Subgroups — C₁₇×#9586 #16867330

Order 4352.

312 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#1574	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×#1543	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1492	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1484	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#532	C₂²	1	1	yes	characteristic
9	C₁₇×#431	C₂²	1	1	yes	characteristic
10	C₁₇×#431	C₂²	1	1	yes	characteristic
11	C₁₇×#431	C₂²	1	1	yes	characteristic
12	C₁₇×#423	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₁₇×#335	—	2	1	no	—
16	C₁₇×#334	—	2	1	no	—
17	C₁₇×#333	C₄	1	1	yes	characteristic
18	C₁₇×#333	C₄	1	1	yes	characteristic
19	C₁₇×#331	C₄	1	1	yes	characteristic
20	C₁₇×#331	C₄	1	1	yes	characteristic
21	C₁₇×D₄⋊C₈	—	2	1	no	—
22	C₁₇×D₄⋊C₈	—	2	1	no	—
23	C₁₇×D₄⋊C₈	—	2	1	no	—
24	C₁₇×D₄⋊C₈	—	2	1	no	—
25	C₁₇×#322	—	2	1	no	—
26	C₈²×C₁₇	—	2	1	no	—
27	C₂×#41×C₁₇	—	2	1	no	—
28	C₂×M₁₆×C₁₇	—	2	1	no	—
29	C₂×M₁₆×C₁₇	—	2	1	no	—
30	C₂²×C₈×C₁₇	—	2	1	no	—
31	C₁₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₁₇×#127	C₂×C₄	1	1	yes	characteristic
33	C₁₇×#124	—	2	1	no	—
34	C₁₇×#121	—	2	1	no	—
35	C₁₇×#121	—	2	1	no	—
36	C₄×D₄×C₁₇	—	2	1	no	—
37	C₄×D₄×C₁₇	—	2	1	no	—
38	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
39	C₁₇×C₄⋊C₈	—	2	1	no	—
40	C₁₇×C₄⋊C₈	—	2	1	no	—
41	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
46	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
47	C₁₇×C₂²⋊C₈	—	2	1	no	—
48	C₁₇×C₂²⋊C₈	—	2	1	no	—
49	C₁₇×C₂²⋊C₈	—	2	1	no	—
50	C₁₇×#97	—	2	1	no	—
51	C₄×C₈×C₁₇	—	2	1	no	—
52	C₄×C₈×C₁₇	—	2	1	no	—
53	C₄×C₈×C₁₇	—	2	1	no	—
54	C₄×C₈×C₁₇	—	2	1	no	—
55	C₄×C₈×C₁₇	—	2	1	no	—
56	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
58	#41×C₁₇	—	8	1	no	—
59	#41×C₁₇	—	8	1	no	—
60	C₂×Q₈×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	—	2	1	no	—
62	C₂×D₄×C₁₇	—	2	1	no	—
63	C₂×D₄×C₁₇	—	2	1	no	—
64	C₂×D₄×C₁₇	—	4	1	no	—
65	C₂²×C₄×C₁₇	—	4	1	no	—
66	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
69	M₁₆×C₁₇	—	2	2	no	—
70	M₁₆×C₁₇	—	4	1	no	—
71	M₁₆×C₁₇	—	8	1	no	—
72	C₂×C₈×C₁₇	—	2	2	no	—
73	C₂×C₈×C₁₇	—	2	2	no	—
74	C₂×C₈×C₁₇	—	2	1	no	—
75	C₂×C₈×C₁₇	—	2	1	no	—
76	C₂×C₈×C₁₇	—	2	1	no	—
77	C₂×C₈×C₁₇	—	2	1	no	—
78	C₂×C₈×C₁₇	—	2	1	no	—
79	C₂×C₈×C₁₇	—	2	1	no	—
80	C₂×C₈×C₁₇	—	2	1	no	—
81	C₂×C₈×C₁₇	—	2	1	no	—
82	C₂×C₈×C₁₇	—	4	1	no	—
83	C₂×C₈×C₁₇	—	4	1	no	—
84	C₄⋊C₄×C₁₇	—	2	1	no	—
85	C₄⋊C₄×C₁₇	—	2	1	no	—
86	C₂²⋊C₄×C₁₇	—	4	1	no	—
87	C₂²⋊C₄×C₁₇	—	4	1	no	—
88	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₇	#41	1	1	yes	characteristic
92	#9586	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₁₇	—	4	1	no	—
94	C₂³×C₁₇	#126	1	1	yes	characteristic
95	Q₈×C₁₇	—	8	1	no	—
96	D₄×C₁₇	—	4	1	no	—
97	D₄×C₁₇	—	4	1	no	—
98	D₄×C₁₇	—	4	1	no	—
99	D₄×C₁₇	—	4	1	no	—
100	D₄×C₁₇	—	8	1	no	—
101	D₄×C₁₇	—	8	2	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	4	1	no	—
109	C₂×C₄×C₁₇	—	8	1	no	—
110	C₂×C₄×C₁₇	—	8	1	no	—
111	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
117	C₈×C₁₇	—	2	2	no	—
118	C₈×C₁₇	—	2	2	no	—
119	C₈×C₁₇	—	4	2	no	—
120	C₈×C₁₇	—	4	1	no	—
121	C₈×C₁₇	—	4	1	no	—
122	C₈×C₁₇	—	4	1	no	—
123	C₈×C₁₇	—	4	1	no	—
124	C₈×C₁₇	—	4	1	no	—
125	C₈×C₁₇	—	4	1	no	—
126	C₈×C₁₇	—	8	1	no	—
127	#1574	C₂×C₁₇	1	1	yes	characteristic, complemented
128	#1543	C₂×C₁₇	1	1	yes	characteristic
129	#1492	C₂×C₁₇	1	1	yes	characteristic
130	#1484	C₂×C₁₇	1	1	yes	characteristic
131	#1450	C₂×C₁₇	1	1	yes	characteristic, complemented
132	#1299	C₂×C₁₇	1	1	yes	characteristic, complemented
133	#1278	C₂×C₁₇	1	1	yes	characteristic, complemented
134	C₂²×C₁₇	—	2	1	no	—
135	C₂²×C₁₇	—	2	1	no	—
136	C₂²×C₁₇	—	2	1	no	—
137	C₂²×C₁₇	—	8	1	no	—
138	C₂²×C₁₇	—	8	1	no	—
139	C₂²×C₁₇	—	8	1	no	—
140	C₂²×C₁₇	#388	1	1	yes	characteristic
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	8	1	no	—
145	C₄×C₁₇	#443	1	1	yes	characteristic
146	C₄×C₁₇	#443	1	1	yes	characteristic
147	C₄×C₁₇	#443	1	1	yes	characteristic
148	C₄×C₁₇	#444	1	1	yes	characteristic
149	C₄×C₁₇	#471	1	1	yes	characteristic
150	C₄×C₁₇	#472	1	1	yes	characteristic
151	#532	C₂²×C₁₇	1	1	yes	characteristic
152	#431	C₂²×C₁₇	1	1	yes	characteristic
153	#431	C₂²×C₁₇	1	1	yes	characteristic
154	#431	C₂²×C₁₇	1	1	yes	characteristic
155	#423	C₂²×C₁₇	1	1	yes	characteristic
156	C₄×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
158	#335	—	2	1	no	—
159	#334	—	2	1	no	—
160	#333	C₄×C₁₇	1	1	yes	characteristic
161	#333	C₄×C₁₇	1	1	yes	characteristic
162	#331	C₄×C₁₇	1	1	yes	characteristic
163	#331	C₄×C₁₇	1	1	yes	characteristic
164	D₄⋊C₈	—	2	1	no	—
165	D₄⋊C₈	—	2	1	no	—
166	D₄⋊C₈	—	2	1	no	—
167	D₄⋊C₈	—	2	1	no	—
168	#322	—	2	1	no	—
169	C₈²	—	2	1	no	—
170	C₂×C₁₇	—	4	1	no	—
171	C₂×C₁₇	—	16	1	no	—
172	C₂×C₁₇	#1929	1	1	yes	characteristic
173	C₂×C₁₇	#1931	1	1	yes	characteristic
174	C₂×C₁₇	#1937	1	1	yes	characteristic
175	C₂×#41	—	2	1	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₄⋊Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
180	#127	C₂×C₄×C₁₇	1	1	yes	characteristic
181	#124	—	2	1	no	—
182	#121	—	2	1	no	—
183	#121	—	2	1	no	—
184	C₄×D₄	—	2	1	no	—
185	C₄×D₄	—	2	1	no	—
186	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
190	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
191	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
192	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
193	C₄⋊C₈	D₄×C₁₇	1	1	yes	characteristic
194	C₄⋊C₈	D₄×C₁₇	1	1	yes	characteristic
195	C₂²⋊C₈	—	2	1	no	—
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	2	1	no	—
198	#97	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₄×C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	D₄×C₁₇	1	1	yes	characteristic
205	C₄×C₈	D₄×C₁₇	1	1	yes	characteristic
206	C₁₇	#9586	1	1	yes	characteristic, direct product factor, Sylow
207	#41	—	8	1	no	—
208	#41	—	8	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
216	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
217	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	4	1	no	—
220	M₁₆	—	8	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₂²⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	4	1	no	—
237	C₄²	C₂²×C₄×C₁₇	1	1	yes	characteristic
238	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
239	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
240	C₄²	#41×C₁₇	1	1	yes	characteristic
241	C₂³	—	4	1	no	—
242	C₂³	C₁₇×#126	1	1	yes	characteristic
243	Q₈	—	8	1	no	—
244	D₄	—	4	1	no	—
245	D₄	—	4	1	no	—
246	D₄	—	4	1	no	—
247	D₄	—	4	1	no	—
248	D₄	—	8	1	no	—
249	D₄	—	8	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	—	8	1	no	—
259	C₂×C₄	C₁₇×#117	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
261	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
262	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
263	C₂×C₄	C₁₇×#123	1	1	yes	characteristic
264	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	4	1	no	—
269	C₈	—	4	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	8	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	8	1	no	—
279	C₂²	—	8	1	no	—
280	C₂²	—	8	1	no	—
281	C₂²	C₁₇×#388	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	8	1	no	—
286	C₄	C₁₇×#443	1	1	yes	characteristic
287	C₄	C₁₇×#443	1	1	yes	characteristic
288	C₄	C₁₇×#443	1	1	yes	characteristic
289	C₄	C₁₇×#444	1	1	yes	characteristic
290	C₄	C₁₇×#471	1	1	yes	characteristic
291	C₄	C₁₇×#472	1	1	yes	characteristic
292	C₂	—	4	1	no	—
293	C₂	—	16	1	no	—
294	C₂	C₁₇×#1929	1	1	yes	characteristic
295	C₂	C₁₇×#1931	1	1	yes	characteristic
296	C₂	C₁₇×#1937	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇	#121
commutator	C₄²	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	#388
prime radical	C₁₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₇×#1574	C₂	semidirect
5. C₁₇×#1450	C₂	semidirect
6. C₁₇×#1299	C₂	semidirect
7. C₁₇×#1278	C₂	semidirect
92. #9586	C₁₇	direct (N × H)
127. #1574	C₂×C₁₇	semidirect
131. #1450	C₂×C₁₇	semidirect
132. #1299	C₂×C₁₇	semidirect
133. #1278	C₂×C₁₇	semidirect
206. C₁₇	#9586	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#1574	2
C₁₇×#1543	2
C₁₇×#1492	2
C₁₇×#1484	2
C₁₇×#1450	2
C₁₇×#1299	2
C₁₇×#1278	2
#9586	17

Subgroups — C17×#9586 #16867330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#9586 #16867330