Subgroups — #14537682

Order 2176.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763606	C₂	1	1	yes	maximal, characteristic
2	#131×D₁₇	C₂	1	2	yes	maximal, complemented
3	#11762931	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762930	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762928	C₂	1	2	yes	maximal, complemented
6	#11762927	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762924	C₂	1	2	yes	maximal, complemented
8	#11762856	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	#10587604	C₄	1	2	yes	complemented
18	#10587601	C₄	1	2	yes	complemented
19	#10587600	C₄	1	2	yes	complemented
20	#10587597	C₄	1	2	yes	complemented
21	#10587593	—	2	2	no	—
22	#10587593	C₂²	1	2	yes	complemented
23	M₁₆×D₁₇	—	2	2	no	—
24	M₁₆×D₁₇	C₂²	1	2	yes	—
25	#10587590	C₂²	1	1	yes	characteristic
26	#10587587	—	2	2	no	—
27	#10587587	C₂²	1	2	yes	complemented
28	C₂×#63218	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₁₇	—	2	2	no	—
30	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
31	#10587558	C₂²	1	1	yes	characteristic
32	C₁₇×#108	C₂²	1	1	yes	characteristic, complemented
33	C₁₇×#104	C₂²	1	2	yes	complemented
34	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
35	#10587508	C₂²	1	2	yes	complemented
36	#10587496	C₂²	1	2	yes	complemented
37	#10587493	C₂²	1	2	yes	complemented
38	#10587490	C₂²	1	1	yes	characteristic, complemented
39	#10587486	C₂²	1	1	yes	characteristic, complemented
40	C₈×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
41	#10587476	C₂²	1	2	yes	complemented
42	#10587474	C₂²	1	1	yes	characteristic, complemented
43	#41×C₁₇	C₂³	1	2	yes	—
44	#63256	C₂×C₄	1	2	yes	—
45	Q₈×D₁₇	C₂×C₄	1	2	yes	—
46	#63254	—	2	2	no	—
47	#63254	C₂×C₄	1	2	yes	—
48	D₄×D₁₇	—	2	2	no	—
49	D₄×D₁₇	C₂×C₄	1	2	yes	—
50	#63252	—	2	2	no	—
51	#63252	C₂³	1	2	yes	—
52	C₂×C₄×D₁₇	—	2	2	no	—
53	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
54	Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
55	SD₁₆×C₁₇	C₂×C₄	1	2	yes	—
56	D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
57	M₁₆×C₁₇	—	2	2	no	—
58	M₁₆×C₁₇	C₂³	1	2	yes	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₄²×C₁₇	C₂³	1	1	yes	characteristic
63	#63231	C₂×C₄	1	2	yes	—
64	#63230	C₂×C₄	1	2	yes	—
65	#63229	C₂×C₄	1	2	yes	—
66	#63228	C₂×C₄	1	2	yes	—
67	#63227	—	2	1	no	—
68	#63225	—	2	1	no	—
69	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
70	#63223	—	2	2	no	—
71	#63223	C₂³	1	2	yes	—
72	C₂×#3622	—	2	2	no	—
73	C₂×#3622	C₂³	1	1	yes	characteristic
74	C₂×#3622	C₂³	1	1	yes	characteristic
75	Dic₆₈	C₂×C₄	1	1	yes	characteristic
76	D₁₃₆	C₂×C₄	1	1	yes	characteristic
77	#63219	C₂×C₄	1	2	yes	—
78	#63218	—	2	2	no	—
79	#63218	—	2	2	no	—
80	#63218	—	2	2	no	—
81	#63218	C₂×C₄	1	2	yes	—
82	#63218	C₂×C₄	1	2	yes	—
83	C₈×D₁₇	—	2	2	no	—
84	C₈×D₁₇	—	2	2	no	—
85	C₈×D₁₇	—	2	2	no	—
86	C₈×D₁₇	D₄	1	2	yes	—
87	C₈×D₁₇	D₄	1	2	yes	—
88	C₂²×D₁₇	—	2	2	no	—
89	C₂²×D₁₇	#41	1	1	yes	characteristic
90	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
91	D₄×C₁₇	—	2	2	no	—
92	D₄×C₁₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₇	—	2	2	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₁₇	—	2	2	no	—
97	C₂²⋊D₁₇	—	2	2	no	—
98	C₂²⋊D₁₇	—	2	2	no	—
99	C₂×Dic₁₇	—	2	2	no	—
100	C₂×Dic₁₇	—	2	1	no	—
101	C₂×Dic₁₇	#41	1	1	yes	characteristic
102	D₆₈	—	2	2	no	—
103	D₆₈	C₂²×C₄	1	2	yes	—
104	C₄×D₁₇	—	2	2	no	—
105	C₄×D₁₇	—	2	2	no	—
106	C₄×D₁₇	—	2	2	no	—
107	C₄×D₁₇	C₂²×C₄	1	2	yes	—
108	C₄×D₁₇	C₂×D₄	1	2	yes	—
109	Dic₃₄	—	2	2	no	—
110	Dic₃₄	C₂²×C₄	1	2	yes	—
111	C₈×C₁₇	—	2	2	no	—
112	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
113	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
116	#3622	—	2	2	no	—
117	#3622	C₂²×C₄	1	2	yes	—
118	#3622	C₂×D₄	1	2	yes	—
119	#2955	—	17	1	no	maximal, Sylow
120	C₂²×C₁₇	—	2	2	no	—
121	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
122	C₂×D₁₇	—	2	2	no	—
123	C₂×D₁₇	—	4	1	no	—
124	C₂×D₁₇	—	4	4	no	—
125	C₂×D₁₇	C₄×D₄	1	2	yes	—
126	C₄×C₁₇	—	2	2	no	—
127	C₄×C₁₇	—	4	1	no	—
128	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
130	Dic₁₇	—	2	2	no	—
131	Dic₁₇	—	4	1	no	—
132	Dic₁₇	C₄×D₄	1	2	yes	—
133	#575	—	17	1	no	—
134	C₂×#131	—	17	2	no	—
135	#444	—	17	1	no	—
136	#444	—	17	1	no	—
137	#444	—	17	2	no	—
138	#443	—	17	2	no	—
139	#443	—	17	1	no	—
140	#443	D₁₇	1	1	yes	characteristic, complemented
141	#430	—	17	1	no	—
142	#421	—	17	2	no	—
143	#405	—	17	1	no	—
144	C₂×C₁₇	—	2	1	no	—
145	C₂×C₁₇	—	4	2	no	—
146	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
147	D₁₇	—	2	2	no	—
148	D₁₇	—	4	2	no	—
149	C₂×#41	—	17	2	no	—
150	#137	—	17	2	no	—
151	#137	—	17	2	no	—
152	AGL(1,ℤ/8)	—	17	2	no	—
153	AGL(1,ℤ/8)	—	17	2	no	—
154	#135	—	17	2	no	—
155	#135	—	17	1	no	—
156	#135	C₂×D₁₇	1	1	yes	characteristic
157	#131	—	17	2	no	—
158	#131	—	17	2	no	—
159	#131	—	17	2	no	—
160	#131	—	34	2	no	—
161	#131	—	34	2	no	—
162	#131	C₂×D₁₇	1	2	yes	complemented
163	C₂×M₁₆	—	17	1	no	—
164	C₂×M₁₆	—	17	2	no	—
165	C₂×M₁₆	—	34	2	no	—
166	C₂²×C₈	—	17	1	no	—
167	C₂²×C₈	—	34	2	no	—
168	#117	—	17	1	no	—
169	#108	—	17	2	no	—
170	#108	—	17	1	no	—
171	#108	C₂×D₁₇	1	1	yes	characteristic, complemented
172	#104	—	17	2	no	—
173	#104	—	17	2	no	—
174	#104	—	17	2	no	—
175	#104	C₂×D₁₇	1	2	yes	complemented
176	#97	—	17	1	no	—
177	#97	—	17	1	no	—
178	C₄×C₈	—	17	1	no	—
179	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
180	C₁₇	#2955	1	1	yes	characteristic, complemented, Sylow
181	#41	—	17	2	no	—
182	#41	—	17	2	no	—
183	#41	—	17	2	no	—
184	#41	—	34	2	no	—
185	#41	—	34	2	no	—
186	#41	C₂²×D₁₇	1	2	yes	—
187	C₂×Q₈	—	17	2	no	—
188	C₂×D₄	—	17	2	no	—
189	C₂×D₄	—	34	2	no	—
190	C₂²×C₄	—	17	1	no	—
191	C₂²×C₄	—	34	2	no	—
192	Q₁₆	—	17	2	no	—
193	Q₁₆	—	17	1	no	—
194	Q₁₆	C₄×D₁₇	1	1	yes	characteristic
195	SD₁₆	—	17	2	no	—
196	SD₁₆	—	17	2	no	—
197	SD₁₆	—	17	2	no	—
198	SD₁₆	C₄×D₁₇	1	2	yes	—
199	D₈	—	17	2	no	—
200	D₈	—	17	1	no	—
201	D₈	C₄×D₁₇	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	17	2	no	—
204	M₁₆	—	17	2	no	—
205	M₁₆	—	17	2	no	—
206	M₁₆	—	34	2	no	—
207	M₁₆	—	34	2	no	—
208	M₁₆	—	34	2	no	—
209	M₁₆	—	34	2	no	—
210	M₁₆	C₂²×D₁₇	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	17	1	no	—
213	C₂×C₈	—	17	1	no	—
214	C₂×C₈	—	17	2	no	—
215	C₂×C₈	—	17	2	no	—
216	C₂×C₈	—	34	2	no	—
217	C₂×C₈	—	34	2	no	—
218	C₂×C₈	—	34	2	no	—
219	C₂×C₈	—	34	2	no	—
220	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
222	C₄⋊C₄	—	34	1	no	—
223	C₂²⋊C₄	—	34	1	no	—
224	C₄²	—	17	1	no	—
225	C₄²	C₂²×D₁₇	1	1	yes	characteristic
226	C₂³	—	17	1	no	—
227	C₂³	—	34	2	no	—
228	Q₈	—	17	2	no	—
229	Q₈	—	34	2	no	—
230	Q₈	C₂×C₄×D₁₇	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	17	2	no	—
233	D₄	—	34	2	no	—
234	D₄	—	34	2	no	—
235	D₄	—	34	2	no	—
236	D₄	—	34	2	no	—
237	D₄	C₂×C₄×D₁₇	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	17	1	no	—
241	C₂×C₄	—	17	2	no	—
242	C₂×C₄	—	17	2	no	—
243	C₂×C₄	—	34	2	no	—
244	C₂×C₄	—	34	2	no	—
245	C₂×C₄	—	34	2	no	—
246	C₂×C₄	—	34	2	no	—
247	C₂×C₄	—	34	1	no	—
248	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	17	2	no	—
251	C₈	—	17	2	no	—
252	C₈	—	34	2	no	—
253	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
254	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
255	C₈	D₄×D₁₇	1	1	yes	characteristic
256	C₈	D₄×D₁₇	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	17	2	no	—
259	C₂²	—	34	2	no	—
260	C₂²	—	68	1	no	—
261	C₂²	—	68	4	no	—
262	C₂²	#41×D₁₇	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	17	2	no	—
266	C₄	—	34	2	no	—
267	C₄	—	68	1	no	—
268	C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
269	C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	34	2	no	—
273	C₂	—	68	2	no	—
274	C₂	C₄×D₄×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#443	C₂
socle	C₂×C₁₇	C₂×C₄×D₄
prime radical	#11763606	C₂
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#443	D₁₇
hypocenter	C₁₇	#2955
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #131×D₁₇	C₂	semidirect
3. #11762931	C₂	semidirect
4. #11762930	C₂	semidirect
5. #11762928	C₂	semidirect
6. #11762927	C₂	semidirect
7. #11762924	C₂	semidirect
8. #11762856	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. #10587604	C₄	semidirect
18. #10587601	C₄	semidirect
19. #10587600	C₄	semidirect
20. #10587597	C₄	semidirect
22. #10587593	C₂²	semidirect
27. #10587587	C₂²	semidirect
32. C₁₇×#108	C₂²	semidirect
33. C₁₇×#104	C₂²	semidirect
34. C₄×C₈×C₁₇	C₂²	semidirect
35. #10587508	C₂²	semidirect
36. #10587496	C₂²	semidirect
37. #10587493	C₂²	semidirect
38. #10587490	C₂²	semidirect
39. #10587486	C₂²	semidirect
40. C₈×Dic₁₇	C₂²	semidirect
41. #10587476	C₂²	semidirect
42. #10587474	C₂²	semidirect
140. #443	D₁₇	semidirect
162. #131	C₂×D₁₇	semidirect
171. #108	C₂×D₁₇	semidirect
175. #104	C₂×D₁₇	semidirect
179. C₄×C₈	C₂×D₁₇	semidirect
180. C₁₇	#2955	semidirect

Maximal subgroups

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

subgroup	index
#11763606	2
#131×D₁₇	2
#11762931	2
#11762930	2
#11762928	2
#11762927	2
#11762924	2
#11762856	2
#11762824	2
#11762763	2
C₁₇×#443	2
#2955	17