Subgroups — C₂×D₁₇×C₄⋊C₁₆ #17187297

Order 4352.

862 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	C₂²×C₁₆×D₁₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#63214	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11762627	C₂	1	2	yes	maximal, complemented
5	D₁₇×C₄⋊C₁₆	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₁₇	C₄	1	1	yes	characteristic
10	C₂²×#63214	C₂²	1	2	yes	—
11	C₂×C₁₆×D₁₇	—	2	8	no	—
12	C₂×C₁₆×D₁₇	C₂²	1	8	yes	—
13	C₂×C₄×#3622	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₁₇	C₄	1	4	yes	—
17	C₄×C₈×D₁₇	C₂²	1	4	yes	—
18	C₄⋊#63214	C₂²	1	4	yes	complemented
19	#11762627	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₁₇	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₁₇	C₈	1	1	yes	characteristic
25	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
30	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
33	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₁₇	—	2	4	no	—
35	C₂×C₈×D₁₇	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₇	D₄	1	4	yes	—
38	C₂×C₈×D₁₇	Q₈	1	4	yes	—
39	C₂×C₈×D₁₇	C₂³	1	2	yes	—
40	C₄²×D₁₇	C₈	1	4	yes	—
41	C₄²×D₁₇	C₂×C₄	1	4	yes	—
42	C₂×C₁₆×C₁₇	—	2	4	no	—
43	C₂×C₁₆×C₁₇	C₂³	1	4	yes	—
44	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₇	C₂³	1	2	yes	—
46	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
47	C₈×Dic₁₇	C₂×C₄	1	4	yes	—
48	C₈×Dic₁₇	C₂³	1	2	yes	—
49	C₂×#63214	—	2	4	no	—
50	C₂×#63214	C₂³	1	4	yes	—
51	C₄×#3622	C₂×C₄	1	4	yes	—
52	C₁₆×D₁₇	—	2	16	no	—
53	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
58	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
60	C₂×C₄×D₁₇	—	2	4	no	—
61	C₂×C₄×D₁₇	C₁₆	1	8	yes	complemented
62	C₂×C₄×D₁₇	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
64	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₁₇	C₂×C₈	1	4	yes	—
66	C₂×C₄×D₁₇	M₁₆	1	4	yes	—
67	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₇	—	2	2	no	—
69	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
74	C₄²×C₁₇	C₂×C₈	1	2	yes	—
75	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
76	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
77	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
78	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
79	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
80	C₂×#3622	—	2	2	no	—
81	C₂×#3622	C₄⋊C₄	1	4	yes	—
82	C₂×#3622	C₂²×C₄	1	2	yes	—
83	C₂×#3622	C₂²×C₄	1	2	yes	—
84	C₈×D₁₇	—	2	8	no	—
85	C₈×D₁₇	C₄⋊C₄	1	8	yes	—
86	C₈×D₁₇	C₂×D₄	1	4	yes	—
87	C₈×D₁₇	C₂×Q₈	1	4	yes	—
88	C₁₆×C₁₇	—	2	4	no	—
89	#63214	—	2	4	no	—
90	C₂²×C₄⋊C₁₆	—	17	1	no	maximal, Sylow
91	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
92	C₂²×D₁₇	C₄⋊C₈	1	4	yes	—
93	C₂²×D₁₇	C₂×C₁₆	1	4	yes	—
94	C₂²×D₁₇	M₃₂	1	4	yes	—
95	C₂²×D₁₇	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₁₇	—	2	2	no	—
97	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	complemented
98	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₁₇	—	2	2	no	—
105	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
106	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	complemented
107	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
108	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
109	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
110	C₄×D₁₇	—	2	8	no	—
111	C₄×D₁₇	C₄⋊C₈	1	8	yes	—
112	C₄×D₁₇	C₂×C₁₆	1	16	yes	complemented
113	C₄×D₁₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₄×D₁₇	C₂×M₁₆	1	4	yes	—
115	C₈×C₁₇	—	2	2	no	—
116	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₈×C₁₇	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
119	#3622	—	2	2	no	—
120	#3622	C₂×C₄⋊C₄	1	4	yes	—
121	C₂³×C₁₆	—	17	2	no	—
122	C₂²×C₄×C₈	—	17	1	no	—
123	C₂×C₄⋊C₁₆	—	17	2	no	—
124	C₂×C₄⋊C₁₆	—	17	1	no	—
125	C₂×C₄⋊C₁₆	—	17	8	no	—
126	C₂×C₄⋊C₁₆	D₁₇	1	1	yes	characteristic, direct product factor
127	C₂²×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
128	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
129	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
130	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
131	C₂×D₁₇	C₄⋊C₁₆	1	16	yes	direct product factor
132	C₂×D₁₇	C₂×C₄⋊C₈	1	4	yes	—
133	C₂×D₁₇	C₂²×C₁₆	1	4	yes	—
134	C₂×D₁₇	C₂×M₃₂	1	4	yes	—
135	C₄×C₁₇	—	2	2	no	—
136	C₄×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
137	C₄×C₁₇	C₂²×C₁₆	1	4	yes	complemented
138	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
140	Dic₁₇	—	2	2	no	—
141	Dic₁₇	C₂×C₄⋊C₈	1	4	yes	—
142	Dic₁₇	C₂²×C₁₆	1	4	yes	complemented
143	C₂³×C₈	—	17	1	no	—
144	C₂³×C₈	—	17	1	no	—
145	C₂²×C₄²	—	17	1	no	—
146	C₂²×C₁₆	—	17	2	no	—
147	C₂²×C₁₆	—	17	8	no	—
148	C₂²×C₁₆	—	34	8	no	—
149	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
150	C₂×C₄×C₈	—	17	1	no	—
151	C₂×C₄×C₈	—	17	1	no	—
152	C₂×C₄×C₈	—	17	1	no	—
153	C₂×C₄×C₈	—	17	4	no	—
154	C₂×C₄×C₈	—	17	4	no	—
155	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
156	C₄⋊C₁₆	—	17	8	no	—
157	C₄⋊C₁₆	—	17	4	no	—
158	C₄⋊C₁₆	C₂×D₁₇	1	4	yes	direct product factor
159	C₂×C₁₇	C₂×C₄⋊C₁₆	1	4	yes	complemented
160	C₂×C₁₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
161	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
162	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
163	D₁₇	C₂×C₄⋊C₁₆	1	8	yes	direct product factor
164	C₂³×C₄	—	17	1	no	—
165	C₂³×C₄	—	17	1	no	—
166	C₂³×C₄	—	17	1	no	—
167	C₂²×C₈	—	17	1	no	—
168	C₂²×C₈	—	17	2	no	—
169	C₂²×C₈	—	17	4	no	—
170	C₂²×C₈	—	17	4	no	—
171	C₂²×C₈	—	17	2	no	—
172	C₂²×C₈	—	17	1	no	—
173	C₂²×C₈	—	17	4	no	—
174	C₂²×C₈	—	34	4	no	—
175	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
177	C₂×C₄²	—	17	1	no	—
178	C₂×C₄²	—	17	1	no	—
179	C₂×C₄²	—	17	1	no	—
180	C₂×C₄²	—	17	4	no	—
181	C₂×C₄²	—	17	4	no	—
182	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
183	C₂×C₁₆	—	2	4	no	—
184	C₂×C₁₆	—	17	4	no	—
185	C₂×C₁₆	—	34	16	no	—
186	C₂×C₁₆	—	34	4	no	—
187	C₂×C₁₆	C₂²×D₁₇	1	4	yes	—
188	C₄×C₈	—	17	4	no	—
189	C₄×C₈	—	17	2	no	—
190	C₄×C₈	—	17	2	no	—
191	C₄×C₈	—	17	4	no	—
192	C₄×C₈	C₄×D₁₇	1	2	yes	—
193	C₄×C₈	C₂²×D₁₇	1	2	yes	—
194	C₁₇	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
195	C₂⁴	—	17	1	no	—
196	C₂²×C₄	—	17	1	no	—
197	C₂²×C₄	—	17	1	no	—
198	C₂²×C₄	—	17	2	no	—
199	C₂²×C₄	—	17	8	no	—
200	C₂²×C₄	—	17	4	no	—
201	C₂²×C₄	—	17	4	no	—
202	C₂²×C₄	—	17	2	no	—
203	C₂²×C₄	—	17	4	no	—
204	C₂²×C₄	—	17	1	no	—
205	C₂²×C₄	—	17	4	no	—
206	C₂²×C₄	—	34	4	no	—
207	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
208	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	17	2	no	—
212	C₂×C₈	—	17	4	no	—
213	C₂×C₈	—	17	4	no	—
214	C₂×C₈	—	17	4	no	—
215	C₂×C₈	—	17	8	no	—
216	C₂×C₈	—	17	2	no	—
217	C₂×C₈	—	34	8	no	—
218	C₂×C₈	—	34	2	no	—
219	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
221	C₂×C₈	D₄×D₁₇	1	2	yes	—
222	C₂×C₈	Q₈×D₁₇	1	2	yes	—
223	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
224	C₄²	—	17	2	no	—
225	C₄²	—	17	4	no	—
226	C₄²	—	17	4	no	—
227	C₄²	—	17	2	no	—
228	C₄²	C₈×D₁₇	1	2	yes	—
229	C₄²	C₂×C₄×D₁₇	1	2	yes	—
230	C₁₆	—	2	4	no	—
231	C₁₆	—	34	4	no	—
232	C₂³	—	17	4	no	—
233	C₂³	—	17	4	no	—
234	C₂³	—	17	2	no	—
235	C₂³	—	17	4	no	—
236	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	17	4	no	—
239	C₂×C₄	—	17	2	no	—
240	C₂×C₄	—	17	2	no	—
241	C₂×C₄	—	17	4	no	—
242	C₂×C₄	—	17	16	no	—
243	C₂×C₄	—	17	4	no	—
244	C₂×C₄	—	17	4	no	—
245	C₂×C₄	—	17	8	no	—
246	C₂×C₄	—	17	2	no	—
247	C₂×C₄	—	34	8	no	—
248	C₂×C₄	—	34	2	no	—
249	C₂×C₄	C₁₆×D₁₇	1	4	yes	complemented
250	C₂×C₄	C₄⋊C₄×D₁₇	1	2	yes	—
251	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
252	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
254	C₂×C₄	M₁₆×D₁₇	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	17	4	no	—
258	C₈	—	34	2	no	—
259	C₈	C₄⋊C₄×D₁₇	1	2	yes	—
260	C₈	C₂×D₄×D₁₇	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
262	C₂²	—	17	4	no	—
263	C₂²	—	17	4	no	—
264	C₂²	—	17	4	no	—
265	C₂²	—	17	16	no	—
266	C₂²	C₄⋊C₈×D₁₇	1	2	yes	—
267	C₂²	C₂×C₁₆×D₁₇	1	2	yes	—
268	C₂²	M₃₂×D₁₇	1	2	yes	—
269	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	17	4	no	—
272	C₄	—	17	4	no	—
273	C₄	—	34	2	no	—
274	C₄	C₄⋊C₈×D₁₇	1	2	yes	—
275	C₄	C₂×C₁₆×D₁₇	1	4	yes	complemented
276	C₄	C₂×C₄⋊C₄×D₁₇	1	1	yes	characteristic
277	C₄	C₂×M₁₆×D₁₇	1	1	yes	characteristic
278	C₂	—	17	8	no	—
279	C₂	D₁₇×C₄⋊C₁₆	1	4	yes	direct product factor
280	C₂	C₂×C₄⋊C₈×D₁₇	1	1	yes	characteristic
281	C₂	C₂²×C₁₆×D₁₇	1	1	yes	characteristic
282	C₂	C₂×M₃₂×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₇
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₁₇×C₄⋊C₁₆	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₇
hypocenter	C₁₇	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊#63214	C₂	semidirect
4. C₂×#11762627	C₂	semidirect
5. D₁₇×C₄⋊C₁₆	C₂	direct (N × H)
6. C₂×C₁₇×C₄⋊C₁₆	C₂	semidirect
18. C₄⋊#63214	C₂²	semidirect
19. #11762627	C₂²	semidirect
22. C₁₇×C₄⋊C₁₆	C₂²	semidirect
61. C₂×C₄×D₁₇	C₁₆	semidirect
97. C₂×C₄×C₁₇	C₂×C₁₆	semidirect
106. C₂×Dic₁₇	C₂×C₁₆	semidirect
112. C₄×D₁₇	C₂×C₁₆	semidirect
126. C₂×C₄⋊C₁₆	D₁₇	direct (N × H)
131. C₂×D₁₇	C₄⋊C₁₆	direct (N × H)
137. C₄×C₁₇	C₂²×C₁₆	semidirect
142. Dic₁₇	C₂²×C₁₆	semidirect
158. C₄⋊C₁₆	C₂×D₁₇	direct (N × H)
159. C₂×C₁₇	C₂×C₄⋊C₁₆	semidirect
163. D₁₇	C₂×C₄⋊C₁₆	direct (N × H)
194. C₁₇	C₂²×C₄⋊C₁₆	semidirect
249. C₂×C₄	C₁₆×D₁₇	semidirect
275. C₄	C₂×C₁₆×D₁₇	semidirect
279. C₂	D₁₇×C₄⋊C₁₆	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×C₄×C₈×D₁₇	2
C₂×C₄⋊#63214	2
C₂×#11762627	2
D₁₇×C₄⋊C₁₆	2
C₂×C₁₇×C₄⋊C₁₆	2
C₂²×C₄⋊C₁₆	17

Subgroups — C2×D17×C4⋊C16 #17187297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₇×C₄⋊C₁₆ #17187297