Subgroups — C₁₇×#33928 #16891672

Order 4352.

498 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#2242	C₂	1	2	yes	maximal, complemented
4	C₁₇×#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#2228	C₂	1	2	yes	maximal, complemented
6	C₁₇×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#2210	C₂	1	2	yes	maximal, complemented
8	C₁₇×#2206	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₇×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₇×D₁₆	C₂	1	2	yes	maximal, complemented
12	C₁₇×#559	—	2	1	no	—
13	C₁₇×#554	—	2	1	no	—
14	C₂×C₁₇×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×D₁₆	C₂²	1	2	yes	—
16	C₁₇×#500	C₂²	1	1	yes	characteristic
17	C₁₇×#499	C₂²	1	2	yes	—
18	C₁₇×#495	C₂²	1	2	yes	—
19	C₁₇×#493	C₂²	1	1	yes	characteristic
20	C₁₇×#478	—	2	2	no	—
21	C₁₇×#476	—	2	1	no	—
22	C₁₇×#463	—	2	2	no	—
23	C₁₇×#460	—	2	1	no	—
24	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
26	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₁₇	C₂²	1	2	yes	—
28	C₄×D₈×C₁₇	C₂²	1	2	yes	—
29	C₁₇×#367	C₂²	1	1	yes	characteristic
30	C₁₇×#366	C₂²	1	2	yes	—
31	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₁₇×#358	C₂²	1	2	yes	—
34	C₁₇×#358	C₂²	1	1	yes	characteristic
35	C₁₇×#357	C₂²	1	2	yes	—
36	C₁₇×#357	C₂²	1	2	yes	—
37	C₁₇×#357	C₂²	1	2	yes	—
38	C₁₇×#357	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×SD₁₆×C₁₇	—	2	2	no	—
43	C₂×D₈×C₁₇	—	2	1	no	—
44	C₂×D₈×C₁₇	C₂³	1	2	yes	—
45	C₂×D₈×C₁₇	C₂³	1	1	yes	characteristic
46	C₁₇×C₄⋊Q₈	—	2	1	no	—
47	C₁₇×#127	—	2	1	no	—
48	C₁₇×#125	—	2	2	no	—
49	C₁₇×#124	—	2	2	no	—
50	C₁₇×#124	—	4	1	no	—
51	C₁₇×#123	—	4	2	no	—
52	C₁₇×C₂²⋊Q₈	—	4	1	no	—
53	C₁₇×C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈×C₁₇	—	2	1	no	—
55	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
56	C₄×D₄×C₁₇	—	2	1	no	—
57	C₄×D₄×C₁₇	—	2	2	no	—
58	C₄×D₄×C₁₇	—	2	2	no	—
59	C₄×D₄×C₁₇	—	4	1	no	—
60	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
61	C₁₇×SD₃₂	—	2	2	no	—
62	C₁₇×D₁₆	—	2	4	no	—
63	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
64	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
66	C₁₇×#107	—	2	1	no	—
67	C₁₇×#107	C₂³	1	1	yes	characteristic
68	C₁₇×#107	C₂³	1	2	yes	—
69	C₁₇×#107	C₂³	1	1	yes	characteristic
70	C₁₇×#106	—	2	2	no	—
71	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
73	C₁₇×#103	—	2	1	no	—
74	C₁₇×#102	—	2	2	no	—
75	C₁₇×#102	—	2	2	no	—
76	C₁₇×#102	—	2	1	no	—
77	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₇	C₂³	1	2	yes	—
79	C₂×Q₈×C₁₇	—	2	1	no	—
80	C₂×Q₈×C₁₇	—	4	1	no	—
81	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
82	C₂×D₄×C₁₇	—	2	1	no	—
83	C₂×D₄×C₁₇	—	2	2	no	—
84	C₂×D₄×C₁₇	—	4	1	no	—
85	C₂×D₄×C₁₇	—	4	1	no	—
86	C₂²×C₄×C₁₇	—	4	1	no	—
87	C₂²×C₄×C₁₇	—	4	2	no	—
88	C₂²×C₄×C₁₇	—	4	2	no	—
89	Q₁₆×C₁₇	—	2	1	no	—
90	Q₁₆×C₁₇	—	2	1	no	—
91	SD₁₆×C₁₇	—	8	2	no	—
92	D₈×C₁₇	—	2	2	no	—
93	D₈×C₁₇	—	2	1	no	—
94	D₈×C₁₇	—	2	2	no	—
95	D₈×C₁₇	—	8	1	no	—
96	D₈×C₁₇	#41	1	2	yes	—
97	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₁₇	—	2	1	no	—
101	C₄⋊C₄×C₁₇	—	2	2	no	—
102	C₄⋊C₄×C₁₇	—	2	1	no	—
103	C₄⋊C₄×C₁₇	—	4	2	no	—
104	C₄⋊C₄×C₁₇	—	4	2	no	—
105	C₄⋊C₄×C₁₇	—	4	1	no	—
106	C₄⋊C₄×C₁₇	—	4	1	no	—
107	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
108	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
109	C₂²⋊C₄×C₁₇	—	4	1	no	—
110	C₂²⋊C₄×C₁₇	—	4	2	no	—
111	C₂²⋊C₄×C₁₇	—	4	2	no	—
112	C₂²⋊C₄×C₁₇	—	4	2	no	—
113	C₂²⋊C₄×C₁₇	—	4	2	no	—
114	C₄²×C₁₇	—	4	1	no	—
115	C₄²×C₁₇	C₂×D₄	1	2	yes	—
116	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₁₇	—	2	2	no	—
118	C₁₆×C₁₇	—	2	1	no	—
119	C₁₆×C₁₇	—	2	1	no	—
120	#33928	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₁₇	—	4	1	no	—
122	C₂³×C₁₇	—	4	2	no	—
123	Q₈×C₁₇	—	2	2	no	—
124	Q₈×C₁₇	—	4	1	no	—
125	Q₈×C₁₇	—	4	1	no	—
126	Q₈×C₁₇	—	16	1	no	—
127	D₄×C₁₇	—	2	2	no	—
128	D₄×C₁₇	—	4	2	no	—
129	D₄×C₁₇	—	4	1	no	—
130	D₄×C₁₇	—	4	2	no	—
131	D₄×C₁₇	—	8	2	no	—
132	D₄×C₁₇	—	16	1	no	—
133	C₂×C₄×C₁₇	—	4	1	no	—
134	C₂×C₄×C₁₇	—	4	1	no	—
135	C₂×C₄×C₁₇	—	4	1	no	—
136	C₂×C₄×C₁₇	—	4	2	no	—
137	C₂×C₄×C₁₇	—	8	2	no	—
138	C₂×C₄×C₁₇	—	16	2	no	—
139	C₂×C₄×C₁₇	—	16	2	no	—
140	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₁₇	—	2	1	no	—
146	C₈×C₁₇	—	2	2	no	—
147	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
149	#3396	C₂×C₁₇	1	1	yes	characteristic, complemented
150	#3380	C₂×C₁₇	1	1	yes	characteristic, complemented
151	#2242	C₂×C₁₇	1	2	yes	complemented
152	#2240	C₂×C₁₇	1	1	yes	characteristic, complemented
153	#2228	C₂×C₁₇	1	2	yes	complemented
154	#2227	C₂×C₁₇	1	1	yes	characteristic, complemented
155	#2210	C₂×C₁₇	1	2	yes	complemented
156	#2206	C₂×C₁₇	1	1	yes	characteristic
157	Q₈×C₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
158	C₄×SD₃₂	C₂×C₁₇	1	1	yes	characteristic, complemented
159	C₄×D₁₆	C₂×C₁₇	1	2	yes	complemented
160	C₂²×C₁₇	—	4	2	no	—
161	C₂²×C₁₇	—	8	1	no	—
162	C₂²×C₁₇	—	8	1	no	—
163	C₂²×C₁₇	—	8	2	no	—
164	C₂²×C₁₇	—	8	2	no	—
165	C₂²×C₁₇	—	8	2	no	—
166	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
167	C₄×C₁₇	—	2	2	no	—
168	C₄×C₁₇	—	2	2	no	—
169	C₄×C₁₇	—	2	1	no	—
170	C₄×C₁₇	—	8	1	no	—
171	C₄×C₁₇	—	8	1	no	—
172	C₄×C₁₇	—	8	1	no	—
173	C₄×C₁₇	—	8	2	no	—
174	C₄×C₁₇	#508	1	2	yes	—
175	C₄×C₁₇	#548	1	1	yes	characteristic
176	C₄×C₁₇	#577	1	1	yes	characteristic
177	#559	—	2	1	no	—
178	#554	—	2	1	no	—
179	C₂×SD₃₂	C₂²×C₁₇	1	1	yes	characteristic
180	C₂×D₁₆	C₂²×C₁₇	1	2	yes	—
181	#500	C₂²×C₁₇	1	1	yes	characteristic
182	#499	C₂²×C₁₇	1	2	yes	—
183	#495	C₂²×C₁₇	1	2	yes	—
184	#493	C₂²×C₁₇	1	1	yes	characteristic
185	#478	—	2	2	no	—
186	#476	—	2	1	no	—
187	#463	—	2	2	no	—
188	#460	—	2	1	no	—
189	C₈×Q₈	C₂²×C₁₇	1	1	yes	characteristic
190	C₄×Q₁₆	C₂²×C₁₇	1	1	yes	characteristic
191	C₄×D₈	C₂²×C₁₇	1	2	yes	—
192	C₄×D₈	C₂²×C₁₇	1	1	yes	characteristic
193	C₄×D₈	C₂²×C₁₇	1	2	yes	—
194	#367	C₂²×C₁₇	1	1	yes	characteristic
195	#366	C₂²×C₁₇	1	2	yes	—
196	C₄⋊C₁₆	C₂²×C₁₇	1	1	yes	characteristic
197	C₄⋊C₁₆	C₂²×C₁₇	1	2	yes	—
198	#358	C₂²×C₁₇	1	1	yes	characteristic
199	#358	C₂²×C₁₇	1	2	yes	—
200	#357	C₂²×C₁₇	1	1	yes	characteristic
201	#357	C₂²×C₁₇	1	2	yes	—
202	#357	C₂²×C₁₇	1	2	yes	—
203	#357	C₂²×C₁₇	1	2	yes	—
204	C₄×C₁₆	C₂²×C₁₇	1	2	yes	—
205	C₄×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
206	C₂×C₁₇	—	8	2	no	—
207	C₂×C₁₇	—	16	2	no	—
208	C₂×C₁₇	#3358	1	1	yes	characteristic
209	C₂×C₁₇	C₂×#508	1	1	yes	characteristic
210	C₂×C₁₇	#3415	1	1	yes	characteristic
211	C₂×Q₁₆	C₂³×C₁₇	1	1	yes	characteristic
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	C₂³×C₁₇	1	2	yes	—
215	C₂×D₈	C₂³×C₁₇	1	1	yes	characteristic
216	C₄⋊Q₈	—	2	1	no	—
217	#127	—	2	1	no	—
218	#125	—	2	2	no	—
219	#124	—	2	2	no	—
220	#124	—	4	1	no	—
221	#123	—	4	2	no	—
222	C₂²⋊Q₈	—	4	1	no	—
223	C₂²⋊Q₈	—	4	2	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×Q₈	D₄×C₁₇	1	1	yes	characteristic
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	4	1	no	—
230	C₂×C₄⋊C₄	—	4	1	no	—
231	SD₃₂	—	2	2	no	—
232	D₁₆	—	2	4	no	—
233	C₂×C₁₆	C₂³×C₁₇	1	1	yes	characteristic
234	C₂×C₁₆	C₂³×C₁₇	1	1	yes	characteristic
235	C₂×C₁₆	C₂³×C₁₇	1	2	yes	—
236	#107	—	2	1	no	—
237	#107	C₂³×C₁₇	1	2	yes	—
238	#107	C₂³×C₁₇	1	1	yes	characteristic
239	#107	C₂³×C₁₇	1	1	yes	characteristic
240	#106	—	2	2	no	—
241	C₄⋊C₈	D₄×C₁₇	1	1	yes	characteristic
242	C₄⋊C₈	D₄×C₁₇	1	2	yes	—
243	#103	—	2	1	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	—	2	1	no	—
247	C₄×C₈	C₂³×C₁₇	1	2	yes	—
248	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
249	C₁₇	#33928	1	1	yes	characteristic, direct product factor, Sylow
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	1	no	—
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	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	1	no	—
262	SD₁₆	—	8	2	no	—
263	D₈	—	2	2	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	2	no	—
266	D₈	—	8	1	no	—
267	D₈	#41×C₁₇	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
270	C₂×C₈	C₂⁴×C₁₇	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	D₈×C₁₇	1	1	yes	characteristic
279	C₄⋊C₄	D₈×C₁₇	1	2	yes	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₄²	—	4	1	no	—
286	C₄²	C₂×D₄×C₁₇	1	2	yes	—
287	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
288	C₁₆	—	2	2	no	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	1	no	—
296	Q₈	—	16	1	no	—
297	D₄	—	2	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	2	no	—
302	D₄	—	16	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	C₂×D₈×C₁₇	1	2	yes	—
311	C₂×C₄	C₂×D₈×C₁₇	1	2	yes	—
312	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
313	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	2	no	—
317	C₈	C₂×#41×C₁₇	1	1	yes	characteristic
318	C₈	C₁₇×2₋¹⁺⁴	1	1	yes	characteristic
319	C₂²	—	4	2	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	C₂²×D₈×C₁₇	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	2	no	—
333	C₄	C₁₇×#508	1	2	yes	—
334	C₄	C₁₇×#548	1	1	yes	characteristic
335	C₄	C₁₇×#577	1	1	yes	characteristic
336	C₂	—	8	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₁₇×#3358	1	1	yes	characteristic
339	C₂	C₂×C₁₇×#508	1	1	yes	characteristic
340	C₂	C₁₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₇
prime radical	C₁₇×#2206	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₄×Q₈×C₁₇ ◁ C₈×Q₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₇×#3396	C₂	semidirect
2. C₁₇×#3380	C₂	semidirect
3. C₁₇×#2242	C₂	semidirect
4. C₁₇×#2240	C₂	semidirect
5. C₁₇×#2228	C₂	semidirect
6. C₁₇×#2227	C₂	semidirect
7. C₁₇×#2210	C₂	semidirect
9. Q₈×C₁₆×C₁₇	C₂	semidirect
10. C₄×C₁₇×SD₃₂	C₂	semidirect
11. C₄×C₁₇×D₁₆	C₂	semidirect
120. #33928	C₁₇	direct (N × H)
149. #3396	C₂×C₁₇	semidirect
150. #3380	C₂×C₁₇	semidirect
151. #2242	C₂×C₁₇	semidirect
152. #2240	C₂×C₁₇	semidirect
153. #2228	C₂×C₁₇	semidirect
154. #2227	C₂×C₁₇	semidirect
155. #2210	C₂×C₁₇	semidirect
157. Q₈×C₁₆	C₂×C₁₇	semidirect
158. C₄×SD₃₂	C₂×C₁₇	semidirect
159. C₄×D₁₆	C₂×C₁₇	semidirect
249. C₁₇	#33928	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×#3396	2
C₁₇×#3380	2
C₁₇×#2242	2
C₁₇×#2240	2
C₁₇×#2228	2
C₁₇×#2227	2
C₁₇×#2210	2
C₁₇×#2206	2
Q₈×C₁₆×C₁₇	2
C₄×C₁₇×SD₃₂	2
C₄×C₁₇×D₁₆	2
#33928	17

Subgroups — C17×#33928 #16891672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#33928 #16891672