Subgroups — C₁₇×#13067 #16870811

Order 4352.

536 conjugacy classes of subgroups.

The subgroup lattice has 330 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×#361	C₂	1	2	yes	maximal, complemented
4	C₁₇×#1810	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1387	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	1	no	—
11	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₁₇	—	2	1	no	—
13	C₁₇×#466	—	2	1	no	—
14	C₁₇×#466	—	2	1	no	—
15	C₁₇×#466	C₄	1	2	yes	—
16	C₁₇×#466	C₄	1	2	yes	—
17	C₂×C₁₇×#108	C₂²	1	2	yes	—
18	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×#102	—	2	1	no	—
20	C₁₇×#407	—	2	1	no	—
21	C₁₇×#361	—	2	4	no	—
22	C₁₇×#341	—	2	1	no	—
23	C₂²×D₄×C₁₇	—	2	1	no	—
24	C₂²×D₄×C₁₇	—	4	1	no	—
25	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
26	C₂×D₈×C₁₇	—	2	2	no	—
27	C₂×D₈×C₁₇	—	2	1	no	—
28	C₂×D₈×C₁₇	—	2	2	no	—
29	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₁₇	—	2	2	no	—
31	C₂²×C₈×C₁₇	—	2	2	no	—
32	C₂²×C₈×C₁₇	—	2	2	no	—
33	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	C₁₇×#121	—	2	2	no	—
40	C₁₇×#121	—	2	2	no	—
41	C₁₇×#121	—	4	1	no	—
42	C₁₇×#121	—	4	1	no	—
43	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₁₇	—	4	1	no	—
45	C₁₇×M₃₂	—	2	4	no	—
46	C₁₇×M₃₂	—	2	2	no	—
47	C₁₇×M₃₂	D₄	1	4	yes	complemented
48	C₂×C₁₆×C₁₇	—	2	4	no	—
49	C₂×C₁₆×C₁₇	—	2	2	no	—
50	C₁₇×#108	—	2	2	no	—
51	C₁₇×#108	—	2	2	no	—
52	C₁₇×#107	—	2	1	no	—
53	C₁₇×#107	C₂×C₄	1	2	yes	—
54	C₁₇×#102	—	2	2	no	—
55	C₁₇×#102	—	2	2	no	—
56	C₁₇×C₂²⋊C₈	—	4	1	no	—
57	C₁₇×C₂²⋊C₈	—	4	1	no	—
58	C₂⁴×C₁₇	—	4	1	no	—
59	C₂⁴×C₁₇	SD₁₆	1	1	yes	characteristic
60	C₂×D₄×C₁₇	—	2	2	no	—
61	C₂×D₄×C₁₇	—	4	2	no	—
62	C₂×D₄×C₁₇	—	4	2	no	—
63	C₂×D₄×C₁₇	—	4	2	no	—
64	C₂×D₄×C₁₇	—	4	1	no	—
65	C₂×D₄×C₁₇	—	4	2	no	—
66	C₂×D₄×C₁₇	—	8	4	no	—
67	C₂²×C₄×C₁₇	—	2	2	no	—
68	C₂²×C₄×C₁₇	—	2	2	no	—
69	C₂²×C₄×C₁₇	—	4	1	no	—
70	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
76	D₈×C₁₇	—	2	4	no	—
77	D₈×C₁₇	—	4	1	no	—
78	D₈×C₁₇	—	4	1	no	—
79	M₁₆×C₁₇	—	4	2	no	—
80	M₁₆×C₁₇	—	4	2	no	—
81	C₂×C₈×C₁₇	—	2	2	no	—
82	C₂×C₈×C₁₇	—	2	2	no	—
83	C₂×C₈×C₁₇	—	2	2	no	—
84	C₂×C₈×C₁₇	—	2	1	no	—
85	C₂×C₈×C₁₇	—	2	1	no	—
86	C₂×C₈×C₁₇	—	2	1	no	—
87	C₂×C₈×C₁₇	—	2	1	no	—
88	C₂×C₈×C₁₇	—	4	2	no	—
89	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	#41	1	2	yes	—
95	C₄⋊C₄×C₁₇	—	2	2	no	—
96	C₄⋊C₄×C₁₇	—	4	1	no	—
97	C₂²⋊C₄×C₁₇	—	4	2	no	—
98	C₂²⋊C₄×C₁₇	—	4	2	no	—
99	C₁₆×C₁₇	—	2	4	no	—
100	#13067	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₁₇	—	2	2	no	—
102	C₂³×C₁₇	—	2	2	no	—
103	C₂³×C₁₇	—	4	1	no	—
104	C₂³×C₁₇	—	4	2	no	—
105	C₂³×C₁₇	—	8	2	no	—
106	C₂³×C₁₇	—	8	2	no	—
107	C₂³×C₁₇	—	8	2	no	—
108	C₂³×C₁₇	#102	1	1	yes	characteristic
109	C₂³×C₁₇	#102	1	1	yes	characteristic
110	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
111	D₄×C₁₇	—	4	2	no	—
112	D₄×C₁₇	—	4	2	no	—
113	D₄×C₁₇	—	8	4	no	—
114	D₄×C₁₇	—	8	4	no	—
115	D₄×C₁₇	—	8	2	no	—
116	C₂×C₄×C₁₇	—	2	2	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	—	2	2	no	—
123	C₂×C₄×C₁₇	—	4	2	no	—
124	C₂×C₄×C₁₇	—	8	2	no	—
125	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
126	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	#121	1	2	yes	—
129	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₇	#135	1	2	yes	—
131	C₈×C₁₇	—	2	1	no	—
132	C₈×C₁₇	—	2	1	no	—
133	C₈×C₁₇	—	8	2	no	—
134	C₈×C₁₇	C₄×D₄	1	2	yes	—
135	C₈×C₁₇	#120	1	1	yes	characteristic
136	C₈×C₁₇	#123	1	1	yes	characteristic
137	C₂²×M₃₂	C₂×C₁₇	1	1	yes	characteristic, complemented
138	C₂×#466	C₂×C₁₇	1	1	yes	characteristic
139	C₂×#361	C₂×C₁₇	1	2	yes	complemented
140	#1810	C₂×C₁₇	1	1	yes	characteristic, complemented
141	#1387	C₂×C₁₇	1	2	yes	complemented
142	C₂²×C₁₇	—	2	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	—	2	1	no	—
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	—	4	4	no	—
150	C₂²×C₁₇	—	8	2	no	—
151	C₂²×C₁₇	—	8	2	no	—
152	C₂²×C₁₇	—	8	2	no	—
153	C₂²×C₁₇	—	16	4	no	—
154	C₂²×C₁₇	#361	1	2	yes	complemented
155	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
156	C₂²×C₁₇	#465	1	2	yes	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	8	2	no	—
160	C₄×C₁₇	#386	1	1	yes	characteristic
161	C₄×C₁₇	#416	1	1	yes	characteristic
162	C₄×C₁₇	#466	1	2	yes	—
163	C₂²×D₈	C₂²×C₁₇	1	1	yes	characteristic
164	C₂³×C₈	C₂²×C₁₇	1	1	yes	characteristic
165	C₂×#121	—	2	1	no	—
166	C₂×M₃₂	—	2	1	no	—
167	C₂×M₃₂	—	2	4	no	—
168	C₂×M₃₂	C₂²×C₁₇	1	2	yes	—
169	C₂²×C₁₆	—	2	1	no	—
170	#466	—	2	1	no	—
171	#466	—	2	1	no	—
172	#466	C₄×C₁₇	1	2	yes	—
173	#466	C₄×C₁₇	1	2	yes	—
174	C₂×#108	C₂²×C₁₇	1	2	yes	—
175	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
176	C₂×#102	—	2	1	no	—
177	#407	—	2	1	no	—
178	#361	—	2	4	no	—
179	#341	—	2	1	no	—
180	C₂×C₁₇	—	2	2	no	—
181	C₂×C₁₇	—	2	2	no	—
182	C₂×C₁₇	—	4	1	no	—
183	C₂×C₁₇	—	16	2	no	—
184	C₂×C₁₇	#1893	1	1	yes	characteristic
185	C₂×C₁₇	C₂×#361	1	1	yes	characteristic
186	C₂×C₁₇	#2148	1	1	yes	characteristic
187	C₂²×D₄	—	2	1	no	—
188	C₂²×D₄	—	4	1	no	—
189	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	2	no	—
193	C₂×D₈	C₂×C₄×C₁₇	1	2	yes	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
198	C₂²×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
199	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
200	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
203	#121	—	2	2	no	—
204	#121	—	2	2	no	—
205	#121	—	4	1	no	—
206	#121	—	4	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₂²⋊C₄	—	4	1	no	—
209	M₃₂	—	2	4	no	—
210	M₃₂	—	2	2	no	—
211	M₃₂	D₄×C₁₇	1	4	yes	complemented
212	C₂×C₁₆	—	2	4	no	—
213	C₂×C₁₆	—	2	2	no	—
214	#108	—	2	2	no	—
215	#108	—	2	2	no	—
216	#107	—	2	1	no	—
217	#107	C₂×C₄×C₁₇	1	2	yes	—
218	#102	—	2	2	no	—
219	#102	—	2	2	no	—
220	C₂²⋊C₈	—	4	1	no	—
221	C₂²⋊C₈	—	4	1	no	—
222	C₁₇	#13067	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴	—	4	1	no	—
224	C₂⁴	SD₁₆×C₁₇	1	1	yes	characteristic
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	8	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
237	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
238	C₂²×C₄	D₈×C₁₇	1	1	yes	characteristic
239	C₂²×C₄	SD₁₆×C₁₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
241	D₈	—	2	4	no	—
242	D₈	—	4	1	no	—
243	D₈	—	4	1	no	—
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
259	C₂×C₈	#41×C₁₇	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₁₆	—	2	4	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	4	1	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	8	2	no	—
270	C₂³	—	8	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	C₁₇×#102	1	1	yes	characteristic
273	C₂³	C₁₇×#102	1	1	yes	characteristic
274	C₂³	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	C₁₇×#102	1	1	yes	characteristic
290	C₂×C₄	C₁₇×#102	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
292	C₂×C₄	C₁₇×#121	1	2	yes	—
293	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
294	C₂×C₄	C₁₇×#135	1	2	yes	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	8	2	no	—
298	C₈	C₄×D₄×C₁₇	1	2	yes	—
299	C₈	C₁₇×#120	1	1	yes	characteristic
300	C₈	C₁₇×#123	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	16	4	no	—
313	C₂²	C₁₇×#361	1	2	yes	complemented
314	C₂²	C₂×C₁₇×#102	1	1	yes	characteristic
315	C₂²	C₁₇×#465	1	2	yes	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	C₁₇×#386	1	1	yes	characteristic
320	C₄	C₁₇×#416	1	1	yes	characteristic
321	C₄	C₁₇×#466	1	2	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	2	2	no	—
324	C₂	—	4	1	no	—
325	C₂	—	16	2	no	—
326	C₂	C₁₇×#1893	1	1	yes	characteristic
327	C₂	C₂×C₁₇×#361	1	1	yes	characteristic
328	C₂	C₁₇×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂×#102
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 ◁ C₂²×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×#361	C₂	semidirect
4. C₁₇×#1810	C₂	semidirect
5. C₁₇×#1387	C₂	semidirect
47. C₁₇×M₃₂	D₄	semidirect
100. #13067	C₁₇	direct (N × H)
137. C₂²×M₃₂	C₂×C₁₇	semidirect
139. C₂×#361	C₂×C₁₇	semidirect
140. #1810	C₂×C₁₇	semidirect
141. #1387	C₂×C₁₇	semidirect
154. C₂²×C₁₇	#361	semidirect
211. M₃₂	D₄×C₁₇	semidirect
222. C₁₇	#13067	direct (N × H)
313. C₂²	C₁₇×#361	semidirect

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×#361	2
C₁₇×#1810	2
C₁₇×#1387	2
#13067	17

Subgroups — C17×#13067 #16870811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13067 #16870811