Subgroups — C₁₇×#10449 #16868193

Order 4352.

408 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#474	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1800	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#1478	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×#474	—	2	2	no	—
11	C₁₇×#474	—	2	2	no	—
12	C₁₇×#431	C₂²	1	1	yes	characteristic
13	C₁₇×#431	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×#107	—	2	1	no	—
15	C₁₇×#423	C₂²	1	1	yes	characteristic
16	C₁₇×#423	C₂²	1	1	yes	characteristic
17	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×#102	—	2	1	no	—
19	C₂×C₁₇×#100	—	2	1	no	—
20	C₁₇×#343	—	2	1	no	—
21	C₁₇×#335	—	2	2	no	—
22	C₁₇×#333	—	2	2	no	—
23	C₁₇×#332	—	2	2	no	—
24	C₁₇×D₄⋊C₈	—	2	2	no	—
25	C₂²×Q₈×C₁₇	—	2	1	no	—
26	C₂²×D₄×C₁₇	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₇	—	2	1	no	—
28	C₂×M₁₆×C₁₇	—	2	1	no	—
29	C₂×M₁₆×C₁₇	—	2	1	no	—
30	C₂²×C₈×C₁₇	—	2	1	no	—
31	C₁₇×C₄⋊Q₈	—	2	1	no	—
32	C₁₇×C₄⋊Q₈	—	2	1	no	—
33	C₁₇×C₄⋊Q₈	—	2	1	no	—
34	C₁₇×C₄⋊Q₈	D₄	1	2	yes	—
35	C₄×D₄×C₁₇	—	2	1	no	—
36	C₄×D₄×C₁₇	—	2	1	no	—
37	C₄×D₄×C₁₇	—	2	1	no	—
38	C₄×D₄×C₁₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
43	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
44	C₁₇×#107	—	2	4	no	—
45	C₁₇×C₄⋊C₈	—	2	1	no	—
46	C₁₇×C₄⋊C₈	—	2	1	no	—
47	C₁₇×C₄⋊C₈	—	2	1	no	—
48	C₁₇×C₄⋊C₈	—	2	1	no	—
49	C₁₇×C₄⋊C₈	—	2	1	no	—
50	C₁₇×C₄⋊C₈	—	2	1	no	—
51	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
52	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
53	C₁₇×#102	—	2	4	no	—
54	C₁₇×#100	—	4	2	no	—
55	C₁₇×C₂²⋊C₈	—	2	1	no	—
56	C₁₇×C₂²⋊C₈	—	2	1	no	—
57	C₄×C₈×C₁₇	—	2	1	no	—
58	C₄×C₈×C₁₇	—	2	1	no	—
59	C₂⁴×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	—	2	2	no	—
61	C₂×Q₈×C₁₇	—	4	1	no	—
62	C₂×Q₈×C₁₇	—	8	2	no	—
63	C₂×D₄×C₁₇	—	2	1	no	—
64	C₂×D₄×C₁₇	—	2	1	no	—
65	C₂×D₄×C₁₇	—	2	1	no	—
66	C₂×D₄×C₁₇	—	2	1	no	—
67	C₂×D₄×C₁₇	—	2	1	no	—
68	C₂×D₄×C₁₇	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₁₇	—	2	1	no	—
70	C₂²×C₄×C₁₇	—	4	1	no	—
71	C₂²×C₄×C₁₇	—	4	1	no	—
72	C₂²×C₄×C₁₇	—	4	1	no	—
73	C₂²×C₄×C₁₇	—	8	2	no	—
74	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
77	M₁₆×C₁₇	—	4	1	no	—
78	M₁₆×C₁₇	—	4	1	no	—
79	M₁₆×C₁₇	—	4	1	no	—
80	M₁₆×C₁₇	—	4	1	no	—
81	C₂×C₈×C₁₇	—	2	2	no	—
82	C₂×C₈×C₁₇	—	2	2	no	—
83	C₂×C₈×C₁₇	—	4	1	no	—
84	C₂×C₈×C₁₇	—	4	1	no	—
85	C₂×C₈×C₁₇	—	4	1	no	—
86	C₂×C₈×C₁₇	—	4	1	no	—
87	C₂×C₈×C₁₇	—	4	1	no	—
88	C₄⋊C₄×C₁₇	—	2	2	no	—
89	C₄⋊C₄×C₁₇	—	2	1	no	—
90	C₄⋊C₄×C₁₇	—	2	2	no	—
91	C₄⋊C₄×C₁₇	—	2	1	no	—
92	C₄⋊C₄×C₁₇	—	4	1	no	—
93	C₄⋊C₄×C₁₇	—	4	1	no	—
94	C₂²⋊C₄×C₁₇	—	4	1	no	—
95	C₂²⋊C₄×C₁₇	—	4	1	no	—
96	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₇	#41	1	1	yes	characteristic
100	#10449	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₁₇	—	2	2	no	—
102	C₂³×C₁₇	—	4	1	no	—
103	C₂³×C₁₇	—	4	1	no	—
104	C₂³×C₁₇	—	8	1	no	—
105	C₂³×C₁₇	—	8	1	no	—
106	C₂³×C₁₇	#123	1	1	yes	characteristic
107	Q₈×C₁₇	—	8	2	no	—
108	Q₈×C₁₇	—	8	2	no	—
109	D₄×C₁₇	—	2	2	no	—
110	D₄×C₁₇	—	2	2	no	—
111	D₄×C₁₇	—	4	1	no	—
112	D₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	—	2	2	no	—
114	C₂×C₄×C₁₇	—	4	1	no	—
115	C₂×C₄×C₁₇	—	4	1	no	—
116	C₂×C₄×C₁₇	—	4	2	no	—
117	C₂×C₄×C₁₇	—	8	2	no	—
118	C₂×C₄×C₁₇	—	8	2	no	—
119	C₂×C₄×C₁₇	—	8	2	no	—
120	C₂×C₄×C₁₇	—	8	1	no	—
121	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
122	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
129	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
131	C₈×C₁₇	—	4	2	no	—
132	C₈×C₁₇	—	4	2	no	—
133	C₈×C₁₇	—	8	1	no	—
134	C₈×C₁₇	—	8	1	no	—
135	C₈×C₁₇	—	8	1	no	—
136	C₂×#474	C₂×C₁₇	1	1	yes	characteristic
137	#1800	C₂×C₁₇	1	1	yes	characteristic
138	#1571	C₂×C₁₇	1	1	yes	characteristic, complemented
139	#1549	C₂×C₁₇	1	1	yes	characteristic, complemented
140	#1548	C₂×C₁₇	1	1	yes	characteristic, complemented
141	#1478	C₂×C₁₇	1	1	yes	characteristic
142	#1301	C₂×C₁₇	1	1	yes	characteristic, complemented
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	4	2	no	—
146	C₂²×C₁₇	—	8	2	no	—
147	C₂²×C₁₇	—	8	2	no	—
148	C₂²×C₁₇	—	8	1	no	—
149	C₂²×C₁₇	—	8	1	no	—
150	C₂²×C₁₇	#394	1	1	yes	characteristic
151	C₂²×C₁₇	#484	1	1	yes	characteristic
152	C₂²×C₁₇	#485	1	1	yes	characteristic
153	C₄×C₁₇	—	2	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	8	1	no	—
157	C₄×C₁₇	—	8	2	no	—
158	C₄×C₁₇	#449	1	1	yes	characteristic
159	C₄×C₁₇	#451	1	1	yes	characteristic
160	C₄×C₁₇	#467	1	1	yes	characteristic
161	C₄×C₁₇	#468	1	1	yes	characteristic
162	C₄×C₁₇	#472	1	2	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
164	C₂×C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
165	#474	—	2	2	no	—
166	#474	—	2	2	no	—
167	#431	C₂²×C₁₇	1	1	yes	characteristic
168	#431	C₂²×C₁₇	1	1	yes	characteristic
169	C₂×#107	—	2	1	no	—
170	#423	C₂²×C₁₇	1	1	yes	characteristic
171	#423	C₂²×C₁₇	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
173	C₂×#102	—	2	1	no	—
174	C₂×#100	—	2	1	no	—
175	#343	—	2	1	no	—
176	#335	—	2	2	no	—
177	#333	—	2	2	no	—
178	#332	—	2	2	no	—
179	D₄⋊C₈	—	2	2	no	—
180	C₂×C₁₇	—	2	1	no	—
181	C₂×C₁₇	—	2	1	no	—
182	C₂×C₁₇	—	8	2	no	—
183	C₂×C₁₇	#2036	1	1	yes	characteristic
184	C₂×C₁₇	#2045	1	1	yes	characteristic
185	C₂×C₁₇	#2049	1	1	yes	characteristic
186	C₂²×Q₈	—	2	1	no	—
187	C₂²×D₄	D₄×C₁₇	1	1	yes	characteristic
188	C₂³×C₄	—	2	1	no	—
189	C₂×M₁₆	—	2	1	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	D₄×C₁₇	1	2	yes	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
205	#107	—	2	4	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	D₄×C₁₇	1	2	yes	—
213	C₄⋊C₈	D₄×C₁₇	1	2	yes	—
214	#102	—	2	4	no	—
215	#100	—	4	2	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₁₇	#10449	1	1	yes	characteristic, direct product factor, Sylow
221	C₂⁴	—	2	1	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	—	8	2	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	Q₁₆×C₁₇	1	2	yes	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	8	2	no	—
236	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
237	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
238	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
239	M₁₆	—	4	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
261	C₄²	#41×C₁₇	1	1	yes	characteristic
262	C₂³	—	2	2	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	8	1	no	—
266	C₂³	—	8	1	no	—
267	C₂³	C₁₇×#123	1	1	yes	characteristic
268	Q₈	—	8	2	no	—
269	Q₈	—	8	2	no	—
270	D₄	—	2	2	no	—
271	D₄	—	2	2	no	—
272	D₄	—	4	1	no	—
273	D₄	—	4	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	8	1	no	—
282	C₂×C₄	C₁₇×#120	1	1	yes	characteristic
283	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
284	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
285	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
286	C₂×C₄	C₁₇×#123	1	1	yes	characteristic
287	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
288	C₂×C₄	C₂×Q₁₆×C₁₇	1	1	yes	characteristic
289	C₂×C₄	C₁₇×#135	1	1	yes	characteristic
290	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂×C₄	C₁₇×#137	1	1	yes	characteristic
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	1	no	—
296	C₈	—	8	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	C₁₇×#394	1	1	yes	characteristic
305	C₂²	C₁₇×#484	1	1	yes	characteristic
306	C₂²	C₁₇×#485	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	2	no	—
312	C₄	C₁₇×#449	1	1	yes	characteristic
313	C₄	C₁₇×#451	1	1	yes	characteristic
314	C₄	C₁₇×#467	1	1	yes	characteristic
315	C₄	C₁₇×#468	1	1	yes	characteristic
316	C₄	C₁₇×#472	1	2	yes	—
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	8	2	no	—
320	C₂	C₁₇×#2036	1	1	yes	characteristic
321	C₂	C₁₇×#2045	1	1	yes	characteristic
322	C₂	C₁₇×#2049	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₂²×D₄×C₁₇	D₄
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
3. C₁₇×#1571	C₂	semidirect
4. C₁₇×#1549	C₂	semidirect
5. C₁₇×#1548	C₂	semidirect
7. C₁₇×#1301	C₂	semidirect
100. #10449	C₁₇	direct (N × H)
138. #1571	C₂×C₁₇	semidirect
139. #1549	C₂×C₁₇	semidirect
140. #1548	C₂×C₁₇	semidirect
142. #1301	C₂×C₁₇	semidirect
220. C₁₇	#10449	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×#474	2
C₁₇×#1800	2
C₁₇×#1571	2
C₁₇×#1549	2
C₁₇×#1548	2
C₁₇×#1478	2
C₁₇×#1301	2
#10449	17

Subgroups — C17×#10449 #16868193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#10449 #16868193