Subgroups — #16925404

Order 4352.

462 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537495	C₂	1	1	yes	maximal, characteristic
2	#14528653	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527431	C₂	1	1	yes	maximal, characteristic
4	#14527278	C₂	1	1	yes	maximal, characteristic
5	#14527077	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526679	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587593	—	2	1	no	—
9	#11763359	C₂²	1	1	yes	characteristic
10	#11762886	—	2	1	no	—
11	#11762886	C₄	1	2	yes	complemented
12	#11762874	—	2	1	no	—
13	#11762874	C₄	1	2	yes	complemented
14	#11762862	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
16	#11762777	—	2	1	no	—
17	C₂×#10587474	—	2	1	no	—
18	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
19	#11762564	—	2	1	no	—
20	#11762554	C₄	1	2	yes	complemented
21	#11762552	C₄	1	2	yes	complemented
22	#11762542	C₄	1	4	yes	—
23	#11762531	—	2	1	no	—
24	#11762530	—	2	1	no	—
25	C₁₇×#431	C₂²	1	1	yes	characteristic
26	C₁₇×#423	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
28	C₂×#63252	—	2	1	no	—
29	C₂×M₁₆×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₂×C₄²×C₁₇	—	2	1	no	—
32	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	#10587616	—	2	1	no	—
34	#10587616	C₂×C₄	1	2	yes	—
35	C₂²×#3622	—	2	1	no	—
36	C₂²×#3622	D₄	1	1	yes	characteristic
37	C₂²×#3622	D₄	1	1	yes	characteristic
38	#10587593	—	4	2	no	—
39	#10587593	—	4	2	no	—
40	C₂×#63218	—	2	2	no	—
41	C₂×C₈×D₁₇	—	2	2	no	—
42	#10587561	—	2	1	no	—
43	#10587560	C₂×C₄	1	1	yes	characteristic
44	C₄×D₆₈	—	2	1	no	—
45	C₄×D₆₈	C₂×C₄	1	2	yes	—
46	#10587555	C₂×C₄	1	1	yes	characteristic
47	C₁₇×C₄⋊C₈	—	2	1	no	—
48	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₇×C₂²⋊C₈	—	2	1	no	—
51	C₄×C₈×C₁₇	—	2	1	no	—
52	C₁₇×#95	C₂×C₄	1	2	yes	—
53	C₂²⋊#3622	C₂×C₄	1	2	yes	—
54	#10587491	—	2	2	no	—
55	#10587491	—	2	2	no	—
56	C₄⋊#3622	—	2	1	no	—
57	C₄⋊#3622	—	2	1	no	—
58	C₄⋊#3622	C₂×C₄	1	2	yes	—
59	#10587474	—	4	2	no	—
60	C₄×#3622	—	2	1	no	—
61	C₄×#3622	C₂×C₄	1	1	yes	characteristic
62	C₄×#3622	C₂×C₄	1	1	yes	characteristic
63	C₄×#3622	D₄	1	1	yes	characteristic
64	C₄×#3622	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	—	2	2	no	—
66	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₇	—	2	2	no	—
70	#63252	—	4	1	no	—
71	#63252	—	4	1	no	—
72	#63252	—	4	1	no	—
73	#63252	—	4	1	no	—
74	C₂×D₆₈	—	2	1	no	—
75	C₂×D₆₈	—	2	1	no	—
76	C₂×D₆₈	C₄²	1	2	yes	—
77	C₂×C₄×D₁₇	—	2	2	no	—
78	C₂×Dic₃₄	—	2	1	no	—
79	M₁₆×C₁₇	—	4	1	no	—
80	M₁₆×C₁₇	—	4	1	no	—
81	C₂×C₈×C₁₇	—	2	2	no	—
82	C₂×C₈×C₁₇	—	2	2	no	—
83	C₄⋊C₄×C₁₇	—	4	2	no	—
84	C₄²×C₁₇	—	4	1	no	—
85	C₄²×C₁₇	—	4	1	no	—
86	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₇	#41	1	1	yes	characteristic
90	#63227	—	2	2	no	—
91	#63227	—	2	2	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
93	C₄⋊Dic₁₇	C₄²	1	2	yes	—
94	C₂×#3622	—	2	1	no	—
95	C₂×#3622	—	2	1	no	—
96	C₂×#3622	—	2	1	no	—
97	C₂×#3622	—	2	1	no	—
98	C₂×#3622	—	2	2	no	—
99	C₂×#3622	—	2	2	no	—
100	C₂×#3622	—	4	1	no	—
101	C₂×#3622	—	4	1	no	—
102	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
106	#63218	—	4	4	no	—
107	C₈×D₁₇	—	4	4	no	—
108	#8253	—	17	1	no	maximal, Sylow
109	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
110	C₂²×D₁₇	—	2	2	no	—
111	C₂×C₄×C₁₇	—	2	2	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	—	4	2	no	—
114	C₂×C₄×C₁₇	—	4	2	no	—
115	C₂×C₄×C₁₇	—	4	2	no	—
116	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
126	C₂²⋊D₁₇	—	4	2	no	—
127	C₂²⋊D₁₇	—	4	2	no	—
128	C₂×Dic₁₇	—	2	2	no	—
129	D₆₈	—	4	2	no	—
130	D₆₈	—	4	1	no	—
131	D₆₈	—	4	1	no	—
132	D₆₈	—	4	2	no	—
133	C₄×D₁₇	—	4	2	no	—
134	C₄×D₁₇	—	4	2	no	—
135	Dic₃₄	—	4	1	no	—
136	Dic₃₄	—	4	1	no	—
137	C₈×C₁₇	—	4	2	no	—
138	C₈×C₁₇	—	4	2	no	—
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3622	—	2	1	no	—
143	#3622	—	4	2	no	—
144	#3622	—	4	2	no	—
145	#2931	—	17	1	no	—
146	#1770	—	17	1	no	—
147	#1540	—	17	1	no	—
148	#1502	—	17	1	no	—
149	#1452	—	17	1	no	—
150	#1303	D₁₇	1	1	yes	characteristic, complemented
151	#1301	—	17	1	no	—
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₇	#440	1	1	yes	characteristic
156	C₂²×C₁₇	#442	1	1	yes	characteristic
157	C₂×D₁₇	—	4	2	no	—
158	C₂×D₁₇	—	4	2	no	—
159	C₂×D₁₇	—	4	2	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	8	2	no	—
164	C₄×C₁₇	#417	1	1	yes	characteristic
165	C₄×C₁₇	#418	1	1	yes	characteristic
166	C₄×C₁₇	#440	1	1	yes	characteristic
167	C₄×C₁₇	#442	1	1	yes	characteristic
168	C₄×C₁₇	#444	1	2	yes	—
169	Dic₁₇	—	4	2	no	—
170	C₂×#131	—	34	1	no	—
171	#532	—	17	1	no	—
172	#436	—	17	2	no	—
173	#436	—	34	1	no	—
174	C₈×D₄	—	17	2	no	—
175	C₈×D₄	—	34	1	no	—
176	#431	—	17	1	no	—
177	#431	C₂×D₁₇	1	1	yes	characteristic
178	#423	C₂×D₁₇	1	1	yes	characteristic
179	C₂×C₄⋊C₈	—	17	1	no	—
180	#408	—	34	1	no	—
181	C₂×#97	—	34	1	no	—
182	C₂×C₄×C₈	—	17	1	no	—
183	C₄×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
184	#336	—	34	1	no	—
185	#332	—	17	2	no	—
186	#331	—	17	2	no	—
187	#327	—	17	4	no	—
188	#322	—	34	1	no	—
189	#322	—	34	1	no	—
190	C₂×C₁₇	—	2	1	no	—
191	C₂×C₁₇	—	2	1	no	—
192	C₂×C₁₇	#1757	1	1	yes	characteristic
193	C₂×C₁₇	#1762	1	1	yes	characteristic
194	C₂×C₁₇	#1765	1	1	yes	characteristic
195	D₁₇	—	8	2	no	—
196	C₂×#41	—	34	1	no	—
197	#131	—	68	2	no	—
198	#131	—	68	2	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	34	2	no	—
201	C₂²×C₈	—	17	1	no	—
202	C₂²×C₈	—	17	1	no	—
203	C₂²×C₈	—	34	2	no	—
204	C₂²×C₈	—	34	1	no	—
205	C₄⋊Q₈	—	17	1	no	—
206	#127	—	17	1	no	—
207	#124	—	34	1	no	—
208	#121	—	17	2	no	—
209	#121	—	34	1	no	—
210	C₄×D₄	—	17	2	no	—
211	C₄×D₄	—	34	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	17	2	no	—
217	C₄⋊C₈	—	34	1	no	—
218	C₄⋊C₈	—	34	1	no	—
219	C₄⋊C₈	C₄×D₁₇	1	2	yes	—
220	C₄⋊C₈	C₄×D₁₇	1	2	yes	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	17	2	no	—
223	C₂²⋊C₈	—	34	2	no	—
224	C₂²⋊C₈	—	34	2	no	—
225	#97	—	68	2	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₄×C₈	—	17	1	no	—
230	C₄×C₈	—	17	1	no	—
231	C₄×C₈	—	34	1	no	—
232	#95	C₄×D₁₇	1	2	yes	—
233	C₁₇	#8253	1	1	yes	characteristic, complemented, Sylow
234	#41	—	68	1	no	—
235	#41	—	68	1	no	—
236	#41	—	68	1	no	—
237	#41	—	68	1	no	—
238	C₂×Q₈	—	34	1	no	—
239	C₂×D₄	—	17	2	no	—
240	C₂×D₄	—	34	2	no	—
241	C₂×D₄	—	34	1	no	—
242	C₂×D₄	—	34	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	34	2	no	—
245	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
247	C₂²×C₄	#63254	1	1	yes	characteristic
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	68	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	17	1	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	17	1	no	—
257	C₂×C₈	—	34	1	no	—
258	C₂×C₈	—	34	1	no	—
259	C₂×C₈	—	34	1	no	—
260	C₂×C₈	—	34	1	no	—
261	C₂×C₈	—	34	2	no	—
262	C₂×C₈	—	34	2	no	—
263	C₂×C₈	—	68	4	no	—
264	C₂×C₈	—	68	1	no	—
265	C₂×C₈	—	68	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	17	2	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	2	no	—
270	C₂²⋊C₄	—	34	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
275	C₄²	D₄×D₁₇	1	1	yes	characteristic
276	C₄²	#63256	1	1	yes	characteristic
277	C₂³	—	34	2	no	—
278	C₂³	#10587567	1	1	yes	characteristic
279	Q₈	—	68	1	no	—
280	Q₈	—	68	1	no	—
281	D₄	—	68	2	no	—
282	D₄	—	68	2	no	—
283	D₄	—	68	2	no	—
284	D₄	—	68	1	no	—
285	D₄	—	68	1	no	—
286	D₄	—	68	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	34	2	no	—
293	C₂×C₄	—	68	2	no	—
294	C₂×C₄	—	68	2	no	—
295	C₂×C₄	C₄²×D₁₇	1	1	yes	characteristic
296	C₂×C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
297	C₂×C₄	#10587567	1	1	yes	characteristic
298	C₂×C₄	#10587578	1	1	yes	characteristic
299	C₂×C₄	#10587579	1	1	yes	characteristic
300	C₂×C₄	#10587579	1	1	yes	characteristic
301	C₂×C₄	#10587597	1	1	yes	characteristic
302	C₂×C₄	#10587600	1	1	yes	characteristic
303	C₂×C₄	#10587600	1	1	yes	characteristic
304	C₂×C₄	#10587604	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	34	1	no	—
308	C₈	—	34	1	no	—
309	C₈	—	34	1	no	—
310	C₈	—	34	1	no	—
311	C₈	—	68	2	no	—
312	C₈	—	68	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	68	2	no	—
316	C₂²	—	68	2	no	—
317	C₂²	—	68	2	no	—
318	C₂²	#11762656	1	1	yes	characteristic
319	C₂²	#11762912	1	1	yes	characteristic
320	C₂²	#11762918	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	68	2	no	—
326	C₄	#11762809	1	1	yes	characteristic
327	C₄	#11762812	1	1	yes	characteristic
328	C₄	#11762908	1	1	yes	characteristic
329	C₄	#11762920	1	1	yes	characteristic
330	C₄	#11762928	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	136	2	no	—
334	C₂	#14528583	1	1	yes	characteristic
335	C₂	#14528608	1	1	yes	characteristic
336	C₂	#14528625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1303	C₂
socle	C₂²×C₁₇	C₄×C₂²⋊C₄
prime radical	#11763359	C₂²
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1303	D₁₇
hypocenter	C₁₇	#8253
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1303

Complements

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

N	H ≅ G/N	type
2. #14528653	C₂	semidirect
5. #14527077	C₂	semidirect
6. #14526679	C₂	semidirect
7. C₁₇×#1303	C₂	semidirect
11. #11762886	C₄	semidirect
13. #11762874	C₄	semidirect
20. #11762554	C₄	semidirect
21. #11762552	C₄	semidirect
150. #1303	D₁₇	semidirect
233. C₁₇	#8253	semidirect

Maximal subgroups

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

subgroup	index
#14537495	2
#14528653	2
#14527431	2
#14527278	2
#14527077	2
#14526679	2
C₁₇×#1303	2
#8253	17