Subgroups — #14541708

Order 2176.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763535	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763474	C₂	1	1	yes	maximal, characteristic
3	#11763217	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763215	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763213	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763112	C₂	1	1	yes	maximal, characteristic
7	#11763110	C₂	1	1	yes	maximal, characteristic
8	#11763102	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763088	C₂	1	1	yes	maximal, characteristic
11	#11762938	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762912	C₂	1	1	yes	maximal, characteristic
13	#11762907	C₂	1	1	yes	maximal, characteristic
14	#11762894	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	—	2	1	no	—
17	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
19	#10587632	—	2	1	no	—
20	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₂×#63230	C₂²	1	1	yes	characteristic
23	#10587582	—	2	1	no	—
24	#10587582	—	2	1	no	—
25	#10587579	C₂²	1	1	yes	characteristic
26	#10587579	C₂²	1	1	yes	characteristic
27	#10587576	—	2	1	no	—
28	#10587576	C₂²	1	1	yes	characteristic
29	#10587574	—	2	1	no	—
30	#10587574	C₂²	1	1	yes	characteristic
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587574	C₂²	1	1	yes	characteristic
33	C₄×D₆₈	C₂²	1	1	yes	characteristic
34	#10587556	—	2	1	no	—
35	#10587556	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₄	—	2	1	no	—
37	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
38	C₁₇×#107	C₂²	1	1	yes	characteristic
39	C₁₇×#107	C₂²	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×#103	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587492	C₂²	1	1	yes	characteristic
45	#10587492	C₂²	1	1	yes	characteristic
46	#10587489	C₂²	1	1	yes	characteristic
47	#10587488	C₂²	1	1	yes	characteristic
48	#10587486	C₂²	1	1	yes	characteristic
49	#10587485	C₂²	1	1	yes	characteristic
50	#10587485	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587479	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587478	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	Q₈×D₁₇	—	4	4	no	—
62	C₂×D₆₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×Dic₃₄	—	2	1	no	—
67	C₂×Dic₃₄	—	2	1	no	—
68	C₂×Dic₃₄	—	2	1	no	—
69	C₂×Dic₃₄	D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	—	2	1	no	—
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₄²×C₁₇	—	2	1	no	—
78	C₄²×C₁₇	C₂³	1	1	yes	characteristic
79	#63230	Q₈	1	4	yes	complemented
80	#63227	—	2	1	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	C₄⋊Dic₁₇	—	2	1	no	—
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
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	#63225	D₄	1	1	yes	characteristic
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂²×D₁₇	—	2	1	no	—
105	Q₈×C₁₇	—	2	1	no	—
106	Q₈×C₁₇	—	4	2	no	—
107	Q₈×C₁₇	C₂×Q₈	1	2	yes	complemented
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	—	2	1	no	—
112	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
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	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
122	D₆₈	—	2	1	no	—
123	D₆₈	C₂×Q₈	1	2	yes	—
124	C₄×D₁₇	—	4	2	no	—
125	C₄×D₁₇	—	4	2	no	—
126	C₄×D₁₇	—	4	2	no	—
127	Dic₃₄	—	2	2	no	—
128	Dic₃₄	—	4	2	no	—
129	Dic₃₄	—	4	2	no	—
130	Dic₃₄	—	4	2	no	—
131	C₈×C₁₇	—	2	1	no	—
132	C₈×C₁₇	—	2	1	no	—
133	#3622	—	2	1	no	—
134	#3622	C₂×Q₈	1	2	yes	complemented
135	#3386	—	17	1	no	maximal, Sylow
136	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₇	—	2	2	no	—
138	C₂×D₁₇	—	4	1	no	—
139	C₂×D₁₇	—	4	1	no	—
140	C₄×C₁₇	—	2	2	no	—
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	C₂²×Q₈	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	AGL(1,ℤ/8)	1	2	yes	—
155	#557	—	17	1	no	—
156	D₄×Q₈	—	17	1	no	—
157	#501	—	17	1	no	—
158	#501	—	17	1	no	—
159	#500	—	17	1	no	—
160	#478	—	17	1	no	—
161	#478	—	17	1	no	—
162	#477	—	17	1	no	—
163	#477	D₁₇	1	1	yes	characteristic, complemented
164	#475	—	17	1	no	—
165	#474	—	17	1	no	—
166	#446	—	17	1	no	—
167	#440	—	17	1	no	—
168	#440	—	17	1	no	—
169	C₄×SD₁₆	—	17	1	no	—
170	C₂×C₁₇	D₄×Q₈	1	1	yes	characteristic
171	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₁₇	#578	1	1	yes	characteristic
173	D₁₇	—	4	2	no	—
174	C₂²×Q₈	—	34	1	no	—
175	C₂×SD₁₆	—	17	1	no	—
176	C₄⋊Q₈	—	17	1	no	—
177	C₄⋊Q₈	—	17	1	no	—
178	C₄⋊Q₈	—	17	1	no	—
179	C₄⋊Q₈	—	34	1	no	—
180	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
181	#125	—	17	1	no	—
182	#125	—	17	1	no	—
183	#125	—	34	1	no	—
184	#125	—	34	1	no	—
185	C₂²⋊Q₈	—	34	1	no	—
186	C₂²⋊Q₈	—	34	1	no	—
187	C₂²⋊Q₈	—	34	1	no	—
188	C₄×Q₈	—	17	1	no	—
189	C₄×Q₈	—	17	1	no	—
190	C₄×Q₈	—	17	1	no	—
191	C₄×Q₈	—	34	1	no	—
192	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
193	C₄×D₄	—	17	1	no	—
194	C₄×D₄	—	17	1	no	—
195	C₄×D₄	—	17	1	no	—
196	#107	—	17	1	no	—
197	#107	—	17	1	no	—
198	#107	—	17	1	no	—
199	#107	—	17	1	no	—
200	#107	C₂×D₁₇	1	1	yes	characteristic
201	#107	C₂×D₁₇	1	1	yes	characteristic
202	#106	—	17	1	no	—
203	#106	—	17	1	no	—
204	#106	—	17	1	no	—
205	C₄⋊C₈	—	17	1	no	—
206	C₄⋊C₈	—	17	1	no	—
207	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
208	#103	—	17	1	no	—
209	#103	C₂×D₁₇	1	1	yes	characteristic
210	#103	C₂×D₁₇	1	1	yes	characteristic
211	#102	—	17	1	no	—
212	#102	—	17	1	no	—
213	#102	—	17	1	no	—
214	#97	—	17	1	no	—
215	#97	—	17	1	no	—
216	C₄×C₈	—	17	1	no	—
217	C₁₇	#3386	1	1	yes	characteristic, complemented, Sylow
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	—	17	1	no	—
220	C₂×Q₈	—	34	1	no	—
221	C₂×Q₈	—	34	1	no	—
222	C₂×Q₈	—	34	1	no	—
223	C₂×Q₈	—	68	4	no	—
224	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
225	C₂×D₄	—	17	1	no	—
226	C₂²×C₄	—	34	1	no	—
227	C₂²×C₄	—	34	1	no	—
228	C₂²×C₄	—	34	1	no	—
229	SD₁₆	—	17	4	no	—
230	C₂×C₈	—	17	1	no	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	17	1	no	—
237	C₄⋊C₄	—	17	1	no	—
238	C₄⋊C₄	—	17	1	no	—
239	C₄⋊C₄	—	17	1	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	17	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	—	34	1	no	—
250	C₄⋊C₄	—	34	1	no	—
251	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
254	C₂²⋊C₄	—	34	1	no	—
255	C₂²⋊C₄	—	34	1	no	—
256	C₂²⋊C₄	—	34	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	17	1	no	—
259	C₄²	—	17	1	no	—
260	C₄²	—	34	1	no	—
261	C₄²	—	34	1	no	—
262	C₄²	C₂²×D₁₇	1	1	yes	characteristic
263	C₂³	—	34	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	4	2	no	—
266	Q₈	—	34	2	no	—
267	Q₈	—	68	2	no	—
268	Q₈	—	68	2	no	—
269	Q₈	—	68	2	no	—
270	Q₈	Q₈×D₁₇	1	2	yes	complemented
271	D₄	—	17	2	no	—
272	D₄	—	34	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	17	1	no	—
278	C₂×C₄	—	17	1	no	—
279	C₂×C₄	—	17	1	no	—
280	C₂×C₄	—	17	1	no	—
281	C₂×C₄	—	34	1	no	—
282	C₂×C₄	—	34	1	no	—
283	C₂×C₄	—	34	1	no	—
284	C₂×C₄	—	68	2	no	—
285	C₂×C₄	—	68	2	no	—
286	C₂×C₄	—	68	2	no	—
287	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
288	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	17	2	no	—
293	C₈	—	34	1	no	—
294	C₂²	—	34	2	no	—
295	C₂²	—	68	1	no	—
296	C₂²	—	68	1	no	—
297	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	17	2	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	C₂×Q₈×D₁₇	1	1	yes	characteristic
312	C₄	#10587697	1	1	yes	characteristic
313	C₂	—	68	2	no	—
314	C₂	#11763472	1	1	yes	characteristic
315	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
316	C₂	#11763622	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×#477	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₆₈	C₂³
hypercenter	#477	D₁₇
hypocenter	C₁₇	#3386
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
1. #11763535	C₂	semidirect
3. #11763217	C₂	semidirect
4. #11763215	C₂	semidirect
5. #11763213	C₂	semidirect
8. #11763102	C₂	semidirect
9. #11763092	C₂	semidirect
11. #11762938	C₂	semidirect
15. C₁₇×#477	C₂	semidirect
79. #63230	Q₈	semidirect
107. Q₈×C₁₇	C₂×Q₈	semidirect
134. #3622	C₂×Q₈	semidirect
163. #477	D₁₇	semidirect
217. C₁₇	#3386	semidirect
270. Q₈	Q₈×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
#11763535	2
#11763474	2
#11763217	2
#11763215	2
#11763213	2
#11763112	2
#11763110	2
#11763102	2
#11763092	2
#11763088	2
#11762938	2
#11762912	2
#11762907	2
#11762894	2
C₁₇×#477	2
#3386	17