Subgroups — #17952284

Order 4352.

462 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587544	C₂	1	1	yes	maximal, characteristic
2	#14544498	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763650	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#3624	C₂	1	1	yes	maximal, characteristic
5	#14543606	C₂	1	1	yes	maximal, characteristic
6	#14543606	C₂	1	1	yes	maximal, characteristic
7	#14528632	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587544	C₄	1	2	yes	—
9	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
10	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
12	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
13	#11763675	C₄	1	2	yes	—
14	#11763675	C₄	1	2	yes	—
15	#11763675	C₄	1	2	yes	—
16	#11763650	—	2	2	no	—
17	C₈×#3624	—	2	2	no	—
18	C₂×#10587485	C₄	1	1	yes	characteristic
19	C₂×#10587485	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
23	#11762564	C₄	1	1	yes	characteristic
24	C₁₇×#336	C₄	1	1	yes	characteristic
25	C₂²×#3624	C₂×C₄	1	2	yes	—
26	C₂²×#3624	C₂×C₄	1	2	yes	—
27	C₂²×#3624	C₂×C₄	1	2	yes	—
28	C₂²×#3624	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
38	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
39	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
40	#10587544	—	2	4	no	—
41	C₄×#3624	—	2	1	no	—
42	C₄×#3624	—	2	1	no	—
43	C₄×#3624	—	2	1	no	—
44	C₄×#3624	—	2	1	no	—
45	C₄×#3624	—	2	1	no	—
46	C₄×#3624	—	2	1	no	—
47	C₄×#3624	D₄	1	2	yes	—
48	C₄×#3624	Q₈	1	2	yes	—
49	#10587485	C₈	1	4	yes	complemented
50	#10587485	C₈	1	4	yes	complemented
51	C₈×Dic₁₇	—	2	1	no	—
52	C₈×Dic₁₇	—	2	1	no	—
53	C₈×Dic₁₇	—	2	1	no	—
54	C₈×Dic₁₇	—	2	1	no	—
55	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
59	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
60	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×#3624	—	2	2	no	—
63	C₂×#3624	—	2	2	no	—
64	C₂×#3624	—	2	2	no	—
65	C₂×#3624	—	2	2	no	—
66	C₂×#3624	—	2	2	no	—
67	C₂×#3624	—	2	2	no	—
68	C₂×#3624	—	2	2	no	—
69	C₂×#3624	—	2	2	no	—
70	C₂×#3624	—	2	2	no	—
71	C₂×#3624	—	2	2	no	—
72	C₂×#3624	C₄⋊C₄	1	4	yes	—
73	C₂×#3624	M₁₆	1	4	yes	—
74	C₂×C₈×C₁₇	—	2	1	no	—
75	C₂×C₈×C₁₇	—	2	1	no	—
76	C₂×C₈×C₁₇	—	2	1	no	—
77	C₂×C₈×C₁₇	—	2	1	no	—
78	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
80	C₄²×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	—	2	1	no	—
82	C₄×Dic₁₇	—	2	2	no	—
83	C₄×Dic₁₇	—	2	2	no	—
84	C₄×Dic₁₇	—	2	2	no	—
85	C₄×Dic₁₇	—	2	2	no	—
86	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
87	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
89	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
90	C₄×Dic₁₇	#41	1	1	yes	characteristic
91	C₄×Dic₁₇	#41	1	1	yes	characteristic
92	C₂×#3622	—	2	1	no	—
93	C₂×#3622	—	2	1	no	—
94	C₂×#3622	—	2	1	no	—
95	C₂×#3622	—	2	1	no	—
96	C₂×#3622	C₂×C₈	1	2	yes	—
97	C₂×#3622	C₂×C₈	1	2	yes	—
98	#8264	—	17	1	no	maximal, Sylow
99	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
111	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
112	C₂×Dic₁₇	—	2	1	no	—
113	C₂×Dic₁₇	—	2	1	no	—
114	C₂×Dic₁₇	—	2	1	no	—
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
117	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
118	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×Dic₁₇	#117	1	1	yes	characteristic
120	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
121	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
122	C₂×Dic₁₇	C₂×M₁₆	1	2	yes	—
123	C₂×Dic₁₇	#131	1	2	yes	—
124	C₂×Dic₁₇	#131	1	2	yes	—
125	C₂×Dic₁₇	#131	1	2	yes	—
126	#3624	—	2	4	no	—
127	#3624	—	2	4	no	—
128	#3624	—	4	2	no	—
129	#3624	—	4	2	no	—
130	C₈×C₁₇	—	2	2	no	—
131	C₈×C₁₇	—	2	2	no	—
132	#3622	—	2	2	no	—
133	#3622	—	2	2	no	—
134	C₄×C₄⋊C₈	—	17	1	no	—
135	C₄×C₄⋊C₈	—	17	1	no	—
136	#1744	—	17	1	no	—
137	C₂×#322	—	17	1	no	—
138	C₂×C₈²	—	17	1	no	—
139	#1273	—	17	1	no	—
140	#1273	—	17	1	no	—
141	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
143	C₂²×C₁₇	C₄×M₁₆	1	1	yes	characteristic
144	C₂²×C₁₇	#405	1	1	yes	characteristic
145	C₂²×C₁₇	#405	1	1	yes	characteristic
146	C₂²×C₁₇	#405	1	1	yes	characteristic
147	C₂²×C₁₇	#405	1	1	yes	characteristic
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
151	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
152	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
153	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
154	Dic₁₇	—	2	2	no	—
155	Dic₁₇	—	2	2	no	—
156	Dic₁₇	#424	1	2	yes	—
157	Dic₁₇	#433	1	2	yes	—
158	Dic₁₇	#435	1	2	yes	—
159	Dic₁₇	#446	1	2	yes	—
160	C₂×C₄⋊C₈	—	17	1	no	—
161	C₂×C₄⋊C₈	—	17	1	no	—
162	C₂×C₄⋊C₈	—	17	2	no	—
163	C₂×C₄×C₈	—	17	1	no	—
164	C₂×C₄×C₈	—	17	1	no	—
165	C₂×C₄×C₈	—	17	1	no	—
166	C₂×C₄×C₈	—	17	1	no	—
167	C₂×C₄×C₈	—	17	1	no	—
168	C₂×C₄×C₈	—	17	1	no	—
169	C₄³	—	17	1	no	—
170	#336	—	17	2	no	—
171	#336	—	17	2	no	—
172	#336	—	17	2	no	—
173	#336	—	17	1	no	—
174	#336	#595	1	1	yes	characteristic
175	#322	—	34	2	no	—
176	C₈²	—	34	2	no	—
177	C₂×C₁₇	C₈×M₁₆	1	1	yes	characteristic
178	C₂×C₁₇	#1450	1	1	yes	characteristic
179	C₂×C₁₇	#1453	1	1	yes	characteristic
180	C₂×C₁₇	#1454	1	1	yes	characteristic
181	C₂×C₁₇	#1768	1	1	yes	characteristic
182	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₁₇	#1771	1	1	yes	characteristic
184	C₂²×C₈	—	17	2	no	—
185	C₂²×C₈	—	17	2	no	—
186	C₂²×C₈	—	17	1	no	—
187	C₂²×C₈	—	17	1	no	—
188	C₂²×C₈	—	17	2	no	—
189	C₂²×C₈	—	17	2	no	—
190	C₂²×C₈	C₂×#595	1	1	yes	characteristic
191	C₂²×C₈	C₂×#595	1	1	yes	characteristic
192	C₂×C₄²	—	17	1	no	—
193	C₂×C₄²	—	17	1	no	—
194	C₂×C₄²	—	17	1	no	—
195	C₂×C₄²	—	17	1	no	—
196	C₂×C₄²	—	17	1	no	—
197	C₂×C₄²	—	17	1	no	—
198	C₂×C₄²	C₂×#595	1	1	yes	characteristic
199	C₄⋊C₈	—	17	4	no	—
200	C₄⋊C₈	—	17	4	no	—
201	C₄⋊C₈	—	34	4	no	—
202	C₄×C₈	—	17	2	no	—
203	C₄×C₈	—	17	2	no	—
204	C₄×C₈	—	34	1	no	—
205	C₄×C₈	—	34	1	no	—
206	C₄×C₈	—	34	1	no	—
207	C₄×C₈	—	34	1	no	—
208	C₄×C₈	—	34	1	no	—
209	C₄×C₈	—	34	1	no	—
210	C₄×C₈	—	34	1	no	—
211	C₄×C₈	—	34	1	no	—
212	C₄×C₈	—	34	1	no	—
213	C₄×C₈	—	34	1	no	—
214	C₁₇	#8264	1	1	yes	characteristic, complemented, Sylow
215	C₂²×C₄	—	17	1	no	—
216	C₂²×C₄	—	17	1	no	—
217	C₂²×C₄	—	17	1	no	—
218	C₂²×C₄	—	17	1	no	—
219	C₂²×C₄	C₄×#595	1	1	yes	characteristic
220	C₂²×C₄	C₄×#595	1	1	yes	characteristic
221	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	17	4	no	—
227	C₂×C₈	—	17	4	no	—
228	C₂×C₈	—	17	2	no	—
229	C₂×C₈	—	17	2	no	—
230	C₂×C₈	—	34	2	no	—
231	C₂×C₈	—	34	2	no	—
232	C₂×C₈	—	34	2	no	—
233	C₂×C₈	—	34	2	no	—
234	C₂×C₈	—	34	1	no	—
235	C₂×C₈	—	34	1	no	—
236	C₂×C₈	—	34	1	no	—
237	C₂×C₈	—	34	1	no	—
238	C₂×C₈	—	34	2	no	—
239	C₂×C₈	—	34	2	no	—
240	C₂×C₈	—	34	2	no	—
241	C₂×C₈	—	34	2	no	—
242	C₂×C₈	—	34	2	no	—
243	C₂×C₈	—	34	2	no	—
244	C₂×C₈	#63242	1	2	yes	—
245	C₂×C₈	#63242	1	2	yes	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	17	2	no	—
249	C₄²	—	17	2	no	—
250	C₄²	—	17	1	no	—
251	C₄²	—	17	1	no	—
252	C₄²	—	17	1	no	—
253	C₄²	—	17	1	no	—
254	C₄²	—	34	2	no	—
255	C₄²	—	34	2	no	—
256	C₄²	—	34	2	no	—
257	C₄²	—	34	2	no	—
258	C₂³	C₂×C₄×#595	1	1	yes	characteristic
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	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	17	1	no	—
266	C₂×C₄	—	17	1	no	—
267	C₂×C₄	—	17	1	no	—
268	C₂×C₄	—	17	1	no	—
269	C₂×C₄	—	17	2	no	—
270	C₂×C₄	—	17	2	no	—
271	C₂×C₄	—	17	2	no	—
272	C₂×C₄	—	17	2	no	—
273	C₂×C₄	—	17	2	no	—
274	C₂×C₄	—	17	2	no	—
275	C₂×C₄	—	34	1	no	—
276	C₂×C₄	—	34	1	no	—
277	C₂×C₄	—	34	1	no	—
278	C₂×C₄	—	34	1	no	—
279	C₂×C₄	C₂×#63242	1	1	yes	characteristic
280	C₂×C₄	C₂×#63242	1	1	yes	characteristic
281	C₂×C₄	#10587678	1	1	yes	characteristic
282	C₂×C₄	#10587678	1	1	yes	characteristic
283	C₂×C₄	D₄×#595	1	1	yes	characteristic
284	C₂×C₄	Q₈×#595	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	—	2	2	no	—
287	C₈	—	34	4	no	—
288	C₈	—	34	4	no	—
289	C₈	—	34	2	no	—
290	C₈	—	34	2	no	—
291	C₈	—	68	2	no	—
292	C₈	—	68	2	no	—
293	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
294	C₂²	C₄×#63242	1	1	yes	characteristic
295	C₂²	M₁₆×#595	1	1	yes	characteristic
296	C₂²	#11763772	1	1	yes	characteristic
297	C₂²	#11763772	1	1	yes	characteristic
298	C₂²	#11763773	1	1	yes	characteristic
299	C₂²	#11763775	1	1	yes	characteristic
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	17	2	no	—
303	C₄	—	17	2	no	—
304	C₄	—	17	2	no	—
305	C₄	—	17	2	no	—
306	C₄	—	34	2	no	—
307	C₄	—	34	2	no	—
308	C₄	#11763836	1	1	yes	characteristic
309	C₄	#11763836	1	1	yes	characteristic
310	C₄	#11763858	1	1	yes	characteristic
311	C₄	#11763858	1	1	yes	characteristic
312	C₂	#14543982	1	1	yes	characteristic
313	C₂	#14543988	1	1	yes	characteristic
314	C₂	#14544001	1	1	yes	characteristic
315	C₂	#14544005	1	1	yes	characteristic
316	C₂	#14544573	1	1	yes	characteristic
317	C₂	#14544576	1	1	yes	characteristic
318	C₂	#14544585	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₁₇×#336	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	#336	#595
hypocenter	C₁₇	#8264
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
49. #10587485	C₈	semidirect
50. #10587485	C₈	semidirect
214. C₁₇	#8264	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587544	2
#14544498	2
C₂×#11763650	2
C₂×C₈×#3624	2
#14543606	2
#14528632	2
#8264	17