Subgroups — #16950419

Order 4352.

400 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762925	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762629	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530513	C₂	1	1	yes	maximal, characteristic
4	#14530511	C₂	1	1	yes	maximal, characteristic
5	#14528954	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526897	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587590	C₂²	1	1	yes	characteristic
9	C₂×#10587593	—	2	1	no	—
10	C₂×#10587483	C₂²	1	1	yes	characteristic
11	#11762925	—	2	1	no	—
12	#11762925	—	2	1	no	—
13	#11762925	C₄	1	2	yes	—
14	#11762925	C₄	1	2	yes	—
15	C₂×#10587490	C₂²	1	1	yes	characteristic
16	C₂×#10587493	—	2	1	no	—
17	#11762832	—	2	1	no	—
18	C₂×#10587496	—	2	1	no	—
19	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
20	#11762629	—	2	1	no	—
21	#11762629	—	2	1	no	—
22	#11762617	—	2	2	no	—
23	#11762615	—	2	2	no	—
24	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
25	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
26	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
27	C₂×#63252	—	2	1	no	—
28	C₂×M₁₆×C₁₇	—	2	1	no	—
29	C₂×M₁₆×C₁₇	—	2	1	no	—
30	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
32	C₂×#63223	—	2	1	no	—
33	C₂²×#3622	D₄	1	1	yes	characteristic
34	#10587593	—	2	2	no	—
35	#10587593	—	2	2	no	—
36	#10587593	—	4	1	no	—
37	#10587593	—	4	1	no	—
38	C₂×Dic₆₈	Q₈	1	1	yes	characteristic
39	#10587590	—	2	1	no	—
40	#10587590	—	2	1	no	—
41	#10587590	C₂×C₄	1	2	yes	—
42	#10587590	D₄	1	2	yes	—
43	C₂×D₁₃₆	Q₈	1	1	yes	characteristic
44	C₂×#63219	—	2	1	no	—
45	C₂×#63218	—	4	1	no	—
46	C₂×C₈×D₁₇	—	4	1	no	—
47	C₁₇×M₃₂	—	2	1	no	—
48	C₁₇×M₃₂	—	2	1	no	—
49	C₂×C₁₆×C₁₇	—	2	1	no	—
50	C₁₇×#108	—	2	1	no	—
51	C₁₇×#108	—	2	1	no	—
52	C₁₇×#108	—	2	1	no	—
53	C₁₇×#108	C₂×C₄	1	2	yes	—
54	#10587496	—	2	2	no	—
55	#10587496	—	2	2	no	—
56	#10587493	—	4	1	no	—
57	#10587493	—	4	1	no	—
58	#10587490	—	2	1	no	—
59	#10587490	—	2	1	no	—
60	#10587485	—	4	2	no	—
61	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₁₇	D₄	1	1	yes	characteristic
64	C₈×Dic₁₇	D₄	1	1	yes	characteristic
65	#10587483	—	2	1	no	—
66	#10587483	—	2	1	no	—
67	C₂×#63214	—	2	1	no	—
68	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₇	—	4	1	no	—
70	C₂²×Dic₁₇	—	2	1	no	—
71	#63252	—	2	2	no	—
72	#63252	—	2	2	no	—
73	#63252	—	4	1	no	—
74	#63252	—	4	1	no	—
75	C₂×D₆₈	—	2	1	no	—
76	C₂×C₄×D₁₇	—	4	1	no	—
77	C₂×Dic₃₄	—	2	1	no	—
78	M₁₆×C₁₇	—	2	2	no	—
79	M₁₆×C₁₇	—	4	1	no	—
80	M₁₆×C₁₇	—	4	1	no	—
81	M₁₆×C₁₇	—	4	1	no	—
82	C₂×C₈×C₁₇	—	4	1	no	—
83	C₂×C₈×C₁₇	—	4	1	no	—
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	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
89	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
90	C₄×Dic₁₇	—	2	1	no	—
91	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
93	#63223	—	4	1	no	—
94	#63223	—	4	1	no	—
95	C₂×#3622	—	4	1	no	—
96	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×#3622	D₈	1	1	yes	characteristic
99	C₂×#3622	D₈	1	1	yes	characteristic
100	C₂×#3622	SD₁₆	1	1	yes	characteristic
101	C₂×#3622	SD₁₆	1	1	yes	characteristic
102	Dic₆₈	—	2	1	no	—
103	Dic₆₈	C₄⋊C₄	1	2	yes	—
104	D₁₃₆	—	2	1	no	—
105	D₁₃₆	C₄⋊C₄	1	2	yes	—
106	#63219	—	2	2	no	—
107	#63219	—	2	2	no	—
108	#63218	—	4	2	no	—
109	#63218	—	4	2	no	—
110	C₈×D₁₇	—	4	2	no	—
111	C₈×D₁₇	—	4	2	no	—
112	C₁₆×C₁₇	—	4	1	no	—
113	#63214	—	4	1	no	—
114	#12822	—	17	1	no	maximal, Sylow
115	C₂³×C₁₇	#137	1	1	yes	characteristic
116	C₂²×D₁₇	—	4	1	no	—
117	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂²⋊D₁₇	—	4	2	no	—
124	C₂²⋊D₁₇	—	4	2	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	4	2	no	—
128	C₂×Dic₁₇	—	4	1	no	—
129	D₆₈	—	2	2	no	—
130	D₆₈	—	4	1	no	—
131	C₄×D₁₇	—	4	2	no	—
132	C₄×D₁₇	—	4	2	no	—
133	Dic₃₄	—	2	2	no	—
134	Dic₃₄	—	4	1	no	—
135	C₈×C₁₇	—	4	2	no	—
136	C₈×C₁₇	—	8	1	no	—
137	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₈×C₁₇	#117	1	1	yes	characteristic
139	C₈×C₁₇	#121	1	1	yes	characteristic
140	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
141	#3622	—	8	1	no	—
142	#3622	#102	1	1	yes	characteristic
143	#3622	#102	1	1	yes	characteristic
144	#3622	#102	1	1	yes	characteristic
145	#3622	#102	1	1	yes	characteristic
146	C₂×#443	—	17	1	no	—
147	C₂×#364	—	17	1	no	—
148	#2143	—	17	1	no	—
149	#2143	—	17	1	no	—
150	#1848	—	17	1	no	—
151	#1388	—	17	1	no	—
152	#1388	D₁₇	1	1	yes	characteristic, complemented
153	C₂²×C₁₇	—	4	1	no	—
154	C₂²×C₁₇	#417	1	1	yes	characteristic
155	C₂²×C₁₇	#461	1	1	yes	characteristic
156	C₂²×C₁₇	#474	1	1	yes	characteristic
157	C₂×D₁₇	—	4	2	no	—
158	C₂×D₁₇	—	8	2	no	—
159	C₄×C₁₇	#380	1	1	yes	characteristic
160	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
161	C₄×C₁₇	#459	1	1	yes	characteristic
162	C₄×C₁₇	#476	1	1	yes	characteristic
163	Dic₁₇	—	2	2	no	—
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	2	no	—
166	C₂×#135	—	17	1	no	—
167	C₂×#131	—	34	1	no	—
168	C₂×M₃₂	—	17	1	no	—
169	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
170	#443	—	17	2	no	—
171	#443	—	17	2	no	—
172	#443	—	34	1	no	—
173	#443	—	34	1	no	—
174	C₂×#108	—	17	1	no	—
175	C₂×#108	—	34	1	no	—
176	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
177	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
178	#423	—	34	1	no	—
179	C₂×#104	—	34	1	no	—
180	C₂×C₄×C₈	—	17	1	no	—
181	#364	—	34	1	no	—
182	#364	—	34	1	no	—
183	#359	—	34	2	no	—
184	#359	—	34	2	no	—
185	C₂×C₁₇	—	2	1	no	—
186	C₂×C₁₇	—	2	1	no	—
187	C₂×C₁₇	—	2	1	no	—
188	C₂×C₁₇	#1802	1	1	yes	characteristic
189	D₁₇	—	8	2	no	—
190	C₂×#41	—	34	1	no	—
191	#135	—	17	2	no	—
192	#135	—	17	2	no	—
193	#135	—	34	1	no	—
194	#135	—	34	1	no	—
195	C₂×Q₁₆	—	17	1	no	—
196	C₂×SD₁₆	—	34	1	no	—
197	C₂×D₈	—	17	1	no	—
198	#131	—	34	2	no	—
199	#131	—	34	2	no	—
200	#131	—	68	1	no	—
201	#131	—	68	1	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	34	1	no	—
205	C₂×M₁₆	—	68	1	no	—
206	C₂²×C₈	—	17	1	no	—
207	C₂²×C₈	—	68	1	no	—
208	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
209	C₂×C₄²	—	17	1	no	—
210	M₃₂	—	2	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	34	1	no	—
213	M₃₂	—	34	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	34	1	no	—
216	#108	—	2	1	no	—
217	#108	—	2	1	no	—
218	#108	—	2	1	no	—
219	#108	—	34	1	no	—
220	#108	—	34	1	no	—
221	#108	—	68	1	no	—
222	#108	—	68	1	no	—
223	#108	C₄×D₁₇	1	2	yes	—
224	C₄⋊C₈	—	68	2	no	—
225	#104	—	34	2	no	—
226	#104	—	34	2	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	C₁₇	#12822	1	1	yes	characteristic, complemented, Sylow
232	#41	—	34	2	no	—
233	#41	—	34	2	no	—
234	#41	—	68	1	no	—
235	#41	—	68	1	no	—
236	C₂×Q₈	—	34	1	no	—
237	C₂×D₄	—	34	1	no	—
238	C₂×D₄	—	68	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	C₂²×C₄	—	68	1	no	—
241	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
242	Q₁₆	—	17	2	no	—
243	Q₁₆	—	34	1	no	—
244	SD₁₆	—	34	2	no	—
245	SD₁₆	—	34	2	no	—
246	D₈	—	17	2	no	—
247	D₈	—	34	1	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	68	2	no	—
253	M₁₆	—	68	2	no	—
254	M₁₆	—	68	1	no	—
255	M₁₆	—	68	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	1	no	—
260	C₂×C₈	—	17	1	no	—
261	C₂×C₈	—	17	1	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	17	1	no	—
264	C₂×C₈	—	68	2	no	—
265	C₂×C₈	—	68	2	no	—
266	C₂×C₈	—	68	1	no	—
267	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
268	C₂×C₈	#63252	1	1	yes	characteristic
269	C₂×C₈	#63252	1	1	yes	characteristic
270	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
271	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
272	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
273	C₄²	—	17	1	no	—
274	C₄²	—	17	1	no	—
275	C₄²	—	34	1	no	—
276	C₁₆	—	4	1	no	—
277	C₁₆	—	68	1	no	—
278	C₂³	—	68	1	no	—
279	C₂³	#10587604	1	1	yes	characteristic
280	Q₈	—	34	2	no	—
281	Q₈	—	68	1	no	—
282	D₄	—	34	2	no	—
283	D₄	—	68	2	no	—
284	D₄	—	68	2	no	—
285	D₄	—	68	1	no	—
286	C₂×C₄	—	34	1	no	—
287	C₂×C₄	—	34	1	no	—
288	C₂×C₄	—	68	2	no	—
289	C₂×C₄	—	68	1	no	—
290	C₂×C₄	—	68	2	no	—
291	C₂×C₄	—	68	2	no	—
292	C₂×C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
293	C₂×C₄	#10587570	1	1	yes	characteristic
294	C₂×C₄	#10587582	1	1	yes	characteristic
295	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
296	C₂×C₄	SD₁₆×D₁₇	1	1	yes	characteristic
297	C₂×C₄	#10587600	1	1	yes	characteristic
298	C₈	—	4	2	no	—
299	C₈	—	8	1	no	—
300	C₈	—	17	1	no	—
301	C₈	—	17	1	no	—
302	C₈	—	17	1	no	—
303	C₈	—	17	1	no	—
304	C₈	—	136	1	no	—
305	C₈	#10587558	1	1	yes	characteristic
306	C₈	#10587570	1	1	yes	characteristic
307	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
308	C₈	#10587582	1	1	yes	characteristic
309	C₂²	—	4	1	no	—
310	C₂²	—	68	2	no	—
311	C₂²	—	136	2	no	—
312	C₂²	#11762809	1	1	yes	characteristic
313	C₂²	#11763013	1	1	yes	characteristic
314	C₂²	#11763088	1	1	yes	characteristic
315	C₄	—	34	2	no	—
316	C₄	—	68	1	no	—
317	C₄	—	68	2	no	—
318	C₄	#11762668	1	1	yes	characteristic
319	C₄	#102×D₁₇	1	1	yes	characteristic
320	C₄	#11763003	1	1	yes	characteristic
321	C₄	#11763098	1	1	yes	characteristic
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	136	2	no	—
326	C₂	#14528783	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763003
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1388	C₂
socle	C₂×C₁₇	#1802
prime radical	C₂×#10587590	C₂²
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1388	D₁₇
hypocenter	C₁₇	#12822
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1388

Complements

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

N	H ≅ G/N	type
2. C₂×#11762629	C₂	semidirect
5. #14528954	C₂	semidirect
6. #14526897	C₂	semidirect
7. C₁₇×#1388	C₂	semidirect
152. #1388	D₁₇	semidirect
231. C₁₇	#12822	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762925	2
C₂×#11762629	2
#14530513	2
#14530511	2
#14528954	2
#14526897	2
C₁₇×#1388	2
#12822	17