Subgroups — C₁₇×#9577 #16867321

Order 4352.

376 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#1962	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1540	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1480	C₂	1	1	yes	maximal, characteristic
6	C₁₇×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#532	C₂²	1	1	yes	characteristic
9	C₁₇×#431	C₂²	1	1	yes	characteristic
10	C₁₇×#431	C₂²	1	1	yes	characteristic
11	C₁₇×#423	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×#104	—	2	1	no	—
15	C₂×C₁₇×#104	—	2	1	no	—
16	C₂×C₄×C₈×C₁₇	—	2	1	no	—
17	C₄³×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#336	—	2	1	no	—
19	C₁₇×#335	—	2	2	no	—
20	C₁₇×#332	C₄	1	2	yes	complemented
21	C₁₇×#331	C₄	1	2	yes	complemented
22	C₁₇×D₄⋊C₈	—	2	2	no	—
23	C₂×#41×C₁₇	—	2	1	no	—
24	C₂×M₁₆×C₁₇	—	2	1	no	—
25	C₂²×C₈×C₁₇	—	2	1	no	—
26	C₂²×C₈×C₁₇	—	2	1	no	—
27	C₁₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
28	C₁₇×#127	C₂×C₄	1	1	yes	characteristic
29	C₁₇×#124	—	2	1	no	—
30	C₁₇×#121	—	2	1	no	—
31	C₁₇×#121	—	2	1	no	—
32	C₄×D₄×C₁₇	—	2	1	no	—
33	C₄×D₄×C₁₇	—	2	1	no	—
34	C₂×C₄²×C₁₇	—	2	1	no	—
35	C₂×C₄²×C₁₇	—	2	1	no	—
36	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
39	C₁₇×C₄⋊C₈	—	2	1	no	—
40	C₁₇×C₄⋊C₈	—	2	1	no	—
41	C₁₇×C₄⋊C₈	—	2	1	no	—
42	C₁₇×C₄⋊C₈	—	2	1	no	—
43	C₁₇×C₄⋊C₈	—	2	1	no	—
44	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
47	C₁₇×#104	—	4	2	no	—
48	C₁₇×#104	—	4	2	no	—
49	C₁₇×C₂²⋊C₈	—	2	1	no	—
50	C₁₇×C₂²⋊C₈	—	2	1	no	—
51	C₄×C₈×C₁₇	—	2	1	no	—
52	C₄×C₈×C₁₇	—	2	1	no	—
53	C₄×C₈×C₁₇	—	4	2	no	—
54	#41×C₁₇	—	4	1	no	—
55	#41×C₁₇	—	4	1	no	—
56	#41×C₁₇	—	4	1	no	—
57	#41×C₁₇	—	4	1	no	—
58	C₂×Q₈×C₁₇	—	2	1	no	—
59	C₂×D₄×C₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	—	2	1	no	—
62	C₂×D₄×C₁₇	—	4	1	no	—
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	—	4	1	no	—
66	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
69	M₁₆×C₁₇	—	4	1	no	—
70	M₁₆×C₁₇	—	4	1	no	—
71	C₂×C₈×C₁₇	—	2	2	no	—
72	C₂×C₈×C₁₇	—	2	2	no	—
73	C₂×C₈×C₁₇	—	2	2	no	—
74	C₂×C₈×C₁₇	—	4	1	no	—
75	C₂×C₈×C₁₇	—	4	1	no	—
76	C₂×C₈×C₁₇	—	4	1	no	—
77	C₂×C₈×C₁₇	—	4	1	no	—
78	C₂×C₈×C₁₇	—	4	1	no	—
79	C₂×C₈×C₁₇	—	4	1	no	—
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	—	2	1	no	—
82	C₂²⋊C₄×C₁₇	—	4	1	no	—
83	C₂²⋊C₄×C₁₇	—	4	1	no	—
84	C₄²×C₁₇	—	2	4	no	—
85	C₄²×C₁₇	—	2	1	no	—
86	C₄²×C₁₇	—	2	1	no	—
87	C₄²×C₁₇	—	2	1	no	—
88	C₄²×C₁₇	—	4	2	no	—
89	C₄²×C₁₇	D₈	1	2	yes	—
90	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₇	#41	1	1	yes	characteristic
93	C₄²×C₁₇	#41	1	1	yes	characteristic
94	#9577	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₇	—	4	1	no	—
96	C₂³×C₁₇	#123	1	1	yes	characteristic
97	Q₈×C₁₇	—	4	1	no	—
98	Q₈×C₁₇	—	4	1	no	—
99	D₄×C₁₇	—	4	1	no	—
100	D₄×C₁₇	—	4	1	no	—
101	D₄×C₁₇	—	4	2	no	—
102	D₄×C₁₇	—	8	1	no	—
103	D₄×C₁₇	—	8	1	no	—
104	D₄×C₁₇	—	8	1	no	—
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	2	2	no	—
107	C₂×C₄×C₁₇	—	2	2	no	—
108	C₂×C₄×C₁₇	—	2	2	no	—
109	C₂×C₄×C₁₇	—	2	2	no	—
110	C₂×C₄×C₁₇	—	4	1	no	—
111	C₂×C₄×C₁₇	—	4	1	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	—	4	1	no	—
114	C₂×C₄×C₁₇	—	8	1	no	—
115	C₂×C₄×C₁₇	—	8	1	no	—
116	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
121	C₂×C₄×C₁₇	#126	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
124	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
126	C₈×C₁₇	—	4	2	no	—
127	C₈×C₁₇	—	4	2	no	—
128	C₈×C₁₇	—	4	2	no	—
129	C₈×C₁₇	—	8	1	no	—
130	C₈×C₁₇	—	8	1	no	—
131	#1962	C₂×C₁₇	1	1	yes	characteristic
132	C₄×C₄⋊C₈	C₂×C₁₇	1	1	yes	characteristic, complemented
133	#1571	C₂×C₁₇	1	1	yes	characteristic, complemented
134	#1540	C₂×C₁₇	1	1	yes	characteristic
135	#1480	C₂×C₁₇	1	1	yes	characteristic
136	#1301	C₂×C₁₇	1	1	yes	characteristic, complemented
137	#1276	C₂×C₁₇	1	1	yes	characteristic, complemented
138	C₂²×C₁₇	—	2	1	no	—
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	8	1	no	—
141	C₂²×C₁₇	—	8	1	no	—
142	C₂²×C₁₇	—	8	1	no	—
143	C₂²×C₁₇	#388	1	1	yes	characteristic
144	C₂²×C₁₇	#480	1	1	yes	characteristic
145	C₂²×C₁₇	#482	1	1	yes	characteristic
146	C₄×C₁₇	—	2	4	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	4	2	no	—
151	C₄×C₁₇	—	8	1	no	—
152	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
153	C₄×C₁₇	#440	1	1	yes	characteristic
154	C₄×C₁₇	#443	1	2	yes	—
155	C₄×C₁₇	#459	1	1	yes	characteristic
156	C₄×C₁₇	#464	1	1	yes	characteristic
157	#532	C₂²×C₁₇	1	1	yes	characteristic
158	#431	C₂²×C₁₇	1	1	yes	characteristic
159	#431	C₂²×C₁₇	1	1	yes	characteristic
160	#423	C₂²×C₁₇	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
163	C₂×#104	—	2	1	no	—
164	C₂×#104	—	2	1	no	—
165	C₂×C₄×C₈	—	2	1	no	—
166	C₄³	C₂²×C₁₇	1	1	yes	characteristic
167	#336	—	2	1	no	—
168	#335	—	2	2	no	—
169	#332	C₄×C₁₇	1	2	yes	complemented
170	#331	C₄×C₁₇	1	2	yes	complemented
171	D₄⋊C₈	—	2	2	no	—
172	C₂×C₁₇	—	2	1	no	—
173	C₂×C₁₇	—	2	1	no	—
174	C₂×C₁₇	—	16	1	no	—
175	C₂×C₁₇	#1925	1	1	yes	characteristic
176	C₂×C₁₇	#1932	1	1	yes	characteristic
177	C₂×C₁₇	#1937	1	1	yes	characteristic
178	C₂×#41	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₄⋊Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
183	#127	C₂×C₄×C₁₇	1	1	yes	characteristic
184	#124	—	2	1	no	—
185	#121	—	2	1	no	—
186	#121	—	2	1	no	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	C₂×C₄×C₁₇	1	2	yes	—
200	C₄⋊C₈	C₂×C₄×C₁₇	1	2	yes	—
201	C₄⋊C₈	D₄×C₁₇	1	2	yes	—
202	#104	—	4	2	no	—
203	#104	—	4	2	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	4	2	no	—
209	C₁₇	#9577	1	1	yes	characteristic, direct product factor, Sylow
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	#41	—	4	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
223	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
224	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	4	1	no	—
239	C₂²⋊C₄	—	4	1	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	4	2	no	—
245	C₄²	D₈×C₁₇	1	2	yes	—
246	C₄²	C₂²×C₄×C₁₇	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
248	C₄²	#41×C₁₇	1	1	yes	characteristic
249	C₄²	#41×C₁₇	1	1	yes	characteristic
250	C₂³	—	4	1	no	—
251	C₂³	C₁₇×#123	1	1	yes	characteristic
252	Q₈	—	4	1	no	—
253	Q₈	—	4	1	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	1	no	—
258	D₄	—	8	1	no	—
259	D₄	—	8	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	C₁₇×#117	1	1	yes	characteristic
272	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
274	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
275	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
276	C₂×C₄	C₁₇×#126	1	1	yes	characteristic
277	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
278	C₂×C₄	C₁₇×#135	1	1	yes	characteristic
279	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂×C₄	C₁₇×#137	1	1	yes	characteristic
281	C₈	—	4	2	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	8	1	no	—
285	C₈	—	8	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	8	1	no	—
289	C₂²	—	8	1	no	—
290	C₂²	—	8	1	no	—
291	C₂²	C₁₇×#388	1	1	yes	characteristic
292	C₂²	C₁₇×#480	1	1	yes	characteristic
293	C₂²	C₁₇×#482	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	8	1	no	—
300	C₄	C₄×D₈×C₁₇	1	1	yes	characteristic
301	C₄	C₁₇×#440	1	1	yes	characteristic
302	C₄	C₁₇×#443	1	2	yes	—
303	C₄	C₁₇×#459	1	1	yes	characteristic
304	C₄	C₁₇×#464	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	16	1	no	—
308	C₂	C₁₇×#1925	1	1	yes	characteristic
309	C₂	C₁₇×#1932	1	1	yes	characteristic
310	C₂	C₁₇×#1937	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇	#121
commutator	C₄²	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	#388
prime radical	C₁₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₇×C₄⋊C₈	C₂	semidirect
3. C₁₇×#1571	C₂	semidirect
6. C₁₇×#1301	C₂	semidirect
7. C₁₇×#1276	C₂	semidirect
20. C₁₇×#332	C₄	semidirect
21. C₁₇×#331	C₄	semidirect
94. #9577	C₁₇	direct (N × H)
132. C₄×C₄⋊C₈	C₂×C₁₇	semidirect
133. #1571	C₂×C₁₇	semidirect
136. #1301	C₂×C₁₇	semidirect
137. #1276	C₂×C₁₇	semidirect
169. #332	C₄×C₁₇	semidirect
170. #331	C₄×C₁₇	semidirect
209. C₁₇	#9577	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×#1962	2
C₄×C₁₇×C₄⋊C₈	2
C₁₇×#1571	2
C₁₇×#1540	2
C₁₇×#1480	2
C₁₇×#1301	2
C₁₇×#1276	2
#9577	17

Subgroups — C17×#9577 #16867321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#9577 #16867321