Subgroups — C₁₇×#7608 #16865352

Order 4352.

484 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2293	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#1745	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1271	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1270	C₂	1	2	yes	maximal
6	C₂×C₄×D₄×C₁₇	C₄	1	1	yes	characteristic
7	C₂×C₁₇×#117	—	2	1	no	—
8	C₂×C₁₇×#97	—	2	1	no	—
9	C₁₇×#390	C₄	1	1	yes	characteristic
10	C₁₇×#390	C₂²	1	1	yes	characteristic
11	C₁₇×#389	C₄	1	1	yes	characteristic
12	C₁₇×#389	C₂²	1	1	yes	characteristic
13	C₁₇×#388	—	2	1	no	—
14	C₁₇×#388	—	2	1	no	—
15	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₁₇	—	2	1	no	—
17	C₄³×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#336	—	2	1	no	—
19	C₁₇×#336	—	2	1	no	—
20	C₁₇×#336	—	2	1	no	—
21	C₁₇×#336	C₂²	1	2	yes	—
22	C₁₇×#336	C₂²	1	2	yes	—
23	C₂²×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₇	—	2	1	no	—
25	C₂²×C₈×C₁₇	—	2	2	no	—
26	C₂²×C₈×C₁₇	—	2	2	no	—
27	C₄×D₄×C₁₇	—	2	2	no	—
28	C₄×D₄×C₁₇	—	2	2	no	—
29	C₁₇×#117	—	4	4	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
40	C₂×C₄²×C₁₇	—	2	1	no	—
41	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
48	C₁₇×#97	—	2	4	no	—
49	C₁₇×#95	—	2	1	no	—
50	C₁₇×#95	—	2	1	no	—
51	C₁₇×#95	—	2	1	no	—
52	C₂⁴×C₁₇	—	2	1	no	—
53	C₂×D₄×C₁₇	—	2	1	no	—
54	C₂×D₄×C₁₇	—	4	1	no	—
55	C₂×D₄×C₁₇	—	4	1	no	—
56	C₂×D₄×C₁₇	M₁₆	1	2	yes	—
57	C₂²×C₄×C₁₇	—	2	4	no	—
58	C₂²×C₄×C₁₇	—	2	1	no	—
59	C₂²×C₄×C₁₇	—	2	1	no	—
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂²×C₄×C₁₇	—	4	2	no	—
62	C₂²×C₄×C₁₇	—	4	2	no	—
63	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	—	2	4	no	—
71	C₂×C₈×C₁₇	—	4	2	no	—
72	C₂×C₈×C₁₇	—	4	2	no	—
73	C₂×C₈×C₁₇	—	4	2	no	—
74	C₂×C₈×C₁₇	—	4	2	no	—
75	C₂×C₈×C₁₇	—	4	2	no	—
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	4	1	no	—
78	C₄⋊C₄×C₁₇	—	4	1	no	—
79	C₄⋊C₄×C₁₇	—	4	2	no	—
80	C₄⋊C₄×C₁₇	—	4	2	no	—
81	C₄⋊C₄×C₁₇	M₁₆	1	2	yes	—
82	C₂²⋊C₄×C₁₇	—	2	4	no	—
83	C₂²⋊C₄×C₁₇	—	4	2	no	—
84	C₂²⋊C₄×C₁₇	—	4	2	no	—
85	C₂²⋊C₄×C₁₇	—	4	2	no	—
86	C₂²⋊C₄×C₁₇	—	4	2	no	—
87	C₄²×C₁₇	—	2	2	no	—
88	C₄²×C₁₇	—	2	1	no	—
89	C₄²×C₁₇	—	2	1	no	—
90	C₄²×C₁₇	—	2	2	no	—
91	C₄²×C₁₇	—	2	2	no	—
92	C₄²×C₁₇	—	4	1	no	—
93	C₄²×C₁₇	—	4	1	no	—
94	C₄²×C₁₇	—	4	1	no	—
95	C₄²×C₁₇	—	4	2	no	—
96	#7608	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₁₇	—	2	2	no	—
98	C₂³×C₁₇	—	4	1	no	—
99	C₂³×C₁₇	—	4	1	no	—
100	C₂³×C₁₇	—	4	1	no	—
101	C₂³×C₁₇	—	4	1	no	—
102	C₂³×C₁₇	—	4	1	no	—
103	C₂³×C₁₇	—	4	1	no	—
104	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	D₄×C₁₇	—	4	2	no	—
106	D₄×C₁₇	—	4	2	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	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	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	4	8	no	—
126	C₂×C₄×C₁₇	—	4	1	no	—
127	C₂×C₄×C₁₇	—	4	1	no	—
128	C₂×C₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	—	4	1	no	—
130	C₂×C₄×C₁₇	—	4	1	no	—
131	C₂×C₄×C₁₇	—	4	1	no	—
132	C₂×C₄×C₁₇	—	4	1	no	—
133	C₂×C₄×C₁₇	—	4	1	no	—
134	C₂×C₄×C₁₇	#104	1	2	yes	—
135	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₇	#135	1	2	yes	—
138	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
139	C₈×C₁₇	—	4	4	no	—
140	C₈×C₁₇	—	8	2	no	—
141	#2293	C₂×C₁₇	1	1	yes	characteristic
142	#1839	C₂×C₁₇	1	1	yes	characteristic, complemented
143	#1745	C₂×C₁₇	1	1	yes	characteristic, complemented
144	#1271	C₂×C₁₇	1	2	yes	complemented
145	#1270	C₂×C₁₇	1	2	yes	—
146	C₂²×C₁₇	—	4	2	no	—
147	C₂²×C₁₇	—	4	2	no	—
148	C₂²×C₁₇	—	4	2	no	—
149	C₂²×C₁₇	—	8	1	no	—
150	C₂²×C₁₇	—	8	1	no	—
151	C₂²×C₁₇	—	8	1	no	—
152	C₂²×C₁₇	—	8	1	no	—
153	C₂²×C₁₇	#407	1	1	yes	characteristic
154	C₂²×C₁₇	#410	1	1	yes	characteristic
155	C₂²×C₁₇	#413	1	1	yes	characteristic
156	C₂²×C₁₇	#416	1	1	yes	characteristic
157	C₂²×C₁₇	#418	1	1	yes	characteristic
158	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
159	C₂²×C₁₇	#421	1	1	yes	characteristic
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	2	no	—
163	C₄×C₁₇	—	2	2	no	—
164	C₄×C₁₇	—	2	2	no	—
165	C₄×C₁₇	—	4	1	no	—
166	C₄×C₁₇	—	4	1	no	—
167	C₄×C₁₇	—	4	2	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	—	8	1	no	—
170	C₄×C₁₇	—	8	1	no	—
171	C₂×C₄×D₄	C₄×C₁₇	1	1	yes	characteristic
172	C₂×#117	—	2	1	no	—
173	C₂×#97	—	2	1	no	—
174	#390	C₄×C₁₇	1	1	yes	characteristic
175	#390	C₂²×C₁₇	1	1	yes	characteristic
176	#389	C₄×C₁₇	1	1	yes	characteristic
177	#389	C₂²×C₁₇	1	1	yes	characteristic
178	#388	—	2	1	no	—
179	#388	—	2	1	no	—
180	C₄×C₄⋊C₄	C₄×C₁₇	1	1	yes	characteristic
181	C₄×C₂²⋊C₄	—	2	1	no	—
182	C₄³	C₂²×C₁₇	1	1	yes	characteristic
183	#336	—	2	1	no	—
184	#336	—	2	1	no	—
185	#336	—	2	1	no	—
186	#336	C₂²×C₁₇	1	2	yes	—
187	#336	C₂²×C₁₇	1	2	yes	—
188	C₂×C₁₇	—	8	2	no	—
189	C₂×C₁₇	#1464	1	1	yes	characteristic
190	C₂×C₁₇	#1488	1	1	yes	characteristic
191	C₂×C₁₇	#1490	1	1	yes	characteristic
192	C₂×C₁₇	#1500	1	1	yes	characteristic
193	C₂×C₁₇	#1514	1	1	yes	characteristic
194	C₂×C₁₇	#1529	1	1	yes	characteristic
195	C₂×C₁₇	#1553	1	1	yes	characteristic
196	C₂²×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
197	C₂³×C₄	—	2	1	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	#117	—	4	4	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	—	2	1	no	—
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	C₂×C₂²⋊C₄	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
215	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
216	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
217	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
221	#97	—	2	4	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	C₁₇	#7608	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴	—	2	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	M₁₆×C₁₇	1	2	yes	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	M₁₆×C₁₇	1	2	yes	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	4	1	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	—	4	1	no	—
277	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	2	1	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	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₁₇×#104	1	2	yes	—
308	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
309	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
310	C₂×C₄	C₁₇×#135	1	2	yes	—
311	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
312	C₈	—	4	4	no	—
313	C₈	—	8	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₁₇×#407	1	1	yes	characteristic
322	C₂²	C₁₇×#410	1	1	yes	characteristic
323	C₂²	C₁₇×#413	1	1	yes	characteristic
324	C₂²	C₁₇×#416	1	1	yes	characteristic
325	C₂²	C₁₇×#418	1	1	yes	characteristic
326	C₂²	C₂×C₁₇×#104	1	1	yes	characteristic
327	C₂²	C₁₇×#421	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₂	—	8	2	no	—
340	C₂	C₁₇×#1464	1	1	yes	characteristic
341	C₂	C₁₇×#1488	1	1	yes	characteristic
342	C₂	C₁₇×#1490	1	1	yes	characteristic
343	C₂	C₁₇×#1500	1	1	yes	characteristic
344	C₂	C₁₇×#1514	1	1	yes	characteristic
345	C₂	C₁₇×#1529	1	1	yes	characteristic
346	C₂	C₁₇×#1553	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₂²×D₄×C₁₇	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₄³×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₇×#1839	C₂	semidirect
3. C₁₇×#1745	C₂	semidirect
4. C₁₇×#1271	C₂	semidirect
96. #7608	C₁₇	direct (N × H)
142. #1839	C₂×C₁₇	semidirect
143. #1745	C₂×C₁₇	semidirect
144. #1271	C₂×C₁₇	semidirect
225. C₁₇	#7608	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×#2293	2
C₁₇×#1839	2
C₁₇×#1745	2
C₁₇×#1271	2
C₁₇×#1270	2
#7608	17

Subgroups — C17×#7608 #16865352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#7608 #16865352