Subgroups — #14541557

Order 2176.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763542	C₂	1	1	yes	maximal, characteristic
2	#11763538	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763172	C₂	1	1	yes	maximal, characteristic
4	#11763169	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763162	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763115	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763112	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763099	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763031	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763029	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763007	C₂	1	1	yes	maximal, characteristic
12	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762913	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762894	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
20	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
21	C₂×#63231	C₂²	1	1	yes	characteristic
22	C₂×#63230	C₂²	1	1	yes	characteristic
23	#10587626	C₂²	1	1	yes	characteristic
24	#10587622	C₂²	1	1	yes	characteristic
25	#10587615	—	2	1	no	—
26	C₄×C₂²⋊D₁₇	—	2	1	no	—
27	#10587579	C₂²	1	1	yes	characteristic
28	#10587576	—	2	1	no	—
29	#10587576	—	2	1	no	—
30	#10587576	C₂²	1	1	yes	characteristic
31	#10587576	C₂²	1	1	yes	characteristic
32	#10587573	C₂²	1	1	yes	characteristic
33	#10587571	—	2	1	no	—
34	#10587569	—	2	1	no	—
35	#10587561	C₂²	1	1	yes	characteristic
36	#10587555	—	2	1	no	—
37	#10587555	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₄	—	2	1	no	—
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#103	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×#97	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587487	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587480	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×D₆₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×Dic₃₄	—	2	1	no	—
67	C₂×Dic₃₄	D₄	1	1	yes	characteristic
68	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
69	SD₁₆×C₁₇	—	2	2	no	—
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	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	—	2	1	no	—
77	C₄²×C₁₇	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₁₇	—	2	1	no	—
79	C₂²⋊Dic₁₇	—	2	1	no	—
80	#63231	—	2	2	no	—
81	#63230	—	2	2	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	—	2	1	no	—
89	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
91	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
101	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
102	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂³×C₁₇	—	2	1	no	—
106	C₂²×D₁₇	—	2	1	no	—
107	Q₈×C₁₇	—	2	1	no	—
108	Q₈×C₁₇	#41	1	2	yes	—
109	D₄×C₁₇	—	2	1	no	—
110	D₄×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	—	8	1	no	—
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₇	—	4	2	no	—
119	C₂²⋊D₁₇	—	8	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	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	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	D₆₈	—	2	1	no	—
128	D₆₈	—	2	1	no	—
129	C₄×D₁₇	—	8	1	no	—
130	Dic₃₄	—	2	1	no	—
131	Dic₃₄	—	2	1	no	—
132	Dic₃₄	—	4	1	no	—
133	Dic₃₄	—	4	2	no	—
134	C₈×C₁₇	—	2	1	no	—
135	C₈×C₁₇	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3371	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	—	4	1	no	—
140	C₂²×C₁₇	—	4	1	no	—
141	C₂²×C₁₇	—	4	1	no	—
142	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₇	—	4	1	no	—
144	C₂×D₁₇	—	4	1	no	—
145	C₂×D₁₇	—	4	1	no	—
146	C₄×C₁₇	—	2	2	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	—	4	1	no	—
151	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
152	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₁₇	—	2	1	no	—
154	Dic₁₇	—	2	1	no	—
155	Dic₁₇	—	2	1	no	—
156	Dic₁₇	—	2	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	#559	—	17	1	no	—
161	#557	—	17	1	no	—
162	#490	—	17	1	no	—
163	#489	—	17	1	no	—
164	#488	—	17	1	no	—
165	#479	—	17	1	no	—
166	#478	—	17	1	no	—
167	#476	—	17	1	no	—
168	#464	—	17	1	no	—
169	#464	—	17	1	no	—
170	#460	—	17	1	no	—
171	#446	—	17	1	no	—
172	#441	—	17	1	no	—
173	#440	D₁₇	1	1	yes	characteristic, complemented
174	C₄×SD₁₆	—	17	1	no	—
175	C₂×C₁₇	—	8	1	no	—
176	C₂×C₁₇	#548	1	1	yes	characteristic
177	C₂×C₁₇	#575	1	1	yes	characteristic
178	C₂×C₁₇	#577	1	1	yes	characteristic
179	D₁₇	—	8	1	no	—
180	C₂×Q₁₆	—	17	1	no	—
181	C₂×SD₁₆	—	17	1	no	—
182	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
183	C₄⋊Q₈	—	17	1	no	—
184	C₄⋊Q₈	—	34	1	no	—
185	#127	—	17	1	no	—
186	#125	—	17	1	no	—
187	#125	—	17	1	no	—
188	#125	—	34	1	no	—
189	#125	—	34	1	no	—
190	#124	—	17	1	no	—
191	#124	—	17	1	no	—
192	#124	—	34	1	no	—
193	#123	—	34	1	no	—
194	#123	—	34	1	no	—
195	C₄×Q₈	—	17	1	no	—
196	C₄×Q₈	—	17	1	no	—
197	C₄×Q₈	—	17	1	no	—
198	C₄×Q₈	—	17	1	no	—
199	C₄×Q₈	—	34	1	no	—
200	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
201	C₄×D₄	—	17	1	no	—
202	C₄×D₄	—	34	1	no	—
203	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
204	#107	C₂×D₁₇	1	1	yes	characteristic
205	#106	—	17	1	no	—
206	#106	—	17	1	no	—
207	C₄⋊C₈	—	17	1	no	—
208	C₄⋊C₈	—	17	1	no	—
209	C₄⋊C₈	—	17	1	no	—
210	#103	—	17	1	no	—
211	#103	—	17	1	no	—
212	#103	—	17	1	no	—
213	#103	C₂×D₁₇	1	1	yes	characteristic
214	#102	—	17	1	no	—
215	#102	—	17	1	no	—
216	#102	—	17	1	no	—
217	#102	—	17	1	no	—
218	#102	—	17	1	no	—
219	#102	C₂×D₁₇	1	1	yes	characteristic
220	#97	—	17	1	no	—
221	#97	C₂×D₁₇	1	1	yes	characteristic
222	C₄×C₈	—	17	1	no	—
223	C₁₇	#3371	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	17	1	no	—
225	C₂×Q₈	—	17	1	no	—
226	C₂×Q₈	—	34	1	no	—
227	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
228	C₂×D₄	—	17	1	no	—
229	C₂×D₄	—	34	1	no	—
230	C₂×D₄	—	34	1	no	—
231	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	34	1	no	—
234	Q₁₆	—	34	2	no	—
235	SD₁₆	—	2	2	no	—
236	SD₁₆	—	34	2	no	—
237	C₂×C₈	—	17	1	no	—
238	C₂×C₈	—	17	1	no	—
239	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	17	1	no	—
243	C₄⋊C₄	—	17	1	no	—
244	C₄⋊C₄	—	17	1	no	—
245	C₄⋊C₄	—	17	1	no	—
246	C₄⋊C₄	—	17	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	—	34	1	no	—
254	C₄⋊C₄	—	34	1	no	—
255	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	34	1	no	—
259	C₂²⋊C₄	—	34	1	no	—
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	17	1	no	—
265	C₄²	—	17	1	no	—
266	C₄²	—	34	1	no	—
267	C₄²	—	34	1	no	—
268	C₄²	—	34	1	no	—
269	C₄²	C₂²×D₁₇	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	34	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	34	1	no	—
274	Q₈	—	34	1	no	—
275	Q₈	—	68	1	no	—
276	Q₈	—	68	2	no	—
277	Q₈	#63252	1	2	yes	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	34	1	no	—
281	D₄	—	34	1	no	—
282	D₄	—	68	2	no	—
283	D₄	—	136	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	17	1	no	—
289	C₂×C₄	—	17	1	no	—
290	C₂×C₄	—	17	1	no	—
291	C₂×C₄	—	17	1	no	—
292	C₂×C₄	—	34	1	no	—
293	C₂×C₄	—	34	1	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	136	1	no	—
296	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	34	1	no	—
302	C₈	—	34	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	68	1	no	—
307	C₂²	—	68	1	no	—
308	C₂²	—	68	1	no	—
309	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	34	1	no	—
316	C₄	—	34	1	no	—
317	C₄	—	34	1	no	—
318	C₄	—	34	1	no	—
319	C₄	—	68	1	no	—
320	C₄	—	68	1	no	—
321	C₄	—	68	1	no	—
322	C₄	C₂×#63252	1	1	yes	characteristic
323	C₄	#10587697	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	136	1	no	—
326	C₂	#11763468	1	1	yes	characteristic
327	C₂	#11763608	1	1	yes	characteristic
328	C₂	#11763619	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×#440	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#10587626	C₂²
hypercenter	#440	D₁₇
hypocenter	C₁₇	#3371
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #11763538	C₂	semidirect
4. #11763169	C₂	semidirect
5. #11763162	C₂	semidirect
6. #11763115	C₂	semidirect
7. #11763112	C₂	semidirect
8. #11763099	C₂	semidirect
9. #11763031	C₂	semidirect
10. #11763029	C₂	semidirect
12. #11762937	C₂	semidirect
13. #11762913	C₂	semidirect
14. #11762894	C₂	semidirect
15. C₁₇×#440	C₂	semidirect
173. #440	D₁₇	semidirect
223. C₁₇	#3371	semidirect

Maximal subgroups

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

subgroup	index
#11763542	2
#11763538	2
#11763172	2
#11763169	2
#11763162	2
#11763115	2
#11763112	2
#11763099	2
#11763031	2
#11763029	2
#11763007	2
#11762937	2
#11762913	2
#11762894	2
C₁₇×#440	2
#3371	17