Subgroups — #17187751

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537701	C₂	1	2	yes	maximal, complemented
2	#14530670	C₂	1	4	yes	maximal, complemented
3	#14530665	C₂	1	4	yes	maximal, complemented
4	C₂×#11762639	C₂	1	1	yes	maximal, characteristic, complemented
5	#14530512	C₂	1	2	yes	maximal, complemented
6	#14530380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#508	C₂	1	1	yes	maximal, characteristic
8	#11763576	—	2	2	no	—
9	C₂×#10587483	C₂²	1	1	yes	characteristic
10	#11762931	—	2	4	no	—
11	#11762926	C₂²	1	8	yes	complemented
12	C₂×#10587490	C₂²	1	2	yes	—
13	#11762826	—	2	2	no	—
14	#11762764	C₂²	1	1	yes	characteristic
15	#11762639	C₂²	1	2	yes	complemented
16	#11762639	C₂²	1	2	yes	complemented
17	#11762616	C₂²	1	8	yes	complemented
18	#11762584	C₂²	1	2	yes	complemented
19	C₁₆×Dic₁₇	C₂²	1	2	yes	complemented
20	C₂×C₁₇×#135	C₂²	1	2	yes	—
21	C₁₇×#508	C₄	1	2	yes	complemented
22	C₁₇×#508	C₄	1	2	yes	complemented
23	C₁₇×#508	C₂²	1	2	yes	—
24	C₁₇×#508	C₂²	1	2	yes	—
25	C₂×C₁₇×Q₃₂	C₄	1	1	yes	characteristic, complemented
26	C₂×C₁₇×SD₃₂	C₄	1	2	yes	complemented
27	C₂×C₁₇×D₁₆	C₄	1	1	yes	characteristic, complemented
28	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
29	C₂×#41×C₁₇	—	2	2	no	—
30	C₁₇×#135	—	2	2	no	—
31	C₁₇×#135	—	2	2	no	—
32	C₁₇×#135	C₂×C₄	1	4	yes	—
33	C₁₇×#135	C₂³	1	4	yes	—
34	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₁₇	—	2	2	no	—
36	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	#10587634	—	2	8	no	—
39	#10587634	—	4	4	no	—
40	#10587612	D₄	1	1	yes	characteristic
41	C₂×#63223	—	2	2	no	—
42	C₂×#63223	—	4	4	no	—
43	C₂×#63223	D₄	1	1	yes	characteristic
44	C₁₇×Q₃₂	C₂×C₄	1	2	yes	—
45	C₁₇×Q₃₂	C₂×C₄	1	2	yes	—
46	C₁₇×SD₃₂	C₂×C₄	1	4	yes	—
47	C₁₇×SD₃₂	C₂×C₄	1	4	yes	—
48	C₁₇×D₁₆	C₂×C₄	1	2	yes	—
49	C₁₇×D₁₆	C₂×C₄	1	2	yes	—
50	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
56	#10587508	—	2	8	no	—
57	#10587490	—	2	2	no	—
58	#10587490	C₂³	1	4	yes	—
59	#10587486	D₄	1	2	yes	—
60	C₈×Dic₁₇	C₂³	1	2	yes	—
61	#10587483	—	2	1	no	—
62	#10587483	—	2	1	no	—
63	C₂×#63214	C₂³	1	2	yes	—
64	#41×C₁₇	—	2	4	no	—
65	#41×C₁₇	—	2	4	no	—
66	#41×C₁₇	—	4	2	no	—
67	#41×C₁₇	—	4	2	no	—
68	C₂×Q₈×C₁₇	—	2	2	no	—
69	C₂×D₄×C₁₇	—	2	2	no	—
70	C₂×D₄×C₁₇	—	4	2	no	—
71	C₂²×C₄×C₁₇	—	4	2	no	—
72	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₁₇	—	2	2	no	—
74	Q₁₆×C₁₇	C₂²×C₄	1	4	yes	—
75	SD₁₆×C₁₇	—	2	4	no	—
76	SD₁₆×C₁₇	—	2	4	no	—
77	D₈×C₁₇	—	2	2	no	—
78	D₈×C₁₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄×Dic₁₇	C₂×D₄	1	2	yes	—
88	#63223	—	2	4	no	—
89	#63223	—	4	2	no	—
90	#63223	—	4	8	no	—
91	#63223	D₈	1	2	yes	—
92	#63223	D₈	1	2	yes	—
93	C₂×#3622	—	4	2	no	—
94	C₂×#3622	—	4	8	no	—
95	C₂×#3622	C₂×D₄	1	2	yes	—
96	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
97	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	#63214	—	2	2	no	—
101	#33465	—	17	1	no	maximal, Sylow
102	C₂³×C₁₇	—	4	2	no	—
103	C₂³×C₁₇	#135	1	1	yes	characteristic
104	Q₈×C₁₇	—	2	4	no	—
105	Q₈×C₁₇	—	4	2	no	—
106	D₄×C₁₇	—	2	4	no	—
107	D₄×C₁₇	—	4	4	no	—
108	D₄×C₁₇	—	4	4	no	—
109	D₄×C₁₇	—	4	2	no	—
110	C₂×C₄×C₁₇	—	4	2	no	—
111	C₂×C₄×C₁₇	—	4	4	no	—
112	C₂×C₄×C₁₇	—	4	4	no	—
113	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₁₇	—	2	2	no	—
120	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
121	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
122	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
123	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
124	#3622	—	4	4	no	—
125	#3622	C₂×D₈	1	4	yes	—
126	C₂×#508	D₁₇	1	1	yes	characteristic
127	#2957	—	17	2	no	—
128	#2179	—	17	4	no	—
129	#2178	—	17	4	no	—
130	C₂×#368	—	17	1	no	—
131	#2143	—	17	2	no	—
132	#2108	—	17	1	no	—
133	C₂²×C₁₇	—	4	1	no	—
134	C₂²×C₁₇	—	4	4	no	—
135	C₂²×C₁₇	—	16	2	no	—
136	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
137	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
138	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
139	C₄×C₁₇	—	4	4	no	—
140	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
141	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
142	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
144	Dic₁₇	—	4	2	no	—
145	C₂×#135	C₂×D₁₇	1	2	yes	—
146	#568	—	34	2	no	—
147	#508	Dic₁₇	1	2	yes	complemented
148	#508	Dic₁₇	1	2	yes	complemented
149	#508	C₂×D₁₇	1	2	yes	—
150	#508	C₂×D₁₇	1	2	yes	—
151	C₂×Q₃₂	Dic₁₇	1	1	yes	characteristic, complemented
152	C₂×SD₃₂	Dic₁₇	1	2	yes	complemented
153	C₂×D₁₆	Dic₁₇	1	1	yes	characteristic, complemented
154	C₂×M₃₂	—	17	1	no	—
155	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
156	#444	—	34	4	no	—
157	#443	—	17	8	no	—
158	C₂×#108	—	17	2	no	—
159	#421	—	34	2	no	—
160	#405	—	17	1	no	—
161	#368	—	17	2	no	—
162	#368	—	17	2	no	—
163	#359	—	17	8	no	—
164	#346	—	17	2	no	—
165	C₄×C₁₆	—	17	2	no	—
166	C₂×C₁₇	—	2	1	no	—
167	C₂×C₁₇	—	2	1	no	—
168	C₂×C₁₇	—	2	1	no	—
169	C₂×C₁₇	—	8	4	no	—
170	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
171	C₂×#41	—	2	2	no	—
172	#135	—	2	2	no	—
173	#135	—	2	2	no	—
174	#135	C₂×Dic₁₇	1	4	yes	—
175	#135	C₂²×D₁₇	1	4	yes	—
176	C₂×Q₁₆	C₂×Dic₁₇	1	2	yes	—
177	C₂×SD₁₆	—	2	2	no	—
178	C₂×D₈	C₂×Dic₁₇	1	2	yes	—
179	#131	—	34	8	no	—
180	#131	—	68	4	no	—
181	C₂×M₁₆	—	17	1	no	—
182	C₂×M₁₆	—	34	2	no	—
183	C₂×M₁₆	—	68	4	no	—
184	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
185	#117	—	17	1	no	—
186	Q₃₂	C₂×Dic₁₇	1	2	yes	—
187	Q₃₂	C₂×Dic₁₇	1	2	yes	—
188	SD₃₂	C₂×Dic₁₇	1	4	yes	—
189	SD₃₂	C₂×Dic₁₇	1	4	yes	—
190	D₁₆	C₂×Dic₁₇	1	2	yes	—
191	D₁₆	C₂×Dic₁₇	1	2	yes	—
192	M₃₂	—	34	1	no	—
193	M₃₂	—	34	1	no	—
194	C₂×C₁₆	—	17	2	no	—
195	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
196	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
197	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
198	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
199	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
200	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
201	#108	—	17	4	no	—
202	#108	—	34	2	no	—
203	#104	—	34	8	no	—
204	#97	—	17	2	no	—
205	C₄×C₈	—	17	2	no	—
206	C₁₇	#33465	1	1	yes	characteristic, complemented, Sylow
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	#41	—	4	2	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×Dic₁₇	1	4	yes	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	2	no	—
221	D₈	C₂²×Dic₁₇	1	4	yes	—
222	M₁₆	—	17	2	no	—
223	M₁₆	—	17	2	no	—
224	M₁₆	—	34	4	no	—
225	M₁₆	—	68	8	no	—
226	M₁₆	—	68	2	no	—
227	C₂×C₈	—	17	2	no	—
228	C₂×C₈	—	68	8	no	—
229	C₂×C₈	—	68	2	no	—
230	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
231	C₂×C₈	#63254	1	1	yes	characteristic
232	C₂×C₈	#63254	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
235	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
236	C₄⋊C₄	—	34	1	no	—
237	C₂²⋊C₄	—	34	1	no	—
238	C₄²	—	17	2	no	—
239	C₁₆	—	34	2	no	—
240	C₁₆	C₂²×Dic₁₇	1	1	yes	characteristic
241	C₁₆	C₂²×Dic₁₇	1	1	yes	characteristic
242	C₁₆	C₂²×Dic₁₇	1	1	yes	characteristic
243	C₁₆	C₂²×Dic₁₇	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	#10587598	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	34	2	no	—
256	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
257	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
258	C₂×C₄	#10587598	1	1	yes	characteristic
259	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
260	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
262	C₈	—	17	4	no	—
263	C₈	—	68	4	no	—
264	C₈	D₄×Dic₁₇	1	1	yes	characteristic
265	C₈	D₄×Dic₁₇	1	1	yes	characteristic
266	C₈	C₂×#63254	1	1	yes	characteristic
267	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	D₈×Dic₁₇	1	1	yes	characteristic
272	C₂²	D₈×Dic₁₇	1	1	yes	characteristic
273	C₂²	C₂×#10587598	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	68	2	no	—
276	C₄	D₈×Dic₁₇	1	1	yes	characteristic
277	C₄	D₈×Dic₁₇	1	1	yes	characteristic
278	C₄	C₂×D₄×Dic₁₇	1	1	yes	characteristic
279	C₄	C₂×D₈×D₁₇	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×D₈×Dic₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₁₇
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting, prime radical	C₂×C₁₇×#508	C₂
socle	C₂×C₁₇	C₂×C₄×D₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#508	D₁₇
hypocenter	C₁₇	#33465
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #14537701	C₂	semidirect
2. #14530670	C₂	semidirect
3. #14530665	C₂	semidirect
4. C₂×#11762639	C₂	semidirect
5. #14530512	C₂	semidirect
6. #14530380	C₂	semidirect
11. #11762926	C₂²	semidirect
15. #11762639	C₂²	semidirect
16. #11762639	C₂²	semidirect
17. #11762616	C₂²	semidirect
18. #11762584	C₂²	semidirect
19. C₁₆×Dic₁₇	C₂²	semidirect
21. C₁₇×#508	C₄	semidirect
22. C₁₇×#508	C₄	semidirect
25. C₂×C₁₇×Q₃₂	C₄	semidirect
26. C₂×C₁₇×SD₃₂	C₄	semidirect
27. C₂×C₁₇×D₁₆	C₄	semidirect
147. #508	Dic₁₇	semidirect
148. #508	Dic₁₇	semidirect
151. C₂×Q₃₂	Dic₁₇	semidirect
152. C₂×SD₃₂	Dic₁₇	semidirect
153. C₂×D₁₆	Dic₁₇	semidirect
206. C₁₇	#33465	semidirect

Maximal subgroups

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

subgroup	index
#14537701	2
#14530670	2
#14530665	2
C₂×#11762639	2
#14530512	2
#14530380	2
C₂×C₁₇×#508	2
#33465	17