Subgroups — #16949252

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×#10587483	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762847	C₂	1	1	yes	maximal, characteristic
3	#14526894	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₇×#108	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587612	—	2	1	no	—
6	C₂×#10587483	—	2	1	no	—
7	C₂×#10587483	—	2	2	no	—
8	C₂×#10587483	C₄	1	4	yes	complemented
9	C₂×#10587483	C₂²	1	2	yes	—
10	C₂²×#63214	—	2	1	no	—
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	C₂²×M₁₆×C₁₇	—	2	1	no	—
18	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×#108	—	2	1	no	—
20	C₂×C₁₇×#108	—	2	2	no	—
21	C₂×C₁₇×#108	C₄	1	4	yes	complemented
22	C₂×C₁₇×#108	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₇	—	2	4	no	—
25	C₂×M₁₆×C₁₇	—	2	2	no	—
26	C₂×M₁₆×C₁₇	—	4	1	no	—
27	C₂×M₁₆×C₁₇	—	4	2	no	—
28	C₂²×C₈×C₁₇	—	4	1	no	—
29	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	D₄	1	2	yes	—
37	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
38	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₁₇	—	4	1	no	—
40	#10587612	—	2	2	no	—
41	#10587612	—	2	2	no	—
42	#10587612	—	4	1	no	—
43	#10587612	—	4	1	no	—
44	C₂×C₄⋊Dic₁₇	—	2	2	no	—
45	C₂×C₄×Dic₁₇	—	4	1	no	—
46	C₁₇×#108	—	2	4	no	—
47	C₁₇×#108	—	2	2	no	—
48	C₁₇×#108	C₂×C₄	1	4	yes	—
49	#10587489	—	2	2	no	—
50	#10587489	C₂×C₄	1	4	yes	—
51	#10587488	—	2	2	no	—
52	#10587488	—	2	2	no	—
53	#10587483	—	2	4	no	—
54	#10587483	—	2	2	no	—
55	#10587483	C₂×C₄	1	4	yes	—
56	C₂×#63214	—	2	4	no	—
57	C₂×#63214	—	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	2	yes	—
66	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
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	M₁₆×C₁₇	—	2	4	no	—
71	M₁₆×C₁₇	—	4	4	no	—
72	M₁₆×C₁₇	—	4	2	no	—
73	C₂×C₈×C₁₇	—	4	4	no	—
74	C₂×C₈×C₁₇	—	4	2	no	—
75	C₂×C₈×C₁₇	C₄²	1	2	yes	—
76	C₂×C₈×C₁₇	C₄²	1	2	yes	—
77	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²⋊Dic₁₇	—	4	2	no	—
94	C₂²⋊Dic₁₇	—	4	2	no	—
95	C₄⋊Dic₁₇	—	2	4	no	—
96	C₄⋊Dic₁₇	—	4	2	no	—
97	C₄×Dic₁₇	—	4	2	no	—
98	C₄×Dic₁₇	—	4	2	no	—
99	#63214	—	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	1	yes	characteristic
111	C₂×C₄×C₁₇	#102	1	2	yes	—
112	C₂×C₄×C₁₇	#102	1	2	yes	—
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	—	4	4	no	—
128	C₈×C₁₇	#95	1	2	yes	—
129	C₈×C₁₇	#95	1	2	yes	—
130	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
131	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×M₃₂	—	17	1	no	—
135	C₂²×#108	D₁₇	1	1	yes	characteristic
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	—	2	1	no	—
156	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
157	C₂×#117	—	34	1	no	—
158	C₂×M₃₂	—	17	4	no	—
159	C₂×M₃₂	—	17	2	no	—
160	C₂×M₃₂	—	34	1	no	—
161	C₂×M₃₂	—	34	2	no	—
162	C₂²×C₁₆	—	34	1	no	—
163	C₂×#108	—	2	1	no	—
164	C₂×#108	—	2	2	no	—
165	C₂×#108	Dic₁₇	1	4	yes	complemented
166	C₂×#108	C₂×D₁₇	1	2	yes	—
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	C₂²⋊D₁₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	4	1	no	—
183	C₂×M₁₆	—	4	2	no	—
184	C₂²×C₈	—	4	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	D₆₈	1	1	yes	characteristic
191	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
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	—	17	4	no	—
203	M₃₂	—	34	2	no	—
204	M₃₂	—	34	4	no	—
205	C₂×C₁₆	—	34	2	no	—
206	C₂×C₁₆	—	34	4	no	—
207	#108	—	2	4	no	—
208	#108	—	2	2	no	—
209	#108	C₂×Dic₁₇	1	4	yes	—
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	#63230	1	1	yes	characteristic
216	C₂²×C₄	—	68	2	no	—
217	C₂²×C₄	#63225	1	2	yes	—
218	C₂²×C₄	#63227	1	1	yes	characteristic
219	C₂²×C₄	#63227	1	1	yes	characteristic
220	C₂²×C₄	#63228	1	1	yes	characteristic
221	C₂²×C₄	#63228	1	1	yes	characteristic
222	C₂²×C₄	#63229	1	2	yes	—
223	C₂²×C₄	#63230	1	1	yes	characteristic
224	C₂²×C₄	#63231	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	4	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	4	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	C₄⋊Dic₁₇	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	#63227	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	C₂×D₆₈	1	1	yes	characteristic
248	C₂×C₈	C₂²×Dic₁₇	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	—	34	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	#10587479	1	2	yes	—
259	C₂³	#10587480	1	1	yes	characteristic
260	C₂³	#10587480	1	1	yes	characteristic
261	C₂³	#10587506	1	2	yes	—
262	C₂³	C₂×#63230	1	1	yes	characteristic
263	C₂×C₄	—	68	4	no	—
264	C₂×C₄	—	136	4	no	—
265	C₂×C₄	#10587478	1	2	yes	—
266	C₂×C₄	#10587478	1	2	yes	—
267	C₂×C₄	#10587479	1	2	yes	—
268	C₂×C₄	#10587480	1	1	yes	characteristic
269	C₂×C₄	#10587480	1	1	yes	characteristic
270	C₂×C₄	#10587481	1	2	yes	—
271	C₂×C₄	#10587481	1	2	yes	—
272	C₂×C₄	#10587502	1	2	yes	—
273	C₂×C₄	#10587502	1	2	yes	—
274	C₂×C₄	#10587503	1	2	yes	—
275	C₂×C₄	#10587503	1	2	yes	—
276	C₂×C₄	#10587506	1	2	yes	—
277	C₂×C₄	C₂×#63225	1	1	yes	characteristic
278	C₂×C₄	C₂×#63227	1	1	yes	characteristic
279	C₂×C₄	C₂×#63228	1	1	yes	characteristic
280	C₂×C₄	C₂×#63229	1	1	yes	characteristic
281	C₂×C₄	C₂×#63231	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
283	C₈	—	4	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×C₄⋊Dic₁₇	1	1	yes	characteristic
289	C₈	C₂×#63227	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	#11762571	1	2	yes	—
296	C₂²	#11762571	1	2	yes	—
297	C₂²	C₂×#10587480	1	1	yes	characteristic
298	C₂²	C₂×#10587506	1	1	yes	characteristic
299	C₂²	C₂×#10587479	1	1	yes	characteristic
300	C₄	—	136	4	no	—
301	C₄	#11762571	1	2	yes	—
302	C₄	#11762571	1	2	yes	—
303	C₄	C₂×#10587502	1	1	yes	characteristic
304	C₄	C₂×#10587503	1	1	yes	characteristic
305	C₄	C₂×#10587481	1	1	yes	characteristic
306	C₄	C₂×#10587478	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×#11762571	1	1	yes	characteristic
312	C₂	#14530531	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10587480
commutator	C₈×C₁₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂²×C₁₇×#108	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂⁴×C₁₇	SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂²×#108	D₁₇
hypocenter	C₁₇	#12569
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #14526894	C₂	semidirect
8. C₂×#10587483	C₄	semidirect
21. C₂×C₁₇×#108	C₄	semidirect
165. C₂×#108	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×#10587483	2
C₂×#11762847	2
#14526894	2
C₂²×C₁₇×#108	2
#12569	17