Subgroups — #17043181

Order 4352.

654 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537725	C₂	1	2	yes	maximal
2	#14529766	C₂	1	2	yes	maximal
3	#14529538	C₂	1	2	yes	maximal
4	#14529533	C₂	1	2	yes	maximal
5	#14528956	C₂	1	1	yes	maximal, characteristic
6	#14528670	C₂	1	2	yes	maximal
7	#14528665	C₂	1	2	yes	maximal
8	#14528470	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#3160	C₂	1	1	yes	maximal, characteristic
10	#11762940	—	2	8	no	—
11	C₂×#10587490	C₂²	1	2	yes	—
12	C₂×#10587490	C₂²	1	2	yes	—
13	#11762836	C₂²	1	4	yes	—
14	#11762833	C₂²	1	2	yes	—
15	#11762765	C₂²	1	4	yes	—
16	C₂×#10587474	C₂²	1	1	yes	characteristic
17	C₂×#10587486	C₂²	1	1	yes	characteristic
18	C₂×#10587486	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
21	#11762655	C₂²	1	1	yes	characteristic
22	#11762568	C₂²	1	4	yes	—
23	#11762568	C₂²	1	4	yes	—
24	C₁₇×#533	C₂²	1	2	yes	—
25	C₁₇×#500	C₄	1	2	yes	complemented
26	C₁₇×#499	C₄	1	2	yes	complemented
27	C₁₇×#499	C₄	1	2	yes	complemented
28	C₁₇×#498	C₄	1	2	yes	complemented
29	C₁₇×#427	C₂²	1	2	yes	—
30	C₁₇×#427	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
35	C₁₇×#125	C₂×C₄	1	4	yes	—
36	C₁₇×C₂²⋊Q₈	—	2	4	no	—
37	C₄×Q₈×C₁₇	—	2	4	no	—
38	C₁₇×#117	C₂³	1	4	yes	—
39	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
42	C₂×#63223	C₂³	1	4	yes	—
43	C₂²×#3622	C₂³	1	1	yes	characteristic
44	C₂²×#3622	C₂³	1	1	yes	characteristic
45	C₁₇×#107	C₂×C₄	1	4	yes	—
46	C₁₇×#107	C₂×C₄	1	4	yes	—
47	C₁₇×#106	C₂×C₄	1	4	yes	—
48	C₁₇×#106	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
53	#10587502	—	2	2	no	—
54	#10587490	—	2	4	no	—
55	#10587490	—	2	4	no	—
56	#10587486	—	2	2	no	—
57	#10587486	Q₈	1	4	yes	complemented
58	C₈×Dic₁₇	—	2	2	no	—
59	C₈×Dic₁₇	Q₈	1	4	yes	complemented
60	C₄⋊#3622	—	2	8	no	—
61	C₄⋊#3622	—	2	4	no	—
62	#10587474	—	2	4	no	—
63	#10587474	D₄	1	4	yes	—
64	C₄×#3622	—	2	4	no	—
65	C₂×Q₈×C₁₇	—	2	4	no	—
66	C₂²×C₄×C₁₇	#41	1	2	yes	—
67	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₇	—	2	1	no	—
69	C₂²×Dic₁₇	—	2	1	no	—
70	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₁₇	#41	1	2	yes	—
77	C₂×C₈×C₁₇	#41	1	2	yes	—
78	C₄⋊C₄×C₁₇	—	2	8	no	—
79	C₄⋊C₄×C₁₇	—	2	4	no	—
80	C₄⋊C₄×C₁₇	C₂²×C₄	1	8	yes	—
81	C₂²⋊C₄×C₁₇	—	2	4	no	—
82	C₄²×C₁₇	—	2	4	no	—
83	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₇	C₂×D₄	1	2	yes	—
86	C₄×Dic₁₇	—	2	1	no	—
87	C₄×Dic₁₇	—	2	1	no	—
88	C₄×Dic₁₇	C₂×Q₈	1	2	yes	—
89	C₄×Dic₁₇	C₂×Q₈	1	2	yes	—
90	#63223	—	2	4	no	—
91	#63223	—	2	4	no	—
92	C₂×#3622	—	2	1	no	—
93	C₂×#3622	—	2	1	no	—
94	C₂×#3622	—	2	4	no	—
95	C₂×#3622	C₂×D₄	1	2	yes	—
96	C₂×#3622	C₂×D₄	1	2	yes	—
97	C₂×#3622	C₂×Q₈	1	2	yes	—
98	C₂×#3622	C₂×Q₈	1	2	yes	—
99	#21733	—	17	1	no	maximal, Sylow
100	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
101	Q₈×C₁₇	—	8	4	no	—
102	C₂×C₄×C₁₇	—	2	8	no	—
103	C₂×C₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₁₇	—	2	2	no	—
113	C₂×Dic₁₇	—	2	2	no	—
114	C₂×Dic₁₇	—	4	2	no	—
115	C₂×Dic₁₇	—	4	2	no	—
116	C₈×C₁₇	C₄×Q₈	1	2	yes	—
117	C₈×C₁₇	C₄×Q₈	1	2	yes	—
118	C₈×C₁₇	C₄×Q₈	1	2	yes	—
119	C₈×C₁₇	C₄×Q₈	1	2	yes	—
120	#3622	—	2	2	no	—
121	#3622	—	4	4	no	—
122	#3622	C₄⋊Q₈	1	4	yes	—
123	#3160	D₁₇	1	1	yes	characteristic
124	#2961	—	17	2	no	—
125	#1995	—	17	2	no	—
126	#1946	—	17	2	no	—
127	#1945	—	17	2	no	—
128	#1848	—	17	1	no	—
129	#1773	—	17	2	no	—
130	#1772	—	17	2	no	—
131	#1726	—	17	1	no	—
132	C₂²×C₁₇	—	2	2	no	—
133	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₁₇	#533	1	1	yes	characteristic
135	C₂²×C₁₇	#533	1	1	yes	characteristic
136	C₄×C₁₇	—	2	2	no	—
137	C₄×C₁₇	—	2	2	no	—
138	C₄×C₁₇	—	4	8	no	—
139	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₄×C₁₇	#532	1	1	yes	characteristic
143	Dic₁₇	—	4	2	no	—
144	Dic₁₇	—	4	2	no	—
145	#533	C₂×D₁₇	1	2	yes	—
146	#500	Dic₁₇	1	2	yes	complemented
147	#499	Dic₁₇	1	2	yes	complemented
148	#499	Dic₁₇	1	2	yes	complemented
149	#498	Dic₁₇	1	2	yes	complemented
150	#446	—	34	8	no	—
151	C₂×#108	—	17	2	no	—
152	C₂×#108	—	17	2	no	—
153	#427	C₂×D₁₇	1	2	yes	—
154	#427	C₂×D₁₇	1	2	yes	—
155	#424	—	17	4	no	—
156	#423	—	17	2	no	—
157	#405	—	17	4	no	—
158	C₂×#97	—	17	1	no	—
159	C₂×#97	—	17	1	no	—
160	C₂×#97	—	17	1	no	—
161	C₂×C₄×C₈	—	17	1	no	—
162	C₂×C₄×C₈	—	17	1	no	—
163	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
164	#376	—	17	1	no	—
165	#339	—	17	4	no	—
166	#339	—	17	4	no	—
167	C₂×C₁₇	—	2	1	no	—
168	C₂×C₁₇	—	2	1	no	—
169	C₂×C₁₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
170	C₂×C₁₇	#2957	1	2	yes	—
171	C₂×M₁₆	—	17	4	no	—
172	C₂²×C₈	—	17	1	no	—
173	C₂²×C₈	—	17	1	no	—
174	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
176	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
177	#125	C₂×Dic₁₇	1	4	yes	—
178	C₂²⋊Q₈	—	2	4	no	—
179	C₄×Q₈	—	2	4	no	—
180	#117	C₂²×D₁₇	1	4	yes	—
181	C₂×C₄²	—	17	1	no	—
182	C₂×C₄²	—	17	1	no	—
183	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
184	#108	—	34	4	no	—
185	#108	—	34	4	no	—
186	#107	C₂×Dic₁₇	1	4	yes	—
187	#107	C₂×Dic₁₇	1	4	yes	—
188	#106	C₂×Dic₁₇	1	4	yes	—
189	#106	C₂×Dic₁₇	1	4	yes	—
190	C₄⋊C₈	—	34	8	no	—
191	C₄⋊C₈	—	34	4	no	—
192	#97	—	17	4	no	—
193	#97	—	17	4	no	—
194	#97	—	34	4	no	—
195	#97	—	34	2	no	—
196	C₄×C₈	—	17	4	no	—
197	C₄×C₈	—	34	4	no	—
198	C₄×C₈	—	34	2	no	—
199	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
200	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
201	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
202	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
203	#95	—	34	2	no	—
204	C₁₇	#21733	1	1	yes	characteristic, complemented, Sylow
205	C₂×Q₈	—	2	4	no	—
206	C₂²×C₄	—	34	1	no	—
207	C₂²×C₄	—	34	1	no	—
208	C₂²×C₄	#63254	1	2	yes	—
209	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
210	M₁₆	—	34	4	no	—
211	M₁₆	—	34	4	no	—
212	C₂×C₈	—	17	2	no	—
213	C₂×C₈	—	17	2	no	—
214	C₂×C₈	—	17	2	no	—
215	C₂×C₈	—	17	2	no	—
216	C₂×C₈	—	34	1	no	—
217	C₂×C₈	—	34	1	no	—
218	C₂×C₈	—	34	4	no	—
219	C₂×C₈	Q₈×D₁₇	1	2	yes	—
220	C₂×C₈	Q₈×D₁₇	1	2	yes	—
221	C₂×C₈	#63256	1	2	yes	—
222	C₂×C₈	#63256	1	2	yes	—
223	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
224	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂²×Dic₁₇	1	8	yes	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	17	2	no	—
233	C₄²	—	17	2	no	—
234	C₄²	—	34	1	no	—
235	C₄²	—	34	1	no	—
236	C₄²	D₄×D₁₇	1	2	yes	—
237	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
238	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
239	C₂³	C₂×#63254	1	1	yes	characteristic
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	34	2	no	—
244	C₂×C₄	—	34	2	no	—
245	C₂×C₄	—	68	2	no	—
246	C₂×C₄	—	68	2	no	—
247	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
248	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
249	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
250	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
252	C₂×C₄	C₂×#63256	1	1	yes	characteristic
253	C₂×C₄	C₂×#63256	1	1	yes	characteristic
254	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
255	C₈	—	17	4	no	—
256	C₈	—	34	2	no	—
257	C₈	—	68	4	no	—
258	C₈	Q₈×Dic₁₇	1	2	yes	—
259	C₈	Q₈×Dic₁₇	1	2	yes	—
260	C₈	Q₈×Dic₁₇	1	2	yes	—
261	C₈	Q₈×Dic₁₇	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×D₄×Dic₁₇	1	1	yes	characteristic
264	C₂²	#11763364	1	1	yes	characteristic
265	C₂²	#11763364	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	8	no	—
269	C₄	—	68	2	no	—
270	C₄	—	68	2	no	—
271	C₄	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
272	C₄	C₂×Q₈×Dic₁₇	1	1	yes	characteristic
273	C₄	C₄⋊Q₈×D₁₇	1	1	yes	characteristic
274	C₄	#11763360	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	C₄⋊Q₈×Dic₁₇	1	1	yes	characteristic
278	C₂	#14537701	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#3160	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂³×C₁₇	C₂×#41
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	#3160	D₁₇
hypocenter	C₁₇	#21733
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
25. C₁₇×#500	C₄	semidirect
26. C₁₇×#499	C₄	semidirect
27. C₁₇×#499	C₄	semidirect
28. C₁₇×#498	C₄	semidirect
57. #10587486	Q₈	semidirect
59. C₈×Dic₁₇	Q₈	semidirect
146. #500	Dic₁₇	semidirect
147. #499	Dic₁₇	semidirect
148. #499	Dic₁₇	semidirect
149. #498	Dic₁₇	semidirect
204. C₁₇	#21733	semidirect

Maximal subgroups

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

subgroup	index
#14537725	2
#14529766	2
#14529538	2
#14529533	2
#14528956	2
#14528670	2
#14528665	2
#14528470	2
C₁₇×#3160	2
#21733	17