Subgroups — #16950830

Order 4352.

486 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587489	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762637	C₂	1	2	yes	maximal, complemented
3	#14528813	C₂	1	1	yes	maximal, characteristic
4	#14526876	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₁₇	—	2	1	no	—
7	C₂×#10587489	—	2	4	no	—
8	C₂×#10587489	—	2	1	no	—
9	C₂×#10587489	C₂²	1	1	yes	characteristic
10	C₂×#10587489	C₂²	1	1	yes	characteristic
11	C₂×#10587489	C₂²	1	2	yes	—
12	#11762669	—	2	1	no	—
13	#11762637	—	2	4	no	—
14	#11762572	—	2	1	no	—
15	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
17	C₁₇×C₂²⋊C₁₆	C₄	1	4	yes	complemented
18	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
19	C₂³×Dic₁₇	—	4	1	no	—
20	C₂²×C₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	—	2	2	no	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₁₇	—	4	1	no	—
27	C₂×C₄⋊Dic₁₇	—	2	4	no	—
28	C₂×C₄⋊Dic₁₇	—	2	2	no	—
29	C₂×C₄⋊Dic₁₇	—	2	1	no	—
30	C₂×C₄⋊Dic₁₇	—	2	1	no	—
31	C₂×C₄⋊Dic₁₇	—	4	2	no	—
32	C₂×C₄⋊Dic₁₇	—	4	1	no	—
33	C₂×C₁₆×C₁₇	—	2	2	no	—
34	C₂×C₁₆×C₁₇	—	2	2	no	—
35	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
36	#10587502	—	4	1	no	—
37	#10587489	—	2	4	no	—
38	#10587489	—	2	2	no	—
39	#10587489	—	2	2	no	—
40	#10587489	—	2	2	no	—
41	#10587489	D₄	1	4	yes	—
42	C₂⁴×C₁₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	—	2	1	no	—
44	C₂²×C₄×C₁₇	—	2	1	no	—
45	C₂²×C₄×C₁₇	—	2	1	no	—
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₇	—	4	4	no	—
52	C₂²×Dic₁₇	—	4	2	no	—
53	C₂²×Dic₁₇	—	4	1	no	—
54	C₂²×Dic₁₇	—	4	1	no	—
55	C₂²×Dic₁₇	—	8	2	no	—
56	C₂²×Dic₁₇	—	8	2	no	—
57	C₂×C₈×C₁₇	—	2	1	no	—
58	C₂×C₈×C₁₇	—	2	1	no	—
59	C₂×C₈×C₁₇	—	2	4	no	—
60	C₂×C₈×C₁₇	—	2	4	no	—
61	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₁₇	#41	1	2	yes	—
66	C₂²⋊Dic₁₇	—	8	2	no	—
67	C₄⋊Dic₁₇	—	2	4	no	—
68	C₄⋊Dic₁₇	—	4	4	no	—
69	C₄⋊Dic₁₇	—	4	2	no	—
70	C₄⋊Dic₁₇	—	4	1	no	—
71	C₄⋊Dic₁₇	—	4	1	no	—
72	C₄⋊Dic₁₇	—	4	1	no	—
73	C₄⋊Dic₁₇	—	4	1	no	—
74	C₁₆×C₁₇	—	2	4	no	—
75	#12920	—	17	1	no	maximal, Sylow
76	C₂³×C₁₇	—	2	1	no	—
77	C₂³×C₁₇	—	2	1	no	—
78	C₂³×C₁₇	—	2	1	no	—
79	C₂³×C₁₇	—	2	1	no	—
80	C₂³×C₁₇	#107	1	2	yes	—
81	C₂³×C₁₇	SD₃₂	1	2	yes	—
82	C₂³×C₁₇	SD₃₂	1	2	yes	—
83	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	—	2	1	no	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	2	no	—
87	C₂×C₄×C₁₇	—	2	2	no	—
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	#107	1	2	yes	—
91	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
94	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
97	C₂×Dic₁₇	—	4	4	no	—
98	C₂×Dic₁₇	—	8	8	no	—
99	C₂×Dic₁₇	—	8	2	no	—
100	C₂×Dic₁₇	—	8	2	no	—
101	C₂×Dic₁₇	—	8	1	no	—
102	C₂×Dic₁₇	—	8	1	no	—
103	C₂×Dic₁₇	—	8	1	no	—
104	C₂×Dic₁₇	—	8	1	no	—
105	C₂×Dic₁₇	—	8	1	no	—
106	C₂×Dic₁₇	—	8	1	no	—
107	C₈×C₁₇	—	2	2	no	—
108	C₈×C₁₇	C₄×D₄	1	2	yes	—
109	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
110	C₂²×#107	—	17	1	no	—
111	C₂×#367	—	17	2	no	—
112	C₂×C₂²⋊C₁₆	D₁₇	1	1	yes	characteristic
113	#1808	—	17	1	no	—
114	#1380	—	17	2	no	—
115	C₂²×C₁₇	—	2	2	no	—
116	C₂²×C₁₇	—	2	2	no	—
117	C₂²×C₁₇	—	2	2	no	—
118	C₂²×C₁₇	—	2	2	no	—
119	C₂²×C₁₇	—	2	2	no	—
120	C₂²×C₁₇	—	2	2	no	—
121	C₂²×C₁₇	#367	1	4	yes	complemented
122	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
123	C₂²×C₁₇	#447	1	1	yes	characteristic
124	C₂²×C₁₇	#480	1	1	yes	characteristic
125	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
126	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
127	C₂²×C₁₇	#509	1	1	yes	characteristic
128	C₂²×C₁₇	#510	1	1	yes	characteristic
129	C₄×C₁₇	—	2	2	no	—
130	C₄×C₁₇	#385	1	1	yes	characteristic
131	C₄×C₁₇	#428	1	1	yes	characteristic
132	C₄×C₁₇	#451	1	1	yes	characteristic
133	C₄×C₁₇	#484	1	1	yes	characteristic
134	Dic₁₇	—	8	4	no	—
135	Dic₁₇	—	16	1	no	—
136	Dic₁₇	—	16	1	no	—
137	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
138	C₂²×C₄⋊C₄	—	34	1	no	—
139	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
140	C₂×#107	—	17	2	no	—
141	C₂×#107	—	17	1	no	—
142	C₂×#107	—	17	1	no	—
143	C₂×#107	—	34	4	no	—
144	C₂×#107	—	34	1	no	—
145	#380	—	34	1	no	—
146	#367	—	34	4	no	—
147	C₂²⋊C₁₆	Dic₁₇	1	4	yes	complemented
148	#340	—	34	1	no	—
149	C₂×C₁₇	—	2	4	no	—
150	C₂×C₁₇	#1852	1	1	yes	characteristic
151	C₂×C₁₇	C₂×#367	1	1	yes	characteristic
152	C₂×C₁₇	#2159	1	1	yes	characteristic
153	C₂×C₁₇	#2185	1	1	yes	characteristic
154	C₂×C₁₇	#2187	1	1	yes	characteristic
155	C₂×C₁₇	#2233	1	1	yes	characteristic
156	C₂×C₁₇	#2235	1	1	yes	characteristic
157	C₂³×C₄	—	68	1	no	—
158	C₂³×C₄	D₆₈	1	1	yes	characteristic
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	Dic₃₄	1	2	yes	—
162	C₂²×C₈	D₆₈	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
164	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
165	C₂×C₄⋊C₄	—	34	4	no	—
166	C₂×C₄⋊C₄	—	34	2	no	—
167	C₂×C₄⋊C₄	—	34	1	no	—
168	C₂×C₄⋊C₄	—	34	1	no	—
169	C₂×C₄⋊C₄	—	68	2	no	—
170	C₂×C₄⋊C₄	—	68	1	no	—
171	C₂×C₂²⋊C₄	—	68	1	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	complemented
175	#107	—	17	4	no	—
176	#107	—	34	4	no	—
177	#107	—	34	2	no	—
178	#107	—	34	2	no	—
179	#107	—	34	2	no	—
180	#95	—	68	1	no	—
181	C₁₇	#12920	1	1	yes	characteristic, complemented, Sylow
182	C₂⁴	D₁₃₆	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	68	4	no	—
188	C₂²×C₄	—	68	2	no	—
189	C₂²×C₄	—	68	1	no	—
190	C₂²×C₄	—	68	1	no	—
191	C₂²×C₄	—	136	2	no	—
192	C₂²×C₄	—	136	2	no	—
193	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
194	C₂²×C₄	Dic₆₈	1	2	yes	—
195	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
196	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
202	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
203	C₂×C₈	D₄×D₁₇	1	2	yes	—
204	C₂×C₈	#63254	1	2	yes	—
205	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
206	C₄⋊C₄	—	34	4	no	—
207	C₄⋊C₄	—	68	4	no	—
208	C₄⋊C₄	—	68	2	no	—
209	C₄⋊C₄	—	68	1	no	—
210	C₄⋊C₄	—	68	1	no	—
211	C₄⋊C₄	—	68	1	no	—
212	C₄⋊C₄	—	68	1	no	—
213	C₂²⋊C₄	—	136	2	no	—
214	C₁₆	—	2	4	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	#10587471	1	2	yes	—
220	C₂³	#10587471	1	2	yes	—
221	C₂³	#10587489	1	2	yes	—
222	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	68	4	no	—
230	C₂×C₄	—	136	1	no	—
231	C₂×C₄	—	136	1	no	—
232	C₂×C₄	—	136	1	no	—
233	C₂×C₄	—	136	1	no	—
234	C₂×C₄	—	136	1	no	—
235	C₂×C₄	—	136	1	no	—
236	C₂×C₄	—	136	8	no	—
237	C₂×C₄	—	136	2	no	—
238	C₂×C₄	—	136	2	no	—
239	C₂×C₄	#10587489	1	2	yes	—
240	C₂×C₄	#10587568	1	1	yes	characteristic
241	C₂×C₄	#10587572	1	1	yes	characteristic
242	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
243	C₂×C₄	#10587594	1	1	yes	characteristic
244	C₂×C₄	#10587595	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	#10587564	1	2	yes	—
248	C₈	D₄×Dic₁₇	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#11762637	1	4	yes	complemented
256	C₂²	C₂×#10587489	1	1	yes	characteristic
257	C₂²	#11762945	1	1	yes	characteristic
258	C₂²	#11763123	1	1	yes	characteristic
259	C₂²	C₂×#10587471	1	1	yes	characteristic
260	C₂²	C₂×#10587471	1	1	yes	characteristic
261	C₂²	#11763247	1	1	yes	characteristic
262	C₂²	#11763252	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	272	1	no	—
265	C₄	—	272	1	no	—
266	C₄	—	136	4	no	—
267	C₄	#11762684	1	1	yes	characteristic
268	C₄	#11762851	1	1	yes	characteristic
269	C₄	#11762967	1	1	yes	characteristic
270	C₄	#11763145	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#14528974	1	1	yes	characteristic
273	C₂	C₂×#11762637	1	1	yes	characteristic
274	C₂	#14530572	1	1	yes	characteristic
275	C₂	#14530703	1	1	yes	characteristic
276	C₂	#14530715	1	1	yes	characteristic
277	C₂	#14530970	1	1	yes	characteristic
278	C₂	#14530982	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×C₂²⋊C₁₆	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂⁴×C₁₇	D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₁₇
hypocenter	C₁₇	#12920
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#11762637	C₂	semidirect
4. #14526876	C₂	semidirect
17. C₁₇×C₂²⋊C₁₆	C₄	semidirect
35. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
121. C₂²×C₁₇	#367	semidirect
147. C₂²⋊C₁₆	Dic₁₇	semidirect
174. C₂×C₁₆	C₂×Dic₁₇	semidirect
181. C₁₇	#12920	semidirect
255. C₂²	#11762637	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587489	2
C₂×#11762637	2
#14528813	2
#14526876	2
C₂×C₁₇×C₂²⋊C₁₆	2
#12920	17