Subgroups — #16927543

Order 4352.

366 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528948	C₂	1	1	yes	maximal, characteristic
2	#14527532	C₂	1	1	yes	maximal, characteristic
3	#14527348	C₂	1	2	yes	maximal
4	#14526644	C₂	1	2	yes	maximal
5	C₁₇×#2960	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	2	yes	—
7	C₂×#10587489	—	2	1	no	—
8	C₂×#10587488	—	2	1	no	—
9	#11762675	C₂²	1	2	yes	—
10	#11762562	—	2	1	no	—
11	#11762560	—	2	1	no	—
12	#11762546	—	2	4	no	—
13	C₁₇×#533	C₂²	1	1	yes	characteristic
14	C₁₇×#446	—	2	1	no	—
15	C₁₇×#446	C₄	1	2	yes	complemented
16	C₈×Q₈×C₁₇	—	2	1	no	—
17	C₈×Q₈×C₁₇	C₄	1	2	yes	complemented
18	C₁₇×#424	—	2	1	no	—
19	C₁₇×#423	C₂²	1	1	yes	characteristic
20	C₁₇×#405	—	2	1	no	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×C₁₇	—	2	1	no	—
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₁₇×C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	C₁₇×C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	C₁₇×#125	—	2	1	no	—
28	C₁₇×C₂²⋊Q₈	—	2	1	no	—
29	C₁₇×C₂²⋊Q₈	D₄	1	2	yes	—
30	C₄×Q₈×C₁₇	—	2	1	no	—
31	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
32	C₁₇×#117	—	2	1	no	—
33	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₇	—	2	2	no	—
35	C₁₇×C₄⋊C₈	—	2	2	no	—
36	C₁₇×C₄⋊C₈	—	2	2	no	—
37	C₁₇×C₄⋊C₈	—	2	1	no	—
38	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₁₇×#97	—	2	2	no	—
40	C₄×C₈×C₁₇	—	2	2	no	—
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
45	#10587502	—	2	2	no	—
46	#10587502	—	2	2	no	—
47	C₂²⋊#3622	—	2	2	no	—
48	#10587489	—	4	2	no	—
49	#10587488	—	4	2	no	—
50	C₄⋊#3622	—	2	2	no	—
51	C₄×#3622	—	2	2	no	—
52	C₂×Q₈×C₁₇	—	2	1	no	—
53	C₂×Q₈×C₁₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
57	C₂²×Dic₁₇	—	4	2	no	—
58	M₁₆×C₁₇	—	4	1	no	—
59	M₁₆×C₁₇	—	4	1	no	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	1	no	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
68	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₁₇	—	2	2	no	—
70	C₄⋊C₄×C₁₇	—	2	2	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	—	2	1	no	—
73	C₄⋊C₄×C₁₇	—	2	1	no	—
74	C₄⋊C₄×C₁₇	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₁₇	—	2	2	no	—
76	C₄²×C₁₇	—	2	2	no	—
77	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₁₇	#41	1	1	yes	characteristic
81	C₄⋊Dic₁₇	—	4	2	no	—
82	C₄⋊Dic₁₇	—	4	2	no	—
83	C₂×#3622	—	4	2	no	—
84	#8677	—	17	1	no	maximal, Sylow
85	C₂³×C₁₇	#121	1	1	yes	characteristic
86	Q₈×C₁₇	—	4	2	no	—
87	Q₈×C₁₇	—	4	2	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	—	4	1	no	—
91	C₂×C₄×C₁₇	—	4	1	no	—
92	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
95	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
102	C₂×Dic₁₇	—	8	2	no	—
103	C₂×Dic₁₇	—	8	2	no	—
104	C₂×Dic₁₇	—	8	2	no	—
105	C₈×C₁₇	—	2	1	no	—
106	C₈×C₁₇	—	2	1	no	—
107	C₈×C₁₇	—	4	2	no	—
108	C₈×C₁₇	#103	1	2	yes	—
109	C₈×C₁₇	#103	1	2	yes	—
110	#3622	—	8	2	no	—
111	#2960	D₁₇	1	1	yes	characteristic
112	#1846	—	17	1	no	—
113	#1564	—	17	1	no	—
114	#1518	—	17	2	no	—
115	#1286	—	17	2	no	—
116	C₂²×C₁₇	—	2	1	no	—
117	C₂²×C₁₇	—	2	1	no	—
118	C₂²×C₁₇	#380	1	1	yes	characteristic
119	C₂²×C₁₇	#460	1	1	yes	characteristic
120	C₂²×C₁₇	#462	1	1	yes	characteristic
121	C₄×C₁₇	—	2	1	no	—
122	C₄×C₁₇	—	2	1	no	—
123	C₄×C₁₇	—	2	1	no	—
124	C₄×C₁₇	—	2	1	no	—
125	C₄×C₁₇	—	4	2	no	—
126	C₄×C₁₇	—	4	2	no	—
127	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
128	C₄×C₁₇	#417	1	1	yes	characteristic
129	C₄×C₁₇	#475	1	1	yes	characteristic
130	C₄×C₁₇	#477	1	1	yes	characteristic
131	Dic₁₇	—	16	2	no	—
132	#533	C₂×D₁₇	1	1	yes	characteristic
133	#446	—	2	1	no	—
134	#446	Dic₁₇	1	2	yes	complemented
135	C₈×Q₈	—	2	1	no	—
136	C₈×Q₈	Dic₁₇	1	2	yes	complemented
137	#431	—	17	2	no	—
138	C₂×#107	—	34	1	no	—
139	C₂×#106	—	34	1	no	—
140	#424	—	2	1	no	—
141	#423	C₂×D₁₇	1	1	yes	characteristic
142	#405	—	2	1	no	—
143	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
144	#382	—	17	2	no	—
145	#335	—	34	1	no	—
146	#334	—	34	1	no	—
147	#328	—	34	4	no	—
148	C₂×C₁₇	—	2	1	no	—
149	C₂×C₁₇	—	2	1	no	—
150	C₂×C₁₇	#1803	1	1	yes	characteristic
151	C₂×C₁₇	#1812	1	1	yes	characteristic
152	C₂×C₁₇	#1814	1	1	yes	characteristic
153	C₂×M₁₆	—	2	1	no	—
154	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
155	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
156	C₄⋊Q₈	Dic₃₄	1	1	yes	characteristic
157	C₄⋊Q₈	Dic₃₄	1	1	yes	characteristic
158	#125	—	2	1	no	—
159	C₂²⋊Q₈	—	2	1	no	—
160	C₂²⋊Q₈	D₆₈	1	2	yes	—
161	C₄×Q₈	—	2	1	no	—
162	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
163	#117	—	2	1	no	—
164	C₂×C₄⋊C₄	—	34	2	no	—
165	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
166	#107	—	68	2	no	—
167	#106	—	68	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	34	2	no	—
172	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
173	C₂²⋊C₈	—	34	2	no	—
174	#97	—	2	2	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	34	2	no	—
177	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
178	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
181	#95	—	34	2	no	—
182	#95	—	34	2	no	—
183	C₁₇	#8677	1	1	yes	characteristic, complemented, Sylow
184	C₂×Q₈	—	2	1	no	—
185	C₂×Q₈	C₄⋊Dic₁₇	1	2	yes	—
186	C₂²×C₄	—	68	2	no	—
187	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
188	C₂²×C₄	#63252	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
190	M₁₆	—	4	1	no	—
191	M₁₆	—	4	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	68	2	no	—
196	C₂×C₈	#63230	1	2	yes	—
197	C₂×C₈	#63231	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	68	2	no	—
208	C₄⋊C₄	—	68	2	no	—
209	C₄⋊C₄	C₄⋊Dic₁₇	1	2	yes	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
213	C₄²	#63252	1	1	yes	characteristic
214	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
216	C₂³	#10587616	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	136	2	no	—
224	C₂×C₄	—	136	2	no	—
225	C₂×C₄	—	136	2	no	—
226	C₂×C₄	#10587610	1	1	yes	characteristic
227	C₂×C₄	#10587610	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
229	C₂×C₄	#10587612	1	1	yes	characteristic
230	C₂×C₄	#10587616	1	1	yes	characteristic
231	C₂×C₄	C₂×#63230	1	1	yes	characteristic
232	C₂×C₄	C₂×#63231	1	1	yes	characteristic
233	C₂×C₄	#10587635	1	1	yes	characteristic
234	C₂×C₄	#10587637	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	4	2	no	—
239	C₈	—	136	2	no	—
240	C₈	#10587506	1	2	yes	—
241	C₈	#10587506	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#11762669	1	1	yes	characteristic
245	C₂²	#11763009	1	1	yes	characteristic
246	C₂²	#11763019	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	272	2	no	—
254	C₄	C₂×#10587506	1	1	yes	characteristic
255	C₄	#11762810	1	1	yes	characteristic
256	C₄	#11763094	1	1	yes	characteristic
257	C₄	#11763104	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#14528789	1	1	yes	characteristic
261	C₂	#14528830	1	1	yes	characteristic
262	C₂	#14528838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762669
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2960	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂³×C₁₇	#121
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2960	D₁₇
hypocenter	C₁₇	#8677
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
15. C₁₇×#446	C₄	semidirect
17. C₈×Q₈×C₁₇	C₄	semidirect
134. #446	Dic₁₇	semidirect
136. C₈×Q₈	Dic₁₇	semidirect
183. C₁₇	#8677	semidirect

Maximal subgroups

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

subgroup	index
#14528948	2
#14527532	2
#14527348	2
#14526644	2
C₁₇×#2960	2
#8677	17