Subgroups — C₂×D₁₇×#368 #17187374

Order 4352.

662 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₁₇	C₂	1	2	yes	maximal
3	C₂×#11762639	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11762638	C₂	1	2	yes	maximal, complemented
5	D₁₇×#368	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₇×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×D₁₇	—	2	2	no	—
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂²×#63214	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₁₇	C₄	1	2	yes	—
11	C₂×C₁₆×D₁₇	C₄	1	4	yes	—
12	C₂×C₁₆×D₁₇	C₄	1	2	yes	—
13	C₂×C₁₆×D₁₇	C₂²	1	4	yes	—
14	C₂×#10587490	C₂²	1	2	yes	—
15	C₂×#10587493	—	2	2	no	—
16	#108×D₁₇	—	2	4	no	—
17	#108×D₁₇	C₂²	1	8	yes	—
18	#11762639	C₂²	1	4	yes	complemented
19	#11762638	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×C₁₇×#108	C₂²	1	2	yes	—
22	C₁₇×#368	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₇	—	2	2	no	—
25	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	C₂×#63223	—	2	2	no	—
27	C₂²×#3622	D₄	1	1	yes	characteristic
28	M₁₆×D₁₇	—	2	8	no	—
29	M₁₆×D₁₇	—	4	4	no	—
30	C₂×#63218	—	4	2	no	—
31	C₂×C₈×D₁₇	—	4	2	no	—
32	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₁₇	D₄	1	2	yes	—
36	C₂×C₈×D₁₇	Q₈	1	2	yes	—
37	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
38	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
39	C₂×C₈×D₁₇	C₂³	1	2	yes	—
40	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
41	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
42	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
43	C₁₇×#108	—	2	2	no	—
44	C₁₇×#108	C₂³	1	4	yes	—
45	#10587493	—	2	8	no	—
46	#10587490	—	2	2	no	—
47	#10587490	C₂³	1	4	yes	—
48	C₂×#63214	C₂×C₄	1	2	yes	—
49	C₂×#63214	C₂×C₄	1	2	yes	—
50	C₂×#63214	C₂³	1	2	yes	—
51	C₁₆×D₁₇	C₂×C₄	1	8	yes	—
52	C₁₆×D₁₇	C₂×C₄	1	8	yes	—
53	C₂³×D₁₇	Q₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₁₇	Q₁₆	1	1	yes	characteristic
56	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
60	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
61	C₂×C₄×D₁₇	D₈	1	2	yes	—
62	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
63	C₂×C₄×D₁₇	C₂×D₄	1	2	yes	—
64	M₁₆×C₁₇	—	2	4	no	—
65	M₁₆×C₁₇	—	4	2	no	—
66	C₂×C₈×C₁₇	—	4	2	no	—
67	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
73	#63223	—	2	4	no	—
74	#63223	—	4	2	no	—
75	C₂×#3622	—	4	2	no	—
76	C₂×#3622	C₄⋊C₄	1	2	yes	—
77	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
78	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
79	C₂×#3622	C₂×D₄	1	2	yes	—
80	#63218	—	4	8	no	—
81	C₈×D₁₇	—	4	8	no	—
82	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
83	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
84	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
85	C₈×D₁₇	C₄⋊C₄	1	2	yes	—
86	C₈×D₁₇	C₂²×C₄	1	2	yes	—
87	C₈×D₁₇	C₂²×C₄	1	2	yes	—
88	C₈×D₁₇	C₂×Q₈	1	2	yes	—
89	C₈×D₁₇	C₂×Q₈	1	2	yes	—
90	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
91	#63214	C₂²×C₄	1	4	yes	—
92	C₂²×#368	—	17	1	no	maximal, Sylow
93	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂²×D₁₇	—	2	4	no	—
95	C₂²×D₁₇	#107	1	1	yes	characteristic
96	C₂²×D₁₇	#107	1	2	yes	—
97	C₂²×D₁₇	#107	1	1	yes	characteristic
98	C₂²×D₁₇	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×Dic₁₇	#107	1	2	yes	—
106	C₂×Dic₁₇	#107	1	1	yes	characteristic
107	C₂×Dic₁₇	#107	1	1	yes	characteristic
108	C₂×Dic₁₇	C₂×Q₁₆	1	2	yes	—
109	C₄×D₁₇	#107	1	2	yes	—
110	C₄×D₁₇	#107	1	2	yes	—
111	C₄×D₁₇	#107	1	2	yes	—
112	C₄×D₁₇	#107	1	2	yes	—
113	C₄×D₁₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₄×D₁₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₄×D₁₇	C₂×D₈	1	2	yes	—
116	C₄×D₁₇	C₂×D₈	1	2	yes	—
117	C₈×C₁₇	—	4	4	no	—
118	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
121	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
122	#3622	—	4	4	no	—
123	#3622	C₂×C₄⋊C₄	1	2	yes	—
124	#3622	C₂×C₄⋊C₄	1	2	yes	—
125	C₂³×C₁₆	—	17	1	no	—
126	C₂²×#108	—	17	2	no	—
127	C₂×#368	—	17	1	no	—
128	C₂×#368	—	17	2	no	—
129	C₂×#368	—	17	8	no	—
130	C₂×#368	D₁₇	1	1	yes	characteristic, direct product factor
131	C₂²×C₁₇	—	2	2	no	—
132	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
133	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
134	C₂²×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
135	C₂×D₁₇	—	2	2	no	—
136	C₂×D₁₇	—	2	4	no	—
137	C₂×D₁₇	—	2	4	no	—
138	C₂×D₁₇	#368	1	4	yes	direct product factor
139	C₂×D₁₇	C₂×#107	1	2	yes	—
140	C₂×D₁₇	C₂×#107	1	2	yes	—
141	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
142	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
143	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
145	Dic₁₇	C₂×#107	1	2	yes	—
146	Dic₁₇	C₂×#107	1	2	yes	—
147	C₂²×M₁₆	—	34	2	no	—
148	C₂³×C₈	—	17	1	no	—
149	C₂²×C₁₆	—	17	1	no	—
150	C₂²×C₁₆	—	17	2	no	—
151	C₂²×C₁₆	—	17	2	no	—
152	C₂²×C₁₆	—	17	4	no	—
153	C₂²×C₁₆	—	17	4	no	—
154	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
155	C₂×#108	—	17	2	no	—
156	C₂×#108	—	17	8	no	—
157	C₂×#108	—	34	4	no	—
158	C₂×#108	—	34	2	no	—
159	C₂×#108	C₂×D₁₇	1	2	yes	—
160	#368	—	17	4	no	—
161	#368	—	17	8	no	—
162	#368	C₂×D₁₇	1	4	yes	direct product factor
163	C₂×C₁₇	—	2	1	no	—
164	C₂×C₁₇	—	2	1	no	—
165	C₂×C₁₇	C₂×#368	1	2	yes	complemented
166	C₂×C₁₇	C₂²×#107	1	1	yes	characteristic
167	D₁₇	—	2	2	no	—
168	D₁₇	C₂×#368	1	4	yes	direct product factor
169	C₂³×C₄	—	17	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	34	8	no	—
172	C₂×M₁₆	—	34	2	no	—
173	C₂×M₁₆	—	68	2	no	—
174	C₂×M₁₆	—	68	4	no	—
175	C₂²×C₈	—	17	1	no	—
176	C₂²×C₈	—	17	1	no	—
177	C₂²×C₈	—	17	1	no	—
178	C₂²×C₈	—	17	2	no	—
179	C₂²×C₈	—	17	2	no	—
180	C₂²×C₈	—	17	1	no	—
181	C₂²×C₈	—	17	1	no	—
182	C₂²×C₈	—	17	2	no	—
183	C₂²×C₈	—	17	2	no	—
184	C₂²×C₈	—	68	2	no	—
185	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
186	C₂×C₁₆	—	17	2	no	—
187	C₂×C₁₆	—	17	2	no	—
188	C₂×C₁₆	—	17	2	no	—
189	C₂×C₁₆	—	17	8	no	—
190	C₂×C₁₆	—	17	8	no	—
191	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
192	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
193	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
194	#108	—	2	2	no	—
195	#108	—	17	4	no	—
196	#108	—	34	8	no	—
197	#108	—	34	2	no	—
198	#108	C₂²×D₁₇	1	4	yes	—
199	C₁₇	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
200	C₂⁴	—	17	1	no	—
201	C₂²×C₄	—	17	1	no	—
202	C₂²×C₄	—	17	2	no	—
203	C₂²×C₄	—	17	1	no	—
204	C₂²×C₄	—	17	2	no	—
205	C₂²×C₄	—	17	1	no	—
206	C₂²×C₄	—	17	2	no	—
207	C₂²×C₄	—	17	1	no	—
208	C₂²×C₄	—	17	1	no	—
209	C₂²×C₄	—	17	2	no	—
210	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	34	4	no	—
214	M₁₆	—	68	8	no	—
215	M₁₆	—	68	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	17	2	no	—
218	C₂×C₈	—	17	1	no	—
219	C₂×C₈	—	17	1	no	—
220	C₂×C₈	—	17	2	no	—
221	C₂×C₈	—	17	2	no	—
222	C₂×C₈	—	17	2	no	—
223	C₂×C₈	—	17	2	no	—
224	C₂×C₈	—	17	2	no	—
225	C₂×C₈	—	17	2	no	—
226	C₂×C₈	—	17	2	no	—
227	C₂×C₈	—	17	2	no	—
228	C₂×C₈	—	17	2	no	—
229	C₂×C₈	—	68	8	no	—
230	C₂×C₈	—	68	2	no	—
231	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
233	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
236	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
237	C₁₆	—	17	4	no	—
238	C₁₆	C₂×C₄×D₁₇	1	4	yes	—
239	C₂³	—	17	1	no	—
240	C₂³	—	17	2	no	—
241	C₂³	—	17	2	no	—
242	C₂³	—	17	1	no	—
243	C₂³	—	34	4	no	—
244	C₂³	Q₁₆×D₁₇	1	1	yes	characteristic
245	C₂×C₄	—	17	2	no	—
246	C₂×C₄	—	17	2	no	—
247	C₂×C₄	—	17	1	no	—
248	C₂×C₄	—	17	1	no	—
249	C₂×C₄	—	17	2	no	—
250	C₂×C₄	—	17	2	no	—
251	C₂×C₄	—	17	2	no	—
252	C₂×C₄	—	17	2	no	—
253	C₂×C₄	—	17	2	no	—
254	C₂×C₄	—	17	2	no	—
255	C₂×C₄	—	17	2	no	—
256	C₂×C₄	—	17	2	no	—
257	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
259	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
260	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
261	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
263	C₈	—	4	4	no	—
264	C₈	—	17	2	no	—
265	C₈	—	17	2	no	—
266	C₈	—	68	4	no	—
267	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
268	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
269	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
270	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
271	C₂²	—	2	2	no	—
272	C₂²	—	17	4	no	—
273	C₂²	—	17	2	no	—
274	C₂²	—	17	2	no	—
275	C₂²	—	34	4	no	—
276	C₂²	—	34	4	no	—
277	C₂²	—	34	2	no	—
278	C₂²	#107×D₁₇	1	1	yes	characteristic
279	C₂²	#107×D₁₇	1	1	yes	characteristic
280	C₂²	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
281	C₄	—	17	2	no	—
282	C₄	—	17	2	no	—
283	C₄	#107×D₁₇	1	1	yes	characteristic
284	C₄	#107×D₁₇	1	1	yes	characteristic
285	C₄	C₂×C₄⋊C₄×D₁₇	1	1	yes	characteristic
286	C₄	C₂×D₈×D₁₇	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	2	1	no	—
289	C₂	—	17	4	no	—
290	C₂	—	34	2	no	—
291	C₂	D₁₇×#368	1	2	yes	direct product factor
292	C₂	C₂×#107×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₇
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×#368	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×D₁₇	Q₁₆
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#368	D₁₇
hypocenter	C₁₇	C₂²×#368
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
3. C₂×#11762639	C₂	semidirect
4. C₂×#11762638	C₂	semidirect
5. D₁₇×#368	C₂	direct (N × H)
6. C₂×C₁₇×#368	C₂	semidirect
18. #11762639	C₂²	semidirect
19. #11762638	C₂²	semidirect
22. C₁₇×#368	C₂²	semidirect
130. C₂×#368	D₁₇	direct (N × H)
138. C₂×D₁₇	#368	direct (N × H)
162. #368	C₂×D₁₇	direct (N × H)
165. C₂×C₁₇	C₂×#368	semidirect
168. D₁₇	C₂×#368	direct (N × H)
199. C₁₇	C₂²×#368	semidirect
291. C₂	D₁₇×#368	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₆×D₁₇	2
C₂×#108×D₁₇	2
C₂×#11762639	2
C₂×#11762638	2
D₁₇×#368	2
C₂×C₁₇×#368	2
C₂²×#368	17

Subgroups — C2×D17×#368 #17187374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₇×#368 #17187374