Subgroups — C₁₇×#12981 #16870725

Order 4352.

488 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×#2929	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#2115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#2112	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#525	C₄	1	1	yes	characteristic
9	C₁₇×#504	—	2	4	no	—
10	C₂×C₁₇×M₃₂	—	2	2	no	—
11	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
12	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
13	C₁₇×#435	—	2	2	no	—
14	C₈×D₄×C₁₇	—	2	2	no	—
15	C₁₇×#424	C₄	1	1	yes	characteristic
16	C₁₇×#408	C₄	1	1	yes	characteristic
17	C₁₇×#407	C₄	1	1	yes	characteristic
18	C₁₇×#405	C₂²	1	1	yes	characteristic
19	C₁₇×#349	—	2	2	no	—
20	C₁₇×C₂²⋊C₁₆	—	2	4	no	—
21	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₁₇×#131	—	2	2	no	—
24	C₁₇×#131	—	2	2	no	—
25	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₁₇	D₄	1	2	yes	—
28	C₂²×C₈×C₁₇	—	2	2	no	—
29	C₂²×C₈×C₁₇	—	2	1	no	—
30	C₂²×C₈×C₁₇	—	2	1	no	—
31	C₂²×C₈×C₁₇	—	2	2	no	—
32	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₇	D₄	1	2	yes	—
34	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₁₇×#123	C₈	1	2	yes	—
36	C₁₇×C₂²⋊Q₈	—	2	1	no	—
37	C₁₇×#121	—	2	1	no	—
38	C₁₇×#120	C₈	1	2	yes	—
39	C₄×D₄×C₁₇	—	2	2	no	—
40	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
41	C₁₇×M₃₂	—	2	4	no	—
42	C₁₇×M₃₂	—	2	2	no	—
43	C₁₇×M₃₂	—	2	2	no	—
44	C₁₇×M₃₂	—	2	2	no	—
45	C₁₇×M₃₂	—	2	2	no	—
46	C₂×C₁₆×C₁₇	—	2	4	no	—
47	C₂×C₁₆×C₁₇	—	2	2	no	—
48	C₂×C₁₆×C₁₇	D₄	1	4	yes	complemented
49	C₁₇×C₄⋊C₈	—	2	2	no	—
50	C₁₇×C₂²⋊C₈	—	2	2	no	—
51	C₁₇×C₂²⋊C₈	—	2	2	no	—
52	C₁₇×#97	—	2	1	no	—
53	C₄×C₈×C₁₇	—	2	1	no	—
54	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
55	#41×C₁₇	—	2	2	no	—
56	#41×C₁₇	—	2	2	no	—
57	C₂×Q₈×C₁₇	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₁₇	—	2	2	no	—
59	C₂×D₄×C₁₇	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₁₇	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂²×C₄×C₁₇	—	2	2	no	—
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂²×C₄×C₁₇	—	2	2	no	—
65	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₁₇	—	2	2	no	—
69	M₁₆×C₁₇	—	2	4	no	—
70	M₁₆×C₁₇	—	2	2	no	—
71	C₂×C₈×C₁₇	—	2	2	no	—
72	C₂×C₈×C₁₇	—	2	2	no	—
73	C₂×C₈×C₁₇	—	2	1	no	—
74	C₂×C₈×C₁₇	—	2	1	no	—
75	C₂×C₈×C₁₇	—	2	2	no	—
76	C₂×C₈×C₁₇	—	4	1	no	—
77	C₂×C₈×C₁₇	—	4	2	no	—
78	C₂×C₈×C₁₇	—	4	1	no	—
79	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₇	#41	1	2	yes	—
84	C₄⋊C₄×C₁₇	—	2	2	no	—
85	C₄⋊C₄×C₁₇	—	2	1	no	—
86	C₂²⋊C₄×C₁₇	—	2	2	no	—
87	C₂²⋊C₄×C₁₇	—	2	2	no	—
88	C₂²⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
89	C₄²×C₁₇	—	2	1	no	—
90	C₁₆×C₁₇	—	2	4	no	—
91	C₁₆×C₁₇	—	4	2	no	—
92	#12981	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₁₇	—	2	1	no	—
94	C₂³×C₁₇	—	2	2	no	—
95	C₂³×C₁₇	—	4	2	no	—
96	C₂³×C₁₇	C₂²⋊C₈	1	2	yes	—
97	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
98	Q₈×C₁₇	—	2	2	no	—
99	D₄×C₁₇	—	2	4	no	—
100	D₄×C₁₇	—	2	2	no	—
101	D₄×C₁₇	—	4	4	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	1	no	—
106	C₂×C₄×C₁₇	—	2	2	no	—
107	C₂×C₄×C₁₇	—	4	1	no	—
108	C₂×C₄×C₁₇	—	4	1	no	—
109	C₂×C₄×C₁₇	—	4	2	no	—
110	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₇	#131	1	2	yes	—
115	C₈×C₁₇	—	2	2	no	—
116	C₈×C₁₇	—	4	1	no	—
117	C₈×C₁₇	—	4	1	no	—
118	C₈×C₁₇	#120	1	1	yes	characteristic
119	C₈×C₁₇	#121	1	2	yes	—
120	C₈×C₁₇	#123	1	1	yes	characteristic
121	#3407	C₂×C₁₇	1	1	yes	characteristic, complemented
122	#2929	C₂×C₁₇	1	1	yes	characteristic
123	#2115	C₂×C₁₇	1	1	yes	characteristic, complemented
124	#2112	C₂×C₁₇	1	2	yes	complemented
125	#1378	C₂×C₁₇	1	2	yes	complemented
126	C₂²×C₁₇	—	2	2	no	—
127	C₂²×C₁₇	—	2	4	no	—
128	C₂²×C₁₇	—	2	2	no	—
129	C₂²×C₁₇	—	4	1	no	—
130	C₂²×C₁₇	—	4	2	no	—
131	C₂²×C₁₇	—	4	1	no	—
132	C₂²×C₁₇	—	4	2	no	—
133	C₂²×C₁₇	—	4	1	no	—
134	C₂²×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₂²×C₁₇	C₈×D₄	1	2	yes	—
136	C₄×C₁₇	—	2	2	no	—
137	C₄×C₁₇	—	4	2	no	—
138	C₄×C₁₇	—	4	1	no	—
139	C₄×C₁₇	#386	1	1	yes	characteristic
140	C₄×C₁₇	#408	1	1	yes	characteristic
141	C₄×C₁₇	#435	1	2	yes	—
142	C₂×#131	C₂²×C₁₇	1	1	yes	characteristic
143	C₂³×C₈	C₂²×C₁₇	1	1	yes	characteristic
144	#525	C₄×C₁₇	1	1	yes	characteristic
145	#504	—	2	4	no	—
146	C₂×M₃₂	—	2	2	no	—
147	C₂×M₃₂	C₂²×C₁₇	1	2	yes	—
148	C₂×M₃₂	C₂²×C₁₇	1	2	yes	—
149	#435	—	2	2	no	—
150	C₈×D₄	—	2	2	no	—
151	#424	C₄×C₁₇	1	1	yes	characteristic
152	#408	C₄×C₁₇	1	1	yes	characteristic
153	#407	C₄×C₁₇	1	1	yes	characteristic
154	#405	C₂²×C₁₇	1	1	yes	characteristic
155	#349	—	2	2	no	—
156	C₂²⋊C₁₆	—	2	4	no	—
157	C₂×C₁₇	—	2	1	no	—
158	C₂×C₁₇	—	2	2	no	—
159	C₂×C₁₇	—	4	2	no	—
160	C₂×C₁₇	—	4	2	no	—
161	C₂×C₁₇	#1869	1	1	yes	characteristic
162	C₂×#41	C₂×C₄×C₁₇	1	1	yes	characteristic
163	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
164	#131	—	2	2	no	—
165	#131	—	2	2	no	—
166	C₂×M₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
167	C₂×M₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
168	C₂×M₁₆	D₄×C₁₇	1	2	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
174	C₂²×C₈	D₄×C₁₇	1	2	yes	—
175	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
176	#123	C₈×C₁₇	1	2	yes	—
177	C₂²⋊Q₈	—	2	1	no	—
178	#121	—	2	1	no	—
179	#120	C₈×C₁₇	1	2	yes	—
180	C₄×D₄	—	2	2	no	—
181	#117	C₂×C₄×C₁₇	1	1	yes	characteristic
182	M₃₂	—	2	4	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	C₂×C₁₆	—	2	4	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	D₄×C₁₇	1	4	yes	complemented
190	C₄⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₁₇	#12981	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₂×C₈×C₁₇	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	C₂×Q₈	M₁₆×C₁₇	1	1	yes	characteristic
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂×C₈×C₁₇	1	2	yes	—
202	C₂×D₄	M₁₆×C₁₇	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
208	C₂²×C₄	C₂×C₈×C₁₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	C₂²⋊C₄×C₁₇	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
225	C₂×C₈	#41×C₁₇	1	2	yes	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	C₂×C₈×C₁₇	1	2	yes	—
231	C₄²	—	2	1	no	—
232	C₁₆	—	2	4	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₁₇×C₂²⋊C₈	1	2	yes	—
238	C₂³	C₂²×C₈×C₁₇	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₁₇×C₂²⋊C₈	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
254	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
255	C₂×C₄	C₁₇×#131	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	C₁₇×#120	1	1	yes	characteristic
260	C₈	C₁₇×#121	1	2	yes	—
261	C₈	C₁₇×#123	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×C₁₇×C₂²⋊C₈	1	1	yes	characteristic
271	C₂²	C₈×D₄×C₁₇	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	C₁₇×#386	1	1	yes	characteristic
276	C₄	C₁₇×#408	1	1	yes	characteristic
277	C₄	C₁₇×#435	1	2	yes	—
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	4	2	no	—
282	C₂	C₁₇×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₇	#120
commutator	C₂³	C₂²×C₈×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
socle	C₂×C₁₇	#1869
prime radical	C₁₇×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₇×#3407	C₂	semidirect
3. C₁₇×#2115	C₂	semidirect
4. C₁₇×#2112	C₂	semidirect
5. C₁₇×#1378	C₂	semidirect
48. C₂×C₁₆×C₁₇	D₄	semidirect
92. #12981	C₁₇	direct (N × H)
121. #3407	C₂×C₁₇	semidirect
123. #2115	C₂×C₁₇	semidirect
124. #2112	C₂×C₁₇	semidirect
125. #1378	C₂×C₁₇	semidirect
189. C₂×C₁₆	D₄×C₁₇	semidirect
195. C₁₇	#12981	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×#3407	2
C₁₇×#2929	2
C₁₇×#2115	2
C₁₇×#2112	2
C₁₇×#1378	2
#12981	17

Subgroups — C17×#12981 #16870725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#12981 #16870725