Subgroups — #16934197

Order 4352.

516 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529762	C₂	1	1	yes	maximal, characteristic, complemented
2	#14529613	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#14527540	C₂	1	1	yes	maximal, characteristic, complemented
5	#14527239	C₂	1	1	yes	maximal, characteristic, complemented
6	#14527235	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#3167	C₂	1	1	yes	maximal, characteristic
8	#11762833	C₂²	1	1	yes	characteristic
9	#11762833	C₂²	1	1	yes	characteristic
10	C₂×#10587508	—	2	2	no	—
11	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₁₇	—	2	2	no	—
13	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
14	#11762568	—	2	2	no	—
15	#11762560	—	2	2	no	—
16	Q₈⋊#3622	—	2	2	no	—
17	D₄⋊#3622	—	2	2	no	—
18	C₁₇×#533	C₂²	1	1	yes	characteristic
19	C₁₇×#532	C₂²	1	1	yes	characteristic
20	C₁₇×#498	C₄	1	1	yes	characteristic, complemented
21	C₁₇×#492	C₄	1	1	yes	characteristic, complemented
22	C₁₇×#487	C₄	1	1	yes	characteristic, complemented
23	C₁₇×#486	C₄	1	1	yes	characteristic, complemented
24	C₁₇×#465	—	2	2	no	—
25	C₄×SD₁₆×C₁₇	—	2	2	no	—
26	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₂×#41×C₁₇	—	2	1	no	—
28	C₂×SD₁₆×C₁₇	—	2	2	no	—
29	C₂²×C₈×C₁₇	D₄	1	2	yes	—
30	C₁₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
31	C₁₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
32	C₁₇×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₇×#127	C₂×C₄	1	1	yes	characteristic
34	C₁₇×#125	—	2	1	no	—
35	C₁₇×#124	—	2	1	no	—
36	C₁₇×C₂²⋊Q₈	—	2	1	no	—
37	C₁₇×C₂²⋊Q₈	—	2	1	no	—
38	C₁₇×#121	—	2	1	no	—
39	C₁₇×#121	—	2	1	no	—
40	C₄×Q₈×C₁₇	—	2	1	no	—
41	C₄×Q₈×C₁₇	—	2	1	no	—
42	C₄×D₄×C₁₇	—	2	1	no	—
43	C₄×D₄×C₁₇	—	2	1	no	—
44	C₁₇×#117	—	2	1	no	—
45	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₁₇	—	2	1	no	—
47	C₂×C₄×Dic₁₇	—	2	1	no	—
48	C₂×C₄×Dic₁₇	D₄	1	2	yes	—
49	C₂×#63223	—	2	1	no	—
50	C₂×#63223	—	2	1	no	—
51	C₂²×#3622	—	2	1	no	—
52	C₁₇×#106	—	2	2	no	—
53	C₁₇×#103	—	2	2	no	—
54	C₁₇×#102	—	2	2	no	—
55	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₇	D₄	1	2	yes	—
58	#10587508	—	4	4	no	—
59	C₈×Dic₁₇	—	2	8	no	—
60	C₄⋊#3622	—	2	1	no	—
61	C₄⋊#3622	—	2	1	no	—
62	C₄⋊#3622	—	2	1	no	—
63	C₄⋊#3622	—	2	1	no	—
64	C₄×#3622	—	2	1	no	—
65	C₄×#3622	D₄	1	2	yes	—
66	#41×C₁₇	—	4	2	no	—
67	#41×C₁₇	—	4	2	no	—
68	C₂×Q₈×C₁₇	—	2	1	no	—
69	C₂×Q₈×C₁₇	—	2	1	no	—
70	C₂×Q₈×C₁₇	—	2	1	no	—
71	C₂×D₄×C₁₇	—	2	1	no	—
72	C₂×D₄×C₁₇	—	2	1	no	—
73	C₂×D₄×C₁₇	—	2	1	no	—
74	C₂×D₄×C₁₇	—	4	1	no	—
75	C₂²×C₄×C₁₇	—	4	1	no	—
76	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
79	C₂²×Dic₁₇	—	2	2	no	—
80	SD₁₆×C₁₇	—	4	4	no	—
81	C₂×C₈×C₁₇	—	2	4	no	—
82	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
84	C₄⋊C₄×C₁₇	—	2	1	no	—
85	C₄⋊C₄×C₁₇	—	2	1	no	—
86	C₄⋊C₄×C₁₇	—	2	1	no	—
87	C₄⋊C₄×C₁₇	—	2	1	no	—
88	C₄⋊C₄×C₁₇	—	2	1	no	—
89	C₄⋊C₄×C₁₇	—	2	1	no	—
90	C₄⋊C₄×C₁₇	—	4	1	no	—
91	C₄⋊C₄×C₁₇	—	4	1	no	—
92	C₂²⋊C₄×C₁₇	—	4	1	no	—
93	C₂²⋊C₄×C₁₇	—	4	1	no	—
94	C₂²⋊C₄×C₁₇	—	4	1	no	—
95	C₄²×C₁₇	—	4	1	no	—
96	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₇	#41	1	1	yes	characteristic
100	C₄×Dic₁₇	—	2	4	no	—
101	C₄×Dic₁₇	—	2	4	no	—
102	C₄×Dic₁₇	—	2	2	no	—
103	C₄×Dic₁₇	SD₁₆	1	4	yes	complemented
104	#63223	—	4	2	no	—
105	#63223	—	4	2	no	—
106	C₂×#3622	—	2	4	no	—
107	C₂×#3622	—	2	2	no	—
108	C₂×#3622	—	4	1	no	—
109	C₂×#3622	—	4	1	no	—
110	#9843	—	17	1	no	maximal, Sylow
111	C₂³×C₁₇	—	4	1	no	—
112	C₂³×C₁₇	#121	1	1	yes	characteristic
113	Q₈×C₁₇	—	4	2	no	—
114	Q₈×C₁₇	—	4	2	no	—
115	Q₈×C₁₇	—	4	2	no	—
116	D₄×C₁₇	—	4	2	no	—
117	D₄×C₁₇	—	4	2	no	—
118	D₄×C₁₇	—	4	2	no	—
119	D₄×C₁₇	—	8	2	no	—
120	C₂×C₄×C₁₇	—	2	2	no	—
121	C₂×C₄×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	—	4	1	no	—
123	C₂×C₄×C₁₇	—	4	1	no	—
124	C₂×C₄×C₁₇	—	4	1	no	—
125	C₂×C₄×C₁₇	—	8	1	no	—
126	C₂×C₄×C₁₇	—	8	1	no	—
127	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₁₇	#135	1	2	yes	—
135	C₂×Dic₁₇	—	2	4	no	—
136	C₂×Dic₁₇	—	2	4	no	—
137	C₂×Dic₁₇	—	2	4	no	—
138	C₈×C₁₇	—	2	2	no	—
139	C₈×C₁₇	#104	1	4	yes	complemented
140	#3622	—	2	4	no	—
141	#3622	—	8	1	no	—
142	#3622	—	8	1	no	—
143	#3167	D₁₇	1	1	yes	characteristic
144	#1994	—	17	1	no	—
145	#1962	—	17	1	no	—
146	C₄²×C₈	—	17	1	no	—
147	#1566	—	17	1	no	—
148	#1493	—	17	1	no	—
149	#1492	—	17	1	no	—
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	8	1	no	—
153	C₂²×C₁₇	—	8	1	no	—
154	C₂²×C₁₇	—	8	1	no	—
155	C₂²×C₁₇	#390	1	1	yes	characteristic
156	C₂²×C₁₇	#465	1	2	yes	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	8	1	no	—
160	C₄×C₁₇	—	8	1	no	—
161	C₄×C₁₇	—	8	1	no	—
162	C₄×C₁₇	C₂×#104	1	2	yes	—
163	C₄×C₁₇	C₄×SD₁₆	1	2	yes	—
164	C₄×C₁₇	#443	1	2	yes	—
165	C₄×C₁₇	#492	1	1	yes	characteristic
166	C₄×C₁₇	#495	1	1	yes	characteristic
167	Dic₁₇	—	2	8	no	—
168	#533	C₂×D₁₇	1	1	yes	characteristic
169	#532	C₂×D₁₇	1	1	yes	characteristic
170	#498	Dic₁₇	1	1	yes	characteristic, complemented
171	#492	Dic₁₇	1	1	yes	characteristic, complemented
172	#487	Dic₁₇	1	1	yes	characteristic, complemented
173	#486	Dic₁₇	1	1	yes	characteristic, complemented
174	#465	—	2	2	no	—
175	C₄×SD₁₆	—	2	2	no	—
176	#423	—	17	1	no	—
177	#423	—	17	1	no	—
178	C₂×#104	—	34	2	no	—
179	C₂×C₄×C₈	—	17	1	no	—
180	C₂×C₄×C₈	—	34	2	no	—
181	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
182	C₄³	—	17	1	no	—
183	#339	—	34	2	no	—
184	#334	—	34	2	no	—
185	Q₈⋊C₈	—	34	2	no	—
186	D₄⋊C₈	—	34	2	no	—
187	C₂×C₁₇	—	2	2	no	—
188	C₂×C₁₇	—	16	1	no	—
189	C₂×C₁₇	#1962	1	1	yes	characteristic
190	C₂×C₁₇	#1968	1	1	yes	characteristic
191	C₂×C₁₇	#1974	1	1	yes	characteristic
192	C₂×#41	—	2	1	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×M₁₆	—	34	1	no	—
195	C₂×M₁₆	—	34	1	no	—
196	C₂²×C₈	—	34	1	no	—
197	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
198	C₄⋊Q₈	C₂×Dic₁₇	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×Dic₁₇	1	1	yes	characteristic
200	C₄⋊Q₈	C₂×Dic₁₇	1	1	yes	characteristic
201	#127	C₂×Dic₁₇	1	1	yes	characteristic
202	#125	—	2	1	no	—
203	#124	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	#121	—	2	1	no	—
207	#121	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	#117	—	2	1	no	—
213	C₂×C₄²	—	17	2	no	—
214	C₂×C₄²	—	34	1	no	—
215	C₂×C₄²	—	34	1	no	—
216	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
217	#106	—	2	2	no	—
218	C₄⋊C₈	—	34	1	no	—
219	C₄⋊C₈	—	34	1	no	—
220	C₄⋊C₈	—	34	1	no	—
221	C₄⋊C₈	—	34	1	no	—
222	#104	—	68	4	no	—
223	#103	—	2	2	no	—
224	#102	—	2	2	no	—
225	C₄×C₈	—	17	2	no	—
226	C₄×C₈	—	34	8	no	—
227	C₄×C₈	—	34	1	no	—
228	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
229	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
230	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
231	C₁₇	#9843	1	1	yes	characteristic, complemented, Sylow
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	34	2	no	—
243	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
244	C₂²×C₄	#63254	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
246	SD₁₆	—	4	4	no	—
247	M₁₆	—	68	2	no	—
248	M₁₆	—	68	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	34	4	no	—
251	C₂×C₈	—	34	2	no	—
252	C₂×C₈	—	68	1	no	—
253	C₂×C₈	—	68	1	no	—
254	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
255	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	17	4	no	—
269	C₄²	—	34	4	no	—
270	C₄²	—	34	4	no	—
271	C₄²	—	34	2	no	—
272	C₄²	D₄×D₁₇	1	1	yes	characteristic
273	C₄²	#63254	1	1	yes	characteristic
274	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	C₂³	#10587624	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	34	4	no	—
293	C₂×C₄	—	34	4	no	—
294	C₂×C₄	—	34	4	no	—
295	C₂×C₄	SD₁₆×D₁₇	1	2	yes	—
296	C₂×C₄	#10587602	1	2	yes	—
297	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
298	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
299	C₂×C₄	#10587622	1	1	yes	characteristic
300	C₂×C₄	#10587624	1	1	yes	characteristic
301	C₂×C₄	#10587626	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	34	4	no	—
305	C₈	—	136	1	no	—
306	C₈	—	136	1	no	—
307	C₈	#10587508	1	4	yes	complemented
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	#11762712	1	1	yes	characteristic
314	C₂²	#11763035	1	2	yes	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	34	8	no	—
321	C₄	C₂×#10587508	1	2	yes	—
322	C₄	SD₁₆×Dic₁₇	1	2	yes	—
323	C₄	#11762926	1	2	yes	—
324	C₄	#11763180	1	1	yes	characteristic
325	C₄	#11763190	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	1	no	—
328	C₂	#14529613	1	1	yes	characteristic
329	C₂	#14529640	1	1	yes	characteristic
330	C₂	#14529673	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#3167	C₂
socle	C₂²×C₁₇	#390
prime radical	C₁₇×#532	C₂²
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#3167	D₁₇
hypocenter	C₁₇	#9843
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3167

Complements

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

N	H ≅ G/N	type
1. #14529762	C₂	semidirect
3. C₄×C₈×Dic₁₇	C₂	semidirect
4. #14527540	C₂	semidirect
5. #14527239	C₂	semidirect
20. C₁₇×#498	C₄	semidirect
21. C₁₇×#492	C₄	semidirect
22. C₁₇×#487	C₄	semidirect
23. C₁₇×#486	C₄	semidirect
103. C₄×Dic₁₇	SD₁₆	semidirect
139. C₈×C₁₇	#104	semidirect
170. #498	Dic₁₇	semidirect
171. #492	Dic₁₇	semidirect
172. #487	Dic₁₇	semidirect
173. #486	Dic₁₇	semidirect
231. C₁₇	#9843	semidirect
307. C₈	#10587508	semidirect

Maximal subgroups

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

subgroup	index
#14529762	2
#14529613	2
C₄×C₈×Dic₁₇	2
#14527540	2
#14527239	2
#14527235	2
C₁₇×#3167	2
#9843	17