Subgroups — #14539950

Order 2176.

334 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763341	C₂	1	1	yes	maximal, characteristic
2	#125×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763210	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763209	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763162	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763162	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763146	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763141	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763126	C₂	1	1	yes	maximal, characteristic
10	#11763121	C₂	1	1	yes	maximal, characteristic
11	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762861	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#125	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
18	#10587624	C₂²	1	1	yes	characteristic
19	#10587622	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₇	—	2	1	no	—
21	#10587612	—	2	1	no	—
22	#10587583	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₇	—	2	1	no	—
24	C₄⋊C₄×D₁₇	—	2	1	no	—
25	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
27	#10587576	C₂²	1	1	yes	characteristic
28	#10587576	C₂²	1	1	yes	characteristic
29	#10587576	C₂²	1	1	yes	characteristic
30	#10587575	—	2	1	no	—
31	#10587575	—	2	1	no	—
32	#10587569	—	2	1	no	—
33	#10587565	—	2	1	no	—
34	C₄²×D₁₇	C₂²	1	1	yes	characteristic
35	#10587556	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
37	C₁₇×#107	C₂²	1	1	yes	characteristic
38	C₁₇×#106	C₂²	1	1	yes	characteristic
39	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₇×#102	C₂²	1	1	yes	characteristic
41	C₁₇×#102	C₂²	1	1	yes	characteristic
42	#10587503	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	#10587491	C₂²	1	1	yes	characteristic
45	#10587491	C₂²	1	1	yes	characteristic
46	#10587489	C₂²	1	1	yes	characteristic
47	#10587488	C₂²	1	1	yes	characteristic
48	#10587487	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	#10587481	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂×C₂²⋊D₁₇	—	2	1	no	—
62	C₂×C₄×D₁₇	—	2	1	no	—
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	C₂³	1	1	yes	characteristic
77	C₂²⋊Dic₁₇	—	2	1	no	—
78	C₂²⋊Dic₁₇	—	2	1	no	—
79	C₂²⋊Dic₁₇	—	2	1	no	—
80	#63227	—	2	1	no	—
81	C₄⋊Dic₁₇	—	2	1	no	—
82	C₄⋊Dic₁₇	—	2	1	no	—
83	C₄⋊Dic₁₇	—	2	1	no	—
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
88	#63225	—	2	1	no	—
89	#63225	—	2	1	no	—
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	C₄×Dic₁₇	—	2	1	no	—
97	C₄×Dic₁₇	D₄	1	1	yes	characteristic
98	C₄×Dic₁₇	D₄	1	1	yes	characteristic
99	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂³×C₁₇	—	2	1	no	—
103	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
104	D₄×C₁₇	—	2	1	no	—
105	D₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	4	2	no	—
110	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
113	C₂²⋊D₁₇	—	4	2	no	—
114	C₂×Dic₁₇	—	2	1	no	—
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
121	C₄×D₁₇	—	2	2	no	—
122	C₄×D₁₇	—	2	1	no	—
123	C₄×D₁₇	—	2	1	no	—
124	C₄×D₁₇	—	4	1	no	—
125	C₄×D₁₇	—	4	2	no	—
126	C₄×D₁₇	—	4	2	no	—
127	Dic₃₄	—	2	1	no	—
128	Dic₃₄	—	2	1	no	—
129	C₈×C₁₇	—	2	1	no	—
130	C₈×C₁₇	—	2	1	no	—
131	#3622	—	2	1	no	—
132	#3622	—	2	1	no	—
133	#3232	—	17	1	no	maximal, Sylow
134	C₂²×C₁₇	—	4	1	no	—
135	C₂²×C₁₇	—	4	1	no	—
136	C₂²×C₁₇	—	4	1	no	—
137	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₇	—	2	1	no	—
139	C₂×D₁₇	—	2	1	no	—
140	C₂×D₁₇	#135	1	2	yes	—
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	4	1	no	—
144	C₄×C₁₇	—	4	1	no	—
145	C₄×C₁₇	—	4	1	no	—
146	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₁₇	—	2	1	no	—
149	Dic₁₇	—	2	1	no	—
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	4	1	no	—
152	Dic₁₇	—	4	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	#529	—	17	1	no	—
156	C₂×#125	—	17	1	no	—
157	#499	—	17	1	no	—
158	#499	—	17	1	no	—
159	#488	—	17	1	no	—
160	#488	—	17	1	no	—
161	#484	—	17	1	no	—
162	#483	—	17	1	no	—
163	#481	—	17	1	no	—
164	#480	—	17	1	no	—
165	#479	—	17	1	no	—
166	#479	—	17	1	no	—
167	#478	—	17	1	no	—
168	#478	D₁₇	1	1	yes	characteristic, complemented
169	#431	—	17	1	no	—
170	C₂×C₁₇	—	8	1	no	—
171	C₂×C₁₇	#536	1	1	yes	characteristic
172	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
173	C₂×C₁₇	#575	1	1	yes	characteristic
174	D₁₇	—	2	2	no	—
175	#126	—	34	1	no	—
176	#125	—	17	1	no	—
177	#125	—	17	1	no	—
178	#125	—	17	1	no	—
179	#125	—	17	1	no	—
180	#125	—	34	1	no	—
181	#125	—	34	1	no	—
182	#125	C₂×D₁₇	1	1	yes	characteristic
183	#124	—	17	1	no	—
184	#123	—	34	1	no	—
185	C₂²⋊Q₈	—	17	1	no	—
186	#121	—	17	1	no	—
187	C₄×Q₈	—	17	1	no	—
188	C₄×D₄	—	34	1	no	—
189	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
190	#117	—	34	1	no	—
191	C₂×C₄⋊C₄	—	17	1	no	—
192	C₂×C₄⋊C₄	—	17	1	no	—
193	C₂×C₄⋊C₄	—	34	1	no	—
194	C₂×C₄⋊C₄	—	34	1	no	—
195	C₂×C₄²	—	17	1	no	—
196	#107	—	17	1	no	—
197	#107	—	17	1	no	—
198	#107	—	17	1	no	—
199	#107	C₂×D₁₇	1	1	yes	characteristic
200	#106	—	17	1	no	—
201	#106	—	17	1	no	—
202	#106	—	17	1	no	—
203	#106	C₂×D₁₇	1	1	yes	characteristic
204	C₄⋊C₈	—	17	1	no	—
205	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
206	#103	—	17	1	no	—
207	#103	—	17	1	no	—
208	#103	—	17	1	no	—
209	#103	—	17	1	no	—
210	#102	—	17	1	no	—
211	#102	—	17	1	no	—
212	#102	C₂×D₁₇	1	1	yes	characteristic
213	#102	C₂×D₁₇	1	1	yes	characteristic
214	C₂²⋊C₈	—	17	1	no	—
215	C₂²⋊C₈	—	17	1	no	—
216	C₄×C₈	—	17	1	no	—
217	C₄×C₈	—	17	1	no	—
218	C₁₇	#3232	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₈	—	17	1	no	—
220	C₂×D₄	—	34	1	no	—
221	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	17	1	no	—
224	C₂²×C₄	—	17	1	no	—
225	C₂²×C₄	—	17	1	no	—
226	C₂²×C₄	—	34	1	no	—
227	C₂²×C₄	—	34	1	no	—
228	C₂×C₈	—	17	1	no	—
229	C₂×C₈	—	17	1	no	—
230	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	17	1	no	—
235	C₄⋊C₄	—	17	1	no	—
236	C₄⋊C₄	—	17	1	no	—
237	C₄⋊C₄	—	34	1	no	—
238	C₄⋊C₄	—	34	1	no	—
239	C₄⋊C₄	—	34	1	no	—
240	C₄⋊C₄	—	34	1	no	—
241	C₄⋊C₄	—	34	1	no	—
242	C₄⋊C₄	—	34	1	no	—
243	C₄⋊C₄	—	34	1	no	—
244	C₄⋊C₄	—	34	1	no	—
245	C₄⋊C₄	—	34	1	no	—
246	C₄⋊C₄	—	34	1	no	—
247	C₄⋊C₄	—	34	1	no	—
248	C₄⋊C₄	—	34	1	no	—
249	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	34	1	no	—
254	C₂²⋊C₄	—	34	1	no	—
255	C₂²⋊C₄	—	34	1	no	—
256	C₂²⋊C₄	—	34	1	no	—
257	C₄²	—	17	1	no	—
258	C₄²	—	17	1	no	—
259	C₄²	—	17	1	no	—
260	C₄²	—	34	1	no	—
261	C₄²	C₂²×D₁₇	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	—	17	1	no	—
264	Q₈	—	34	1	no	—
265	Q₈	—	34	1	no	—
266	D₄	—	2	1	no	—
267	D₄	—	2	1	no	—
268	D₄	—	68	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	17	1	no	—
274	C₂×C₄	—	17	1	no	—
275	C₂×C₄	—	17	1	no	—
276	C₂×C₄	—	34	2	no	—
277	C₂×C₄	—	34	1	no	—
278	C₂×C₄	—	34	1	no	—
279	C₂×C₄	—	34	1	no	—
280	C₂×C₄	—	34	1	no	—
281	C₂×C₄	—	34	1	no	—
282	C₂×C₄	—	34	1	no	—
283	C₂×C₄	—	68	1	no	—
284	C₂×C₄	—	68	2	no	—
285	C₂×C₄	—	68	2	no	—
286	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
287	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	34	1	no	—
292	C₈	—	34	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	17	2	no	—
297	C₂²	—	34	1	no	—
298	C₂²	—	34	1	no	—
299	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	34	1	no	—
306	C₄	—	34	1	no	—
307	C₄	—	34	1	no	—
308	C₄	—	68	1	no	—
309	C₄	—	68	1	no	—
310	C₄	—	68	1	no	—
311	C₄	—	68	1	no	—
312	C₄	#10587694	1	1	yes	characteristic
313	C₄	#10587697	1	1	yes	characteristic
314	C₂	—	8	1	no	—
315	C₂	—	34	2	no	—
316	C₂	#11763382	1	1	yes	characteristic
317	C₂	#135×D₁₇	1	1	yes	characteristic
318	C₂	#11763605	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#478	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#10587624	C₂²
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3232
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
2. #125×D₁₇	C₂	semidirect
3. #11763210	C₂	semidirect
4. #11763209	C₂	semidirect
5. #11763162	C₂	semidirect
6. #11763162	C₂	semidirect
7. #11763146	C₂	semidirect
8. #11763141	C₂	semidirect
11. #11763117	C₂	semidirect
12. #11763113	C₂	semidirect
13. #11763109	C₂	semidirect
14. #11762861	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
168. #478	D₁₇	semidirect
218. C₁₇	#3232	semidirect

Maximal subgroups

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

subgroup	index
#11763341	2
#125×D₁₇	2
#11763210	2
#11763209	2
#11763162	2
#11763146	2
#11763141	2
#11763126	2
#11763121	2
#11763117	2
#11763113	2
#11763109	2
#11762861	2
C₁₇×#478	2
#3232	17