Subgroups — C₁₇×#13073 #16870817

Order 4352.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₇×#472	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₇×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1816	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂²×D₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
11	C₁₇×#504	—	2	2	no	—
12	C₁₇×#504	—	2	2	no	—
13	C₂×C₁₇×M₃₂	—	2	1	no	—
14	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₇	—	2	1	no	—
16	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#472	—	2	2	no	—
18	C₁₇×#472	C₄	1	4	yes	complemented
19	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
20	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
21	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
22	C₁₇×#417	—	2	1	no	—
23	C₂×C₁₇×#100	—	2	1	no	—
24	C₁₇×#361	—	2	2	no	—
25	C₁₇×#357	—	2	1	no	—
26	C₁₇×#357	—	2	1	no	—
27	C₁₇×#343	—	2	1	no	—
28	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₁₇	—	2	1	no	—
30	C₁₇×AGL(1,ℤ/8)	—	2	4	no	—
31	C₁₇×AGL(1,ℤ/8)	—	4	2	no	—
32	C₂×SD₁₆×C₁₇	—	4	1	no	—
33	C₂×D₈×C₁₇	—	2	1	no	—
34	C₂×D₈×C₁₇	—	2	1	no	—
35	C₂×D₈×C₁₇	—	2	1	no	—
36	C₂×D₈×C₁₇	—	2	1	no	—
37	C₂×D₈×C₁₇	—	2	1	no	—
38	C₂×D₈×C₁₇	—	4	1	no	—
39	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
40	C₁₇×#131	—	2	1	no	—
41	C₁₇×#131	—	2	1	no	—
42	C₁₇×#131	C₂×C₄	1	2	yes	—
43	C₁₇×#131	D₄	1	2	yes	—
44	C₂×M₁₆×C₁₇	—	2	1	no	—
45	C₂×M₁₆×C₁₇	—	2	1	no	—
46	C₂×M₁₆×C₁₇	—	2	1	no	—
47	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₇	—	2	1	no	—
49	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
51	C₁₇×M₃₂	—	2	4	no	—
52	C₁₇×M₃₂	—	2	1	no	—
53	C₁₇×M₃₂	D₄	1	2	yes	—
54	C₂×C₁₆×C₁₇	—	2	4	no	—
55	C₂×C₁₆×C₁₇	—	2	1	no	—
56	C₂×C₁₆×C₁₇	—	2	1	no	—
57	C₂×C₁₆×C₁₇	—	2	1	no	—
58	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
60	C₁₇×#108	—	2	1	no	—
61	C₁₇×#108	—	2	1	no	—
62	C₁₇×#108	—	2	1	no	—
63	C₁₇×#108	C₂×C₄	1	2	yes	—
64	C₁₇×#107	—	2	1	no	—
65	C₁₇×#107	—	2	1	no	—
66	C₁₇×#103	—	4	1	no	—
67	C₁₇×#102	—	4	1	no	—
68	C₁₇×#100	—	2	4	no	—
69	C₂⁴×C₁₇	—	4	1	no	—
70	#41×C₁₇	—	2	1	no	—
71	#41×C₁₇	—	2	1	no	—
72	#41×C₁₇	C₂²⋊C₄	1	2	yes	—
73	#41×C₁₇	SD₁₆	1	2	yes	—
74	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
75	C₂×D₄×C₁₇	—	2	1	no	—
76	C₂×D₄×C₁₇	—	2	2	no	—
77	C₂×D₄×C₁₇	—	4	1	no	—
78	C₂×D₄×C₁₇	—	4	1	no	—
79	C₂×D₄×C₁₇	—	4	1	no	—
80	C₂×D₄×C₁₇	—	4	1	no	—
81	C₂×D₄×C₁₇	—	4	1	no	—
82	C₂×D₄×C₁₇	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₇	—	2	1	no	—
84	C₂²×C₄×C₁₇	—	4	1	no	—
85	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
86	SD₁₆×C₁₇	—	4	4	no	—
87	D₈×C₁₇	—	2	2	no	—
88	D₈×C₁₇	—	2	2	no	—
89	D₈×C₁₇	—	4	4	no	—
90	D₈×C₁₇	—	4	1	no	—
91	D₈×C₁₇	—	4	1	no	—
92	M₁₆×C₁₇	—	2	1	no	—
93	M₁₆×C₁₇	—	2	2	no	—
94	M₁₆×C₁₇	—	2	2	no	—
95	M₁₆×C₁₇	—	2	2	no	—
96	M₁₆×C₁₇	—	4	1	no	—
97	M₁₆×C₁₇	—	4	1	no	—
98	M₁₆×C₁₇	—	4	1	no	—
99	M₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
100	C₂×C₈×C₁₇	—	2	1	no	—
101	C₂×C₈×C₁₇	—	2	2	no	—
102	C₂×C₈×C₁₇	—	2	2	no	—
103	C₂×C₈×C₁₇	—	4	1	no	—
104	C₂×C₈×C₁₇	—	4	1	no	—
105	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
111	C₄⋊C₄×C₁₇	—	4	1	no	—
112	C₄⋊C₄×C₁₇	—	4	1	no	—
113	C₁₆×C₁₇	—	2	1	no	—
114	C₁₆×C₁₇	—	2	2	no	—
115	C₁₆×C₁₇	—	2	1	no	—
116	#13073	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₁₇	—	2	1	no	—
118	C₂³×C₁₇	—	4	2	no	—
119	C₂³×C₁₇	—	8	1	no	—
120	C₂³×C₁₇	—	8	1	no	—
121	C₂³×C₁₇	—	16	1	no	—
122	C₂³×C₁₇	—	16	1	no	—
123	C₂³×C₁₇	#135	1	1	yes	characteristic
124	Q₈×C₁₇	—	2	1	no	—
125	Q₈×C₁₇	#102	1	2	yes	—
126	D₄×C₁₇	—	2	2	no	—
127	D₄×C₁₇	—	2	2	no	—
128	D₄×C₁₇	—	2	1	no	—
129	D₄×C₁₇	—	4	2	no	—
130	D₄×C₁₇	—	4	2	no	—
131	D₄×C₁₇	—	8	1	no	—
132	D₄×C₁₇	—	8	1	no	—
133	D₄×C₁₇	#102	1	2	yes	—
134	C₂×C₄×C₁₇	—	2	1	no	—
135	C₂×C₄×C₁₇	—	2	2	no	—
136	C₂×C₄×C₁₇	—	2	2	no	—
137	C₂×C₄×C₁₇	—	8	1	no	—
138	C₂×C₄×C₁₇	—	16	1	no	—
139	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
145	C₈×C₁₇	—	2	2	no	—
146	C₈×C₁₇	—	4	2	no	—
147	C₈×C₁₇	—	8	1	no	—
148	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
149	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
150	C₈×C₁₇	#120	1	1	yes	characteristic
151	C₈×C₁₇	#123	1	1	yes	characteristic
152	C₂×#504	C₂×C₁₇	1	1	yes	characteristic, complemented
153	C₂×#472	C₂×C₁₇	1	1	yes	characteristic
154	C₂×#361	C₂×C₁₇	1	1	yes	characteristic, complemented
155	C₂×#357	C₂×C₁₇	1	1	yes	characteristic, complemented
156	#1816	C₂×C₁₇	1	1	yes	characteristic, complemented
157	#1387	C₂×C₁₇	1	1	yes	characteristic, complemented
158	#1381	C₂×C₁₇	1	1	yes	characteristic, complemented
159	C₂²×C₁₇	—	2	1	no	—
160	C₂²×C₁₇	—	2	1	no	—
161	C₂²×C₁₇	—	2	2	no	—
162	C₂²×C₁₇	—	4	1	no	—
163	C₂²×C₁₇	—	4	1	no	—
164	C₂²×C₁₇	—	8	2	no	—
165	C₂²×C₁₇	—	16	2	no	—
166	C₂²×C₁₇	—	16	2	no	—
167	C₂²×C₁₇	—	16	1	no	—
168	C₂²×C₁₇	—	16	1	no	—
169	C₂²×C₁₇	#416	1	1	yes	characteristic
170	C₂²×C₁₇	#465	1	1	yes	characteristic
171	C₂²×C₁₇	#466	1	1	yes	characteristic
172	C₄×C₁₇	—	2	2	no	—
173	C₄×C₁₇	—	16	1	no	—
174	C₄×C₁₇	#386	1	1	yes	characteristic
175	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
176	C₄×C₁₇	#465	1	1	yes	characteristic
177	C₄×C₁₇	#466	1	1	yes	characteristic
178	C₂×AGL(1,ℤ/8)	—	2	1	no	—
179	C₂²×D₈	C₂²×C₁₇	1	1	yes	characteristic
180	C₂×#131	C₂²×C₁₇	1	1	yes	characteristic
181	#504	—	2	2	no	—
182	#504	—	2	2	no	—
183	C₂×M₃₂	—	2	1	no	—
184	C₂×M₃₂	C₂²×C₁₇	1	1	yes	characteristic
185	C₂²×C₁₆	—	2	1	no	—
186	C₂²×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
187	#472	—	2	2	no	—
188	#472	C₄×C₁₇	1	4	yes	complemented
189	C₂×#108	C₂²×C₁₇	1	1	yes	characteristic
190	C₂×#108	C₂²×C₁₇	1	1	yes	characteristic
191	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
192	#417	—	2	1	no	—
193	C₂×#100	—	2	1	no	—
194	#361	—	2	2	no	—
195	#357	—	2	1	no	—
196	#357	—	2	1	no	—
197	#343	—	2	1	no	—
198	C₂×C₁₇	—	2	1	no	—
199	C₂×C₁₇	—	2	1	no	—
200	C₂×C₁₇	—	4	2	no	—
201	C₂×C₁₇	—	16	2	no	—
202	C₂×C₁₇	#1893	1	1	yes	characteristic
203	C₂×C₁₇	#2221	1	1	yes	characteristic
204	C₂×C₁₇	#2223	1	1	yes	characteristic
205	C₂×#41	D₄×C₁₇	1	1	yes	characteristic
206	C₂²×D₄	—	2	1	no	—
207	AGL(1,ℤ/8)	—	2	4	no	—
208	AGL(1,ℤ/8)	—	4	2	no	—
209	C₂×SD₁₆	—	4	1	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	—	4	1	no	—
216	C₂×D₈	C₂×C₄×C₁₇	1	2	yes	—
217	#131	—	2	1	no	—
218	#131	—	2	1	no	—
219	#131	C₂×C₄×C₁₇	1	2	yes	—
220	#131	D₄×C₁₇	1	2	yes	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	—	2	1	no	—
224	C₂×M₁₆	D₄×C₁₇	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	2	1	no	—
228	M₃₂	—	2	4	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	D₄×C₁₇	1	2	yes	—
231	C₂×C₁₆	—	2	4	no	—
232	C₂×C₁₆	—	2	1	no	—
233	C₂×C₁₆	—	2	1	no	—
234	C₂×C₁₆	—	2	1	no	—
235	C₂×C₁₆	D₄×C₁₇	1	1	yes	characteristic
236	C₂×C₁₆	D₄×C₁₇	1	1	yes	characteristic
237	#108	—	2	1	no	—
238	#108	—	2	1	no	—
239	#108	—	2	1	no	—
240	#108	C₂×C₄×C₁₇	1	2	yes	—
241	#107	—	2	1	no	—
242	#107	—	2	1	no	—
243	#103	—	4	1	no	—
244	#102	—	4	1	no	—
245	#100	—	2	4	no	—
246	C₁₇	#13073	1	1	yes	characteristic, direct product factor, Sylow
247	C₂⁴	—	4	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	C₂²⋊C₄×C₁₇	1	2	yes	—
251	#41	SD₁₆×C₁₇	1	2	yes	—
252	C₂×Q₈	D₈×C₁₇	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	D₈×C₁₇	1	1	yes	characteristic
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
264	SD₁₆	—	4	4	no	—
265	D₈	—	2	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	4	4	no	—
268	D₈	—	4	1	no	—
269	D₈	—	4	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	C₂²⋊C₄×C₁₇	1	2	yes	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
287	C₂×C₈	#41×C₁₇	1	1	yes	characteristic
288	C₂×C₈	#41×C₁₇	1	1	yes	characteristic
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	1	no	—
297	C₂³	—	8	1	no	—
298	C₂³	—	16	1	no	—
299	C₂³	—	16	1	no	—
300	C₂³	C₁₇×#135	1	1	yes	characteristic
301	Q₈	—	2	1	no	—
302	Q₈	C₁₇×#102	1	2	yes	—
303	D₄	—	2	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	1	no	—
309	D₄	—	8	1	no	—
310	D₄	C₁₇×#102	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	16	1	no	—
316	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
317	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
318	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
319	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
320	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
321	C₂×C₄	C₁₇×#135	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	4	2	no	—
324	C₈	—	8	1	no	—
325	C₈	C₄×D₄×C₁₇	1	1	yes	characteristic
326	C₈	C₄×D₄×C₁₇	1	1	yes	characteristic
327	C₈	C₁₇×#120	1	1	yes	characteristic
328	C₈	C₁₇×#123	1	1	yes	characteristic
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	16	2	no	—
336	C₂²	—	16	2	no	—
337	C₂²	—	16	1	no	—
338	C₂²	—	16	1	no	—
339	C₂²	C₁₇×#416	1	1	yes	characteristic
340	C₂²	C₁₇×#465	1	1	yes	characteristic
341	C₂²	C₁₇×#466	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	16	1	no	—
344	C₄	C₁₇×#386	1	1	yes	characteristic
345	C₄	C₂×C₁₇×#102	1	1	yes	characteristic
346	C₄	C₁₇×#465	1	1	yes	characteristic
347	C₄	C₁₇×#466	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	4	2	no	—
351	C₂	—	16	2	no	—
352	C₂	C₁₇×#1893	1	1	yes	characteristic
353	C₂	C₁₇×#2221	1	1	yes	characteristic
354	C₂	C₁₇×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	#466
commutator	C₂×C₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
prime radical	C₂×C₁₇×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₂×C₁₇×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×#504	C₂	semidirect
3. C₂×C₁₇×#361	C₂	semidirect
4. C₂×C₁₇×#357	C₂	semidirect
5. C₁₇×#1816	C₂	semidirect
6. C₁₇×#1387	C₂	semidirect
7. C₁₇×#1381	C₂	semidirect
18. C₁₇×#472	C₄	semidirect
116. #13073	C₁₇	direct (N × H)
152. C₂×#504	C₂×C₁₇	semidirect
154. C₂×#361	C₂×C₁₇	semidirect
155. C₂×#357	C₂×C₁₇	semidirect
156. #1816	C₂×C₁₇	semidirect
157. #1387	C₂×C₁₇	semidirect
158. #1381	C₂×C₁₇	semidirect
188. #472	C₄×C₁₇	semidirect
246. C₁₇	#13073	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×#504	2
C₂×C₁₇×#472	2
C₂×C₁₇×#361	2
C₂×C₁₇×#357	2
C₁₇×#1816	2
C₁₇×#1387	2
C₁₇×#1381	2
#13073	17

Subgroups — C17×#13073 #16870817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13073 #16870817