Subgroups — #16949251

Order 4352.

570 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587490	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762847	C₂	1	1	yes	maximal, characteristic
3	#14526893	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63223	—	2	1	no	—
6	C₂×#10587612	—	2	1	no	—
7	C₂×#10587490	—	2	1	no	—
8	C₂×#10587490	—	2	2	no	—
9	C₂×#10587490	C₄	1	4	yes	complemented
10	C₂×#10587490	C₂²	1	2	yes	—
11	#11762847	—	2	1	no	—
12	#11762847	—	2	1	no	—
13	#11762847	C₄	1	2	yes	—
14	#11762847	C₄	1	2	yes	—
15	C₂×#10587489	C₂²	1	2	yes	—
16	C₂×#10587488	—	2	1	no	—
17	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×M₃₂	—	2	1	no	—
19	C₂×C₁₇×M₃₂	—	2	2	no	—
20	C₂×C₁₇×M₃₂	C₄	1	4	yes	complemented
21	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
22	C₂²×C₁₆×C₁₇	—	2	1	no	—
23	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	D₄	1	2	yes	—
32	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
33	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₇	—	4	1	no	—
35	#10587612	—	2	2	no	—
36	#10587612	—	2	2	no	—
37	#10587612	—	4	1	no	—
38	#10587612	—	4	1	no	—
39	C₂×C₄⋊Dic₁₇	—	2	2	no	—
40	C₂×C₄×Dic₁₇	—	4	1	no	—
41	C₂×#63223	—	2	4	no	—
42	C₂×#63223	—	2	2	no	—
43	C₂×#63223	—	4	1	no	—
44	C₂×#63223	—	4	2	no	—
45	C₂²×#3622	—	4	1	no	—
46	C₁₇×M₃₂	—	2	4	no	—
47	C₁₇×M₃₂	—	2	2	no	—
48	C₁₇×M₃₂	C₂×C₄	1	4	yes	—
49	C₂×C₁₆×C₁₇	—	2	4	no	—
50	C₂×C₁₆×C₁₇	—	2	2	no	—
51	#10587490	—	2	4	no	—
52	#10587490	—	2	2	no	—
53	#10587490	C₂×C₄	1	4	yes	—
54	#10587489	—	2	2	no	—
55	#10587489	C₂×C₄	1	4	yes	—
56	#10587488	—	2	2	no	—
57	#10587488	—	2	2	no	—
58	C₂⁴×C₁₇	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
68	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₁₇	—	4	2	no	—
70	C₂×C₈×C₁₇	C₄²	1	2	yes	—
71	C₂×C₈×C₁₇	C₄²	1	2	yes	—
72	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
88	C₂²⋊Dic₁₇	—	4	2	no	—
89	C₂²⋊Dic₁₇	—	4	2	no	—
90	C₄⋊Dic₁₇	—	2	4	no	—
91	C₄⋊Dic₁₇	—	4	2	no	—
92	C₄×Dic₁₇	—	4	2	no	—
93	C₄×Dic₁₇	—	4	2	no	—
94	#63223	—	2	4	no	—
95	#63223	—	4	4	no	—
96	#63223	—	4	2	no	—
97	C₂×#3622	—	4	4	no	—
98	C₂×#3622	—	4	2	no	—
99	C₁₆×C₁₇	—	2	4	no	—
100	#12569	—	17	1	no	maximal, Sylow
101	C₂³×C₁₇	—	4	2	no	—
102	C₂³×C₁₇	#102	1	1	yes	characteristic
103	C₂³×C₁₇	#102	1	1	yes	characteristic
104	C₂³×C₁₇	#103	1	2	yes	—
105	C₂³×C₁₇	#106	1	2	yes	—
106	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₇	#95	1	2	yes	—
108	C₂×C₄×C₁₇	#95	1	2	yes	—
109	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#102	1	2	yes	—
111	C₂×C₄×C₁₇	#102	1	2	yes	—
112	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₁₇	#103	1	2	yes	—
114	C₂×C₄×C₁₇	#103	1	2	yes	—
115	C₂×C₄×C₁₇	#103	1	2	yes	—
116	C₂×C₄×C₁₇	#106	1	2	yes	—
117	C₂×C₄×C₁₇	#107	1	2	yes	—
118	C₂×C₄×C₁₇	#107	1	2	yes	—
119	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
125	C₂×Dic₁₇	—	4	4	no	—
126	C₂×Dic₁₇	—	8	4	no	—
127	C₈×C₁₇	#95	1	2	yes	—
128	C₈×C₁₇	#95	1	2	yes	—
129	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
130	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	#3622	—	4	4	no	—
134	C₂²×M₃₂	D₁₇	1	1	yes	characteristic
135	C₂²×#108	—	17	1	no	—
136	C₂×#427	—	17	1	no	—
137	#1387	—	17	4	no	—
138	C₂²×C₁₇	—	2	2	no	—
139	C₂²×C₁₇	—	2	2	no	—
140	C₂²×C₁₇	—	2	2	no	—
141	C₂²×C₁₇	—	4	2	no	—
142	C₂²×C₁₇	—	4	2	no	—
143	C₂²×C₁₇	#340	1	2	yes	—
144	C₂²×C₁₇	#340	1	2	yes	—
145	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
146	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
147	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
148	C₄×C₁₇	#340	1	2	yes	—
149	C₄×C₁₇	#340	1	2	yes	—
150	C₄×C₁₇	C₂×#95	1	1	yes	characteristic
151	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
152	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
153	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
154	Dic₁₇	—	8	4	no	—
155	C₂²×M₁₆	—	34	1	no	—
156	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
157	C₂×#117	—	34	1	no	—
158	C₂×M₃₂	—	2	2	no	—
159	C₂×M₃₂	—	2	1	no	—
160	C₂×M₃₂	Dic₁₇	1	4	yes	complemented
161	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
162	C₂²×C₁₆	—	2	1	no	—
163	C₂×#108	—	17	4	no	—
164	C₂×#108	—	17	2	no	—
165	C₂×#108	—	34	1	no	—
166	C₂×#108	—	34	2	no	—
167	#427	—	17	2	no	—
168	#427	—	17	2	no	—
169	#427	—	34	1	no	—
170	#427	—	34	1	no	—
171	C₂×#107	—	17	2	no	—
172	C₂×#106	—	34	1	no	—
173	C₂×C₁₇	—	2	2	no	—
174	C₂×C₁₇	—	2	2	no	—
175	C₂×C₁₇	—	2	1	no	—
176	C₂×C₁₇	—	2	1	no	—
177	C₂×C₁₇	C₂×#340	1	1	yes	characteristic
178	C₂×C₁₇	#2148	1	2	yes	—
179	C₂³×C₄	D₆₈	1	1	yes	characteristic
180	C₂×M₁₆	—	34	2	no	—
181	C₂×M₁₆	—	34	4	no	—
182	C₂×M₁₆	—	68	2	no	—
183	C₂×M₁₆	—	68	1	no	—
184	C₂²×C₈	—	68	1	no	—
185	C₂²×C₈	Dic₃₄	1	2	yes	—
186	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
187	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₁₇	1	2	yes	—
189	C₂²×C₈	D₆₈	1	1	yes	characteristic
190	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
194	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
195	#117	—	34	2	no	—
196	#117	—	34	2	no	—
197	#117	—	68	1	no	—
198	#117	—	68	1	no	—
199	C₂×C₄⋊C₄	—	34	2	no	—
200	C₂×C₂²⋊C₄	—	68	1	no	—
201	C₂×C₄²	—	68	1	no	—
202	M₃₂	—	2	4	no	—
203	M₃₂	—	2	2	no	—
204	M₃₂	C₂×Dic₁₇	1	4	yes	—
205	C₂×C₁₆	—	2	4	no	—
206	C₂×C₁₆	—	2	2	no	—
207	#108	—	17	4	no	—
208	#108	—	34	2	no	—
209	#108	—	34	4	no	—
210	#107	—	17	4	no	—
211	#107	—	34	2	no	—
212	#106	—	34	2	no	—
213	#106	—	34	2	no	—
214	C₁₇	#12569	1	1	yes	characteristic, complemented, Sylow
215	C₂⁴	#63219	1	1	yes	characteristic
216	C₂²×C₄	—	68	2	no	—
217	C₂²×C₄	#63219	1	1	yes	characteristic
218	C₂²×C₄	#63219	1	2	yes	—
219	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
220	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
221	C₂²×C₄	Dic₆₈	1	2	yes	—
222	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
223	C₂²×C₄	#63227	1	1	yes	characteristic
224	C₂²×C₄	#63227	1	1	yes	characteristic
225	C₂²×C₄	#63227	1	2	yes	—
226	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
227	M₁₆	—	34	4	no	—
228	M₁₆	—	68	4	no	—
229	M₁₆	—	68	2	no	—
230	C₂×C₈	—	68	4	no	—
231	C₂×C₈	—	68	2	no	—
232	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
233	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
234	C₂×C₈	#63225	1	2	yes	—
235	C₂×C₈	#63225	1	2	yes	—
236	C₂×C₈	#63225	1	2	yes	—
237	C₂×C₈	#63225	1	2	yes	—
238	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
239	C₂×C₈	#63227	1	1	yes	characteristic
240	C₂×C₈	#63227	1	1	yes	characteristic
241	C₂×C₈	#63227	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
243	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
244	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
250	C₄⋊C₄	—	34	4	no	—
251	C₄⋊C₄	—	68	2	no	—
252	C₂²⋊C₄	—	68	2	no	—
253	C₂²⋊C₄	—	68	2	no	—
254	C₄²	—	68	2	no	—
255	C₄²	—	68	2	no	—
256	C₁₆	—	2	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	#10587487	1	2	yes	—
259	C₂³	#10587488	1	2	yes	—
260	C₂³	#10587492	1	1	yes	characteristic
261	C₂³	#10587492	1	1	yes	characteristic
262	C₂³	C₂×#63219	1	1	yes	characteristic
263	C₂×C₄	—	68	4	no	—
264	C₂×C₄	—	136	4	no	—
265	C₂×C₄	#10587487	1	2	yes	—
266	C₂×C₄	#10587487	1	2	yes	—
267	C₂×C₄	#10587487	1	2	yes	—
268	C₂×C₄	#10587488	1	2	yes	—
269	C₂×C₄	#10587489	1	2	yes	—
270	C₂×C₄	#10587489	1	2	yes	—
271	C₂×C₄	#10587492	1	2	yes	—
272	C₂×C₄	#10587492	1	2	yes	—
273	C₂×C₄	#10587492	1	1	yes	characteristic
274	C₂×C₄	#10587492	1	1	yes	characteristic
275	C₂×C₄	#10587502	1	2	yes	—
276	C₂×C₄	#10587502	1	2	yes	—
277	C₂×C₄	C₂×#63219	1	1	yes	characteristic
278	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
279	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
281	C₂×C₄	C₂×#63227	1	1	yes	characteristic
282	C₂×C₄	C₂×#63227	1	1	yes	characteristic
283	C₈	—	68	4	no	—
284	C₈	#10587502	1	2	yes	—
285	C₈	#10587502	1	2	yes	—
286	C₈	C₂×C₄×Dic₁₇	1	1	yes	characteristic
287	C₈	C₂×#63225	1	1	yes	characteristic
288	C₈	C₂×#63225	1	1	yes	characteristic
289	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	#11762572	1	2	yes	—
296	C₂²	#11762572	1	2	yes	—
297	C₂²	C₂×#10587492	1	1	yes	characteristic
298	C₂²	C₂×#10587487	1	1	yes	characteristic
299	C₂²	C₂×#10587488	1	1	yes	characteristic
300	C₄	—	136	4	no	—
301	C₄	#11762572	1	2	yes	—
302	C₄	#11762572	1	2	yes	—
303	C₄	C₂×#10587502	1	1	yes	characteristic
304	C₄	C₂×#10587492	1	1	yes	characteristic
305	C₄	C₂×#10587487	1	1	yes	characteristic
306	C₄	C₂×#10587489	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	C₂×#11762572	1	1	yes	characteristic
312	C₂	#14530534	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10587492
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂²×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂⁴×C₁₇	SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂²×M₃₂	D₁₇
hypocenter	C₁₇	#12569
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #14526893	C₂	semidirect
9. C₂×#10587490	C₄	semidirect
20. C₂×C₁₇×M₃₂	C₄	semidirect
160. C₂×M₃₂	Dic₁₇	semidirect
214. C₁₇	#12569	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587490	2
C₂×#11762847	2
#14526893	2
C₂²×C₁₇×M₃₂	2
#12569	17