Subgroups — #16927260

Order 4352.

454 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528776	C₂	1	2	yes	maximal
2	#14527543	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527540	C₂	1	1	yes	maximal, characteristic, complemented
4	#14526659	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×#436	C₂	1	1	yes	maximal, characteristic
6	#11762833	C₂²	1	2	yes	—
7	#11762833	C₂²	1	2	yes	—
8	C₂×#10587504	—	2	2	no	—
9	#11762578	—	2	2	no	—
10	#11762562	—	2	1	no	—
11	#11762562	—	2	1	no	—
12	#11762560	—	2	1	no	—
13	#11762560	—	2	1	no	—
14	C₂²×M₁₆×C₁₇	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#436	—	2	1	no	—
17	C₁₇×#436	—	2	1	no	—
18	C₁₇×#436	—	2	1	no	—
19	C₁₇×#436	—	2	1	no	—
20	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₇×C₂²⋊C₈	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂²×D₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₇	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₇	—	2	2	no	—
26	C₂×M₁₆×C₁₇	—	2	2	no	—
27	C₂×M₁₆×C₁₇	—	4	4	no	—
28	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×D₄×C₁₇	—	2	1	no	—
32	C₄×D₄×C₁₇	—	2	1	no	—
33	C₄×D₄×C₁₇	—	2	1	no	—
34	C₄×D₄×C₁₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
38	C₂×#63223	—	2	2	no	—
39	C₂×#63223	—	2	2	no	—
40	C₁₇×C₄⋊C₈	—	2	1	no	—
41	C₁₇×C₄⋊C₈	—	2	1	no	—
42	C₁₇×C₂²⋊C₈	—	2	2	no	—
43	C₁₇×C₂²⋊C₈	—	2	2	no	—
44	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
48	#10587504	—	4	4	no	—
49	C₄⋊#3622	—	2	2	no	—
50	C₄⋊#3622	—	2	2	no	—
51	C₄⋊#3622	—	2	2	no	—
52	C₄⋊#3622	—	2	2	no	—
53	C₂⁴×C₁₇	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₁₇	—	2	1	no	—
55	C₂×D₄×C₁₇	—	2	1	no	—
56	C₂×D₄×C₁₇	—	2	2	no	—
57	C₂×D₄×C₁₇	—	2	2	no	—
58	C₂²×C₄×C₁₇	—	2	2	no	—
59	C₂²×C₄×C₁₇	—	2	2	no	—
60	C₂²×C₄×C₁₇	—	4	2	no	—
61	C₂²×C₄×C₁₇	—	4	2	no	—
62	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
66	M₁₆×C₁₇	—	4	4	no	—
67	M₁₆×C₁₇	—	4	4	no	—
68	C₂×C₈×C₁₇	—	2	2	no	—
69	C₂×C₈×C₁₇	—	2	2	no	—
70	C₂×C₈×C₁₇	—	4	1	no	—
71	C₂×C₈×C₁₇	—	4	2	no	—
72	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₇	—	2	2	no	—
79	C₂²⋊C₄×C₁₇	—	2	2	no	—
80	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₇	#41	1	1	yes	characteristic
83	C₄²×C₁₇	#41	1	1	yes	characteristic
84	#63223	—	4	2	no	—
85	#63223	—	4	2	no	—
86	#63223	—	4	2	no	—
87	#63223	—	4	2	no	—
88	C₂×#3622	—	4	2	no	—
89	C₂×#3622	—	4	2	no	—
90	#8618	—	17	1	no	maximal, Sylow
91	C₂³×C₁₇	—	2	2	no	—
92	C₂³×C₁₇	—	2	4	no	—
93	C₂³×C₁₇	—	4	2	no	—
94	C₂³×C₁₇	—	4	2	no	—
95	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	D₄×C₁₇	—	4	2	no	—
97	D₄×C₁₇	—	4	1	no	—
98	D₄×C₁₇	—	4	1	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	2	2	no	—
102	C₂×C₄×C₁₇	—	4	1	no	—
103	C₂×C₄×C₁₇	—	4	4	no	—
104	C₂×C₄×C₁₇	—	4	2	no	—
105	C₂×C₄×C₁₇	—	4	2	no	—
106	C₂×C₄×C₁₇	—	4	2	no	—
107	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₁₇	#137	1	2	yes	—
115	C₈×C₁₇	—	2	1	no	—
116	C₈×C₁₇	—	2	1	no	—
117	C₈×C₁₇	—	2	1	no	—
118	C₈×C₁₇	—	4	2	no	—
119	C₈×C₁₇	#100	1	2	yes	complemented
120	#3622	—	8	2	no	—
121	#3622	—	8	2	no	—
122	C₂×#436	D₁₇	1	1	yes	characteristic
123	#1800	—	17	2	no	—
124	#1567	—	17	1	no	—
125	#1566	—	17	1	no	—
126	#1292	—	17	2	no	—
127	C₂²×C₁₇	—	2	1	no	—
128	C₂²×C₁₇	—	2	1	no	—
129	C₂²×C₁₇	—	4	2	no	—
130	C₂²×C₁₇	—	4	2	no	—
131	C₂²×C₁₇	—	4	2	no	—
132	C₂²×C₁₇	—	8	4	no	—
133	C₂²×C₁₇	#380	1	1	yes	characteristic
134	C₂²×C₁₇	#428	1	2	yes	—
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	4	2	no	—
139	C₄×C₁₇	C₂×#100	1	1	yes	characteristic
140	C₄×C₁₇	#413	1	1	yes	characteristic
141	C₄×C₁₇	#468	1	2	yes	—
142	C₄×C₁₇	#469	1	1	yes	characteristic
143	C₄×C₁₇	#470	1	1	yes	characteristic
144	C₂²×M₁₆	Dic₁₇	1	2	yes	—
145	C₂×C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
146	#436	—	2	1	no	—
147	#436	—	2	1	no	—
148	#436	—	2	1	no	—
149	#436	—	2	1	no	—
150	#423	—	17	2	no	—
151	#423	—	17	2	no	—
152	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
153	C₂×#100	—	34	2	no	—
154	C₂×C₂²⋊C₈	Dic₁₇	1	2	yes	—
155	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
156	#343	—	34	2	no	—
157	#335	—	34	1	no	—
158	#335	—	34	1	no	—
159	#334	—	34	1	no	—
160	#334	—	34	1	no	—
161	C₂×C₁₇	—	2	2	no	—
162	C₂×C₁₇	—	8	2	no	—
163	C₂×C₁₇	#1800	1	1	yes	characteristic
164	C₂×C₁₇	#1809	1	1	yes	characteristic
165	C₂×C₁₇	#1813	1	1	yes	characteristic
166	C₂²×D₄	D₆₈	1	1	yes	characteristic
167	C₂³×C₄	C₂×Dic₁₇	1	2	yes	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	4	4	no	—
171	C₂×M₁₆	—	34	2	no	—
172	C₂×M₁₆	—	34	2	no	—
173	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
174	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
175	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
176	C₄×D₄	—	2	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	Dic₃₄	1	2	yes	—
182	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	34	2	no	—
186	C₄⋊C₈	—	34	2	no	—
187	C₄⋊C₈	—	34	2	no	—
188	C₄⋊C₈	—	34	2	no	—
189	#100	—	68	4	no	—
190	C₂²⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
193	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
196	C₁₇	#8618	1	1	yes	characteristic, complemented, Sylow
197	C₂⁴	C₄⋊Dic₁₇	1	2	yes	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	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	C₄⋊Dic₁₇	1	2	yes	—
207	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
210	M₁₆	—	4	4	no	—
211	M₁₆	—	4	4	no	—
212	M₁₆	—	68	2	no	—
213	M₁₆	—	68	2	no	—
214	M₁₆	—	68	2	no	—
215	M₁₆	—	68	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	68	2	no	—
221	C₂×C₈	—	68	2	no	—
222	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
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	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₄²	#63252	1	1	yes	characteristic
231	C₄²	#63252	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	4	2	no	—
238	C₂³	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	D₄	—	4	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	#10587594	1	2	yes	—
251	C₂×C₄	#10587595	1	2	yes	—
252	C₂×C₄	#10587610	1	1	yes	characteristic
253	C₂×C₄	#10587610	1	1	yes	characteristic
254	C₂×C₄	#10587612	1	1	yes	characteristic
255	C₂×C₄	#10587616	1	1	yes	characteristic
256	C₂×C₄	#10587616	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	4	2	no	—
262	C₈	—	136	2	no	—
263	C₈	—	136	2	no	—
264	C₈	#10587504	1	2	yes	complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	8	4	no	—
271	C₂²	#11762669	1	1	yes	characteristic
272	C₂²	#11762851	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	C₂×#10587504	1	1	yes	characteristic
278	C₄	#11762793	1	1	yes	characteristic
279	C₄	#11763055	1	2	yes	—
280	C₄	#11763062	1	1	yes	characteristic
281	C₄	#11763067	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	8	2	no	—
284	C₂	#14528776	1	1	yes	characteristic
285	C₂	#14528817	1	1	yes	characteristic
286	C₂	#14528834	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762669
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#436	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂²×D₄×C₁₇	D₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#436	D₁₇
hypocenter	C₁₇	#8618
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. #14527543	C₂	semidirect
3. #14527540	C₂	semidirect
4. #14526659	C₂	semidirect
119. C₈×C₁₇	#100	semidirect
196. C₁₇	#8618	semidirect
264. C₈	#10587504	semidirect

Maximal subgroups

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

subgroup	index
#14528776	2
#14527543	2
#14527540	2
#14526659	2
C₂×C₁₇×#436	2
#8618	17