Subgroups — C₁₇×#10172 #16867916

Order 4352.

562 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#3227	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#3152	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#1492	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×#137	—	2	2	no	—
9	C₁₇×#532	C₂²	1	1	yes	characteristic
10	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₁₇×#498	—	2	2	no	—
13	C₁₇×#498	—	2	1	no	—
14	C₁₇×#498	—	2	2	no	—
15	C₁₇×#497	—	2	2	no	—
16	C₁₇×#494	—	2	1	no	—
17	C₁₇×#492	—	2	1	no	—
18	C₁₇×#486	—	2	1	no	—
19	C₁₇×#481	—	2	2	no	—
20	C₁₇×#476	—	2	2	no	—
21	C₁₇×#431	C₂²	1	1	yes	characteristic
22	C₂×C₁₇×#106	—	2	2	no	—
23	C₁₇×#423	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
26	C₁₇×#343	—	2	2	no	—
27	C₁₇×#332	—	2	2	no	—
28	C₁₇×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₁₇	—	2	1	no	—
30	C₂²×Q₈×C₁₇	—	2	1	no	—
31	C₂²×Q₈×C₁₇	—	2	1	no	—
32	C₁₇×#137	—	4	4	no	—
33	C₁₇×#137	—	4	4	no	—
34	C₂×Q₁₆×C₁₇	—	2	4	no	—
35	C₂×SD₁₆×C₁₇	—	2	4	no	—
36	C₂×M₁₆×C₁₇	—	2	1	no	—
37	C₂×M₁₆×C₁₇	D₄	1	2	yes	—
38	C₂²×C₈×C₁₇	—	2	1	no	—
39	C₁₇×C₄⋊Q₈	—	2	1	no	—
40	C₁₇×C₄⋊Q₈	—	2	1	no	—
41	C₁₇×C₄⋊Q₈	—	2	1	no	—
42	C₁₇×C₄⋊Q₈	—	2	1	no	—
43	C₁₇×C₄⋊Q₈	—	2	1	no	—
44	C₁₇×C₄⋊Q₈	—	2	1	no	—
45	C₁₇×C₄⋊Q₈	—	2	1	no	—
46	C₁₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
48	C₁₇×#127	D₄	1	1	yes	characteristic
49	C₁₇×#124	—	2	1	no	—
50	C₁₇×#121	—	2	1	no	—
51	C₁₇×#121	—	2	1	no	—
52	C₄×D₄×C₁₇	—	2	1	no	—
53	C₄×D₄×C₁₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₇	D₄	1	2	yes	—
58	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
59	C₁₇×#106	—	2	8	no	—
60	C₁₇×#106	—	2	4	no	—
61	C₁₇×C₄⋊C₈	—	2	1	no	—
62	C₁₇×C₄⋊C₈	—	2	1	no	—
63	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
64	C₁₇×#102	—	2	4	no	—
65	C₁₇×C₂²⋊C₈	—	2	1	no	—
66	C₁₇×#97	—	2	1	no	—
67	C₄×C₈×C₁₇	—	2	1	no	—
68	C₄×C₈×C₁₇	—	2	1	no	—
69	C₄×C₈×C₁₇	—	2	1	no	—
70	C₄×C₈×C₁₇	D₄	1	2	yes	—
71	#41×C₁₇	—	4	2	no	—
72	#41×C₁₇	—	4	2	no	—
73	C₂×Q₈×C₁₇	—	2	1	no	—
74	C₂×Q₈×C₁₇	—	2	2	no	—
75	C₂×Q₈×C₁₇	—	4	1	no	—
76	C₂×Q₈×C₁₇	—	4	1	no	—
77	C₂×Q₈×C₁₇	—	4	1	no	—
78	C₂×Q₈×C₁₇	—	4	2	no	—
79	C₂×Q₈×C₁₇	—	4	2	no	—
80	C₂×Q₈×C₁₇	—	8	2	no	—
81	C₂×D₄×C₁₇	—	2	1	no	—
82	C₂×D₄×C₁₇	—	2	1	no	—
83	C₂×D₄×C₁₇	—	2	1	no	—
84	C₂×D₄×C₁₇	—	4	1	no	—
85	C₂²×C₄×C₁₇	—	2	2	no	—
86	C₂²×C₄×C₁₇	—	4	1	no	—
87	C₂²×C₄×C₁₇	—	4	1	no	—
88	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
91	Q₁₆×C₁₇	—	4	8	no	—
92	SD₁₆×C₁₇	—	4	8	no	—
93	M₁₆×C₁₇	—	2	2	no	—
94	M₁₆×C₁₇	—	4	2	no	—
95	M₁₆×C₁₇	SD₁₆	1	4	yes	—
96	C₂×C₈×C₁₇	—	2	4	no	—
97	C₂×C₈×C₁₇	—	2	2	no	—
98	C₂×C₈×C₁₇	—	2	2	no	—
99	C₂×C₈×C₁₇	—	2	2	no	—
100	C₂×C₈×C₁₇	—	4	1	no	—
101	C₄⋊C₄×C₁₇	—	2	2	no	—
102	C₄⋊C₄×C₁₇	—	2	2	no	—
103	C₄⋊C₄×C₁₇	—	2	2	no	—
104	C₄⋊C₄×C₁₇	—	2	1	no	—
105	C₄⋊C₄×C₁₇	—	2	2	no	—
106	C₄⋊C₄×C₁₇	—	2	1	no	—
107	C₄⋊C₄×C₁₇	—	4	1	no	—
108	C₄⋊C₄×C₁₇	—	4	1	no	—
109	C₄⋊C₄×C₁₇	—	4	1	no	—
110	C₄⋊C₄×C₁₇	—	4	1	no	—
111	C₂²⋊C₄×C₁₇	—	4	1	no	—
112	C₂²⋊C₄×C₁₇	—	4	1	no	—
113	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
117	#10172	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₁₇	—	4	1	no	—
119	C₂³×C₁₇	#121	1	1	yes	characteristic
120	Q₈×C₁₇	—	4	4	no	—
121	Q₈×C₁₇	—	4	2	no	—
122	Q₈×C₁₇	—	8	2	no	—
123	Q₈×C₁₇	—	8	2	no	—
124	Q₈×C₁₇	—	8	1	no	—
125	Q₈×C₁₇	—	8	1	no	—
126	D₄×C₁₇	—	4	2	no	—
127	D₄×C₁₇	—	4	2	no	—
128	D₄×C₁₇	—	4	2	no	—
129	D₄×C₁₇	—	8	2	no	—
130	C₂×C₄×C₁₇	—	2	2	no	—
131	C₂×C₄×C₁₇	—	2	2	no	—
132	C₂×C₄×C₁₇	—	4	1	no	—
133	C₂×C₄×C₁₇	—	4	2	no	—
134	C₂×C₄×C₁₇	—	4	2	no	—
135	C₂×C₄×C₁₇	—	8	2	no	—
136	C₂×C₄×C₁₇	—	8	1	no	—
137	C₂×C₄×C₁₇	—	8	1	no	—
138	C₂×C₄×C₁₇	—	8	2	no	—
139	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
145	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
147	C₈×C₁₇	—	2	4	no	—
148	C₈×C₁₇	—	2	4	no	—
149	C₈×C₁₇	—	4	2	no	—
150	C₈×C₁₇	—	8	1	no	—
151	#3227	C₂×C₁₇	1	1	yes	characteristic
152	C₂×#498	C₂×C₁₇	1	1	yes	characteristic, complemented
153	#3152	C₂×C₁₇	1	1	yes	characteristic
154	#1997	C₂×C₁₇	1	1	yes	characteristic, complemented
155	#1546	C₂×C₁₇	1	1	yes	characteristic, complemented
156	#1492	C₂×C₁₇	1	1	yes	characteristic
157	#1278	C₂×C₁₇	1	1	yes	characteristic, complemented
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	—	2	1	no	—
160	C₂²×C₁₇	—	8	1	no	—
161	C₂²×C₁₇	—	8	1	no	—
162	C₂²×C₁₇	—	8	1	no	—
163	C₂²×C₁₇	#392	1	1	yes	characteristic
164	C₂²×C₁₇	#460	1	2	yes	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	8	1	no	—
168	C₄×C₁₇	—	8	2	no	—
169	C₄×C₁₇	—	8	2	no	—
170	C₄×C₁₇	#450	1	2	yes	—
171	C₄×C₁₇	#456	1	2	yes	—
172	C₄×C₁₇	#471	1	2	yes	—
173	C₄×C₁₇	#492	1	1	yes	characteristic
174	C₄×C₁₇	#496	1	1	yes	characteristic
175	C₂×#137	—	2	2	no	—
176	#532	C₂²×C₁₇	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
179	#498	—	2	2	no	—
180	#498	—	2	1	no	—
181	#498	—	2	2	no	—
182	#497	—	2	2	no	—
183	#494	—	2	1	no	—
184	#492	—	2	1	no	—
185	#486	—	2	1	no	—
186	#481	—	2	2	no	—
187	#476	—	2	2	no	—
188	#431	C₂²×C₁₇	1	1	yes	characteristic
189	C₂×#106	—	2	2	no	—
190	#423	C₂²×C₁₇	1	1	yes	characteristic
191	C₄×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
193	#343	—	2	2	no	—
194	#332	—	2	2	no	—
195	D₄⋊C₈	—	2	2	no	—
196	C₂×C₁₇	—	2	2	no	—
197	C₂×C₁₇	—	16	1	no	—
198	C₂×C₁₇	#2005	1	1	yes	characteristic
199	C₂×C₁₇	#2014	1	1	yes	characteristic
200	C₂×C₁₇	#2019	1	1	yes	characteristic
201	C₂×#41	—	2	1	no	—
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×Q₈	—	2	1	no	—
204	#137	—	4	4	no	—
205	#137	—	4	4	no	—
206	C₂×Q₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	D₄×C₁₇	1	2	yes	—
210	C₂²×C₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	D₄×C₁₇	1	2	yes	—
219	C₄⋊Q₈	D₄×C₁₇	1	1	yes	characteristic
220	#127	D₄×C₁₇	1	1	yes	characteristic
221	#124	—	2	1	no	—
222	#121	—	2	1	no	—
223	#121	—	2	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	D₄×C₁₇	1	2	yes	—
230	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
231	#106	—	2	8	no	—
232	#106	—	2	4	no	—
233	C₄⋊C₈	—	2	1	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	D₄×C₁₇	1	2	yes	—
236	#102	—	2	4	no	—
237	C₂²⋊C₈	—	2	1	no	—
238	#97	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	D₄×C₁₇	1	2	yes	—
243	C₁₇	#10172	1	1	yes	characteristic, direct product factor, Sylow
244	#41	—	4	2	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	8	2	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
263	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
264	Q₁₆	—	4	8	no	—
265	SD₁₆	—	4	8	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	4	2	no	—
268	M₁₆	SD₁₆×C₁₇	1	4	yes	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
290	C₂³	—	4	1	no	—
291	C₂³	C₁₇×#121	1	1	yes	characteristic
292	Q₈	—	4	4	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	8	1	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	8	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	C₁₇×#120	1	1	yes	characteristic
312	C₂×C₄	C₁₇×#120	1	1	yes	characteristic
313	C₂×C₄	C₁₇×#120	1	1	yes	characteristic
314	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
315	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
316	C₂×C₄	C₁₇×#127	1	1	yes	characteristic
317	C₂×C₄	C₂×SD₁₆×C₁₇	1	2	yes	—
318	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	8	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	C₁₇×#392	1	1	yes	characteristic
329	C₂²	C₁₇×#460	1	2	yes	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	C₁₇×#450	1	2	yes	—
336	C₄	C₁₇×#456	1	2	yes	—
337	C₄	C₁₇×#471	1	2	yes	—
338	C₄	C₁₇×#492	1	1	yes	characteristic
339	C₄	C₁₇×#496	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	16	1	no	—
342	C₂	C₁₇×#2005	1	1	yes	characteristic
343	C₂	C₁₇×#2014	1	1	yes	characteristic
344	C₂	C₁₇×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₁₇×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₇×#498	C₂	semidirect
4. C₁₇×#1997	C₂	semidirect
5. C₁₇×#1546	C₂	semidirect
7. C₁₇×#1278	C₂	semidirect
117. #10172	C₁₇	direct (N × H)
152. C₂×#498	C₂×C₁₇	semidirect
154. #1997	C₂×C₁₇	semidirect
155. #1546	C₂×C₁₇	semidirect
157. #1278	C₂×C₁₇	semidirect
243. C₁₇	#10172	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×#3227	2
C₂×C₁₇×#498	2
C₁₇×#3152	2
C₁₇×#1997	2
C₁₇×#1546	2
C₁₇×#1492	2
C₁₇×#1278	2
#10172	17

Subgroups — C17×#10172 #16867916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10172 #16867916