Subgroups — #16927398

Order 4352.

526 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528912	C₂	1	1	yes	maximal, characteristic, complemented
2	#14528784	C₂	1	2	yes	maximal
3	C₂×#11762562	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526668	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×#10587503	—	2	2	no	—
8	#11762679	C₂²	1	2	yes	—
9	#11762572	—	2	1	no	—
10	#11762572	—	2	1	no	—
11	#11762571	—	2	2	no	—
12	#11762562	—	2	2	no	—
13	C₂³×C₈×C₁₇	—	2	1	no	—
14	C₂×C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₁₇	—	2	2	no	—
16	C₈×D₄×C₁₇	C₄	1	4	yes	complemented
17	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×C₂²⋊C₈	—	2	1	no	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×D₄×C₁₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₇	—	2	1	no	—
22	C₂²×C₈×C₁₇	—	2	1	no	—
23	C₂²×C₈×C₁₇	—	2	4	no	—
24	C₂²×C₈×C₁₇	—	4	4	no	—
25	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₄×D₄×C₁₇	—	2	1	no	—
28	C₄×D₄×C₁₇	—	2	1	no	—
29	C₄×D₄×C₁₇	C₂×C₄	1	2	yes	—
30	C₄×D₄×C₁₇	Q₈	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
33	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₇	—	2	2	no	—
35	C₂×C₄⋊Dic₁₇	—	2	2	no	—
36	C₂×C₄⋊Dic₁₇	—	2	2	no	—
37	C₂²×#3622	—	2	2	no	—
38	C₁₇×C₄⋊C₈	—	2	1	no	—
39	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₁₇×C₂²⋊C₈	—	2	4	no	—
41	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₇	D₄	1	2	yes	—
43	#10587503	—	4	4	no	—
44	C₄⋊#3622	—	2	2	no	—
45	C₄⋊#3622	—	2	2	no	—
46	C₂⁴×C₁₇	—	2	1	no	—
47	C₂×D₄×C₁₇	—	2	1	no	—
48	C₂×D₄×C₁₇	—	2	1	no	—
49	C₂×D₄×C₁₇	—	2	1	no	—
50	C₂×D₄×C₁₇	C₄⋊C₄	1	2	yes	—
51	C₂×D₄×C₁₇	D₈	1	2	yes	—
52	C₂×D₄×C₁₇	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₁₇	—	2	1	no	—
54	C₂²×C₄×C₁₇	—	2	2	no	—
55	C₂²×C₄×C₁₇	—	2	2	no	—
56	C₂²×C₄×C₁₇	—	4	2	no	—
57	C₂²×C₄×C₁₇	—	4	2	no	—
58	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
61	C₂²×Dic₁₇	—	4	2	no	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	—	2	2	no	—
64	C₂×C₈×C₁₇	—	2	2	no	—
65	C₂×C₈×C₁₇	—	4	8	no	—
66	C₂×C₈×C₁₇	—	4	2	no	—
67	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	C₄⋊C₄	1	2	yes	—
71	C₂²⋊C₄×C₁₇	—	2	2	no	—
72	C₂²⋊C₄×C₁₇	—	2	2	no	—
73	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₁₇	#41	1	1	yes	characteristic
77	C₄⋊Dic₁₇	—	4	2	no	—
78	C₄⋊Dic₁₇	—	4	2	no	—
79	C₄⋊Dic₁₇	—	4	2	no	—
80	C₄⋊Dic₁₇	—	4	2	no	—
81	C₄⋊Dic₁₇	—	8	2	no	—
82	C₂×#3622	—	4	2	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	#8648	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	—	2	2	no	—
87	C₂³×C₁₇	—	2	2	no	—
88	C₂³×C₁₇	—	2	2	no	—
89	C₂³×C₁₇	—	4	2	no	—
90	C₂³×C₁₇	—	4	2	no	—
91	C₂³×C₁₇	#121	1	1	yes	characteristic
92	D₄×C₁₇	—	2	2	no	—
93	D₄×C₁₇	—	2	2	no	—
94	D₄×C₁₇	—	2	2	no	—
95	D₄×C₁₇	#107	1	4	yes	—
96	C₂×C₄×C₁₇	—	2	2	no	—
97	C₂×C₄×C₁₇	—	2	2	no	—
98	C₂×C₄×C₁₇	—	4	2	no	—
99	C₂×C₄×C₁₇	—	4	2	no	—
100	C₂×C₄×C₁₇	—	4	2	no	—
101	C₂×C₄×C₁₇	—	4	4	no	—
102	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
111	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
112	C₂×C₄×C₁₇	#135	1	2	yes	—
113	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×Dic₁₇	—	8	2	no	—
115	C₂×Dic₁₇	—	8	2	no	—
116	C₂×Dic₁₇	—	8	2	no	—
117	C₈×C₁₇	—	2	2	no	—
118	C₈×C₁₇	—	2	2	no	—
119	C₈×C₁₇	—	4	2	no	—
120	#3622	—	8	2	no	—
121	C₂×C₈×D₄	D₁₇	1	1	yes	characteristic
122	#1834	—	17	1	no	—
123	#1802	—	17	2	no	—
124	C₂×#335	—	17	1	no	—
125	#1296	—	17	2	no	—
126	C₂²×C₁₇	—	2	4	no	—
127	C₂²×C₁₇	—	4	2	no	—
128	C₂²×C₁₇	—	4	2	no	—
129	C₂²×C₁₇	—	4	4	no	—
130	C₂²×C₁₇	—	4	4	no	—
131	C₂²×C₁₇	#380	1	1	yes	characteristic
132	C₂²×C₁₇	#463	1	2	yes	—
133	C₂²×C₁₇	#466	1	1	yes	characteristic
134	C₂²×C₁₇	#467	1	1	yes	characteristic
135	C₂²×C₁₇	#471	1	2	yes	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	4	2	no	—
139	C₄×C₁₇	#416	1	1	yes	characteristic
140	C₄×C₁₇	#418	1	1	yes	characteristic
141	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
142	C₄×C₁₇	#427	1	1	yes	characteristic
143	C₄×C₁₇	#466	1	1	yes	characteristic
144	C₄×C₁₇	#467	1	1	yes	characteristic
145	C₄×C₁₇	#478	1	2	yes	—
146	Dic₁₇	—	16	2	no	—
147	C₂³×C₈	—	2	1	no	—
148	C₂×C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
149	C₈×D₄	—	2	2	no	—
150	C₈×D₄	Dic₁₇	1	4	yes	complemented
151	C₂×C₄⋊C₈	—	17	2	no	—
152	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
153	C₂×#102	—	34	2	no	—
154	C₂×C₂²⋊C₈	—	2	1	no	—
155	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
156	#384	—	17	2	no	—
157	#340	—	34	1	no	—
158	#340	—	34	1	no	—
159	#340	—	34	2	no	—
160	#335	—	34	2	no	—
161	C₂×C₁₇	—	4	4	no	—
162	C₂×C₁₇	#1681	1	2	yes	—
163	C₂×C₁₇	#1683	1	2	yes	—
164	C₂×C₁₇	#1804	1	1	yes	characteristic
165	C₂×C₁₇	#1808	1	1	yes	characteristic
166	C₂×C₁₇	#1815	1	1	yes	characteristic
167	C₂²×D₄	D₆₈	1	1	yes	characteristic
168	C₂³×C₄	—	2	1	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	4	4	no	—
172	C₂²×C₈	—	34	2	no	—
173	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
175	C₄×D₄	—	2	1	no	—
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	Dic₃₄	1	2	yes	—
178	C₄×D₄	C₂×Dic₁₇	1	2	yes	—
179	C₂×C₄⋊C₄	—	34	2	no	—
180	C₂×C₄⋊C₄	—	34	2	no	—
181	C₂×C₄⋊C₄	—	34	2	no	—
182	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	—	2	1	no	—
184	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
185	C₄⋊C₈	—	2	1	no	—
186	C₄⋊C₈	—	34	2	no	—
187	C₄⋊C₈	—	34	2	no	—
188	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
189	#102	—	68	4	no	—
190	C₂²⋊C₈	—	2	4	no	—
191	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
192	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
193	C₁₇	#8648	1	1	yes	characteristic, complemented, Sylow
194	C₂⁴	—	2	1	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	D₁₃₆	1	2	yes	—
199	C₂×D₄	Dic₆₈	1	2	yes	—
200	C₂×D₄	C₄⋊Dic₁₇	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	68	2	no	—
207	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
208	C₂²×C₄	#63252	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	8	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	68	2	no	—
216	C₂×C₈	—	68	2	no	—
217	C₂×C₈	—	68	2	no	—
218	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
219	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
220	C₄⋊C₄	—	2	1	no	—
221	C₄⋊C₄	—	68	2	no	—
222	C₄⋊C₄	—	68	2	no	—
223	C₄⋊C₄	—	68	2	no	—
224	C₄⋊C₄	—	68	2	no	—
225	C₄⋊C₄	—	136	2	no	—
226	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
230	C₄²	#63252	1	1	yes	characteristic
231	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	#10587616	1	1	yes	characteristic
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	2	2	no	—
242	D₄	#10587489	1	4	yes	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	136	2	no	—
250	C₂×C₄	—	136	2	no	—
251	C₂×C₄	—	136	2	no	—
252	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
253	C₂×C₄	#10587590	1	1	yes	characteristic
254	C₂×C₄	#10587590	1	1	yes	characteristic
255	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
256	C₂×C₄	#10587610	1	1	yes	characteristic
257	C₂×C₄	#10587610	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
259	C₂×C₄	#10587612	1	1	yes	characteristic
260	C₂×C₄	#10587616	1	1	yes	characteristic
261	C₂×C₄	#10587618	1	2	yes	—
262	C₂×C₄	#10587636	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	136	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	4	4	no	—
273	C₂²	#11762669	1	1	yes	characteristic
274	C₂²	#11763024	1	2	yes	—
275	C₂²	#11763045	1	1	yes	characteristic
276	C₂²	#11763050	1	1	yes	characteristic
277	C₂²	#11763072	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	272	2	no	—
282	C₄	#11762805	1	1	yes	characteristic
283	C₄	#11762814	1	1	yes	characteristic
284	C₄	C₂×#10587489	1	1	yes	characteristic
285	C₄	#11762847	1	1	yes	characteristic
286	C₄	#11763045	1	1	yes	characteristic
287	C₄	#11763050	1	1	yes	characteristic
288	C₄	#11763109	1	2	yes	—
289	C₂	—	4	4	no	—
290	C₂	#14528221	1	2	yes	—
291	C₂	#14528235	1	2	yes	—
292	C₂	#14528794	1	1	yes	characteristic
293	C₂	#14528813	1	1	yes	characteristic
294	C₂	#14528843	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587616
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₈×D₄×C₁₇	C₂
socle	C₂³×C₁₇	#121
prime radical	C₂²×D₄×C₁₇	D₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×C₈×D₄	D₁₇
hypocenter	C₁₇	#8648
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #14528912	C₂	semidirect
3. C₂×#11762562	C₂	semidirect
4. #14526668	C₂	semidirect
16. C₈×D₄×C₁₇	C₄	semidirect
150. C₈×D₄	Dic₁₇	semidirect
193. C₁₇	#8648	semidirect

Maximal subgroups

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

subgroup	index
#14528912	2
#14528784	2
C₂×#11762562	2
#14526668	2
C₂×C₈×D₄×C₁₇	2
#8648	17