Subgroups — #17189827

Order 4352.

474 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539317	C₂	1	2	yes	maximal
2	#14531068	C₂	1	2	yes	maximal, complemented
3	#14531067	C₂	1	2	yes	maximal, complemented
4	#14531063	C₂	1	1	yes	maximal, characteristic, complemented
5	#14531061	C₂	1	1	yes	maximal, characteristic, complemented
6	#14531058	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530569	C₂	1	2	yes	maximal
8	#14530565	C₂	1	1	yes	maximal, characteristic
9	D₁₇×#366	C₂	1	1	yes	maximal, characteristic
10	#14530379	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763436	—	2	2	no	—
13	C₂×#10587469	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
15	#11763217	—	2	2	no	—
16	#11763215	—	2	2	no	—
17	#11763213	C₂²	1	2	yes	—
18	#11763211	C₂²	1	2	yes	—
19	#11763210	C₂²	1	2	yes	—
20	#11762850	—	2	2	no	—
21	#11762846	C₂²	1	2	yes	—
22	#107×D₁₇	C₂²	1	2	yes	—
23	#11762763	C₂²	1	1	yes	characteristic
24	#11762637	C₂²	1	2	yes	—
25	#11762636	C₂²	1	2	yes	—
26	#11762636	C₂²	1	2	yes	—
27	#11762635	C₂²	1	1	yes	characteristic
28	#11762635	C₂²	1	1	yes	characteristic
29	#11762634	C₂²	1	2	yes	—
30	#11762634	C₂²	1	2	yes	—
31	#11762585	C₂²	1	1	yes	characteristic
32	#11762584	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
34	C₁₇×#500	C₂²	1	2	yes	—
35	C₁₇×#366	C₂²	1	1	yes	characteristic
36	C₁₇×#366	C₂²	1	2	yes	—
37	C₁₇×#366	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
39	C₁₇×C₄⋊Q₈	—	2	2	no	—
40	#10587632	—	4	2	no	—
41	#10587630	—	4	2	no	—
42	C₂×#63218	D₄	1	1	yes	characteristic
43	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
44	#10587582	—	4	2	no	—
45	#10587578	—	2	2	no	—
46	C₄⋊C₄×D₁₇	—	2	2	no	—
47	#10587576	—	2	2	no	—
48	#10587575	—	4	2	no	—
49	#10587574	—	2	2	no	—
50	#10587558	D₄	1	1	yes	characteristic
51	#10587556	—	2	2	no	—
52	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
54	C₁₇×#107	—	2	2	no	—
55	C₁₇×#107	C₂³	1	2	yes	—
56	C₁₇×#107	C₂³	1	2	yes	—
57	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
58	#10587489	C₂³	1	2	yes	—
59	#10587489	C₂³	1	2	yes	—
60	#10587488	—	2	2	no	—
61	#10587486	D₄	1	1	yes	characteristic
62	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
63	C₂×#63214	C₂³	1	1	yes	characteristic
64	C₂×#63214	C₂³	1	1	yes	characteristic
65	#10587479	—	2	2	no	—
66	#10587479	—	2	2	no	—
67	#10587478	—	2	2	no	—
68	#10587478	—	2	2	no	—
69	#10587474	D₄	1	1	yes	characteristic
70	#10587469	Q₈	1	4	yes	complemented
71	C₁₆×D₁₇	Q₈	1	4	yes	complemented
72	C₂×Q₈×C₁₇	—	4	2	no	—
73	C₂×C₄×D₁₇	—	4	2	no	—
74	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂×Dic₃₄	—	4	2	no	—
76	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	—	2	2	no	—
79	C₄⋊C₄×C₁₇	—	2	2	no	—
80	C₄⋊C₄×C₁₇	—	4	2	no	—
81	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
82	#63227	—	4	2	no	—
83	#63227	D₈	1	1	yes	characteristic
84	#63227	D₈	1	1	yes	characteristic
85	C₄⋊Dic₁₇	—	2	2	no	—
86	C₄⋊Dic₁₇	—	2	2	no	—
87	C₄⋊Dic₁₇	—	4	2	no	—
88	#63225	—	4	2	no	—
89	#63225	—	4	2	no	—
90	#63225	—	4	2	no	—
91	#63225	—	4	2	no	—
92	#63225	—	4	2	no	—
93	#63225	D₈	1	1	yes	characteristic
94	#63225	D₈	1	1	yes	characteristic
95	C₄×Dic₁₇	—	4	2	no	—
96	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×#3622	C₂×D₄	1	1	yes	characteristic
98	C₂×#3622	C₂×D₄	1	1	yes	characteristic
99	#63218	Q₁₆	1	4	yes	—
100	C₈×D₁₇	C₂×Q₈	1	2	yes	—
101	C₈×D₁₇	C₂×Q₈	1	2	yes	—
102	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
103	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
104	#63214	C₂×Q₈	1	2	yes	—
105	#63214	C₂×Q₈	1	2	yes	—
106	#33729	—	17	1	no	maximal, Sylow
107	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₁₇	—	16	2	no	—
109	C₂×C₄×C₁₇	—	4	2	no	—
110	C₂×C₄×C₁₇	—	4	2	no	—
111	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₁₇	—	4	2	no	—
115	C₂×Dic₁₇	—	4	2	no	—
116	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
117	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
118	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
119	C₄×D₁₇	—	16	2	no	—
120	C₄×D₁₇	C₄⋊Q₈	1	2	yes	—
121	C₄×D₁₇	C₂×Q₁₆	1	2	yes	—
122	Dic₃₄	—	16	2	no	—
123	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
124	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
125	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
126	#3622	C₄⋊Q₈	1	2	yes	—
127	#3622	C₂×Q₁₆	1	2	yes	—
128	#3164	—	17	2	no	—
129	#2255	—	17	2	no	—
130	#2255	—	17	2	no	—
131	#2253	—	17	1	no	—
132	#2253	—	17	1	no	—
133	#2252	—	17	1	no	—
134	#2252	D₁₇	1	1	yes	characteristic, complemented
135	#2159	—	17	2	no	—
136	#2158	—	17	1	no	—
137	C₂×#366	—	17	1	no	—
138	#2108	—	17	1	no	—
139	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₁₇	—	2	1	no	—
141	C₂×D₁₇	—	2	1	no	—
142	C₂×D₁₇	#500	1	2	yes	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	8	2	no	—
146	C₄×C₁₇	—	8	2	no	—
147	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
149	Dic₁₇	—	2	1	no	—
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	8	2	no	—
152	Dic₁₇	—	8	2	no	—
153	Dic₁₇	#500	1	2	yes	—
154	#544	—	34	2	no	—
155	C₂×M₃₂	—	17	1	no	—
156	C₂²×C₁₆	—	17	1	no	—
157	#501	—	34	2	no	—
158	#501	—	34	2	no	—
159	#500	—	17	2	no	—
160	#500	C₂×D₁₇	1	2	yes	—
161	#499	—	17	2	no	—
162	#499	—	17	2	no	—
163	#428	—	34	2	no	—
164	#427	—	17	2	no	—
165	C₂×#107	—	17	2	no	—
166	#405	—	17	1	no	—
167	#367	—	17	2	no	—
168	#367	—	17	2	no	—
169	#367	—	17	2	no	—
170	#366	—	17	1	no	—
171	#366	—	17	1	no	—
172	#366	—	17	2	no	—
173	#366	—	17	2	no	—
174	#366	C₂×D₁₇	1	1	yes	characteristic
175	#366	C₂×D₁₇	1	1	yes	characteristic
176	#366	C₂×D₁₇	1	2	yes	—
177	#346	—	17	1	no	—
178	#346	—	17	1	no	—
179	C₄×C₁₆	—	17	1	no	—
180	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
181	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
182	C₂×C₁₇	#3415	1	1	yes	characteristic
183	C₂×C₁₇	#3417	1	1	yes	characteristic
184	D₁₇	—	2	2	no	—
185	C₂×M₁₆	—	17	1	no	—
186	C₂²×C₈	—	17	1	no	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	34	2	no	—
189	C₄⋊Q₈	—	68	2	no	—
190	#125	—	34	2	no	—
191	#125	—	34	2	no	—
192	#125	—	68	2	no	—
193	C₂²⋊Q₈	—	68	2	no	—
194	C₂²⋊Q₈	—	68	2	no	—
195	#117	—	17	1	no	—
196	#117	—	34	2	no	—
197	C₂×C₄⋊C₄	—	34	2	no	—
198	M₃₂	—	17	4	no	—
199	C₂×C₁₆	—	17	1	no	—
200	C₂×C₁₆	—	17	1	no	—
201	C₂×C₁₆	—	17	4	no	—
202	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
203	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
204	#107	—	2	2	no	—
205	#107	—	17	2	no	—
206	#107	—	17	2	no	—
207	#107	—	34	2	no	—
208	#107	—	34	2	no	—
209	#107	C₂²×D₁₇	1	2	yes	—
210	#107	C₂²×D₁₇	1	2	yes	—
211	#106	—	34	2	no	—
212	#106	—	34	2	no	—
213	#106	—	34	2	no	—
214	#97	—	17	1	no	—
215	#97	—	17	1	no	—
216	C₄×C₈	—	17	1	no	—
217	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
218	C₁₇	#33729	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	68	2	no	—
221	C₂²×C₄	—	17	1	no	—
222	C₂²×C₄	—	68	2	no	—
223	M₁₆	—	17	4	no	—
224	C₂×C₈	—	17	1	no	—
225	C₂×C₈	—	17	1	no	—
226	C₂×C₈	—	17	2	no	—
227	C₂×C₈	—	17	2	no	—
228	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	17	1	no	—
234	C₄⋊C₄	—	17	1	no	—
235	C₄⋊C₄	—	34	2	no	—
236	C₄⋊C₄	—	34	2	no	—
237	C₄⋊C₄	—	68	2	no	—
238	C₄⋊C₄	—	68	2	no	—
239	C₄⋊C₄	—	68	2	no	—
240	C₄⋊C₄	—	68	2	no	—
241	C₄⋊C₄	—	68	2	no	—
242	C₄⋊C₄	—	68	2	no	—
243	C₂²⋊C₄	—	17	1	no	—
244	C₂²⋊C₄	—	17	1	no	—
245	C₂²⋊C₄	—	68	2	no	—
246	C₄²	—	17	1	no	—
247	C₄²	—	68	2	no	—
248	C₄²	D₄×D₁₇	1	1	yes	characteristic
249	C₁₆	—	17	2	no	—
250	C₁₆	—	17	2	no	—
251	C₁₆	Q₈×D₁₇	1	2	yes	—
252	C₁₆	Q₈×D₁₇	1	2	yes	—
253	C₂³	—	17	1	no	—
254	Q₈	—	16	2	no	—
255	Q₈	—	272	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	272	2	no	—
259	C₂×C₄	—	17	1	no	—
260	C₂×C₄	—	17	2	no	—
261	C₂×C₄	—	17	2	no	—
262	C₂×C₄	—	17	1	no	—
263	C₂×C₄	—	17	1	no	—
264	C₂×C₄	—	68	2	no	—
265	C₂×C₄	—	68	2	no	—
266	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
267	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
269	C₈	—	17	2	no	—
270	C₈	—	17	2	no	—
271	C₈	Q₁₆×D₁₇	1	2	yes	—
272	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
274	C₂²	—	17	2	no	—
275	C₂²	—	34	1	no	—
276	C₂²	—	34	1	no	—
277	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	17	2	no	—
283	C₄	—	34	1	no	—
284	C₄	—	34	1	no	—
285	C₄	—	136	2	no	—
286	C₄	—	136	2	no	—
287	C₄	C₄⋊Q₈×D₁₇	1	1	yes	characteristic
288	C₄	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
289	C₂	—	34	2	no	—
290	C₂	D₁₇×#500	1	1	yes	characteristic
291	C₂	#14541921	1	1	yes	characteristic
292	C₂	#14541933	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2252	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂²×D₁₇	C₂×D₈
hypercenter	#2252	D₁₇
hypocenter	C₁₇	#33729
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. #14531068	C₂	semidirect
3. #14531067	C₂	semidirect
4. #14531063	C₂	semidirect
5. #14531061	C₂	semidirect
6. #14531058	C₂	semidirect
11. C₁₇×#2252	C₂	semidirect
70. #10587469	Q₈	semidirect
71. C₁₆×D₁₇	Q₈	semidirect
134. #2252	D₁₇	semidirect
218. C₁₇	#33729	semidirect

Maximal subgroups

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

subgroup	index
#14539317	2
#14531068	2
#14531067	2
#14531063	2
#14531061	2
#14531058	2
#14530569	2
#14530565	2
D₁₇×#366	2
#14530379	2
C₁₇×#2252	2
#33729	17