Subgroups — #14540047

Order 2176.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763439	C₂	1	1	yes	maximal, characteristic
2	#11763420	C₂	1	1	yes	maximal, characteristic
3	#11763218	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763208	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763159	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763149	C₂	1	1	yes	maximal, characteristic
7	#11763148	C₂	1	1	yes	maximal, characteristic
8	#11763143	C₂	1	1	yes	maximal, characteristic
9	#11763138	C₂	1	1	yes	maximal, characteristic
10	#11763118	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763114	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763102	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762871	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#491	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	#10587632	C₂²	1	1	yes	characteristic
19	#10587632	C₂²	1	1	yes	characteristic
20	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	#10587584	—	2	1	no	—
23	#10587584	—	2	1	no	—
24	#10587583	—	2	1	no	—
25	#10587582	—	2	1	no	—
26	#10587578	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
28	#10587576	C₂²	1	1	yes	characteristic
29	#10587575	—	2	1	no	—
30	#10587575	—	2	1	no	—
31	#10587575	—	2	1	no	—
32	#10587574	—	2	1	no	—
33	#10587574	C₂²	1	1	yes	characteristic
34	#10587558	C₂²	1	1	yes	characteristic
35	#10587556	C₂²	1	1	yes	characteristic
36	#10587555	C₂²	1	1	yes	characteristic
37	C₁₇×#103	C₂²	1	1	yes	characteristic
38	C₁₇×#103	C₂²	1	1	yes	characteristic
39	C₁₇×#103	C₂²	1	1	yes	characteristic
40	C₁₇×#103	C₂²	1	1	yes	characteristic
41	C₁₇×#97	C₂²	1	1	yes	characteristic
42	#10587506	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587488	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	#10587485	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×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₇	—	2	1	no	—
62	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₄	—	2	1	no	—
64	C₂×Dic₃₄	—	2	1	no	—
65	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₇	—	2	1	no	—
68	C₄⋊C₄×C₁₇	—	2	1	no	—
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₄²×C₁₇	C₂³	1	1	yes	characteristic
74	#63227	—	2	1	no	—
75	#63227	—	2	1	no	—
76	#63227	—	2	1	no	—
77	#63227	D₄	1	1	yes	characteristic
78	#63227	D₄	1	1	yes	characteristic
79	C₄⋊Dic₁₇	—	2	1	no	—
80	C₄⋊Dic₁₇	—	2	1	no	—
81	C₄⋊Dic₁₇	—	2	1	no	—
82	C₄⋊Dic₁₇	—	2	1	no	—
83	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
87	#63225	—	2	1	no	—
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	D₄	1	1	yes	characteristic
95	#63225	D₄	1	1	yes	characteristic
96	C₄×Dic₁₇	—	2	1	no	—
97	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
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	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₁₇	—	2	1	no	—
104	Q₈×C₁₇	—	2	1	no	—
105	Q₈×C₁₇	—	2	1	no	—
106	Q₈×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
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	1	no	—
122	C₄×D₁₇	—	2	1	no	—
123	C₄×D₁₇	—	8	1	no	—
124	Dic₃₄	—	8	1	no	—
125	Dic₃₄	—	8	1	no	—
126	C₈×C₁₇	—	2	1	no	—
127	C₈×C₁₇	—	2	1	no	—
128	#3622	—	2	1	no	—
129	#3622	—	2	1	no	—
130	#3242	—	17	1	no	maximal, Sylow
131	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₇	—	2	1	no	—
133	C₂×D₁₇	—	2	1	no	—
134	C₂×D₁₇	—	2	1	no	—
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	4	1	no	—
138	C₄×C₁₇	—	4	1	no	—
139	C₄×C₁₇	—	4	1	no	—
140	C₄×C₁₇	—	4	1	no	—
141	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
142	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
143	Dic₁₇	—	2	1	no	—
144	Dic₁₇	—	2	1	no	—
145	Dic₁₇	—	2	1	no	—
146	Dic₁₇	—	4	1	no	—
147	Dic₁₇	—	4	1	no	—
148	Dic₁₇	—	4	1	no	—
149	Dic₁₇	—	4	1	no	—
150	#544	—	17	1	no	—
151	#541	—	17	1	no	—
152	#501	—	17	1	no	—
153	#499	—	17	1	no	—
154	#491	D₁₇	1	1	yes	characteristic, complemented
155	#487	—	17	1	no	—
156	#485	—	17	1	no	—
157	#485	—	17	1	no	—
158	#484	—	17	1	no	—
159	#483	—	17	1	no	—
160	#479	—	17	1	no	—
161	#479	—	17	1	no	—
162	#477	—	17	1	no	—
163	#475	—	17	1	no	—
164	#433	—	17	1	no	—
165	C₂×C₁₇	#536	1	1	yes	characteristic
166	C₂×C₁₇	#577	1	1	yes	characteristic
167	C₂×C₁₇	#578	1	1	yes	characteristic
168	D₁₇	—	4	1	no	—
169	C₄⋊Q₈	—	17	1	no	—
170	C₄⋊Q₈	—	17	1	no	—
171	C₄⋊Q₈	—	34	1	no	—
172	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
173	#126	—	34	1	no	—
174	#126	—	34	1	no	—
175	#125	—	17	1	no	—
176	#125	—	17	1	no	—
177	#125	—	34	1	no	—
178	#125	—	34	1	no	—
179	#125	—	34	1	no	—
180	#125	C₂×D₁₇	1	1	yes	characteristic
181	C₂²⋊Q₈	—	17	1	no	—
182	C₂²⋊Q₈	—	17	1	no	—
183	C₂²⋊Q₈	—	34	1	no	—
184	C₂²⋊Q₈	—	34	1	no	—
185	C₄×Q₈	—	17	1	no	—
186	C₄×Q₈	—	17	1	no	—
187	#117	—	17	1	no	—
188	#117	—	17	1	no	—
189	C₂×C₄⋊C₄	—	17	1	no	—
190	#107	—	17	1	no	—
191	#107	—	17	1	no	—
192	#107	—	17	1	no	—
193	#107	—	17	1	no	—
194	#106	—	17	1	no	—
195	#106	—	17	1	no	—
196	#106	—	17	1	no	—
197	#106	—	17	1	no	—
198	C₄⋊C₈	—	17	1	no	—
199	C₄⋊C₈	—	17	1	no	—
200	#103	—	17	1	no	—
201	#103	—	17	1	no	—
202	#103	—	17	1	no	—
203	#103	—	17	1	no	—
204	#103	C₂×D₁₇	1	1	yes	characteristic
205	#103	C₂×D₁₇	1	1	yes	characteristic
206	#103	C₂×D₁₇	1	1	yes	characteristic
207	#103	C₂×D₁₇	1	1	yes	characteristic
208	C₂²⋊C₈	—	17	1	no	—
209	C₂²⋊C₈	—	17	1	no	—
210	#97	C₂×D₁₇	1	1	yes	characteristic
211	C₄×C₈	—	17	1	no	—
212	C₁₇	#3242	1	1	yes	characteristic, complemented, Sylow
213	C₂×Q₈	—	34	1	no	—
214	C₂×Q₈	—	34	1	no	—
215	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
216	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
217	C₂²×C₄	—	17	1	no	—
218	C₂²×C₄	—	34	1	no	—
219	C₂×C₈	—	17	1	no	—
220	C₂×C₈	—	17	1	no	—
221	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	17	1	no	—
228	C₄⋊C₄	—	17	1	no	—
229	C₄⋊C₄	—	17	1	no	—
230	C₄⋊C₄	—	17	1	no	—
231	C₄⋊C₄	—	17	1	no	—
232	C₄⋊C₄	—	17	1	no	—
233	C₄⋊C₄	—	34	1	no	—
234	C₄⋊C₄	—	34	1	no	—
235	C₄⋊C₄	—	34	1	no	—
236	C₄⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
246	C₂²⋊C₄	—	17	1	no	—
247	C₂²⋊C₄	—	17	1	no	—
248	C₂²⋊C₄	—	34	1	no	—
249	C₂²⋊C₄	—	34	1	no	—
250	C₂²⋊C₄	—	34	1	no	—
251	C₄²	—	17	1	no	—
252	C₄²	—	34	1	no	—
253	C₄²	—	34	1	no	—
254	C₄²	—	34	1	no	—
255	C₄²	C₂²×D₁₇	1	1	yes	characteristic
256	C₂³	—	17	1	no	—
257	Q₈	—	2	1	no	—
258	Q₈	—	2	1	no	—
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	136	1	no	—
262	Q₈	—	136	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	17	1	no	—
268	C₂×C₄	—	17	1	no	—
269	C₂×C₄	—	17	1	no	—
270	C₂×C₄	—	34	1	no	—
271	C₂×C₄	—	34	1	no	—
272	C₂×C₄	—	34	1	no	—
273	C₂×C₄	—	34	1	no	—
274	C₂×C₄	—	34	1	no	—
275	C₂×C₄	—	34	1	no	—
276	C₂×C₄	—	136	1	no	—
277	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
278	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	34	1	no	—
283	C₈	—	34	1	no	—
284	C₂²	—	34	1	no	—
285	C₂²	—	34	1	no	—
286	C₂²	—	34	1	no	—
287	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	34	1	no	—
295	C₄	—	34	1	no	—
296	C₄	—	34	1	no	—
297	C₄	—	68	1	no	—
298	C₄	—	68	1	no	—
299	C₄	—	68	1	no	—
300	C₄	—	68	1	no	—
301	C₄	#10587697	1	1	yes	characteristic
302	C₄	#10587697	1	1	yes	characteristic
303	C₂	—	68	1	no	—
304	C₂	#11763379	1	1	yes	characteristic
305	C₂	#11763618	1	1	yes	characteristic
306	C₂	#11763625	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×#491	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂²×D₁₇	C₂×D₄
hypercenter	#491	D₁₇
hypocenter	C₁₇	#3242
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

Complements

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

N	H ≅ G/N	type
3. #11763218	C₂	semidirect
4. #11763208	C₂	semidirect
5. #11763159	C₂	semidirect
10. #11763118	C₂	semidirect
11. #11763114	C₂	semidirect
12. #11763102	C₂	semidirect
13. #11763092	C₂	semidirect
15. C₁₇×#491	C₂	semidirect
154. #491	D₁₇	semidirect
212. C₁₇	#3242	semidirect

Maximal subgroups

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

subgroup	index
#11763439	2
#11763420	2
#11763218	2
#11763208	2
#11763159	2
#11763149	2
#11763148	2
#11763143	2
#11763138	2
#11763118	2
#11763114	2
#11763102	2
#11763092	2
#11762871	2
C₁₇×#491	2
#3242	17