Subgroups — C₁₇×#7707 #16865451

Order 4352.

500 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2857	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1817	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1749	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1272	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₁₇×C₂²⋊Q₈	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₁₇×C₂²⋊C₈	—	2	1	no	—
9	C₂×C₁₇×C₂²⋊C₈	—	2	1	no	—
10	C₂×C₁₇×#97	—	2	1	no	—
11	C₁₇×#391	C₂²	1	2	yes	—
12	C₁₇×#389	—	2	1	no	—
13	C₁₇×#389	—	2	1	no	—
14	C₁₇×#384	C₄	1	1	yes	characteristic
15	C₁₇×#380	—	2	1	no	—
16	C₁₇×#380	—	2	1	no	—
17	C₁₇×#336	—	2	1	no	—
18	C₁₇×#336	C₂²	1	2	yes	—
19	C₁₇×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	—	2	2	no	—
25	C₂²×C₈×C₁₇	—	2	2	no	—
26	C₁₇×C₄⋊Q₈	—	4	2	no	—
27	C₁₇×C₂²⋊Q₈	—	4	4	no	—
28	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
35	C₂×C₄²×C₁₇	—	4	2	no	—
36	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
40	C₁₇×C₂²⋊C₈	—	2	4	no	—
41	C₁₇×C₂²⋊C₈	—	4	2	no	—
42	C₁₇×#97	—	4	2	no	—
43	C₁₇×#95	—	2	2	no	—
44	C₁₇×#95	—	2	2	no	—
45	C₂⁴×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×Q₈×C₁₇	—	2	4	no	—
47	C₂×Q₈×C₁₇	—	4	2	no	—
48	C₂×Q₈×C₁₇	—	4	2	no	—
49	C₂²×C₄×C₁₇	—	2	2	no	—
50	C₂²×C₄×C₁₇	—	2	2	no	—
51	C₂²×C₄×C₁₇	—	2	2	no	—
52	C₂²×C₄×C₁₇	—	2	2	no	—
53	C₂²×C₄×C₁₇	—	2	2	no	—
54	C₂²×C₄×C₁₇	—	2	2	no	—
55	C₂²×C₄×C₁₇	—	2	2	no	—
56	C₂²×C₄×C₁₇	—	2	2	no	—
57	C₂²×C₄×C₁₇	—	4	1	no	—
58	C₂²×C₄×C₁₇	—	4	1	no	—
59	C₂²×C₄×C₁₇	—	4	1	no	—
60	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	—	2	4	no	—
67	C₂×C₈×C₁₇	—	4	2	no	—
68	C₂×C₈×C₁₇	—	4	2	no	—
69	C₂×C₈×C₁₇	—	4	2	no	—
70	C₂×C₈×C₁₇	—	4	2	no	—
71	C₂×C₈×C₁₇	—	4	2	no	—
72	C₄⋊C₄×C₁₇	—	4	2	no	—
73	C₄⋊C₄×C₁₇	—	4	4	no	—
74	C₄⋊C₄×C₁₇	—	4	2	no	—
75	C₄⋊C₄×C₁₇	—	4	2	no	—
76	C₄⋊C₄×C₁₇	—	4	2	no	—
77	C₄⋊C₄×C₁₇	—	4	2	no	—
78	C₂²⋊C₄×C₁₇	—	4	4	no	—
79	C₄²×C₁₇	—	2	2	no	—
80	C₄²×C₁₇	—	2	2	no	—
81	C₄²×C₁₇	—	4	1	no	—
82	C₄²×C₁₇	—	4	1	no	—
83	#7707	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₁₇	—	2	1	no	—
85	C₂³×C₁₇	—	2	1	no	—
86	C₂³×C₁₇	—	2	1	no	—
87	C₂³×C₁₇	—	2	1	no	—
88	C₂³×C₁₇	—	2	1	no	—
89	C₂³×C₁₇	—	2	1	no	—
90	C₂³×C₁₇	#100	1	2	yes	—
91	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	Q₈×C₁₇	—	8	4	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	4	4	no	—
107	C₂×C₄×C₁₇	—	4	4	no	—
108	C₂×C₄×C₁₇	—	4	4	no	—
109	C₂×C₄×C₁₇	—	4	2	no	—
110	C₂×C₄×C₁₇	—	4	2	no	—
111	C₂×C₄×C₁₇	—	4	2	no	—
112	C₂×C₄×C₁₇	—	4	2	no	—
113	C₂×C₄×C₁₇	—	4	2	no	—
114	C₂×C₄×C₁₇	—	4	2	no	—
115	C₂×C₄×C₁₇	#99	1	2	yes	—
116	C₂×C₄×C₁₇	#104	1	2	yes	—
117	C₂×C₄×C₁₇	#131	1	2	yes	—
118	C₂×C₄×C₁₇	#137	1	2	yes	—
119	C₂×C₄×C₁₇	#137	1	2	yes	—
120	C₈×C₁₇	—	4	4	no	—
121	C₈×C₁₇	—	8	2	no	—
122	#2857	C₂×C₁₇	1	1	yes	characteristic
123	#1817	C₂×C₁₇	1	1	yes	characteristic
124	#1749	C₂×C₁₇	1	1	yes	characteristic
125	#1272	C₂×C₁₇	1	4	yes	complemented
126	C₂²×C₁₇	—	2	2	no	—
127	C₂²×C₁₇	—	2	2	no	—
128	C₂²×C₁₇	—	2	2	no	—
129	C₂²×C₁₇	—	4	1	no	—
130	C₂²×C₁₇	—	4	1	no	—
131	C₂²×C₁₇	—	4	1	no	—
132	C₂²×C₁₇	—	4	1	no	—
133	C₂²×C₁₇	#408	1	1	yes	characteristic
134	C₂²×C₁₇	C₂×#99	1	1	yes	characteristic
135	C₂²×C₁₇	C₂×#100	1	1	yes	characteristic
136	C₂²×C₁₇	#419	1	1	yes	characteristic
137	C₂²×C₁₇	#419	1	1	yes	characteristic
138	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
139	C₂²×C₁₇	#421	1	1	yes	characteristic
140	C₄×C₁₇	—	2	2	no	—
141	C₄×C₁₇	—	2	2	no	—
142	C₄×C₁₇	—	2	2	no	—
143	C₄×C₁₇	—	2	2	no	—
144	C₄×C₁₇	—	2	2	no	—
145	C₄×C₁₇	—	4	1	no	—
146	C₄×C₁₇	—	8	2	no	—
147	C₄×C₁₇	—	8	2	no	—
148	C₂×C₄⋊Q₈	C₄×C₁₇	1	1	yes	characteristic
149	C₂×C₂²⋊Q₈	C₄×C₁₇	1	2	yes	—
150	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
151	C₂×C₂²⋊C₈	—	2	1	no	—
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×#97	—	2	1	no	—
154	#391	C₂²×C₁₇	1	2	yes	—
155	#389	—	2	1	no	—
156	#389	—	2	1	no	—
157	#384	C₄×C₁₇	1	1	yes	characteristic
158	#380	—	2	1	no	—
159	#380	—	2	1	no	—
160	#336	—	2	1	no	—
161	#336	C₂²×C₁₇	1	2	yes	—
162	#336	C₂²×C₁₇	1	2	yes	—
163	C₂×C₁₇	—	4	2	no	—
164	C₂×C₁₇	#1471	1	1	yes	characteristic
165	C₂×C₁₇	#1485	1	1	yes	characteristic
166	C₂×C₁₇	#1497	1	1	yes	characteristic
167	C₂×C₁₇	#1508	1	1	yes	characteristic
168	C₂×C₁₇	#1511	1	1	yes	characteristic
169	C₂×C₁₇	#1530	1	1	yes	characteristic
170	C₂×C₁₇	#1558	1	1	yes	characteristic
171	C₂²×Q₈	C₂×C₄×C₁₇	1	2	yes	—
172	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
174	C₂³×C₄	D₄×C₁₇	1	1	yes	characteristic
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₄⋊Q₈	—	4	2	no	—
178	C₂²⋊Q₈	—	4	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	—	4	2	no	—
187	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
191	C₂²⋊C₈	—	2	4	no	—
192	C₂²⋊C₈	—	4	2	no	—
193	#97	—	4	2	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	C₁₇	#7707	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊C₄×C₁₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	C₁₇×#100	1	2	yes	—
242	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
243	Q₈	—	8	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	C₁₇×#99	1	2	yes	—
267	C₂×C₄	C₁₇×#104	1	2	yes	—
268	C₂×C₄	C₁₇×#131	1	2	yes	—
269	C₂×C₄	C₁₇×#137	1	2	yes	—
270	C₂×C₄	C₁₇×#137	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₈	—	8	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	C₁₇×#408	1	1	yes	characteristic
281	C₂²	C₂×C₁₇×#99	1	1	yes	characteristic
282	C₂²	C₂×C₁₇×#100	1	1	yes	characteristic
283	C₂²	C₁₇×#419	1	1	yes	characteristic
284	C₂²	C₁₇×#419	1	1	yes	characteristic
285	C₂²	C₂×C₁₇×#104	1	1	yes	characteristic
286	C₂²	C₁₇×#421	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	1	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₂	—	4	2	no	—
296	C₂	C₁₇×#1471	1	1	yes	characteristic
297	C₂	C₁₇×#1485	1	1	yes	characteristic
298	C₂	C₁₇×#1497	1	1	yes	characteristic
299	C₂	C₁₇×#1508	1	1	yes	characteristic
300	C₂	C₁₇×#1511	1	1	yes	characteristic
301	C₂	C₁₇×#1530	1	1	yes	characteristic
302	C₂	C₁₇×#1558	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	C₂⁴×C₁₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂²×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₇×#1272	C₂	semidirect
83. #7707	C₁₇	direct (N × H)
125. #1272	C₂×C₁₇	semidirect
196. C₁₇	#7707	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×#2857	2
C₁₇×#1817	2
C₁₇×#1749	2
C₁₇×#1272	2
#7707	17

Subgroups — C17×#7707 #16865451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#7707 #16865451