Subgroups — C₁₇×#13407 #16871151

Order 4352.

566 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₇×#357	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₇×#357	C₂	1	2	yes	maximal, complemented
4	C₁₇×#1942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₇	—	2	1	no	—
7	C₂²×D₈×C₁₇	C₂²	1	2	yes	—
8	C₂×C₁₇×#127	—	2	1	no	—
9	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
10	C₁₇×#493	—	2	4	no	—
11	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#389	—	2	1	no	—
15	C₁₇×#357	—	2	4	no	—
16	C₁₇×#357	—	2	4	no	—
17	C₁₇×#340	—	2	1	no	—
18	C₂²×D₄×C₁₇	—	2	2	no	—
19	C₂²×D₄×C₁₇	—	4	1	no	—
20	C₂²×D₄×C₁₇	—	4	1	no	—
21	C₂×D₈×C₁₇	—	2	4	no	—
22	C₂×D₈×C₁₇	—	2	2	no	—
23	C₂×D₈×C₁₇	—	2	2	no	—
24	C₂×D₈×C₁₇	—	2	2	no	—
25	C₂×D₈×C₁₇	—	2	2	no	—
26	C₂×D₈×C₁₇	—	2	2	no	—
27	C₂×D₈×C₁₇	—	4	2	no	—
28	C₂×D₈×C₁₇	—	4	2	no	—
29	C₂×D₈×C₁₇	D₄	1	4	yes	—
30	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
32	C₁₇×#127	—	2	4	no	—
33	C₁₇×#127	—	4	2	no	—
34	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
37	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₁₇	—	2	2	no	—
39	C₂×C₁₆×C₁₇	—	2	2	no	—
40	C₂×C₁₆×C₁₇	—	2	2	no	—
41	C₁₇×#107	—	2	1	no	—
42	C₁₇×#107	—	2	2	no	—
43	C₁₇×#107	D₄	1	2	yes	complemented
44	C₄×C₈×C₁₇	—	2	2	no	—
45	C₁₇×#95	—	4	1	no	—
46	C₂⁴×C₁₇	—	4	2	no	—
47	C₂×D₄×C₁₇	—	2	4	no	—
48	C₂×D₄×C₁₇	—	4	4	no	—
49	C₂×D₄×C₁₇	—	4	4	no	—
50	C₂×D₄×C₁₇	—	4	2	no	—
51	C₂×D₄×C₁₇	—	4	2	no	—
52	C₂×D₄×C₁₇	—	4	2	no	—
53	C₂×D₄×C₁₇	—	4	2	no	—
54	C₂×D₄×C₁₇	—	4	2	no	—
55	C₂×D₄×C₁₇	—	8	2	no	—
56	C₂×D₄×C₁₇	—	8	2	no	—
57	C₂×D₄×C₁₇	—	8	2	no	—
58	C₂×D₄×C₁₇	—	8	2	no	—
59	C₂²×C₄×C₁₇	—	4	1	no	—
60	C₂²×C₄×C₁₇	—	4	1	no	—
61	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
64	D₈×C₁₇	—	2	4	no	—
65	D₈×C₁₇	—	2	4	no	—
66	D₈×C₁₇	—	4	8	no	—
67	D₈×C₁₇	—	4	2	no	—
68	D₈×C₁₇	—	4	2	no	—
69	C₂×C₈×C₁₇	—	2	1	no	—
70	C₂×C₈×C₁₇	—	2	1	no	—
71	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
72	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₇	#41	1	2	yes	—
75	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	—	2	2	no	—
77	C₄⋊C₄×C₁₇	—	4	1	no	—
78	C₄⋊C₄×C₁₇	—	4	1	no	—
79	C₄⋊C₄×C₁₇	—	4	1	no	—
80	C₄⋊C₄×C₁₇	—	8	2	no	—
81	C₄²×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	—	2	1	no	—
83	C₁₆×C₁₇	—	4	2	no	—
84	#13407	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₇	—	4	4	no	—
86	C₂³×C₁₇	—	8	2	no	—
87	C₂³×C₁₇	—	8	2	no	—
88	C₂³×C₁₇	—	8	2	no	—
89	C₂³×C₁₇	—	8	2	no	—
90	C₂³×C₁₇	—	8	2	no	—
91	C₂³×C₁₇	—	8	2	no	—
92	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
93	D₄×C₁₇	—	4	4	no	—
94	D₄×C₁₇	—	4	4	no	—
95	D₄×C₁₇	—	8	8	no	—
96	D₄×C₁₇	—	8	8	no	—
97	D₄×C₁₇	—	8	2	no	—
98	D₄×C₁₇	—	8	2	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	—	8	1	no	—
105	C₂×C₄×C₁₇	—	8	1	no	—
106	C₂×C₄×C₁₇	—	8	1	no	—
107	C₂×C₄×C₁₇	—	8	1	no	—
108	C₂×C₄×C₁₇	—	8	1	no	—
109	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
110	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
111	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
112	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₈×C₁₇	—	2	2	no	—
118	C₈×C₁₇	#121	1	2	yes	—
119	C₈×C₁₇	#123	1	2	yes	—
120	C₂×#493	C₂×C₁₇	1	1	yes	characteristic
121	C₂×#357	C₂×C₁₇	1	2	yes	complemented
122	C₂×#357	C₂×C₁₇	1	2	yes	complemented
123	#1942	C₂×C₁₇	1	1	yes	characteristic, complemented
124	#1374	C₂×C₁₇	1	1	yes	characteristic, complemented
125	C₂²×C₁₇	—	8	4	no	—
126	C₂²×C₁₇	—	8	4	no	—
127	C₂²×C₁₇	—	8	4	no	—
128	C₂²×C₁₇	—	16	2	no	—
129	C₂²×C₁₇	—	16	2	no	—
130	C₂²×C₁₇	—	16	2	no	—
131	C₂²×C₁₇	—	16	2	no	—
132	C₂²×C₁₇	#447	1	1	yes	characteristic
133	C₂²×C₁₇	#459	1	1	yes	characteristic
134	C₂²×C₁₇	#486	1	1	yes	characteristic
135	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
136	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
137	C₂²×C₁₇	#509	1	1	yes	characteristic
138	C₂²×C₁₇	#509	1	1	yes	characteristic
139	C₄×C₁₇	—	2	2	no	—
140	C₄×C₁₇	—	2	2	no	—
141	C₄×C₁₇	—	8	2	no	—
142	C₄×C₁₇	—	16	1	no	—
143	C₄×C₁₇	#394	1	1	yes	characteristic
144	C₄×C₁₇	#450	1	1	yes	characteristic
145	C₄×C₁₇	#460	1	1	yes	characteristic
146	C₄×C₁₇	#490	1	1	yes	characteristic
147	C₂²×D₈	—	2	1	no	—
148	C₂²×D₈	C₂²×C₁₇	1	2	yes	—
149	C₂×#127	—	2	1	no	—
150	C₂²×C₁₆	C₂²×C₁₇	1	2	yes	—
151	#493	—	2	4	no	—
152	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
153	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
155	#389	—	2	1	no	—
156	#357	—	2	4	no	—
157	#357	—	2	4	no	—
158	#340	—	2	1	no	—
159	C₂×C₁₇	—	16	4	no	—
160	C₂×C₁₇	#2041	1	1	yes	characteristic
161	C₂×C₁₇	#2184	1	1	yes	characteristic
162	C₂×C₁₇	#2187	1	1	yes	characteristic
163	C₂×C₁₇	#2206	1	1	yes	characteristic
164	C₂×C₁₇	#2207	1	1	yes	characteristic
165	C₂×C₁₇	#2240	1	1	yes	characteristic
166	C₂×C₁₇	#2244	1	1	yes	characteristic
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	4	1	no	—
169	C₂²×D₄	—	4	1	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	—	2	2	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	—	4	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂×D₈	D₄×C₁₇	1	4	yes	—
179	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
181	#127	—	2	4	no	—
182	#127	—	4	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	4	1	no	—
186	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	2	2	no	—
190	#107	—	2	1	no	—
191	#107	—	2	2	no	—
192	#107	D₄×C₁₇	1	2	yes	complemented
193	C₄×C₈	—	2	2	no	—
194	#95	—	4	1	no	—
195	C₁₇	#13407	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	—	4	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	8	2	no	—
206	C₂×D₄	—	8	2	no	—
207	C₂×D₄	—	8	2	no	—
208	C₂×D₄	—	8	2	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
212	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
214	D₈	—	2	4	no	—
215	D₈	—	2	4	no	—
216	D₈	—	4	8	no	—
217	D₈	—	4	2	no	—
218	D₈	—	4	2	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	SD₁₆×C₁₇	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
224	C₂×C₈	#41×C₁₇	1	1	yes	characteristic
225	C₂×C₈	#41×C₁₇	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	8	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₂×D₈×C₁₇	1	1	yes	characteristic
242	D₄	—	4	4	no	—
243	D₄	—	4	4	no	—
244	D₄	—	8	8	no	—
245	D₄	—	8	8	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	C₁₇×D₁₆	1	2	yes	—
259	C₂×C₄	C₁₇×SD₃₂	1	2	yes	—
260	C₂×C₄	C₁₇×#120	1	1	yes	characteristic
261	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
262	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
263	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
264	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	C₁₇×#121	1	2	yes	—
268	C₈	C₁₇×#123	1	2	yes	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	8	4	no	—
272	C₂²	—	16	2	no	—
273	C₂²	—	16	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₁₇×#447	1	1	yes	characteristic
277	C₂²	C₁₇×#459	1	1	yes	characteristic
278	C₂²	C₁₇×#486	1	1	yes	characteristic
279	C₂²	C₂×C₁₇×D₁₆	1	1	yes	characteristic
280	C₂²	C₂×C₁₇×SD₃₂	1	1	yes	characteristic
281	C₂²	C₁₇×#509	1	1	yes	characteristic
282	C₂²	C₁₇×#509	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	8	2	no	—
286	C₄	—	16	1	no	—
287	C₄	C₁₇×#394	1	1	yes	characteristic
288	C₄	C₁₇×#450	1	1	yes	characteristic
289	C₄	C₁₇×#460	1	1	yes	characteristic
290	C₄	C₁₇×#490	1	1	yes	characteristic
291	C₂	—	16	4	no	—
292	C₂	C₁₇×#2041	1	1	yes	characteristic
293	C₂	C₁₇×#2184	1	1	yes	characteristic
294	C₂	C₁₇×#2187	1	1	yes	characteristic
295	C₂	C₁₇×#2206	1	1	yes	characteristic
296	C₂	C₁₇×#2207	1	1	yes	characteristic
297	C₂	C₁₇×#2240	1	1	yes	characteristic
298	C₂	C₁₇×#2244	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
prime radical	C₂×C₁₇×#493	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ C₂×C₄×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₇×#357	C₂	semidirect
3. C₂×C₁₇×#357	C₂	semidirect
4. C₁₇×#1942	C₂	semidirect
5. C₁₇×#1374	C₂	semidirect
43. C₁₇×#107	D₄	semidirect
84. #13407	C₁₇	direct (N × H)
121. C₂×#357	C₂×C₁₇	semidirect
122. C₂×#357	C₂×C₁₇	semidirect
123. #1942	C₂×C₁₇	semidirect
124. #1374	C₂×C₁₇	semidirect
192. #107	D₄×C₁₇	semidirect
195. C₁₇	#13407	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×#493	2
C₂×C₁₇×#357	2
C₁₇×#1942	2
C₁₇×#1374	2
#13407	17

Subgroups — C17×#13407 #16871151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13407 #16871151