Subgroups — C₁₇×#21989 #16879733

Order 4352.

786 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#432	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#2283	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1949	C₂	1	4	yes	maximal, complemented
5	C₁₇×#1854	C₂	1	4	yes	maximal
6	C₁₇×#1847	C₂	1	2	yes	maximal, complemented
7	C₁₇×#1836	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₇×#117	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×#432	C₄	1	8	yes	complemented
11	C₂×C₁₇×#107	C₂²	1	4	yes	—
12	C₂×C₁₇×#106	C₂²	1	4	yes	—
13	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₇×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₇×#97	C₂²	1	2	yes	—
16	C₁₇×#384	C₂²	1	2	yes	—
17	C₁₇×#383	—	2	1	no	—
18	C₁₇×#382	C₂²	1	2	yes	—
19	C₁₇×#382	C₂²	1	1	yes	characteristic
20	C₁₇×#382	C₂²	1	1	yes	characteristic
21	C₁₇×#381	—	2	1	no	—
22	C₁₇×#380	—	2	1	no	—
23	C₁₇×#380	—	2	1	no	—
24	C₁₇×#380	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
26	C₄×C₄⋊C₄×C₁₇	—	2	1	no	—
27	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₁₇	—	2	1	no	—
29	C₁₇×#376	—	2	1	no	—
30	C₁₇×#340	C₂²	1	8	yes	—
31	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	C₂³	1	4	yes	—
35	C₁₇×#117	—	4	2	no	—
36	C₁₇×#117	—	4	2	no	—
37	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
44	C₂×C₄²×C₁₇	—	2	2	no	—
45	C₂×C₄²×C₁₇	—	2	2	no	—
46	C₂×C₄²×C₁₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₇	Q₈	1	4	yes	—
50	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
52	C₁₇×#107	—	4	4	no	—
53	C₁₇×#106	—	4	4	no	—
54	C₁₇×C₄⋊C₈	C₂×C₄	1	8	yes	—
55	C₁₇×C₂²⋊C₈	C₂×C₄	1	8	yes	—
56	C₁₇×#97	C₂×C₄	1	8	yes	—
57	C₁₇×#95	—	2	2	no	—
58	C₁₇×#95	—	2	1	no	—
59	C₁₇×#95	—	2	1	no	—
60	C₁₇×#95	—	2	1	no	—
61	C₁₇×#95	—	2	1	no	—
62	C₁₇×#95	—	2	2	no	—
63	C₁₇×#95	—	2	2	no	—
64	C₁₇×#95	—	2	2	no	—
65	C₂⁴×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	—	2	2	no	—
67	C₂²×C₄×C₁₇	—	2	2	no	—
68	C₂²×C₄×C₁₇	—	2	2	no	—
69	C₂²×C₄×C₁₇	—	2	2	no	—
70	C₂²×C₄×C₁₇	—	2	4	no	—
71	C₂²×C₄×C₁₇	—	4	2	no	—
72	C₂²×C₄×C₁₇	—	4	2	no	—
73	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₁₇	—	2	4	no	—
84	C₂×C₈×C₁₇	—	2	4	no	—
85	C₂×C₈×C₁₇	C₂²×C₄	1	8	yes	—
86	C₄⋊C₄×C₁₇	—	2	4	no	—
87	C₄⋊C₄×C₁₇	—	2	4	no	—
88	C₄⋊C₄×C₁₇	—	2	4	no	—
89	C₄⋊C₄×C₁₇	—	2	4	no	—
90	C₄⋊C₄×C₁₇	—	4	4	no	—
91	C₄⋊C₄×C₁₇	—	4	4	no	—
92	C₂²⋊C₄×C₁₇	—	4	4	no	—
93	C₂²⋊C₄×C₁₇	—	4	4	no	—
94	C₄²×C₁₇	—	4	4	no	—
95	C₄²×C₁₇	—	4	4	no	—
96	C₄²×C₁₇	C₄⋊C₄	1	4	yes	—
97	C₄²×C₁₇	C₄⋊C₄	1	4	yes	—
98	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
101	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
102	#21989	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₇	—	2	1	no	—
104	C₂³×C₁₇	—	2	1	no	—
105	C₂³×C₁₇	—	2	2	no	—
106	C₂³×C₁₇	—	2	2	no	—
107	C₂³×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	2	no	—
112	C₂×C₄×C₁₇	—	2	2	no	—
113	C₂×C₄×C₁₇	—	2	2	no	—
114	C₂×C₄×C₁₇	—	2	2	no	—
115	C₂×C₄×C₁₇	—	2	4	no	—
116	C₂×C₄×C₁₇	—	2	4	no	—
117	C₂×C₄×C₁₇	—	4	4	no	—
118	C₂×C₄×C₁₇	—	4	4	no	—
119	C₂×C₄×C₁₇	—	4	4	no	—
120	C₂×C₄×C₁₇	—	4	2	no	—
121	C₂×C₄×C₁₇	—	4	2	no	—
122	C₂×C₄×C₁₇	—	4	2	no	—
123	C₂×C₄×C₁₇	—	4	2	no	—
124	C₂×C₄×C₁₇	—	4	2	no	—
125	C₂×C₄×C₁₇	—	4	2	no	—
126	C₂×C₄×C₁₇	—	4	4	no	—
127	C₂×C₄×C₁₇	—	4	4	no	—
128	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₇	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
137	C₈×C₁₇	—	2	8	no	—
138	C₂×#432	C₂×C₁₇	1	1	yes	characteristic
139	#2285	C₂×C₁₇	1	1	yes	characteristic
140	#2283	C₂×C₁₇	1	1	yes	characteristic
141	#1949	C₂×C₁₇	1	4	yes	complemented
142	#1854	C₂×C₁₇	1	4	yes	—
143	#1847	C₂×C₁₇	1	2	yes	complemented
144	#1836	C₂×C₁₇	1	2	yes	complemented
145	C₂²×C₁₇	—	2	2	no	—
146	C₂²×C₁₇	—	2	2	no	—
147	C₂²×C₁₇	—	2	2	no	—
148	C₂²×C₁₇	—	4	2	no	—
149	C₂²×C₁₇	—	4	2	no	—
150	C₂²×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₁₇	#535	1	1	yes	characteristic
152	C₂²×C₁₇	#536	1	1	yes	characteristic
153	C₂²×C₁₇	#575	1	2	yes	—
154	C₂²×C₁₇	#575	1	2	yes	—
155	C₄×C₁₇	—	2	2	no	—
156	C₄×C₁₇	—	2	2	no	—
157	C₄×C₁₇	—	2	2	no	—
158	C₄×C₁₇	—	2	2	no	—
159	C₄×C₁₇	—	2	2	no	—
160	C₄×C₁₇	—	8	4	no	—
161	C₄×C₁₇	—	8	2	no	—
162	C₄×C₁₇	—	8	2	no	—
163	C₄×C₁₇	#520	1	2	yes	—
164	C₄×C₁₇	#544	1	2	yes	—
165	C₂×#117	C₂²×C₁₇	1	2	yes	—
166	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
167	#432	C₄×C₁₇	1	8	yes	complemented
168	C₂×#107	C₂²×C₁₇	1	4	yes	—
169	C₂×#106	C₂²×C₁₇	1	4	yes	—
170	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
171	C₂×C₂²⋊C₈	C₂²×C₁₇	1	2	yes	—
172	C₂×#97	C₂²×C₁₇	1	2	yes	—
173	#384	C₂²×C₁₇	1	2	yes	—
174	#383	—	2	1	no	—
175	#382	C₂²×C₁₇	1	1	yes	characteristic
176	#382	C₂²×C₁₇	1	1	yes	characteristic
177	#382	C₂²×C₁₇	1	2	yes	—
178	#381	—	2	1	no	—
179	#380	—	2	1	no	—
180	#380	—	2	1	no	—
181	#380	C₂²×C₁₇	1	2	yes	—
182	C₄×C₄⋊C₄	—	2	1	no	—
183	C₄×C₄⋊C₄	—	2	1	no	—
184	C₄×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
185	C₄×C₂²⋊C₄	—	2	1	no	—
186	#376	—	2	1	no	—
187	#340	C₂²×C₁₇	1	8	yes	—
188	C₂×C₁₇	—	4	2	no	—
189	C₂×C₁₇	#2317	1	1	yes	characteristic
190	C₂×C₁₇	#2911	1	1	yes	characteristic
191	C₂×C₁₇	#2911	1	1	yes	characteristic
192	C₂×C₁₇	#3219	1	2	yes	—
193	C₂×C₁₇	#3240	1	2	yes	—
194	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
195	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
196	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₇	1	4	yes	—
198	#117	—	4	2	no	—
199	#117	—	4	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₂×C₄×C₁₇	1	4	yes	—
210	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
212	C₂×C₄²	Q₈×C₁₇	1	4	yes	—
213	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
215	#107	—	4	4	no	—
216	#106	—	4	4	no	—
217	C₄⋊C₈	C₂×C₄×C₁₇	1	8	yes	—
218	C₂²⋊C₈	C₂×C₄×C₁₇	1	8	yes	—
219	#97	C₂×C₄×C₁₇	1	8	yes	—
220	#95	—	2	2	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	C₁₇	#21989	1	1	yes	characteristic, direct product factor, Sylow
229	C₂⁴	C₂×D₄×C₁₇	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₄⋊C₄×C₁₇	1	4	yes	—
238	C₂²×C₄	C₄⋊C₄×C₁₇	1	4	yes	—
239	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×Q₈×C₁₇	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	C₂²×C₄×C₁₇	1	8	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₄²	—	4	4	no	—
259	C₄²	—	4	4	no	—
260	C₄²	C₄⋊C₄×C₁₇	1	4	yes	—
261	C₄²	C₄⋊C₄×C₁₇	1	4	yes	—
262	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
263	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
264	C₄²	C₂×Q₈×C₁₇	1	2	yes	—
265	C₄²	C₂×Q₈×C₁₇	1	2	yes	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
271	C₂³	C₂²×D₄×C₁₇	1	1	yes	characteristic
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
294	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	2	yes	—
296	C₂×C₄	C₂³×C₄×C₁₇	1	1	yes	characteristic
297	C₂×C₄	C₂²×Q₈×C₁₇	1	1	yes	characteristic
298	C₂×C₄	C₁₇×H(5,𝔽₂)	1	2	yes	—
299	C₂×C₄	C₁₇×2₋¹⁺⁴	1	2	yes	—
300	C₈	—	2	8	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂²×C₄⋊C₄×C₁₇	1	1	yes	characteristic
307	C₂²	C₁₇×#535	1	1	yes	characteristic
308	C₂²	C₁₇×#536	1	1	yes	characteristic
309	C₂²	C₁₇×#575	1	2	yes	—
310	C₂²	C₁₇×#575	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	8	4	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	C₁₇×#520	1	2	yes	—
320	C₄	C₁₇×#544	1	2	yes	—
321	C₂	—	4	2	no	—
322	C₂	C₁₇×#2317	1	1	yes	characteristic
323	C₂	C₁₇×#2911	1	1	yes	characteristic
324	C₂	C₁₇×#2911	1	1	yes	characteristic
325	C₂	C₁₇×#3219	1	2	yes	—
326	C₂	C₁₇×#3240	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₇
prime radical	C₂⁴×C₁₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂²×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₇×#1949	C₂	semidirect
6. C₁₇×#1847	C₂	semidirect
7. C₁₇×#1836	C₂	semidirect
10. C₁₇×#432	C₄	semidirect
102. #21989	C₁₇	direct (N × H)
141. #1949	C₂×C₁₇	semidirect
143. #1847	C₂×C₁₇	semidirect
144. #1836	C₂×C₁₇	semidirect
167. #432	C₄×C₁₇	semidirect
228. C₁₇	#21989	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#432	2
C₁₇×#2285	2
C₁₇×#2283	2
C₁₇×#1949	2
C₁₇×#1854	2
C₁₇×#1847	2
C₁₇×#1836	2
#21989	17

Subgroups — C17×#21989 #16879733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#21989 #16879733