Subgroups — #14541397

Order 2176.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763527	C₂	1	1	yes	maximal, characteristic
2	#11763517	C₂	1	1	yes	maximal, characteristic
3	#11763206	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763148	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763141	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763140	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763103	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763066	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763065	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762901	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762884	C₂	1	1	yes	maximal, characteristic
13	#11762852	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₇×#117	C₂²	1	1	yes	characteristic
18	#10587632	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
20	C₂×#63231	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
23	#10587610	C₂²	1	1	yes	characteristic
24	C₂×#63223	C₂²	1	1	yes	characteristic
25	C₂×#63218	C₂²	1	1	yes	characteristic
26	#10587583	—	2	1	no	—
27	#10587583	C₂²	1	1	yes	characteristic
28	#10587578	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
30	#10587575	—	2	1	no	—
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587574	C₂²	1	1	yes	characteristic
33	#10587573	C₂²	1	1	yes	characteristic
34	#10587571	—	2	1	no	—
35	#10587567	—	2	1	no	—
36	#10587565	—	2	1	no	—
37	#10587564	—	2	1	no	—
38	#10587562	—	2	1	no	—
39	C₄×Dic₃₄	—	2	1	no	—
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587488	C₂²	1	1	yes	characteristic
48	#10587488	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₇	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	#10587479	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
62	C₂²×Dic₁₇	—	2	1	no	—
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
65	C₂×Dic₃₄	—	2	1	no	—
66	C₂×Dic₃₄	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₇	—	2	1	no	—
75	C₂²⋊C₄×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	—	2	1	no	—
77	C₂²⋊Dic₁₇	—	2	1	no	—
78	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
79	#63231	—	2	2	no	—
80	#63227	—	2	1	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	#63225	D₄	1	1	yes	characteristic
97	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	#63223	—	2	2	no	—
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	#63218	—	2	2	no	—
105	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₁₇	—	2	1	no	—
108	Q₈×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₁₇	—	2	2	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	8	1	no	—
123	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₇	—	2	1	no	—
127	C₄×D₁₇	—	2	1	no	—
128	C₄×D₁₇	—	8	1	no	—
129	Dic₃₄	—	2	1	no	—
130	Dic₃₄	—	2	1	no	—
131	Dic₃₄	—	4	2	no	—
132	Dic₃₄	—	8	1	no	—
133	C₈×C₁₇	—	2	1	no	—
134	C₈×C₁₇	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3622	#41	1	2	yes	—
137	#3356	—	17	1	no	maximal, Sylow
138	C₂²×C₁₇	—	2	1	no	—
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₇	—	2	1	no	—
143	C₂×D₁₇	—	2	1	no	—
144	C₂×D₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	4	1	no	—
147	C₄×C₁₇	—	4	1	no	—
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
151	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₁₇	—	2	1	no	—
153	Dic₁₇	—	2	1	no	—
154	Dic₁₇	—	4	1	no	—
155	Dic₁₇	—	4	1	no	—
156	Dic₁₇	—	4	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	#137	1	2	yes	—
159	#556	—	17	1	no	—
160	#554	—	17	1	no	—
161	#498	—	17	1	no	—
162	#485	—	17	1	no	—
163	#485	D₁₇	1	1	yes	characteristic, complemented
164	#483	—	17	1	no	—
165	#483	—	17	1	no	—
166	#477	—	17	1	no	—
167	#475	—	17	1	no	—
168	#470	—	17	1	no	—
169	#470	—	17	1	no	—
170	C₄×Q₁₆	—	17	1	no	—
171	#436	—	17	1	no	—
172	#428	—	17	1	no	—
173	#428	—	17	1	no	—
174	C₂×C₁₇	—	4	1	no	—
175	C₂×C₁₇	#547	1	1	yes	characteristic
176	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
177	C₂×C₁₇	#577	1	1	yes	characteristic
178	D₁₇	—	4	1	no	—
179	C₂×Q₁₆	—	17	1	no	—
180	C₂×M₁₆	—	17	1	no	—
181	C₂×M₁₆	—	17	1	no	—
182	C₄⋊Q₈	—	17	1	no	—
183	C₄⋊Q₈	—	17	1	no	—
184	#126	—	34	1	no	—
185	#126	—	34	1	no	—
186	#125	—	34	1	no	—
187	#124	—	34	1	no	—
188	C₂²⋊Q₈	—	17	1	no	—
189	C₂²⋊Q₈	—	17	1	no	—
190	C₂²⋊Q₈	—	17	1	no	—
191	C₂²⋊Q₈	—	34	1	no	—
192	C₂²⋊Q₈	—	34	1	no	—
193	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
194	C₄×Q₈	—	17	1	no	—
195	C₄×Q₈	—	17	1	no	—
196	C₄×Q₈	—	34	1	no	—
197	C₄×D₄	—	17	1	no	—
198	C₄×D₄	—	34	1	no	—
199	#117	—	17	1	no	—
200	#117	C₂×D₁₇	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	17	1	no	—
202	C₂×C₄⋊C₄	—	17	1	no	—
203	#107	—	17	1	no	—
204	#107	—	17	1	no	—
205	#107	C₂×D₁₇	1	1	yes	characteristic
206	#106	—	17	1	no	—
207	#106	—	17	1	no	—
208	#106	—	17	1	no	—
209	#106	—	17	1	no	—
210	#106	—	17	1	no	—
211	#106	C₂×D₁₇	1	1	yes	characteristic
212	C₄⋊C₈	—	17	1	no	—
213	#103	—	17	1	no	—
214	#103	—	17	1	no	—
215	#103	—	17	1	no	—
216	#103	—	17	1	no	—
217	#103	C₂×D₁₇	1	1	yes	characteristic
218	#103	C₂×D₁₇	1	1	yes	characteristic
219	C₂²⋊C₈	—	17	1	no	—
220	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
221	C₄×C₈	—	17	1	no	—
222	C₁₇	#3356	1	1	yes	characteristic, complemented, Sylow
223	C₂×Q₈	—	17	1	no	—
224	C₂×Q₈	—	34	1	no	—
225	C₂×Q₈	—	34	1	no	—
226	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
227	C₂×D₄	—	17	1	no	—
228	C₂²×C₄	—	17	1	no	—
229	C₂²×C₄	—	34	1	no	—
230	C₂²×C₄	—	34	1	no	—
231	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
232	Q₁₆	—	34	2	no	—
233	M₁₆	—	34	2	no	—
234	M₁₆	—	34	2	no	—
235	C₂×C₈	—	17	1	no	—
236	C₂×C₈	—	17	1	no	—
237	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	17	1	no	—
242	C₄⋊C₄	—	17	1	no	—
243	C₄⋊C₄	—	17	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	—	34	1	no	—
250	C₄⋊C₄	—	34	1	no	—
251	C₄⋊C₄	—	34	1	no	—
252	C₄⋊C₄	—	34	1	no	—
253	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	17	1	no	—
259	C₂²⋊C₄	—	17	1	no	—
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	—	34	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	17	1	no	—
266	C₄²	—	34	1	no	—
267	C₄²	—	34	1	no	—
268	C₄²	—	34	1	no	—
269	C₂³	—	17	1	no	—
270	C₂³	D₄×D₁₇	1	1	yes	characteristic
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	34	1	no	—
274	Q₈	—	34	1	no	—
275	Q₈	—	68	2	no	—
276	Q₈	—	136	1	no	—
277	D₄	—	34	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	17	1	no	—
285	C₂×C₄	—	17	1	no	—
286	C₂×C₄	—	17	1	no	—
287	C₂×C₄	—	34	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	34	1	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	—	34	1	no	—
293	C₂×C₄	—	136	1	no	—
294	C₂×C₄	—	136	1	no	—
295	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	17	2	no	—
300	C₈	—	34	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	34	1	no	—
305	C₂²	—	34	1	no	—
306	C₂²	—	34	1	no	—
307	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	17	2	no	—
314	C₄	—	34	1	no	—
315	C₄	—	34	1	no	—
316	C₄	—	68	1	no	—
317	C₄	—	68	1	no	—
318	C₄	—	68	1	no	—
319	C₄	—	68	1	no	—
320	C₄	#41×D₁₇	1	1	yes	characteristic
321	C₄	#10587697	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	68	1	no	—
324	C₂	#11763457	1	1	yes	characteristic
325	C₂	#137×D₁₇	1	1	yes	characteristic
326	C₂	#11763616	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×#485	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×C₂²⋊D₁₇	D₄
hypercenter	#485	D₁₇
hypocenter	C₁₇	#3356
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #11763206	C₂	semidirect
4. #11763148	C₂	semidirect
5. #11763141	C₂	semidirect
6. #11763140	C₂	semidirect
7. #11763103	C₂	semidirect
8. #11763092	C₂	semidirect
9. #11763066	C₂	semidirect
10. #11763065	C₂	semidirect
11. #11762901	C₂	semidirect
13. #11762852	C₂	semidirect
14. #11762850	C₂	semidirect
15. C₁₇×#485	C₂	semidirect
163. #485	D₁₇	semidirect
222. C₁₇	#3356	semidirect

Maximal subgroups

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

subgroup	index
#11763527	2
#11763517	2
#11763206	2
#11763148	2
#11763141	2
#11763140	2
#11763103	2
#11763092	2
#11763066	2
#11763065	2
#11762901	2
#11762884	2
#11762852	2
#11762850	2
C₁₇×#485	2
#3356	17