Subgroups — C₁₇×#3274 #14526280

Order 2176.

556 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₁₇×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#469	C₂	1	2	yes	maximal, complemented
6	C₁₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#449	C₂	1	2	yes	maximal, complemented
9	C₁₇×#447	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₇	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₁₇	—	2	1	no	—
14	C₂²×D₄×C₁₇	C₂²	1	2	yes	—
15	C₁₇×AGL(1,ℤ/8)	—	2	8	no	—
16	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₇	C₂²	1	2	yes	complemented
18	C₂×D₈×C₁₇	C₂²	1	2	yes	—
19	C₂×D₈×C₁₇	C₂²	1	2	yes	complemented
20	C₂×D₈×C₁₇	C₂²	1	2	yes	complemented
21	C₂×M₁₆×C₁₇	C₂²	1	2	yes	complemented
22	C₁₇×#127	—	2	1	no	—
23	C₁₇×#127	C₂²	1	1	yes	characteristic
24	C₁₇×#127	C₂²	1	1	yes	characteristic, complemented
25	C₁₇×#121	—	2	1	no	—
26	C₁₇×#121	—	2	2	no	—
27	C₁₇×#121	C₂²	1	2	yes	—
28	C₁₇×#121	C₂²	1	2	yes	complemented
29	C₁₇×#120	—	2	2	no	—
30	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
31	C₄×D₄×C₁₇	—	2	1	no	—
32	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
34	C₁₇×#106	C₂²	1	1	yes	characteristic, complemented
35	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₁₇×#103	C₂²	1	1	yes	characteristic, complemented
37	C₁₇×#102	C₂²	1	1	yes	characteristic
38	C₁₇×#102	C₂²	1	2	yes	complemented
39	C₁₇×#102	C₂²	1	2	yes	complemented
40	C₁₇×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
42	C₂⁴×C₁₇	—	2	2	no	—
43	#41×C₁₇	—	2	4	no	—
44	#41×C₁₇	—	4	2	no	—
45	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₇	—	2	4	no	—
47	C₂×D₄×C₁₇	—	2	2	no	—
48	C₂×D₄×C₁₇	—	2	2	no	—
49	C₂×D₄×C₁₇	—	2	2	no	—
50	C₂×D₄×C₁₇	—	2	1	no	—
51	C₂×D₄×C₁₇	—	2	1	no	—
52	C₂×D₄×C₁₇	—	2	2	no	—
53	C₂×D₄×C₁₇	—	2	1	no	—
54	C₂×D₄×C₁₇	—	2	1	no	—
55	C₂×D₄×C₁₇	—	4	2	no	—
56	C₂×D₄×C₁₇	—	4	4	no	—
57	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
58	C₂×D₄×C₁₇	C₂³	1	2	yes	—
59	C₂×D₄×C₁₇	C₂³	1	2	yes	—
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂²×C₄×C₁₇	—	2	2	no	—
63	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
64	SD₁₆×C₁₇	—	2	4	no	—
65	SD₁₆×C₁₇	D₄	1	4	yes	complemented
66	D₈×C₁₇	—	2	4	no	—
67	D₈×C₁₇	—	2	4	no	—
68	D₈×C₁₇	—	4	2	no	—
69	M₁₆×C₁₇	—	2	4	no	—
70	C₂×C₈×C₁₇	C₂³	1	2	yes	—
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₇	—	2	2	no	—
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	2	no	—
80	C₂²⋊C₄×C₁₇	D₄	1	2	yes	—
81	C₄²×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	C₂³	1	1	yes	characteristic
83	C₂³×C₁₇	—	2	1	no	—
84	C₂³×C₁₇	—	2	2	no	—
85	C₂³×C₁₇	—	2	2	no	—
86	C₂³×C₁₇	—	4	4	no	—
87	C₂³×C₁₇	—	4	4	no	—
88	C₂³×C₁₇	—	8	2	no	—
89	C₂³×C₁₇	C₂×D₄	1	2	yes	—
90	Q₈×C₁₇	—	2	1	no	—
91	Q₈×C₁₇	C₂×D₄	1	2	yes	complemented
92	D₄×C₁₇	—	2	2	no	—
93	D₄×C₁₇	—	2	2	no	—
94	D₄×C₁₇	—	2	2	no	—
95	D₄×C₁₇	—	2	1	no	—
96	D₄×C₁₇	—	2	2	no	—
97	D₄×C₁₇	—	2	2	no	—
98	D₄×C₁₇	—	4	4	no	—
99	D₄×C₁₇	—	4	4	no	—
100	D₄×C₁₇	—	4	4	no	—
101	D₄×C₁₇	—	4	2	no	—
102	D₄×C₁₇	—	4	2	no	—
103	D₄×C₁₇	—	8	1	no	—
104	D₄×C₁₇	—	8	1	no	—
105	D₄×C₁₇	—	8	2	no	—
106	D₄×C₁₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	2	no	—
109	C₂×C₄×C₁₇	—	2	2	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	4	4	no	—
113	C₂×C₄×C₁₇	—	4	2	no	—
114	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
118	C₈×C₁₇	—	2	2	no	—
119	C₈×C₁₇	—	2	1	no	—
120	C₈×C₁₇	C₂×D₄	1	2	yes	complemented
121	#3274	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₁₇	—	2	2	no	—
123	C₂²×C₁₇	—	2	2	no	—
124	C₂²×C₁₇	—	2	2	no	—
125	C₂²×C₁₇	—	2	2	no	—
126	C₂²×C₁₇	—	4	1	no	—
127	C₂²×C₁₇	—	4	1	no	—
128	C₂²×C₁₇	—	4	2	no	—
129	C₂²×C₁₇	—	4	2	no	—
130	C₂²×C₁₇	—	4	2	no	—
131	C₂²×C₁₇	—	4	2	no	—
132	C₂²×C₁₇	—	4	2	no	—
133	C₂²×C₁₇	—	4	2	no	—
134	C₂²×C₁₇	—	8	8	no	—
135	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₁₇	—	2	2	no	—
137	C₄×C₁₇	—	2	2	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	4	1	no	—
140	C₄×C₁₇	—	4	1	no	—
141	C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	complemented
142	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×AGL(1,ℤ/8)	C₂×C₁₇	1	2	yes	complemented
145	#550	C₂×C₁₇	1	1	yes	characteristic, complemented
146	D₄²	C₂×C₁₇	1	1	yes	characteristic, complemented
147	#493	C₂×C₁₇	1	1	yes	characteristic, complemented
148	#469	C₂×C₁₇	1	2	yes	complemented
149	#460	C₂×C₁₇	1	1	yes	characteristic, complemented
150	#459	C₂×C₁₇	1	1	yes	characteristic, complemented
151	#449	C₂×C₁₇	1	2	yes	complemented
152	#447	C₂×C₁₇	1	2	yes	complemented
153	C₄×SD₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
154	#436	C₂×C₁₇	1	1	yes	characteristic, complemented
155	C₂×C₁₇	—	4	2	no	—
156	C₂×C₁₇	—	4	2	no	—
157	C₂×C₁₇	—	8	2	no	—
158	C₂×C₁₇	—	8	2	no	—
159	C₂×C₁₇	D₄²	1	1	yes	characteristic
160	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₁₇	#576	1	1	yes	characteristic
162	C₂×#41	C₂²×C₁₇	1	2	yes	complemented
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	C₂²×C₁₇	1	2	yes	—
165	AGL(1,ℤ/8)	—	2	8	no	—
166	C₂×SD₁₆	C₂²×C₁₇	1	1	yes	characteristic
167	C₂×SD₁₆	C₂²×C₁₇	1	2	yes	complemented
168	C₂×D₈	C₂²×C₁₇	1	2	yes	—
169	C₂×D₈	C₂²×C₁₇	1	2	yes	complemented
170	C₂×D₈	C₂²×C₁₇	1	2	yes	complemented
171	C₂×M₁₆	C₂²×C₁₇	1	2	yes	complemented
172	#127	—	2	1	no	—
173	#127	C₂²×C₁₇	1	1	yes	characteristic
174	#127	C₂²×C₁₇	1	1	yes	characteristic, complemented
175	#121	—	2	1	no	—
176	#121	—	2	2	no	—
177	#121	C₂²×C₁₇	1	2	yes	—
178	#121	C₂²×C₁₇	1	2	yes	complemented
179	#120	—	2	2	no	—
180	C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
181	C₄×D₄	—	2	1	no	—
182	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
183	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
184	#106	C₂²×C₁₇	1	1	yes	characteristic, complemented
185	C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
186	#103	C₂²×C₁₇	1	1	yes	characteristic, complemented
187	#102	C₂²×C₁₇	1	1	yes	characteristic
188	#102	C₂²×C₁₇	1	2	yes	complemented
189	#102	C₂²×C₁₇	1	2	yes	complemented
190	C₂²⋊C₈	C₂²×C₁₇	1	2	yes	complemented
191	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
192	C₁₇	#3274	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	—	2	2	no	—
194	#41	—	2	4	no	—
195	#41	—	4	2	no	—
196	C₂×Q₈	C₂³×C₁₇	1	1	yes	characteristic
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	D₄×C₁₇	1	1	yes	characteristic
209	C₂×D₄	C₂³×C₁₇	1	2	yes	—
210	C₂×D₄	C₂³×C₁₇	1	2	yes	—
211	C₂×D₄	C₂³×C₁₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂³×C₁₇	1	2	yes	—
215	SD₁₆	—	2	4	no	—
216	SD₁₆	D₄×C₁₇	1	4	yes	complemented
217	D₈	—	2	4	no	—
218	D₈	—	2	4	no	—
219	D₈	—	4	2	no	—
220	M₁₆	—	2	4	no	—
221	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
222	C₂×C₈	C₂³×C₁₇	1	2	yes	—
223	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
226	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
227	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	D₄×C₁₇	1	2	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	C₂³×C₁₇	1	1	yes	characteristic
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	C₂×D₄×C₁₇	1	2	yes	—
241	Q₈	—	2	1	no	—
242	Q₈	C₂×D₄×C₁₇	1	2	yes	complemented
243	D₄	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	1	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	8	1	no	—
255	D₄	—	8	1	no	—
256	D₄	—	8	2	no	—
257	D₄	C₂×D₄×C₁₇	1	2	yes	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
267	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
268	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	2	1	no	—
271	C₈	C₂×D₄×C₁₇	1	2	yes	complemented
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	8	8	no	—
285	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	complemented
292	C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
293	C₄	C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
294	C₂	—	4	2	no	—
295	C₂	—	4	2	no	—
296	C₂	—	8	2	no	—
297	C₂	—	8	2	no	—
298	C₂	D₄²×C₁₇	1	1	yes	characteristic
299	C₂	C₂×C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
300	C₂	C₁₇×#576	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×AGL(1,ℤ/8)	C₂	semidirect
2. C₁₇×#550	C₂	semidirect
3. D₄²×C₁₇	C₂	semidirect
4. C₁₇×#493	C₂	semidirect
5. C₁₇×#469	C₂	semidirect
6. C₁₇×#460	C₂	semidirect
7. C₁₇×#459	C₂	semidirect
8. C₁₇×#449	C₂	semidirect
9. C₁₇×#447	C₂	semidirect
10. C₄×SD₁₆×C₁₇	C₂	semidirect
11. C₁₇×#436	C₂	semidirect
12. C₂×#41×C₁₇	C₂²	semidirect
17. C₂×SD₁₆×C₁₇	C₂²	semidirect
19. C₂×D₈×C₁₇	C₂²	semidirect
20. C₂×D₈×C₁₇	C₂²	semidirect
21. C₂×M₁₆×C₁₇	C₂²	semidirect
24. C₁₇×#127	C₂²	semidirect
28. C₁₇×#121	C₂²	semidirect
30. C₄×Q₈×C₁₇	C₂²	semidirect
34. C₁₇×#106	C₂²	semidirect
35. C₁₇×C₄⋊C₈	C₂²	semidirect
36. C₁₇×#103	C₂²	semidirect
38. C₁₇×#102	C₂²	semidirect
39. C₁₇×#102	C₂²	semidirect
40. C₁₇×C₂²⋊C₈	C₂²	semidirect
41. C₄×C₈×C₁₇	C₂²	semidirect
65. SD₁₆×C₁₇	D₄	semidirect
91. Q₈×C₁₇	C₂×D₄	semidirect
120. C₈×C₁₇	C₂×D₄	semidirect
121. #3274	C₁₇	direct (N × H)
141. C₄×C₁₇	AGL(1,ℤ/8)	semidirect
144. C₂×AGL(1,ℤ/8)	C₂×C₁₇	semidirect
145. #550	C₂×C₁₇	semidirect
146. D₄²	C₂×C₁₇	semidirect
147. #493	C₂×C₁₇	semidirect
148. #469	C₂×C₁₇	semidirect
149. #460	C₂×C₁₇	semidirect
150. #459	C₂×C₁₇	semidirect
151. #449	C₂×C₁₇	semidirect
152. #447	C₂×C₁₇	semidirect
153. C₄×SD₁₆	C₂×C₁₇	semidirect
154. #436	C₂×C₁₇	semidirect
162. C₂×#41	C₂²×C₁₇	semidirect
167. C₂×SD₁₆	C₂²×C₁₇	semidirect
169. C₂×D₈	C₂²×C₁₇	semidirect
170. C₂×D₈	C₂²×C₁₇	semidirect
171. C₂×M₁₆	C₂²×C₁₇	semidirect
174. #127	C₂²×C₁₇	semidirect
178. #121	C₂²×C₁₇	semidirect
180. C₄×Q₈	C₂²×C₁₇	semidirect
184. #106	C₂²×C₁₇	semidirect
185. C₄⋊C₈	C₂²×C₁₇	semidirect
186. #103	C₂²×C₁₇	semidirect
188. #102	C₂²×C₁₇	semidirect
189. #102	C₂²×C₁₇	semidirect
190. C₂²⋊C₈	C₂²×C₁₇	semidirect
191. C₄×C₈	C₂²×C₁₇	semidirect
192. C₁₇	#3274	direct (N × H)
216. SD₁₆	D₄×C₁₇	semidirect
242. Q₈	C₂×D₄×C₁₇	semidirect
271. C₈	C₂×D₄×C₁₇	semidirect
291. C₄	C₁₇×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₇×AGL(1,ℤ/8)	2
C₁₇×#550	2
D₄²×C₁₇	2
C₁₇×#493	2
C₁₇×#469	2
C₁₇×#460	2
C₁₇×#459	2
C₁₇×#449	2
C₁₇×#447	2
C₄×SD₁₆×C₁₇	2
C₁₇×#436	2
#3274	17

Subgroups — C17×#3274 #14526280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3274 #14526280