C₄	1	4	yes	complemented
51	C₂×#63214	C₂³	1	2	yes	—
52	#10587479	—	2	4	no	—
53	#10587478	—	2	4	no	—
54	#10587469	C₂×C₄	1	8	yes	complemented
55	#10587469	C₂×C₄	1	8	yes	complemented
56	C₂³×D₁₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
59	C₂×C₄×D₁₇	—	4	4	no	—
60	C₂×C₄×D₁₇	C₄⋊C₄	1	4	yes	—
61	C₂×C₄×D₁₇	D₈	1	2	yes	—
62	C₂×C₄×D₁₇	Q₁₆	1	4	yes	—
63	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₁₇	—	2	4	no	—
69	C₂²⋊C₄×C₁₇	—	4	2	no	—
70	C₄²×C₁₇	—	4	2	no	—
71	C₂²⋊Dic₁₇	—	4	2	no	—
72	#63227	—	4	4	no	—
73	C₄⋊Dic₁₇	—	2	4	no	—
74	#63225	—	4	4	no	—
75	C₄×Dic₁₇	—	4	2	no	—
76	C₄×Dic₁₇	—	4	4	no	—
77	C₂×#3622	C₄⋊C₄	1	4	yes	—
78	C₂×#3622	C₂×D₄	1	2	yes	—
79	C₈×D₁₇	C₄⋊C₄	1	8	yes	—
80	C₈×D₁₇	C₂²×C₄	1	4	yes	—
81	C₈×D₁₇	C₂×Q₈	1	4	yes	—
82	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
83	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
84	#63214	C₂²×C₄	1	4	yes	—
85	#33409	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
87	C₂²×D₁₇	—	4	2	no	—
88	C₂²×D₁₇	#107	1	4	yes	—
89	C₂²×D₁₇	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₁₇	—	4	4	no	—
91	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
94	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×Dic₁₇	—	4	4	no	—
96	C₂×Dic₁₇	#107	1	4	yes	—
97	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
98	C₄×D₁₇	—	8	8	no	—
99	C₄×D₁₇	#107	1	8	yes	—
100	C₄×D₁₇	C₂×C₄⋊C₄	1	4	yes	—
101	C₄×D₁₇	C₂×Q₁₆	1	4	yes	—
102	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
105	#3622	C₂×C₄⋊C₄	1	4	yes	—
106	C₂²×M₃₂	—	17	1	no	—
107	C₂×#427	—	17	2	no	—
108	#2159	—	17	4	no	—
109	#2159	—	17	4	no	—
110	#2158	—	17	1	no	—
111	#2158	—	17	2	no	—
112	#2158	D₁₇	1	1	yes	characteristic, complemented
113	C₂²×C₁₇	—	2	2	no	—
114	C₂²×C₁₇	C₂×#107	1	2	yes	—
115	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
116	C₂×D₁₇	—	2	4	no	—
117	C₂×D₁₇	—	4	4	no	—
118	C₂×D₁₇	C₂×#107	1	4	yes	—
119	C₄×C₁₇	—	8	4	no	—
120	C₄×C₁₇	C₂×#107	1	2	yes	—
121	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
123	Dic₁₇	—	8	4	no	—
124	Dic₁₇	C₂×#107	1	4	yes	—
125	C₂³×C₈	—	17	1	no	—
126	C₂×#117	—	34	2	no	—
127	C₂×M₃₂	—	17	8	no	—
128	C₂×M₃₂	—	17	2	no	—
129	C₂×M₃₂	—	17	2	no	—
130	C₂²×C₁₆	—	17	1	no	—
131	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
132	#427	—	17	2	no	—
133	#427	—	17	4	no	—
134	#427	—	34	2	no	—
135	#427	—	34	2	no	—
136	#427	C₂×D₁₇	1	2	yes	—
137	C₂×#107	—	17	4	no	—
138	C₂×#106	—	34	2	no	—
139	#367	—	17	2	no	—
140	#367	—	17	4	no	—
141	#367	C₂×D₁₇	1	2	yes	complemented
142	#366	—	17	2	no	—
143	#366	—	17	4	no	—
144	#366	C₂×D₁₇	1	2	yes	complemented
145	C₂×C₁₇	—	2	2	no	—
146	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
147	C₂×C₁₇	#3414	1	2	yes	—
148	D₁₇	—	2	4	no	—
149	C₂³×C₄	—	17	1	no	—
150	C₂²×C₈	—	17	1	no	—
151	C₂²×C₈	—	17	4	no	—
152	C₂²×C₈	—	17	2	no	—
153	C₂²×C₈	—	17	4	no	—
154	C₂²×C₈	—	17	2	no	—
155	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
156	#117	—	2	2	no	—
157	#117	—	34	4	no	—
158	#117	—	34	2	no	—
159	#117	—	68	2	no	—
160	#117	—	68	2	no	—
161	C₂×C₄⋊C₄	—	34	4	no	—
162	C₂×C₂²⋊C₄	—	68	2	no	—
163	C₂×C₄²	—	68	2	no	—
164	M₃₂	—	17	8	no	—
165	M₃₂	—	17	8	no	—
166	C₂×C₁₆	—	17	4	no	—
167	C₂×C₁₆	—	17	2	no	—
168	C₂×C₁₆	C₄×D₁₇	1	4	yes	complemented
169	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
170	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
171	#107	—	17	4	no	—
172	#107	—	34	4	no	—
173	#107	C₂²×D₁₇	1	4	yes	—
174	#106	—	2	2	no	—
175	#106	—	34	4	no	—
176	#106	—	34	2	no	—
177	C₁₇	#33409	1	1	yes	characteristic, complemented, Sylow
178	C₂⁴	—	17	1	no	—
179	C₂²×C₄	—	17	1	no	—
180	C₂²×C₄	—	17	2	no	—
181	C₂²×C₄	—	17	4	no	—
182	C₂²×C₄	—	17	2	no	—
183	C₂²×C₄	—	17	4	no	—
184	C₂²×C₄	—	68	4	no	—
185	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
186	C₂×C₈	—	17	2	no	—
187	C₂×C₈	—	17	4	no	—
188	C₂×C₈	—	17	4	no	—
189	C₂×C₈	—	17	4	no	—
190	C₂×C₈	—	17	8	no	—
191	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
192	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
193	C₂×C₈	Q₈×D₁₇	1	2	yes	—
194	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	34	4	no	—
197	C₄⋊C₄	—	68	4	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	68	4	no	—
200	C₂²⋊C₄	—	68	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	68	4	no	—
203	C₄²	—	68	2	no	—
204	C₁₆	—	17	4	no	—
205	C₁₆	C₂×C₄×D₁₇	1	2	yes	—
206	C₁₆	C₂×C₄×D₁₇	1	2	yes	—
207	C₂³	—	17	2	no	—
208	C₂³	—	17	4	no	—
209	C₂³	—	68	2	no	—
210	C₂³	D₈×D₁₇	1	1	yes	characteristic
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	17	2	no	—
213	C₂×C₄	—	17	4	no	—
214	C₂×C₄	—	17	4	no	—
215	C₂×C₄	—	17	4	no	—
216	C₂×C₄	—	17	8	no	—
217	C₂×C₄	—	68	4	no	—
218	C₂×C₄	—	136	8	no	—
219	C₂×C₄	C₄⋊C₄×D₁₇	1	2	yes	—
220	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
221	C₂×C₄	Q₁₆×D₁₇	1	2	yes	—
222	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
223	C₈	—	17	4	no	—
224	C₈	C₄⋊C₄×D₁₇	1	2	yes	—
225	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
226	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	17	4	no	—
229	C₂²	—	34	4	no	—
230	C₂²	—	68	4	no	—
231	C₂²	#107×D₁₇	1	2	yes	—
232	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
233	C₄	—	8	4	no	—
234	C₄	—	17	4	no	—
235	C₄	—	136	4	no	—
236	C₄	#107×D₁₇	1	2	yes	—
237	C₄	C₂×C₄⋊C₄×D₁₇	1	1	yes	characteristic
238	C₄	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
239	C₂	—	2	2	no	—
240	C₂	—	34	4	no	—
241	C₂	C₂×#107×D₁₇	1	1	yes	characteristic
242	C₂	#14541917	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₇
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2158	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×D₁₇	D₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2158	D₁₇
hypocenter	C₁₇	#33409
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #14530571	C₂	semidirect
4. #14530569	C₂	semidirect
5. #14530568	C₂	semidirect
6. #14530566	C₂	semidirect
7. C₁₇×#2158	C₂	semidirect
10. C₂×#10587469	C₄	semidirect
20. #11762637	C₂²	semidirect
21. #11762636	C₂²	semidirect
22. #11762635	C₂²	semidirect
23. #11762634	C₂²	semidirect
26. C₁₇×#367	C₂²	semidirect
27. C₁₇×#366	C₂²	semidirect
43. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
50. C₂×#63214	C₂×C₄	semidirect
54. #10587469	C₂×C₄	semidirect
55. #10587469	C₂×C₄	semidirect
112. #2158	D₁₇	semidirect
141. #367	C₂×D₁₇	semidirect
144. #366	C₂×D₁₇	semidirect
168. C₂×C₁₆	C₄×D₁₇	semidirect
177. C₁₇	#33409	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587469	2
D₁₇×#427	2
#14530571	2
#14530569	2
#14530568	2
#14530566	2
C₁₇×#2158	2
#33409	17