Subgroups — C₁₇×#8021 #16865765

Order 4352.

786 conjugacy classes of subgroups.

The subgroup lattice has 324 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×#2283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1294	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₁₇	—	2	1	no	—
6	C₂²×C₄²×C₁₇	—	2	1	no	—
7	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#432	—	2	2	no	—
9	C₁₇×#432	C₄	1	4	yes	complemented
10	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
11	C₂×C₁₇×C₂²⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₇×#97	—	2	1	no	—
13	C₁₇×#383	—	2	1	no	—
14	C₁₇×#382	C₄	1	2	yes	complemented
15	C₁₇×#381	—	2	1	no	—
16	C₁₇×#380	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
18	C₄×C₄⋊C₄×C₁₇	C₄	1	2	yes	complemented
19	C₄×C₄⋊C₄×C₁₇	C₄	1	4	yes	complemented
20	C₄×C₂²⋊C₄×C₁₇	—	2	1	no	—
21	C₁₇×#376	—	2	1	no	—
22	C₂×C₁₇×#95	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₇	—	2	2	no	—
24	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	8	no	—
29	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
32	C₂×C₄²×C₁₇	—	2	4	no	—
33	C₂×C₄²×C₁₇	—	2	2	no	—
34	C₂×C₄²×C₁₇	—	4	4	no	—
35	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₇	D₄	1	2	yes	—
41	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₇	Q₈	1	2	yes	—
43	C₁₇×C₄⋊C₈	—	2	4	no	—
44	C₁₇×C₂²⋊C₈	—	2	2	no	—
45	C₁₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
46	C₁₇×#97	—	2	4	no	—
47	C₁₇×#95	—	2	2	no	—
48	C₁₇×#95	—	2	2	no	—
49	C₁₇×#95	—	2	2	no	—
50	C₁₇×#95	—	2	2	no	—
51	C₁₇×#95	—	2	2	no	—
52	C₁₇×#95	C₂×C₄	1	4	yes	—
53	C₂⁴×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	—	2	2	no	—
55	C₂²×C₄×C₁₇	—	2	2	no	—
56	C₂²×C₄×C₁₇	—	2	4	no	—
57	C₂²×C₄×C₁₇	—	2	8	no	—
58	C₂²×C₄×C₁₇	—	2	4	no	—
59	C₂²×C₄×C₁₇	—	2	4	no	—
60	C₂²×C₄×C₁₇	—	4	2	no	—
61	C₂²×C₄×C₁₇	—	4	4	no	—
62	C₂²×C₄×C₁₇	—	4	4	no	—
63	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₁₇	—	2	4	no	—
78	C₂×C₈×C₁₇	—	4	4	no	—
79	C₄⋊C₄×C₁₇	—	2	16	no	—
80	C₄⋊C₄×C₁₇	—	4	4	no	—
81	C₂²⋊C₄×C₁₇	—	4	2	no	—
82	C₂²⋊C₄×C₁₇	—	4	2	no	—
83	C₄²×C₁₇	—	4	8	no	—
84	C₄²×C₁₇	—	4	4	no	—
85	C₄²×C₁₇	C₄²	1	2	yes	—
86	C₄²×C₁₇	C₄²	1	2	yes	—
87	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
90	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
91	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
92	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
93	#8021	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₁₇	—	2	2	no	—
95	C₂³×C₁₇	—	2	2	no	—
96	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
100	C₂³×C₁₇	#137	1	2	yes	—
101	C₂×C₄×C₁₇	—	2	2	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	2	no	—
104	C₂×C₄×C₁₇	—	2	2	no	—
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	2	2	no	—
107	C₂×C₄×C₁₇	—	2	2	no	—
108	C₂×C₄×C₁₇	—	2	4	no	—
109	C₂×C₄×C₁₇	—	2	4	no	—
110	C₂×C₄×C₁₇	—	2	4	no	—
111	C₂×C₄×C₁₇	—	2	8	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	—	4	1	no	—
114	C₂×C₄×C₁₇	—	4	1	no	—
115	C₂×C₄×C₁₇	—	4	1	no	—
116	C₂×C₄×C₁₇	—	4	4	no	—
117	C₂×C₄×C₁₇	—	4	16	no	—
118	C₂×C₄×C₁₇	—	4	4	no	—
119	C₂×C₄×C₁₇	—	4	4	no	—
120	C₂×C₄×C₁₇	—	4	4	no	—
121	C₂×C₄×C₁₇	—	4	4	no	—
122	C₂×C₄×C₁₇	#95	1	2	yes	—
123	C₂×C₄×C₁₇	#95	1	2	yes	—
124	C₂×C₄×C₁₇	#95	1	2	yes	—
125	C₂×C₄×C₁₇	#95	1	2	yes	—
126	C₂×C₄×C₁₇	#104	1	4	yes	—
127	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₈×C₁₇	—	4	4	no	—
132	C₂×#432	C₂×C₁₇	1	1	yes	characteristic
133	#2283	C₂×C₁₇	1	1	yes	characteristic
134	C₂×C₄×C₄⋊C₄	C₂×C₁₇	1	1	yes	characteristic
135	#1294	C₂×C₁₇	1	4	yes	—
136	C₂²×C₁₇	—	2	1	no	—
137	C₂²×C₁₇	—	2	1	no	—
138	C₂²×C₁₇	—	2	1	no	—
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	4	no	—
141	C₂²×C₁₇	—	2	4	no	—
142	C₂²×C₁₇	C₂×#95	1	1	yes	characteristic
143	C₂²×C₁₇	#410	1	2	yes	—
144	C₂²×C₁₇	#417	1	2	yes	—
145	C₂²×C₁₇	#418	1	1	yes	characteristic
146	C₂²×C₁₇	#419	1	1	yes	characteristic
147	C₂²×C₁₇	C₂×#104	1	2	yes	—
148	C₂²×C₁₇	#428	1	2	yes	—
149	C₄×C₁₇	—	2	2	no	—
150	C₄×C₁₇	—	2	2	no	—
151	C₄×C₁₇	—	2	2	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	4	8	no	—
157	C₄×C₁₇	—	8	4	no	—
158	C₄×C₁₇	#339	1	4	yes	—
159	C₂²×C₄⋊C₄	—	2	1	no	—
160	C₂²×C₄²	—	2	1	no	—
161	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
162	#432	—	2	2	no	—
163	#432	C₄×C₁₇	1	4	yes	complemented
164	C₂×C₄⋊C₈	—	2	1	no	—
165	C₂×C₂²⋊C₈	C₂²×C₁₇	1	2	yes	—
166	C₂×#97	—	2	1	no	—
167	#383	—	2	1	no	—
168	#382	C₄×C₁₇	1	2	yes	complemented
169	#381	—	2	1	no	—
170	#380	C₂²×C₁₇	1	2	yes	—
171	C₄×C₄⋊C₄	—	2	2	no	—
172	C₄×C₄⋊C₄	C₄×C₁₇	1	2	yes	complemented
173	C₄×C₄⋊C₄	C₄×C₁₇	1	4	yes	complemented
174	C₄×C₂²⋊C₄	—	2	1	no	—
175	#376	—	2	1	no	—
176	C₂×#95	C₂²×C₁₇	1	2	yes	—
177	C₂×C₁₇	—	2	1	no	—
178	C₂×C₁₇	—	2	1	no	—
179	C₂×C₁₇	—	2	2	no	—
180	C₂×C₁₇	#1509	1	4	yes	—
181	C₂×C₁₇	C₂×#339	1	1	yes	characteristic
182	C₂×C₁₇	#1736	1	1	yes	characteristic
183	C₂×C₁₇	#1740	1	1	yes	characteristic
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
186	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
188	C₂²×C₈	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	8	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	4	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	—	4	4	no	—
196	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
199	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₁₇	1	2	yes	—
202	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
203	C₂×C₄²	Q₈×C₁₇	1	2	yes	—
204	C₄⋊C₈	—	2	4	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₂²⋊C₈	C₂×C₄×C₁₇	1	4	yes	—
207	#97	—	2	4	no	—
208	#95	—	2	2	no	—
209	#95	—	2	2	no	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	C₂×C₄×C₁₇	1	4	yes	—
214	C₁₇	#8021	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	C₂×D₄×C₁₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
229	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
230	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
231	C₂²×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
232	C₂²×C₄	C₄⋊C₄×C₁₇	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₈×C₁₇	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	4	no	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	—	4	8	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₄²×C₁₇	1	2	yes	—
248	C₄²	C₄²×C₁₇	1	2	yes	—
249	C₄²	C₂²⋊C₄×C₁₇	1	2	yes	—
250	C₄²	C₂²⋊C₄×C₁₇	1	2	yes	—
251	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
252	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
253	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
254	C₄²	C₄⋊C₄×C₁₇	1	2	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
258	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
259	C₂³	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
260	C₂³	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
261	C₂³	C₁₇×#137	1	2	yes	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	16	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	C₁₇×#95	1	2	yes	—
284	C₂×C₄	C₁₇×#95	1	2	yes	—
285	C₂×C₄	C₁₇×#95	1	2	yes	—
286	C₂×C₄	C₁₇×#95	1	2	yes	—
287	C₂×C₄	C₁₇×#104	1	4	yes	—
288	C₂×C₄	C₂×C₄²×C₁₇	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
292	C₈	—	4	4	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	C₂×C₁₇×#95	1	1	yes	characteristic
300	C₂²	C₁₇×#410	1	2	yes	—
301	C₂²	C₁₇×#417	1	2	yes	—
302	C₂²	C₁₇×#418	1	1	yes	characteristic
303	C₂²	C₁₇×#419	1	1	yes	characteristic
304	C₂²	C₂×C₁₇×#104	1	2	yes	—
305	C₂²	C₁₇×#428	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	8	no	—
314	C₄	—	8	4	no	—
315	C₄	C₁₇×#339	1	4	yes	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	C₁₇×#1509	1	4	yes	—
320	C₂	C₂×C₁₇×#339	1	1	yes	characteristic
321	C₂	C₁₇×#1736	1	1	yes	characteristic
322	C₂	C₁₇×#1740	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₇
prime radical	C₂⁴×C₁₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
9. C₁₇×#432	C₄	semidirect
14. C₁₇×#382	C₄	semidirect
18. C₄×C₄⋊C₄×C₁₇	C₄	semidirect
19. C₄×C₄⋊C₄×C₁₇	C₄	semidirect
93. #8021	C₁₇	direct (N × H)
163. #432	C₄×C₁₇	semidirect
168. #382	C₄×C₁₇	semidirect
172. C₄×C₄⋊C₄	C₄×C₁₇	semidirect
173. C₄×C₄⋊C₄	C₄×C₁₇	semidirect
214. C₁₇	#8021	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×#2283	2
C₂×C₄×C₄⋊C₄×C₁₇	2
C₁₇×#1294	2
#8021	17

Subgroups — C17×#8021 #16865765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#8021 #16865765