Subgroups — #14538821

Order 2176.

384 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763363	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
3	#11763146	C₂	1	1	yes	maximal, characteristic
4	#11763143	C₂	1	1	yes	maximal, characteristic
5	#11763103	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763102	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763018	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763017	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762966	C₂	1	1	yes	maximal, characteristic
10	#11762963	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#63231	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#63231	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762860	C₂	1	1	yes	maximal, characteristic
15	C₄×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	#10587632	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	#10587630	C₂²	1	1	yes	characteristic
23	C₂×#63231	C₂²	1	1	yes	characteristic
24	C₂×#63231	C₂²	1	1	yes	characteristic
25	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
26	#10587583	C₂²	1	1	yes	characteristic
27	#10587582	—	2	1	no	—
28	C₄⋊C₄×D₁₇	—	2	1	no	—
29	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
30	#10587575	—	2	1	no	—
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587573	—	2	1	no	—
33	#10587573	C₂²	1	1	yes	characteristic
34	#10587558	—	2	1	no	—
35	C₄²×D₁₇	—	2	1	no	—
36	C₄²×D₁₇	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₄	—	2	1	no	—
38	C₄×Dic₃₄	—	2	1	no	—
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
41	C₁₇×#107	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	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	Q₈×D₁₇	—	2	2	no	—
62	Q₈×D₁₇	—	2	2	no	—
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
67	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
68	C₂×Dic₃₄	—	2	1	no	—
69	C₂×Dic₃₄	—	2	1	no	—
70	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
71	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
72	Q₁₆×C₁₇	—	2	2	no	—
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₄²×C₁₇	—	2	1	no	—
80	C₄²×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	C₂³	1	1	yes	characteristic
82	#63231	—	2	2	no	—
83	#63231	—	2	2	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
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	—	2	1	no	—
97	#63225	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	D₄	1	1	yes	characteristic
103	C₄×Dic₁₇	D₄	1	1	yes	characteristic
104	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	Dic₆₈	—	2	2	no	—
108	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	#41	1	2	yes	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
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	C₄×D₁₇	—	2	4	no	—
128	C₄×D₁₇	—	2	1	no	—
129	C₄×D₁₇	—	2	1	no	—
130	C₄×D₁₇	—	4	1	no	—
131	C₄×D₁₇	—	4	2	no	—
132	C₄×D₁₇	Q₁₆	1	4	yes	complemented
133	Dic₃₄	—	2	4	no	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	2	1	no	—
136	Dic₃₄	—	2	1	no	—
137	Dic₃₄	—	4	2	no	—
138	Dic₃₄	#41	1	2	yes	—
139	C₈×C₁₇	—	2	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3622	—	2	1	no	—
143	#3104	—	17	1	no	maximal, Sylow
144	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₇	—	2	1	no	—
146	C₂×D₁₇	—	2	1	no	—
147	C₂×D₁₇	C₂×Q₁₆	1	2	yes	—
148	C₄×C₁₇	—	2	2	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	4	1	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
154	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
156	Dic₁₇	—	2	1	no	—
157	Dic₁₇	—	2	2	no	—
158	Dic₁₇	—	2	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	Dic₁₇	C₂×Q₁₆	1	2	yes	—
163	#533	—	17	1	no	—
164	C₂×C₄×Q₈	—	17	1	no	—
165	#484	—	17	1	no	—
166	#484	—	17	1	no	—
167	#477	—	17	1	no	—
168	#477	—	17	1	no	—
169	#462	—	17	1	no	—
170	#462	—	17	1	no	—
171	#451	—	17	1	no	—
172	#451	—	17	1	no	—
173	C₄×Q₁₆	—	17	1	no	—
174	C₄×Q₁₆	—	17	1	no	—
175	C₄×Q₁₆	—	17	1	no	—
176	C₄×Q₁₆	D₁₇	1	1	yes	characteristic, complemented
177	#431	—	17	1	no	—
178	C₂×C₁₇	#525	1	1	yes	characteristic
179	C₂×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
180	C₂×C₁₇	#575	1	1	yes	characteristic
181	D₁₇	—	2	2	no	—
182	C₂²×Q₈	—	17	1	no	—
183	C₂×Q₁₆	—	17	1	no	—
184	C₂×Q₁₆	—	17	1	no	—
185	C₂×Q₁₆	—	17	1	no	—
186	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
187	C₄⋊Q₈	—	17	1	no	—
188	C₄⋊Q₈	—	17	1	no	—
189	#125	—	34	1	no	—
190	C₂²⋊Q₈	—	17	1	no	—
191	C₂²⋊Q₈	—	17	1	no	—
192	C₂²⋊Q₈	—	34	1	no	—
193	C₄×Q₈	—	17	1	no	—
194	C₄×Q₈	—	17	1	no	—
195	C₄×Q₈	—	17	1	no	—
196	C₄×Q₈	—	17	1	no	—
197	C₄×Q₈	—	34	1	no	—
198	C₄×Q₈	—	34	1	no	—
199	C₄×Q₈	—	34	1	no	—
200	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
201	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
202	#117	—	34	1	no	—
203	C₂×C₄⋊C₄	—	17	1	no	—
204	C₂×C₄⋊C₄	—	34	1	no	—
205	C₂×C₄²	—	17	1	no	—
206	C₂×C₄²	—	34	1	no	—
207	#107	—	17	1	no	—
208	#107	—	17	1	no	—
209	#107	—	17	1	no	—
210	#107	C₂×D₁₇	1	1	yes	characteristic
211	C₄⋊C₈	—	17	1	no	—
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	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	C₂×D₁₇	1	1	yes	characteristic
220	#103	C₂×D₁₇	1	1	yes	characteristic
221	C₂²⋊C₈	—	17	1	no	—
222	C₂²⋊C₈	—	17	1	no	—
223	C₄×C₈	—	17	1	no	—
224	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
225	C₁₇	#3104	1	1	yes	characteristic, complemented, Sylow
226	C₂×Q₈	—	17	1	no	—
227	C₂×Q₈	—	17	1	no	—
228	C₂×Q₈	—	34	1	no	—
229	C₂×Q₈	—	34	1	no	—
230	C₂×Q₈	—	34	2	no	—
231	C₂×Q₈	—	34	2	no	—
232	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂²×C₄	—	17	1	no	—
235	C₂²×C₄	—	17	1	no	—
236	C₂²×C₄	—	17	1	no	—
237	C₂²×C₄	—	34	1	no	—
238	C₂²×C₄	—	34	1	no	—
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	34	2	no	—
241	Q₁₆	—	34	2	no	—
242	Q₁₆	—	34	2	no	—
243	C₂×C₈	—	17	1	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	17	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	—	34	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	17	1	no	—
268	C₄²	—	17	1	no	—
269	C₄²	—	17	1	no	—
270	C₄²	—	34	1	no	—
271	C₄²	—	34	1	no	—
272	C₄²	—	34	1	no	—
273	C₄²	—	34	1	no	—
274	C₄²	C₂²×D₁₇	1	1	yes	characteristic
275	C₂³	—	17	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	17	2	no	—
280	Q₈	—	34	4	no	—
281	Q₈	—	34	1	no	—
282	Q₈	—	34	1	no	—
283	Q₈	—	34	1	no	—
284	Q₈	—	68	2	no	—
285	Q₈	#63252	1	2	yes	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	17	1	no	—
291	C₂×C₄	—	17	4	no	—
292	C₂×C₄	—	17	1	no	—
293	C₂×C₄	—	17	1	no	—
294	C₂×C₄	—	34	4	no	—
295	C₂×C₄	—	34	1	no	—
296	C₂×C₄	—	34	1	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	—	34	1	no	—
301	C₂×C₄	—	68	1	no	—
302	C₂×C₄	—	68	2	no	—
303	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
304	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	34	1	no	—
309	C₈	—	34	1	no	—
310	C₂²	—	17	2	no	—
311	C₂²	—	34	1	no	—
312	C₂²	—	34	1	no	—
313	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	17	2	no	—
320	C₄	—	34	1	no	—
321	C₄	—	34	2	no	—
322	C₄	—	34	1	no	—
323	C₄	—	68	1	no	—
324	C₄	—	68	1	no	—
325	C₄	—	68	1	no	—
326	C₄	Q₁₆×D₁₇	1	2	yes	—
327	C₄	C₂×#63252	1	1	yes	characteristic
328	C₄	#41×D₁₇	1	1	yes	characteristic
329	C₂	—	34	2	no	—
330	C₂	#11763318	1	1	yes	characteristic
331	C₂	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
332	C₂	#11763606	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₄×Q₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂²×D₁₇	C₂×D₄
hypercenter	C₄×Q₁₆	D₁₇
hypocenter	C₁₇	#3104
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
5. #11763103	C₂	semidirect
6. #11763102	C₂	semidirect
7. #11763018	C₂	semidirect
8. #11763017	C₂	semidirect
11. C₄×Dic₆₈	C₂	semidirect
12. C₄×#63231	C₂	semidirect
13. C₄×#63231	C₂	semidirect
15. C₄×Q₁₆×C₁₇	C₂	semidirect
132. C₄×D₁₇	Q₁₆	semidirect
176. C₄×Q₁₆	D₁₇	semidirect
225. C₁₇	#3104	semidirect

Maximal subgroups

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

subgroup	index
#11763363	2
C₄×Q₈×D₁₇	2
#11763146	2
#11763143	2
#11763103	2
#11763102	2
#11763018	2
#11763017	2
#11762966	2
#11762963	2
C₄×Dic₆₈	2
C₄×#63231	2
#11762860	2
C₄×Q₁₆×C₁₇	2
#3104	17