Subgroups — #17034699

Order 4352.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528933	C₂	1	2	yes	maximal, complemented
2	#14528908	C₂	1	2	yes	maximal, complemented
3	#14528888	C₂	1	2	yes	maximal
4	#14528834	C₂	1	1	yes	maximal, characteristic
5	#14528830	C₂	1	1	yes	maximal, characteristic
6	#14528668	C₂	1	2	yes	maximal, complemented
7	#14528584	C₂	1	2	yes	maximal
8	#14528539	C₂	1	2	yes	maximal, complemented
9	C₁₇×#2875	C₂	1	1	yes	maximal, characteristic
10	C₂×#10587490	C₂²	1	2	yes	—
11	#11762847	C₂²	1	2	yes	—
12	#11762793	C₂²	1	2	yes	—
13	#11762783	C₂²	1	2	yes	—
14	#11762764	C₂²	1	4	yes	—
15	#11762580	C₂²	1	4	yes	—
16	#11762569	C₂²	1	4	yes	—
17	#11762567	C₂²	1	2	yes	—
18	#11762566	C₂²	1	2	yes	—
19	#11762565	C₂²	1	4	yes	—
20	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
21	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
22	C₄×#41×C₁₇	C₂²	1	1	yes	characteristic
23	C₁₇×#405	C₄	1	2	yes	complemented
24	C₁₇×#405	C₄	1	2	yes	complemented
25	C₁₇×#405	C₂²	1	1	yes	characteristic
26	C₁₇×#405	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₁₇	C₄	1	2	yes	complemented
28	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
29	C₂×C₁₇×#97	C₄	1	2	yes	complemented
30	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
31	C₂×#41×C₁₇	C₂³	1	1	yes	characteristic
32	C₁₇×#131	—	2	4	no	—
33	C₁₇×#131	—	2	2	no	—
34	C₁₇×#131	—	2	2	no	—
35	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₁₇	—	2	1	no	—
44	C₄×D₄×C₁₇	—	2	2	no	—
45	C₄×D₄×C₁₇	—	2	1	no	—
46	C₁₇×#117	C₂×C₄	1	2	yes	—
47	C₁₇×#117	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
50	#10587612	C₂³	1	4	yes	—
51	C₂×#63223	C₂³	1	4	yes	—
52	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
53	C₁₇×#97	C₂×C₄	1	2	yes	—
54	C₁₇×#97	C₂×C₄	1	2	yes	—
55	C₁₇×#97	C₂×C₄	1	2	yes	—
56	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
57	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
58	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
62	#10587507	—	2	2	no	—
63	#10587505	—	2	4	no	—
64	#10587504	—	2	2	no	—
65	#10587490	—	2	2	no	—
66	#10587490	—	2	2	no	—
67	#10587489	—	2	2	no	—
68	#10587488	—	2	2	no	—
69	#10587486	—	2	4	no	—
70	C₈×Dic₁₇	—	2	4	no	—
71	#41×C₁₇	—	2	2	no	—
72	#41×C₁₇	—	2	2	no	—
73	C₂×Q₈×C₁₇	#41	1	1	yes	characteristic
74	C₂×D₄×C₁₇	#41	1	2	yes	—
75	C₂×D₄×C₁₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	—	2	2	no	—
77	C₂²×C₄×C₁₇	—	2	1	no	—
78	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
80	M₁₆×C₁₇	—	2	2	no	—
81	M₁₆×C₁₇	—	2	2	no	—
82	M₁₆×C₁₇	—	2	4	no	—
83	M₁₆×C₁₇	—	2	4	no	—
84	C₂×C₈×C₁₇	—	2	2	no	—
85	C₂×C₈×C₁₇	—	2	2	no	—
86	C₂×C₈×C₁₇	—	2	4	no	—
87	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₇	#41	1	4	yes	—
94	C₂×C₈×C₁₇	#41	1	2	yes	—
95	C₂×C₈×C₁₇	#41	1	2	yes	—
96	C₄⋊C₄×C₁₇	—	2	2	no	—
97	C₄⋊C₄×C₁₇	—	2	1	no	—
98	C₂²⋊C₄×C₁₇	—	2	2	no	—
99	C₂²⋊C₄×C₁₇	—	2	1	no	—
100	C₄²×C₁₇	—	2	2	no	—
101	C₄²×C₁₇	—	2	1	no	—
102	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²⋊Dic₁₇	—	2	4	no	—
106	C₄⋊Dic₁₇	—	2	4	no	—
107	C₄×Dic₁₇	—	2	4	no	—
108	#63223	—	2	4	no	—
109	#63223	—	2	4	no	—
110	C₂×#3622	—	2	4	no	—
111	#20788	—	17	1	no	maximal, Sylow
112	C₂³×C₁₇	#117	1	2	yes	—
113	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
114	Q₈×C₁₇	—	2	2	no	—
115	D₄×C₁₇	—	2	4	no	—
116	D₄×C₁₇	—	2	2	no	—
117	C₂×C₄×C₁₇	—	2	2	no	—
118	C₂×C₄×C₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	2	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	8	1	no	—
123	C₂×C₄×C₁₇	—	8	2	no	—
124	C₂×C₄×C₁₇	#117	1	2	yes	—
125	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
127	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₁₇	—	4	4	no	—
130	C₈×C₁₇	—	2	2	no	—
131	C₈×C₁₇	—	2	2	no	—
132	C₈×C₁₇	#117	1	1	yes	characteristic
133	C₈×C₁₇	#117	1	1	yes	characteristic
134	C₈×C₁₇	#117	1	2	yes	—
135	C₈×C₁₇	#117	1	2	yes	—
136	C₈×C₁₇	#117	1	2	yes	—
137	#3622	—	4	4	no	—
138	#2875	D₁₇	1	1	yes	characteristic
139	#1841	—	17	2	no	—
140	#1833	—	17	2	no	—
141	#1827	—	17	2	no	—
142	#1813	—	17	1	no	—
143	#1812	—	17	1	no	—
144	#1773	—	17	2	no	—
145	#1757	—	17	2	no	—
146	#1747	—	17	2	no	—
147	C₂²×C₁₇	—	2	2	no	—
148	C₂²×C₁₇	—	4	1	no	—
149	C₂²×C₁₇	—	4	2	no	—
150	C₂²×C₁₇	C₂×#117	1	1	yes	characteristic
151	C₂²×C₁₇	#529	1	2	yes	—
152	C₄×C₁₇	—	2	2	no	—
153	C₄×C₁₇	—	4	2	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
157	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
158	C₄×C₁₇	#529	1	2	yes	—
159	Dic₁₇	—	8	4	no	—
160	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
161	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
162	C₄×#41	C₂×D₁₇	1	1	yes	characteristic
163	C₂×#108	—	17	2	no	—
164	#427	—	17	2	no	—
165	#413	—	17	2	no	—
166	#410	—	17	2	no	—
167	#405	—	17	4	no	—
168	#405	Dic₁₇	1	2	yes	complemented
169	#405	Dic₁₇	1	2	yes	complemented
170	#405	C₂×D₁₇	1	1	yes	characteristic
171	#405	C₂×D₁₇	1	1	yes	characteristic
172	C₄×M₁₆	Dic₁₇	1	2	yes	complemented
173	C₄×M₁₆	C₂×D₁₇	1	1	yes	characteristic
174	C₂×#97	Dic₁₇	1	2	yes	complemented
175	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
176	#344	—	17	4	no	—
177	#339	—	17	4	no	—
178	#338	—	17	2	no	—
179	H(3,ℤ/4)	—	17	2	no	—
180	H(3,ℤ/4)	—	17	4	no	—
181	C₂×C₁₇	—	2	2	no	—
182	C₂×C₁₇	—	2	1	no	—
183	C₂×C₁₇	—	4	2	no	—
184	C₂×C₁₇	#2283	1	1	yes	characteristic
185	C₂×#41	C₂²×D₁₇	1	1	yes	characteristic
186	#131	—	2	4	no	—
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	C₂×M₁₆	—	17	4	no	—
190	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
191	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
192	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
193	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
194	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
195	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
196	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
198	C₄×Q₈	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	2	no	—
201	#117	—	17	4	no	—
202	#117	C₂×Dic₁₇	1	2	yes	—
203	#117	C₂²×D₁₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
205	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
206	#108	—	34	2	no	—
207	#108	—	34	2	no	—
208	#107	—	34	2	no	—
209	#106	—	34	2	no	—
210	#101	—	34	2	no	—
211	#100	—	34	2	no	—
212	#99	—	34	4	no	—
213	#97	—	34	4	no	—
214	#97	C₂×Dic₁₇	1	1	yes	characteristic
215	#97	C₂×Dic₁₇	1	2	yes	—
216	#97	C₂×Dic₁₇	1	2	yes	—
217	#97	C₂×Dic₁₇	1	2	yes	—
218	#97	C₂×Dic₁₇	1	1	yes	characteristic
219	#97	C₂×Dic₁₇	1	1	yes	characteristic
220	#97	C₂×Dic₁₇	1	1	yes	characteristic
221	C₄×C₈	—	34	4	no	—
222	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
223	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
224	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
225	C₁₇	#20788	1	1	yes	characteristic, complemented, Sylow
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#63252	1	1	yes	characteristic
229	C₂×D₄	#63252	1	1	yes	characteristic
230	C₂×D₄	#63252	1	2	yes	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂²×Dic₁₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	34	4	no	—
240	M₁₆	—	34	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	34	4	no	—
245	C₂×C₈	#63252	1	4	yes	—
246	C₂×C₈	#63252	1	2	yes	—
247	C₂×C₈	#63252	1	2	yes	—
248	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
249	C₂×C₈	C₂²×Dic₁₇	1	2	yes	—
250	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	34	4	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	34	4	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	34	4	no	—
263	C₄²	C₂²×Dic₁₇	1	2	yes	—
264	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
265	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
266	C₂³	#10587612	1	2	yes	—
267	C₂³	C₂×#63252	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	8	1	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	68	4	no	—
279	C₂×C₄	#10587612	1	2	yes	—
280	C₂×C₄	C₂×#63252	1	2	yes	—
281	C₂×C₄	C₂×#63252	1	2	yes	—
282	C₂×C₄	C₂×#63252	1	1	yes	characteristic
283	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	68	4	no	—
287	C₈	#10587612	1	1	yes	characteristic
288	C₈	#10587612	1	1	yes	characteristic
289	C₈	#10587612	1	2	yes	—
290	C₈	#10587612	1	2	yes	—
291	C₈	#10587612	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×#10587612	1	1	yes	characteristic
296	C₂²	#11763343	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	136	4	no	—
302	C₄	C₂×#10587612	1	1	yes	characteristic
303	C₄	C₂×#10587612	1	1	yes	characteristic
304	C₄	#11763343	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#14531182	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#10587612
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#2875	C₂
socle	C₂×C₁₇	#2283
prime radical	C₂×#41×C₁₇	C₂³
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	#2875	D₁₇
hypocenter	C₁₇	#20788
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2875

Complements

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

N	H ≅ G/N	type
1. #14528933	C₂	semidirect
2. #14528908	C₂	semidirect
6. #14528668	C₂	semidirect
8. #14528539	C₂	semidirect
23. C₁₇×#405	C₄	semidirect
24. C₁₇×#405	C₄	semidirect
27. C₄×M₁₆×C₁₇	C₄	semidirect
29. C₂×C₁₇×#97	C₄	semidirect
168. #405	Dic₁₇	semidirect
169. #405	Dic₁₇	semidirect
172. C₄×M₁₆	Dic₁₇	semidirect
174. C₂×#97	Dic₁₇	semidirect
225. C₁₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#14528933	2
#14528908	2
#14528888	2
#14528834	2
#14528830	2
#14528668	2
#14528584	2
#14528539	2
C₁₇×#2875	2
#20788	17