Subgroups — #16927628

Order 4352.

384 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587488	C₂	1	1	yes	maximal, characteristic, complemented
2	#14527547	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527309	C₂	1	2	yes	maximal
4	#14526641	C₂	1	2	yes	maximal, complemented
5	C₁₇×#2966	C₂	1	1	yes	maximal, characteristic
6	#11762866	C₂²	1	2	yes	—
7	C₂×#10587488	—	2	1	no	—
8	C₄×C₄⋊Dic₁₇	C₂²	1	2	yes	—
9	#11762572	—	2	1	no	—
10	#11762562	—	2	1	no	—
11	#11762560	—	2	1	no	—
12	#11762546	—	2	2	no	—
13	#11762543	—	2	2	no	—
14	C₁₇×#529	C₂²	1	1	yes	characteristic
15	C₁₇×#446	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₁₇	C₄	1	1	yes	characteristic, complemented
17	C₁₇×#436	C₄	1	1	yes	characteristic, complemented
18	C₁₇×#435	—	2	1	no	—
19	C₈×D₄×C₁₇	—	2	1	no	—
20	C₈×D₄×C₁₇	C₄	1	1	yes	characteristic, complemented
21	C₁₇×#433	—	2	1	no	—
22	C₁₇×#432	C₂²	1	1	yes	characteristic
23	C₁₇×#405	—	2	1	no	—
24	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
25	C₂×M₁₆×C₁₇	—	2	1	no	—
26	C₂²×C₈×C₁₇	—	2	1	no	—
27	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₁₇×#126	—	2	1	no	—
30	C₁₇×#125	Q₈	1	1	yes	characteristic
31	C₁₇×#124	Q₈	1	1	yes	characteristic
32	C₁₇×#123	—	2	1	no	—
33	C₁₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
34	C₁₇×#121	D₄	1	1	yes	characteristic
35	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₁₇	—	2	1	no	—
37	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₁₇×#117	—	2	1	no	—
39	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₇	—	2	2	no	—
41	C₂×C₄×Dic₁₇	—	2	2	no	—
42	C₁₇×C₄⋊C₈	—	2	1	no	—
43	C₁₇×C₄⋊C₈	—	2	1	no	—
44	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₇×C₂²⋊C₈	—	2	1	no	—
47	C₁₇×C₂²⋊C₈	—	2	1	no	—
48	C₁₇×C₂²⋊C₈	—	2	1	no	—
49	C₁₇×#97	—	2	1	no	—
50	C₁₇×#97	—	2	1	no	—
51	C₄×C₈×C₁₇	—	2	1	no	—
52	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
56	#10587502	—	2	2	no	—
57	C₂²⋊#3622	—	2	2	no	—
58	#10587488	—	2	4	no	—
59	C₄⋊#3622	—	2	2	no	—
60	#10587474	—	2	2	no	—
61	C₂×Q₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₄×C₁₇	—	2	1	no	—
63	C₂×D₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
68	C₂²×Dic₁₇	—	4	2	no	—
69	M₁₆×C₁₇	—	4	2	no	—
70	C₂×C₈×C₁₇	—	2	2	no	—
71	C₂×C₈×C₁₇	—	2	1	no	—
72	C₂×C₈×C₁₇	—	2	1	no	—
73	C₂×C₈×C₁₇	—	4	1	no	—
74	C₂×C₈×C₁₇	—	4	2	no	—
75	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	—	2	1	no	—
82	C₄⋊C₄×C₁₇	—	2	1	no	—
83	C₄⋊C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₇	—	2	1	no	—
86	C₂²⋊C₄×C₁₇	—	2	1	no	—
87	C₂²⋊C₄×C₁₇	—	2	1	no	—
88	C₂²⋊C₄×C₁₇	—	2	1	no	—
89	C₂²⋊C₄×C₁₇	—	2	1	no	—
90	C₄²×C₁₇	—	2	1	no	—
91	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
94	C₄²×C₁₇	#41	1	1	yes	characteristic
95	C₄⋊Dic₁₇	—	2	4	no	—
96	C₄⋊Dic₁₇	—	4	2	no	—
97	C₄×Dic₁₇	—	8	2	no	—
98	C₂×#3622	—	4	2	no	—
99	#8694	—	17	1	no	maximal, Sylow
100	C₂³×C₁₇	—	2	1	no	—
101	C₂³×C₁₇	#121	1	1	yes	characteristic
102	Q₈×C₁₇	—	4	1	no	—
103	D₄×C₁₇	—	4	1	no	—
104	D₄×C₁₇	—	4	2	no	—
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
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	—	4	1	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	—	4	1	no	—
114	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×Dic₁₇	—	4	4	no	—
121	C₂×Dic₁₇	—	8	2	no	—
122	C₂×Dic₁₇	—	8	2	no	—
123	C₈×C₁₇	—	2	1	no	—
124	C₈×C₁₇	—	2	1	no	—
125	C₈×C₁₇	—	2	1	no	—
126	C₈×C₁₇	—	4	1	no	—
127	C₈×C₁₇	—	4	1	no	—
128	C₈×C₁₇	#104	1	2	yes	—
129	#3622	—	8	2	no	—
130	#2966	D₁₇	1	1	yes	characteristic
131	C₄×#106	—	17	1	no	—
132	#1568	—	17	1	no	—
133	#1509	—	17	2	no	—
134	#1285	—	17	2	no	—
135	C₂²×C₁₇	—	2	1	no	—
136	C₂²×C₁₇	—	2	1	no	—
137	C₂²×C₁₇	—	4	1	no	—
138	C₂²×C₁₇	—	4	1	no	—
139	C₂²×C₁₇	—	4	1	no	—
140	C₂²×C₁₇	#380	1	1	yes	characteristic
141	C₂²×C₁₇	#471	1	2	yes	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	4	1	no	—
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
151	C₄×C₁₇	#421	1	1	yes	characteristic
152	Dic₁₇	—	8	4	no	—
153	#529	C₂×D₁₇	1	1	yes	characteristic
154	#446	Dic₁₇	1	1	yes	characteristic, complemented
155	C₈×Q₈	Dic₁₇	1	1	yes	characteristic, complemented
156	#436	Dic₁₇	1	1	yes	characteristic, complemented
157	#435	—	2	1	no	—
158	C₈×D₄	—	2	1	no	—
159	C₈×D₄	Dic₁₇	1	1	yes	characteristic, complemented
160	#433	—	2	1	no	—
161	#432	—	17	2	no	—
162	#432	C₂×D₁₇	1	1	yes	characteristic
163	C₂×#106	—	34	1	no	—
164	#405	—	2	1	no	—
165	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
166	C₄×C₄⋊C₄	—	17	2	no	—
167	#340	—	34	1	no	—
168	#335	—	34	1	no	—
169	#334	—	34	1	no	—
170	#328	—	34	2	no	—
171	#327	—	34	2	no	—
172	C₂×C₁₇	—	2	2	no	—
173	C₂×C₁₇	—	8	1	no	—
174	C₂×C₁₇	#1806	1	1	yes	characteristic
175	C₂×C₁₇	#1812	1	1	yes	characteristic
176	C₂×C₁₇	#1815	1	1	yes	characteristic
177	C₂×M₁₆	—	2	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
181	#126	—	2	1	no	—
182	#125	Dic₃₄	1	1	yes	characteristic
183	#124	Dic₃₄	1	1	yes	characteristic
184	#123	—	2	1	no	—
185	C₂²⋊Q₈	D₆₈	1	1	yes	characteristic
186	#121	D₆₈	1	1	yes	characteristic
187	C₄×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	C₂×Dic₁₇	1	1	yes	characteristic
190	#117	—	2	1	no	—
191	C₂×C₄⋊C₄	—	34	2	no	—
192	C₂×C₄²	—	34	2	no	—
193	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
194	#106	—	34	4	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	34	2	no	—
198	C₄⋊C₈	C₂×Dic₁₇	1	1	yes	characteristic
199	C₄⋊C₈	C₂×Dic₁₇	1	1	yes	characteristic
200	C₂²⋊C₈	—	2	1	no	—
201	C₂²⋊C₈	—	2	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	34	2	no	—
204	#97	—	2	1	no	—
205	#97	—	2	1	no	—
206	#97	—	34	2	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
209	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
211	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
212	#95	—	34	2	no	—
213	C₁₇	#8694	1	1	yes	characteristic, complemented, Sylow
214	C₂×Q₈	C₄⋊Dic₁₇	1	1	yes	characteristic
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	C₄⋊Dic₁₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	68	2	no	—
219	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
220	C₂²×C₄	#63252	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	68	2	no	—
229	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	34	4	no	—
238	C₄⋊C₄	—	68	2	no	—
239	C₄⋊C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
240	C₄⋊C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	2	1	no	—
243	C₂²⋊C₄	—	2	1	no	—
244	C₂²⋊C₄	—	2	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	136	2	no	—
248	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
249	C₄²	#63252	1	1	yes	characteristic
250	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
251	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
252	C₂³	—	2	1	no	—
253	C₂³	#10587616	1	1	yes	characteristic
254	Q₈	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	68	4	no	—
267	C₂×C₄	—	136	2	no	—
268	C₂×C₄	—	136	2	no	—
269	C₂×C₄	#10587610	1	1	yes	characteristic
270	C₂×C₄	#10587610	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
272	C₂×C₄	#10587612	1	1	yes	characteristic
273	C₂×C₄	#10587616	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	4	1	no	—
280	C₈	—	136	2	no	—
281	C₈	#10587508	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	#11762669	1	1	yes	characteristic
288	C₂²	#11763072	1	2	yes	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	136	4	no	—
298	C₄	C₂×#10587508	1	1	yes	characteristic
299	C₄	#11762826	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	8	1	no	—
302	C₂	#14528804	1	1	yes	characteristic
303	C₂	#14528830	1	1	yes	characteristic
304	C₂	#14528843	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762669
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2966	C₂
socle	C₂²×C₁₇	#380
prime radical	C₁₇×#121	D₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2966	D₁₇
hypocenter	C₁₇	#8694
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966

Complements

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

N	H ≅ G/N	type
1. C₄×#10587488	C₂	semidirect
2. #14527547	C₂	semidirect
4. #14526641	C₂	semidirect
15. C₁₇×#446	C₄	semidirect
16. C₈×Q₈×C₁₇	C₄	semidirect
17. C₁₇×#436	C₄	semidirect
20. C₈×D₄×C₁₇	C₄	semidirect
154. #446	Dic₁₇	semidirect
155. C₈×Q₈	Dic₁₇	semidirect
156. #436	Dic₁₇	semidirect
159. C₈×D₄	Dic₁₇	semidirect
213. C₁₇	#8694	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587488	2
#14527547	2
#14527309	2
#14526641	2
C₁₇×#2966	2
#8694	17