Subgroups — #14537696

Order 2176.

452 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#11763574	C₂	1	2	yes	maximal, complemented
3	#11762929	C₂	1	2	yes	maximal, complemented
4	#11762928	C₂	1	2	yes	maximal, complemented
5	#11762927	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762926	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762925	C₂	1	1	yes	maximal, characteristic, complemented
8	#108×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762824	C₂	1	2	yes	maximal, complemented
10	#11762763	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₁₇	C₂²	1	2	yes	—
13	C₁₇×#135	C₂²	1	1	yes	characteristic
14	C₁₇×#131	C₂²	1	2	yes	complemented
15	#10587636	C₂²	1	2	yes	—
16	#10587634	C₂²	1	2	yes	complemented
17	#10587605	C₄	1	1	yes	characteristic, complemented
18	Q₁₆×D₁₇	C₄	1	1	yes	characteristic, complemented
19	#10587602	C₄	1	2	yes	complemented
20	SD₁₆×D₁₇	C₄	1	2	yes	complemented
21	#10587598	C₄	1	1	yes	characteristic, complemented
22	D₈×D₁₇	C₄	1	1	yes	characteristic, complemented
23	#10587593	—	2	2	no	—
24	#10587593	C₂²	1	2	yes	complemented
25	M₁₆×D₁₇	—	2	2	no	—
26	M₁₆×D₁₇	C₂²	1	2	yes	—
27	#10587590	C₂²	1	1	yes	characteristic
28	#10587587	—	2	2	no	—
29	#10587587	C₂²	1	2	yes	complemented
30	C₂×#63218	—	2	2	no	—
31	C₂×#63218	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
33	#10587558	C₂²	1	1	yes	characteristic
34	C₁₇×#108	C₂²	1	1	yes	characteristic, complemented
35	C₁₇×#104	C₂²	1	2	yes	complemented
36	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
37	#10587508	C₂²	1	2	yes	complemented
38	#10587496	C₂²	1	2	yes	complemented
39	#10587493	C₂²	1	2	yes	complemented
40	#10587490	C₂²	1	1	yes	characteristic, complemented
41	#10587486	C₂²	1	1	yes	characteristic, complemented
42	C₈×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
43	#10587476	C₂²	1	2	yes	complemented
44	#10587474	C₂²	1	1	yes	characteristic, complemented
45	#41×C₁₇	C₂³	1	2	yes	—
46	#63256	C₂×C₄	1	2	yes	—
47	Q₈×D₁₇	C₂×C₄	1	2	yes	—
48	#63254	—	2	2	no	—
49	#63254	C₂×C₄	1	2	yes	—
50	D₄×D₁₇	—	2	2	no	—
51	D₄×D₁₇	C₂×C₄	1	2	yes	—
52	#63252	—	2	2	no	—
53	#63252	C₂³	1	2	yes	—
54	C₂×C₄×D₁₇	—	2	2	no	—
55	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
56	Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
57	SD₁₆×C₁₇	C₂×C₄	1	2	yes	—
58	D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
59	M₁₆×C₁₇	—	2	2	no	—
60	M₁₆×C₁₇	C₂³	1	2	yes	—
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
64	C₄²×C₁₇	C₂³	1	1	yes	characteristic
65	#63231	C₂×C₄	1	2	yes	—
66	#63230	C₂×C₄	1	2	yes	—
67	#63229	C₂×C₄	1	2	yes	—
68	#63228	C₂×C₄	1	2	yes	—
69	#63227	—	2	1	no	—
70	#63225	—	2	1	no	—
71	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
72	#63223	—	2	2	no	—
73	#63223	C₂³	1	2	yes	—
74	C₂×#3622	—	2	2	no	—
75	C₂×#3622	C₂³	1	1	yes	characteristic
76	C₂×#3622	C₂³	1	1	yes	characteristic
77	Dic₆₈	C₂×C₄	1	1	yes	characteristic
78	D₁₃₆	C₂×C₄	1	1	yes	characteristic
79	#63219	C₂×C₄	1	2	yes	—
80	#63218	—	2	2	no	—
81	#63218	—	2	2	no	—
82	#63218	—	2	2	no	—
83	#63218	D₄	1	1	yes	characteristic
84	#63218	D₄	1	1	yes	characteristic
85	#63218	D₄	1	1	yes	characteristic
86	#63218	D₄	1	1	yes	characteristic
87	C₈×D₁₇	—	2	2	no	—
88	C₈×D₁₇	—	2	2	no	—
89	C₈×D₁₇	—	2	2	no	—
90	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
91	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
92	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
93	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
94	C₂²×D₁₇	—	2	2	no	—
95	C₂²×D₁₇	#41	1	1	yes	characteristic
96	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
97	D₄×C₁₇	—	2	2	no	—
98	D₄×C₁₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₇	—	2	2	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₁₇	—	2	2	no	—
103	C₂²⋊D₁₇	—	2	2	no	—
104	C₂²⋊D₁₇	—	2	2	no	—
105	C₂×Dic₁₇	—	2	2	no	—
106	C₂×Dic₁₇	—	2	1	no	—
107	C₂×Dic₁₇	#41	1	1	yes	characteristic
108	D₆₈	—	2	2	no	—
109	D₆₈	C₂²×C₄	1	2	yes	—
110	C₄×D₁₇	—	2	2	no	—
111	C₄×D₁₇	—	2	2	no	—
112	C₄×D₁₇	—	2	2	no	—
113	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
114	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
116	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
117	Dic₃₄	—	2	2	no	—
118	Dic₃₄	C₂²×C₄	1	2	yes	—
119	C₈×C₁₇	—	2	2	no	—
120	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
121	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
122	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
124	#3622	—	2	2	no	—
125	#3622	C₂²×C₄	1	1	yes	characteristic
126	#3622	C₂²×C₄	1	1	yes	characteristic
127	#3622	C₂×D₄	1	1	yes	characteristic
128	#3622	C₂×D₄	1	1	yes	characteristic
129	#2957	—	17	1	no	maximal, Sylow
130	C₂²×C₁₇	—	2	2	no	—
131	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
132	C₂×D₁₇	—	2	2	no	—
133	C₂×D₁₇	—	4	1	no	—
134	C₂×D₁₇	—	8	2	no	—
135	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
136	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
137	C₄×C₁₇	—	2	2	no	—
138	C₄×C₁₇	—	4	1	no	—
139	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
141	Dic₁₇	—	2	2	no	—
142	Dic₁₇	—	4	1	no	—
143	Dic₁₇	C₄×D₄	1	1	yes	characteristic
144	Dic₁₇	C₄×D₄	1	1	yes	characteristic
145	C₂×#135	—	17	1	no	—
146	#568	—	17	2	no	—
147	#444	—	17	2	no	—
148	#444	—	17	2	no	—
149	#443	—	17	1	no	—
150	#443	—	17	1	no	—
151	#443	—	17	1	no	—
152	#443	D₁₇	1	1	yes	characteristic, complemented
153	C₂×#108	—	17	1	no	—
154	#421	—	17	2	no	—
155	#405	—	17	1	no	—
156	C₂×C₁₇	—	2	1	no	—
157	C₂×C₁₇	—	4	2	no	—
158	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
159	D₁₇	—	2	1	no	—
160	D₁₇	—	2	1	no	—
161	D₁₇	—	4	2	no	—
162	C₂×#41	—	17	2	no	—
163	#135	—	17	2	no	—
164	#135	—	17	1	no	—
165	#135	—	17	2	no	—
166	#135	—	17	1	no	—
167	#135	—	17	1	no	—
168	#135	C₂×D₁₇	1	1	yes	characteristic
169	C₂×Q₁₆	—	17	1	no	—
170	C₂×SD₁₆	—	17	2	no	—
171	C₂×D₈	—	17	1	no	—
172	#131	—	17	2	no	—
173	#131	—	17	2	no	—
174	#131	—	17	2	no	—
175	#131	—	34	2	no	—
176	#131	—	34	2	no	—
177	#131	C₂×D₁₇	1	2	yes	complemented
178	C₂×M₁₆	—	17	1	no	—
179	C₂×M₁₆	—	17	2	no	—
180	C₂×M₁₆	—	34	2	no	—
181	C₂×M₁₆	—	34	2	no	—
182	C₂²×C₈	—	17	1	no	—
183	#117	—	17	1	no	—
184	#108	—	17	2	no	—
185	#108	—	17	1	no	—
186	#108	C₂×D₁₇	1	1	yes	characteristic, complemented
187	#104	—	17	2	no	—
188	#104	—	17	2	no	—
189	#104	—	17	2	no	—
190	#104	C₂×D₁₇	1	2	yes	complemented
191	#97	—	17	1	no	—
192	#97	—	17	1	no	—
193	C₄×C₈	—	17	1	no	—
194	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
195	C₁₇	#2957	1	1	yes	characteristic, complemented, Sylow
196	#41	—	17	2	no	—
197	#41	—	17	2	no	—
198	#41	—	17	2	no	—
199	#41	—	34	2	no	—
200	#41	—	34	2	no	—
201	#41	C₂²×D₁₇	1	2	yes	—
202	C₂×Q₈	—	17	2	no	—
203	C₂×D₄	—	17	2	no	—
204	C₂×D₄	—	34	2	no	—
205	C₂²×C₄	—	17	1	no	—
206	C₂²×C₄	—	34	2	no	—
207	Q₁₆	—	17	2	no	—
208	Q₁₆	—	17	1	no	—
209	Q₁₆	C₄×D₁₇	1	1	yes	characteristic
210	SD₁₆	—	17	2	no	—
211	SD₁₆	—	17	2	no	—
212	SD₁₆	—	17	2	no	—
213	SD₁₆	C₄×D₁₇	1	2	yes	—
214	D₈	—	17	2	no	—
215	D₈	—	17	1	no	—
216	D₈	C₄×D₁₇	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	17	1	no	—
219	M₁₆	—	17	1	no	—
220	M₁₆	—	17	1	no	—
221	M₁₆	—	17	1	no	—
222	M₁₆	—	17	2	no	—
223	M₁₆	—	34	2	no	—
224	M₁₆	—	34	2	no	—
225	M₁₆	—	34	2	no	—
226	M₁₆	—	34	2	no	—
227	M₁₆	C₂²×D₁₇	1	2	yes	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	17	1	no	—
230	C₂×C₈	—	17	1	no	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	—	17	1	no	—
233	C₂×C₈	—	17	1	no	—
234	C₂×C₈	—	17	1	no	—
235	C₂×C₈	—	34	2	no	—
236	C₂×C₈	—	34	2	no	—
237	C₂×C₈	—	34	2	no	—
238	C₂×C₈	—	34	2	no	—
239	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₄⋊C₄	—	34	1	no	—
242	C₂²⋊C₄	—	34	1	no	—
243	C₄²	—	17	1	no	—
244	C₄²	C₂²×D₁₇	1	1	yes	characteristic
245	C₂³	—	17	1	no	—
246	C₂³	—	34	2	no	—
247	Q₈	—	17	2	no	—
248	Q₈	—	34	2	no	—
249	Q₈	C₂×C₄×D₁₇	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	17	2	no	—
252	D₄	—	34	2	no	—
253	D₄	—	34	2	no	—
254	D₄	—	34	2	no	—
255	D₄	—	34	2	no	—
256	D₄	C₂×C₄×D₁₇	1	2	yes	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	17	1	no	—
260	C₂×C₄	—	17	1	no	—
261	C₂×C₄	—	17	1	no	—
262	C₂×C₄	—	17	1	no	—
263	C₂×C₄	—	17	1	no	—
264	C₂×C₄	—	34	2	no	—
265	C₂×C₄	—	34	2	no	—
266	C₂×C₄	—	34	2	no	—
267	C₂×C₄	—	34	2	no	—
268	C₂×C₄	—	34	1	no	—
269	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	17	1	no	—
272	C₈	—	17	1	no	—
273	C₈	—	17	1	no	—
274	C₈	—	17	1	no	—
275	C₈	—	34	2	no	—
276	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
277	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
278	C₈	D₄×D₁₇	1	1	yes	characteristic
279	C₈	D₄×D₁₇	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	17	1	no	—
282	C₂²	—	17	1	no	—
283	C₂²	—	34	2	no	—
284	C₂²	—	68	1	no	—
285	C₂²	—	136	2	no	—
286	C₂²	#41×D₁₇	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	4	1	no	—
289	C₄	—	17	1	no	—
290	C₄	—	17	1	no	—
291	C₄	—	34	2	no	—
292	C₄	—	68	1	no	—
293	C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
294	C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	4	2	no	—
297	C₂	—	34	1	no	—
298	C₂	—	34	1	no	—
299	C₂	—	68	2	no	—
300	C₂	C₄×D₄×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#443	C₂
socle	C₂×C₁₇	C₂×C₄×D₄
prime radical	#135×D₁₇	C₂
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#443	D₁₇
hypocenter	C₁₇	#2957
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #11763574	C₂	semidirect
3. #11762929	C₂	semidirect
4. #11762928	C₂	semidirect
5. #11762927	C₂	semidirect
6. #11762926	C₂	semidirect
7. #11762925	C₂	semidirect
8. #108×D₁₇	C₂	semidirect
9. #11762824	C₂	semidirect
10. #11762763	C₂	semidirect
11. C₁₇×#443	C₂	semidirect
14. C₁₇×#131	C₂²	semidirect
16. #10587634	C₂²	semidirect
17. #10587605	C₄	semidirect
18. Q₁₆×D₁₇	C₄	semidirect
19. #10587602	C₄	semidirect
20. SD₁₆×D₁₇	C₄	semidirect
21. #10587598	C₄	semidirect
22. D₈×D₁₇	C₄	semidirect
24. #10587593	C₂²	semidirect
29. #10587587	C₂²	semidirect
34. C₁₇×#108	C₂²	semidirect
35. C₁₇×#104	C₂²	semidirect
36. C₄×C₈×C₁₇	C₂²	semidirect
37. #10587508	C₂²	semidirect
38. #10587496	C₂²	semidirect
39. #10587493	C₂²	semidirect
40. #10587490	C₂²	semidirect
41. #10587486	C₂²	semidirect
42. C₈×Dic₁₇	C₂²	semidirect
43. #10587476	C₂²	semidirect
44. #10587474	C₂²	semidirect
152. #443	D₁₇	semidirect
177. #131	C₂×D₁₇	semidirect
186. #108	C₂×D₁₇	semidirect
190. #104	C₂×D₁₇	semidirect
194. C₄×C₈	C₂×D₁₇	semidirect
195. C₁₇	#2957	semidirect

Maximal subgroups

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

subgroup	index
#135×D₁₇	2
#11763574	2
#11762929	2
#11762928	2
#11762927	2
#11762926	2
#11762925	2
#108×D₁₇	2
#11762824	2
#11762763	2
C₁₇×#443	2
#2957	17