Subgroups — C₁₇×#13061 #16870805

Order 4352.

558 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₇×#466	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₇×#357	C₂	1	2	yes	maximal, complemented
4	C₁₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₇×#121	—	2	1	no	—
9	C₂×C₁₇×M₃₂	—	2	4	no	—
10	C₂×C₁₇×M₃₂	—	2	2	no	—
11	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
12	C₁₇×#466	—	2	2	no	—
13	C₁₇×#466	C₄	1	4	yes	complemented
14	C₂×C₁₇×#107	C₂²	1	2	yes	—
15	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×#102	—	2	1	no	—
17	C₁₇×#380	—	2	1	no	—
18	C₁₇×#357	—	2	4	no	—
19	C₁₇×#340	—	2	1	no	—
20	C₂²×D₄×C₁₇	—	2	1	no	—
21	C₂²×D₄×C₁₇	—	4	1	no	—
22	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
23	C₂×D₈×C₁₇	—	2	2	no	—
24	C₂×D₈×C₁₇	—	2	1	no	—
25	C₂×D₈×C₁₇	—	2	2	no	—
26	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₇	—	2	2	no	—
28	C₂²×C₈×C₁₇	—	2	2	no	—
29	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	D₄	1	2	yes	—
32	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
33	C₁₇×#121	—	2	4	no	—
34	C₁₇×#121	—	4	2	no	—
35	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₁₇	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₁₇	—	4	1	no	—
39	C₁₇×M₃₂	—	2	8	no	—
40	C₂×C₁₆×C₁₇	—	2	2	no	—
41	C₂×C₁₆×C₁₇	—	2	2	no	—
42	C₂×C₁₆×C₁₇	D₄	1	4	yes	complemented
43	C₁₇×#107	—	2	1	no	—
44	C₁₇×#107	—	2	2	no	—
45	C₁₇×#107	—	2	2	no	—
46	C₁₇×#107	C₂×C₄	1	2	yes	—
47	C₁₇×#102	—	2	4	no	—
48	C₁₇×#95	—	4	1	no	—
49	C₂⁴×C₁₇	—	4	1	no	—
50	C₂⁴×C₁₇	D₈	1	1	yes	characteristic
51	C₂×D₄×C₁₇	—	2	2	no	—
52	C₂×D₄×C₁₇	—	4	4	no	—
53	C₂×D₄×C₁₇	—	4	2	no	—
54	C₂×D₄×C₁₇	—	4	1	no	—
55	C₂×D₄×C₁₇	—	4	2	no	—
56	C₂×D₄×C₁₇	—	8	4	no	—
57	C₂²×C₄×C₁₇	—	2	2	no	—
58	C₂²×C₄×C₁₇	—	2	2	no	—
59	C₂²×C₄×C₁₇	—	4	1	no	—
60	C₂²×C₄×C₁₇	—	4	2	no	—
61	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	D₈×C₁₇	—	2	4	no	—
66	D₈×C₁₇	—	4	1	no	—
67	D₈×C₁₇	—	4	1	no	—
68	C₂×C₈×C₁₇	—	2	2	no	—
69	C₂×C₈×C₁₇	—	2	4	no	—
70	C₂×C₈×C₁₇	—	2	2	no	—
71	C₂×C₈×C₁₇	—	2	2	no	—
72	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₇	#41	1	2	yes	—
77	C₄⋊C₄×C₁₇	—	2	2	no	—
78	C₄⋊C₄×C₁₇	—	4	1	no	—
79	C₄⋊C₄×C₁₇	—	4	2	no	—
80	C₄⋊C₄×C₁₇	—	4	2	no	—
81	C₂²⋊C₄×C₁₇	—	4	4	no	—
82	C₂²⋊C₄×C₁₇	—	8	2	no	—
83	C₁₆×C₁₇	—	2	4	no	—
84	#13061	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₇	—	2	2	no	—
86	C₂³×C₁₇	—	2	1	no	—
87	C₂³×C₁₇	—	2	2	no	—
88	C₂³×C₁₇	—	2	1	no	—
89	C₂³×C₁₇	—	4	2	no	—
90	C₂³×C₁₇	—	8	2	no	—
91	C₂³×C₁₇	—	8	2	no	—
92	C₂³×C₁₇	—	8	1	no	—
93	C₂³×C₁₇	—	8	1	no	—
94	C₂³×C₁₇	#102	1	2	yes	—
95	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
96	D₄×C₁₇	—	4	2	no	—
97	D₄×C₁₇	—	4	2	no	—
98	D₄×C₁₇	—	8	8	no	—
99	D₄×C₁₇	—	8	2	no	—
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	2	2	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	4	no	—
104	C₂×C₄×C₁₇	—	4	2	no	—
105	C₂×C₄×C₁₇	—	8	2	no	—
106	C₂×C₄×C₁₇	—	8	2	no	—
107	C₂×C₄×C₁₇	—	8	2	no	—
108	C₂×C₄×C₁₇	—	8	2	no	—
109	C₂×C₄×C₁₇	#102	1	2	yes	—
110	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	#121	1	2	yes	—
112	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₇	#135	1	2	yes	—
114	C₈×C₁₇	—	2	2	no	—
115	C₈×C₁₇	C₄×D₄	1	2	yes	—
116	C₈×C₁₇	#120	1	1	yes	characteristic
117	C₈×C₁₇	#123	1	1	yes	characteristic
118	C₂²×M₃₂	C₂×C₁₇	1	1	yes	characteristic, complemented
119	C₂×#466	C₂×C₁₇	1	1	yes	characteristic
120	C₂×#357	C₂×C₁₇	1	2	yes	complemented
121	#1808	C₂×C₁₇	1	1	yes	characteristic, complemented
122	#1380	C₂×C₁₇	1	2	yes	complemented
123	C₂²×C₁₇	—	2	2	no	—
124	C₂²×C₁₇	—	2	2	no	—
125	C₂²×C₁₇	—	2	2	no	—
126	C₂²×C₁₇	—	4	2	no	—
127	C₂²×C₁₇	—	4	2	no	—
128	C₂²×C₁₇	—	8	2	no	—
129	C₂²×C₁₇	—	8	2	no	—
130	C₂²×C₁₇	—	8	2	no	—
131	C₂²×C₁₇	—	16	2	no	—
132	C₂²×C₁₇	—	16	2	no	—
133	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
134	C₂²×C₁₇	#466	1	2	yes	—
135	C₂²×C₁₇	#509	1	1	yes	characteristic
136	C₂²×C₁₇	#509	1	2	yes	—
137	C₂²×C₁₇	#510	1	1	yes	characteristic
138	C₄×C₁₇	—	2	2	no	—
139	C₄×C₁₇	—	8	2	no	—
140	C₄×C₁₇	—	16	2	no	—
141	C₄×C₁₇	#386	1	1	yes	characteristic
142	C₄×C₁₇	#416	1	1	yes	characteristic
143	C₄×C₁₇	#465	1	2	yes	—
144	C₂²×D₈	C₂²×C₁₇	1	1	yes	characteristic
145	C₂³×C₈	C₂²×C₁₇	1	1	yes	characteristic
146	C₂×#121	—	2	1	no	—
147	C₂×M₃₂	—	2	2	no	—
148	C₂×M₃₂	—	2	4	no	—
149	C₂²×C₁₆	C₂²×C₁₇	1	2	yes	—
150	#466	—	2	2	no	—
151	#466	C₄×C₁₇	1	4	yes	complemented
152	C₂×#107	C₂²×C₁₇	1	2	yes	—
153	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
154	C₂×#102	—	2	1	no	—
155	#380	—	2	1	no	—
156	#357	—	2	4	no	—
157	#340	—	2	1	no	—
158	C₂×C₁₇	—	4	2	no	—
159	C₂×C₁₇	—	16	2	no	—
160	C₂×C₁₇	#1893	1	1	yes	characteristic
161	C₂×C₁₇	#2139	1	1	yes	characteristic
162	C₂×C₁₇	#2140	1	1	yes	characteristic
163	C₂×C₁₇	#2218	1	2	yes	—
164	C₂×C₁₇	#2220	1	2	yes	—
165	C₂²×D₄	—	2	1	no	—
166	C₂²×D₄	—	4	1	no	—
167	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	2	no	—
171	C₂×D₈	C₂×C₄×C₁₇	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	C₂×C₄×C₁₇	1	2	yes	—
175	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₁₇	1	2	yes	—
177	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
178	#121	—	2	4	no	—
179	#121	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	C₂×C₂²⋊C₄	—	4	1	no	—
184	M₃₂	—	2	8	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₂×C₁₆	D₄×C₁₇	1	4	yes	complemented
188	#107	—	2	1	no	—
189	#107	—	2	2	no	—
190	#107	—	2	2	no	—
191	#107	C₂×C₄×C₁₇	1	2	yes	—
192	#102	—	2	4	no	—
193	#95	—	4	1	no	—
194	C₁₇	#13061	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	—	4	1	no	—
196	C₂⁴	D₈×C₁₇	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	8	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
208	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
209	C₂²×C₄	SD₁₆×C₁₇	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
211	D₈	—	2	4	no	—
212	D₈	—	4	1	no	—
213	D₈	—	4	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	C₂²⋊C₄×C₁₇	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
220	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
222	C₂×C₈	#41×C₁₇	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₂²⋊C₄	—	8	2	no	—
229	C₁₆	—	2	4	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	8	2	no	—
236	C₂³	—	8	1	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₁₇×#102	1	2	yes	—
240	C₂³	C₂×D₈×C₁₇	1	1	yes	characteristic
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	8	8	no	—
244	D₄	—	8	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₁₇×#102	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
256	C₂×C₄	C₁₇×#121	1	2	yes	—
257	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
258	C₂×C₄	C₁₇×#135	1	2	yes	—
259	C₈	—	2	2	no	—
260	C₈	C₄×D₄×C₁₇	1	2	yes	—
261	C₈	C₁₇×#120	1	1	yes	characteristic
262	C₈	C₁₇×#123	1	1	yes	characteristic
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	16	2	no	—
272	C₂²	—	16	2	no	—
273	C₂²	C₂×C₁₇×#102	1	1	yes	characteristic
274	C₂²	C₁₇×#466	1	2	yes	—
275	C₂²	C₁₇×#509	1	1	yes	characteristic
276	C₂²	C₁₇×#509	1	2	yes	—
277	C₂²	C₁₇×#510	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	16	2	no	—
281	C₄	C₁₇×#386	1	1	yes	characteristic
282	C₄	C₁₇×#416	1	1	yes	characteristic
283	C₄	C₁₇×#465	1	2	yes	—
284	C₂	—	4	2	no	—
285	C₂	—	16	2	no	—
286	C₂	C₁₇×#1893	1	1	yes	characteristic
287	C₂	C₁₇×#2139	1	1	yes	characteristic
288	C₂	C₁₇×#2140	1	1	yes	characteristic
289	C₂	C₁₇×#2218	1	2	yes	—
290	C₂	C₁₇×#2220	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
prime radical	C₂×C₁₇×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂³×C₄×C₁₇ ◁ C₂³×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₇×M₃₂	C₂	semidirect
3. C₂×C₁₇×#357	C₂	semidirect
4. C₁₇×#1808	C₂	semidirect
5. C₁₇×#1380	C₂	semidirect
13. C₁₇×#466	C₄	semidirect
42. C₂×C₁₆×C₁₇	D₄	semidirect
84. #13061	C₁₇	direct (N × H)
118. C₂²×M₃₂	C₂×C₁₇	semidirect
120. C₂×#357	C₂×C₁₇	semidirect
121. #1808	C₂×C₁₇	semidirect
122. #1380	C₂×C₁₇	semidirect
151. #466	C₄×C₁₇	semidirect
187. C₂×C₁₆	D₄×C₁₇	semidirect
194. C₁₇	#13061	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₇×M₃₂	2
C₂×C₁₇×#466	2
C₂×C₁₇×#357	2
C₁₇×#1808	2
C₁₇×#1380	2
#13061	17

Subgroups — C17×#13061 #16870805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13061 #16870805