Subgroups — C₁₇×#9704 #16867448

Order 4352.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2858	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×#332	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×#331	C₂	1	2	yes	maximal
6	C₂×C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₇×#127	C₂²	1	1	yes	characteristic
8	C₂×C₁₇×#124	—	2	1	no	—
9	C₂×C₁₇×#124	—	2	1	no	—
10	C₂×C₁₇×C₄⋊C₈	—	2	2	no	—
11	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₇×#97	—	2	1	no	—
14	C₁₇×#390	—	2	2	no	—
15	C₁₇×#390	C₄	1	4	yes	—
16	C₄³×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×#336	—	2	1	no	—
18	C₁₇×#332	—	2	4	no	—
19	C₁₇×#331	—	2	4	no	—
20	C₂²×Q₈×C₁₇	—	2	1	no	—
21	C₂²×D₄×C₁₇	—	2	1	no	—
22	C₂²×D₄×C₁₇	—	2	1	no	—
23	C₂²×D₄×C₁₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₇	—	2	2	no	—
25	C₂²×C₈×C₁₇	—	2	2	no	—
26	C₁₇×C₄⋊Q₈	—	2	2	no	—
27	C₁₇×C₄⋊Q₈	—	2	2	no	—
28	C₁₇×#127	—	2	2	no	—
29	C₁₇×#127	—	2	2	no	—
30	C₁₇×#124	—	4	4	no	—
31	C₁₇×#124	—	4	2	no	—
32	C₁₇×#124	—	4	2	no	—
33	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₇	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₁₇	—	2	4	no	—
37	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₇	D₄	1	2	yes	—
40	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	—	2	2	no	—
43	C₁₇×C₄⋊C₈	—	2	2	no	—
44	C₁₇×C₄⋊C₈	—	2	2	no	—
45	C₁₇×C₄⋊C₈	—	4	4	no	—
46	C₁₇×C₄⋊C₈	D₄	1	4	yes	—
47	C₁₇×#97	—	4	2	no	—
48	C₂⁴×C₁₇	—	2	2	no	—
49	C₂×Q₈×C₁₇	—	4	2	no	—
50	C₂×Q₈×C₁₇	—	4	1	no	—
51	C₂×Q₈×C₁₇	—	4	1	no	—
52	C₂×Q₈×C₁₇	—	4	2	no	—
53	C₂×D₄×C₁₇	—	2	4	no	—
54	C₂×D₄×C₁₇	—	4	4	no	—
55	C₂×D₄×C₁₇	—	4	2	no	—
56	C₂×D₄×C₁₇	—	4	1	no	—
57	C₂×D₄×C₁₇	—	4	1	no	—
58	C₂×D₄×C₁₇	—	4	1	no	—
59	C₂×D₄×C₁₇	—	4	1	no	—
60	C₂×D₄×C₁₇	—	4	2	no	—
61	C₂×D₄×C₁₇	—	4	2	no	—
62	C₂×D₄×C₁₇	—	4	2	no	—
63	C₂²×C₄×C₁₇	—	2	2	no	—
64	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	—	2	4	no	—
70	C₂×C₈×C₁₇	—	4	2	no	—
71	C₂×C₈×C₁₇	—	4	2	no	—
72	C₂×C₈×C₁₇	—	4	2	no	—
73	C₂×C₈×C₁₇	—	4	4	no	—
74	C₄⋊C₄×C₁₇	—	2	4	no	—
75	C₄⋊C₄×C₁₇	—	4	1	no	—
76	C₄⋊C₄×C₁₇	—	4	1	no	—
77	C₂²⋊C₄×C₁₇	—	4	8	no	—
78	C₂²⋊C₄×C₁₇	—	4	8	no	—
79	C₄²×C₁₇	—	2	2	no	—
80	C₄²×C₁₇	—	2	2	no	—
81	C₄²×C₁₇	—	2	2	no	—
82	C₄²×C₁₇	—	4	2	no	—
83	C₄²×C₁₇	D₈	1	2	yes	—
84	C₄²×C₁₇	SD₁₆	1	2	yes	—
85	C₄²×C₁₇	C₂×D₄	1	2	yes	—
86	C₄²×C₁₇	#41	1	2	yes	—
87	#9704	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₁₇	—	4	2	no	—
89	C₂³×C₁₇	—	4	4	no	—
90	C₂³×C₁₇	—	4	2	no	—
91	C₂³×C₁₇	—	4	4	no	—
92	C₂³×C₁₇	—	4	2	no	—
93	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	Q₈×C₁₇	—	8	2	no	—
95	Q₈×C₁₇	—	8	2	no	—
96	D₄×C₁₇	—	8	4	no	—
97	D₄×C₁₇	—	8	2	no	—
98	D₄×C₁₇	—	8	2	no	—
99	D₄×C₁₇	—	8	2	no	—
100	D₄×C₁₇	—	8	1	no	—
101	D₄×C₁₇	—	8	1	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	2	no	—
104	C₂×C₄×C₁₇	—	2	2	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	—	4	2	no	—
109	C₂×C₄×C₁₇	—	4	2	no	—
110	C₂×C₄×C₁₇	—	4	2	no	—
111	C₂×C₄×C₁₇	#102	1	4	yes	—
112	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₇	#121	1	2	yes	—
114	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₁₇	#137	1	2	yes	—
121	C₈×C₁₇	—	4	4	no	—
122	C₈×C₁₇	—	8	2	no	—
123	#2858	C₂×C₁₇	1	1	yes	characteristic
124	#1842	C₂×C₁₇	1	1	yes	characteristic, complemented
125	#1767	C₂×C₁₇	1	1	yes	characteristic, complemented
126	C₂×#332	C₂×C₁₇	1	2	yes	complemented
127	C₂×#331	C₂×C₁₇	1	2	yes	—
128	C₂²×C₁₇	—	8	4	no	—
129	C₂²×C₁₇	—	8	4	no	—
130	C₂²×C₁₇	—	8	2	no	—
131	C₂²×C₁₇	—	8	2	no	—
132	C₂²×C₁₇	—	8	2	no	—
133	C₂²×C₁₇	#390	1	1	yes	characteristic
134	C₂²×C₁₇	#413	1	2	yes	—
135	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
136	C₂²×C₁₇	#417	1	2	yes	—
137	C₂²×C₁₇	#418	1	1	yes	characteristic
138	C₄×C₁₇	—	2	4	no	—
139	C₄×C₁₇	—	2	2	no	—
140	C₄×C₁₇	—	4	2	no	—
141	C₄×C₁₇	—	8	2	no	—
142	C₄×C₁₇	#459	1	2	yes	—
143	C₄×C₁₇	#460	1	2	yes	—
144	C₄×C₁₇	#489	1	2	yes	—
145	C₄×C₁₇	#496	1	1	yes	characteristic
146	C₄×C₁₇	#497	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
148	C₂×#127	C₂²×C₁₇	1	1	yes	characteristic
149	C₂×#124	—	2	1	no	—
150	C₂×#124	—	2	1	no	—
151	C₂×C₄⋊C₈	—	2	2	no	—
152	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	—
154	C₂×#97	—	2	1	no	—
155	#390	—	2	2	no	—
156	#390	C₄×C₁₇	1	4	yes	—
157	C₄³	C₂²×C₁₇	1	1	yes	characteristic
158	#336	—	2	1	no	—
159	#332	—	2	4	no	—
160	#331	—	2	4	no	—
161	C₂×C₁₇	—	16	2	no	—
162	C₂×C₁₇	#1547	1	2	yes	—
163	C₂×C₁₇	#1557	1	2	yes	—
164	C₂×C₁₇	#1959	1	1	yes	characteristic
165	C₂×C₁₇	#1967	1	1	yes	characteristic
166	C₂×C₁₇	#1982	1	1	yes	characteristic
167	C₂²×Q₈	—	2	1	no	—
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	C₂×C₄×C₁₇	1	2	yes	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	#127	—	2	2	no	—
176	#127	—	2	2	no	—
177	#124	—	4	4	no	—
178	#124	—	4	2	no	—
179	#124	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₂²⋊C₄	—	2	4	no	—
184	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
185	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₁₇	1	2	yes	—
187	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	4	4	no	—
193	C₄⋊C₈	D₄×C₁₇	1	4	yes	—
194	#97	—	4	2	no	—
195	C₁₇	#9704	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
213	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₂²⋊C₄	—	4	8	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	D₈×C₁₇	1	2	yes	—
232	C₄²	SD₁₆×C₁₇	1	2	yes	—
233	C₄²	C₂×D₄×C₁₇	1	2	yes	—
234	C₄²	#41×C₁₇	1	2	yes	—
235	C₂³	—	4	2	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	2	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	4	2	no	—
240	C₂³	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
241	Q₈	—	8	2	no	—
242	Q₈	—	8	2	no	—
243	D₄	—	8	4	no	—
244	D₄	—	8	1	no	—
245	D₄	—	8	1	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	2	no	—
248	D₄	—	8	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	C₁₇×#102	1	4	yes	—
259	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
260	C₂×C₄	C₁₇×#121	1	2	yes	—
261	C₂×C₄	C₁₇×#124	1	1	yes	characteristic
262	C₂×C₄	C₁₇×#127	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₁₇	1	1	yes	characteristic
264	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
265	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
266	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
267	C₂×C₄	C₁₇×#137	1	2	yes	—
268	C₈	—	4	4	no	—
269	C₈	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	C₁₇×#390	1	1	yes	characteristic
276	C₂²	C₁₇×#413	1	2	yes	—
277	C₂²	C₂×C₁₇×#102	1	1	yes	characteristic
278	C₂²	C₁₇×#417	1	2	yes	—
279	C₂²	C₁₇×#418	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	8	2	no	—
284	C₄	C₁₇×#459	1	2	yes	—
285	C₄	C₁₇×#460	1	2	yes	—
286	C₄	C₁₇×#489	1	2	yes	—
287	C₄	C₁₇×#496	1	1	yes	characteristic
288	C₄	C₁₇×#497	1	1	yes	characteristic
289	C₂	—	16	2	no	—
290	C₂	C₁₇×#1547	1	2	yes	—
291	C₂	C₁₇×#1557	1	2	yes	—
292	C₂	C₁₇×#1959	1	1	yes	characteristic
293	C₂	C₁₇×#1967	1	1	yes	characteristic
294	C₂	C₁₇×#1982	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	C₂×C₁₇×#127	C₂²
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×#1842	C₂	semidirect
3. C₁₇×#1767	C₂	semidirect
4. C₂×C₁₇×#332	C₂	semidirect
87. #9704	C₁₇	direct (N × H)
124. #1842	C₂×C₁₇	semidirect
125. #1767	C₂×C₁₇	semidirect
126. C₂×#332	C₂×C₁₇	semidirect
195. C₁₇	#9704	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×#2858	2
C₁₇×#1842	2
C₁₇×#1767	2
C₂×C₁₇×#332	2
C₂×C₁₇×#331	2
#9704	17

Subgroups — C17×#9704 #16867448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#9704 #16867448