Subgroups — #17005168

Order 4352.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539363	C₂	1	1	yes	maximal, characteristic
2	#14537859	C₂	1	2	yes	maximal, complemented
3	#14527650	C₂	1	4	yes	maximal, complemented
4	#14527615	C₂	1	4	yes	maximal, complemented
5	#14527547	C₂	1	1	yes	maximal, characteristic, complemented
6	#14527254	C₂	1	2	yes	maximal, complemented
7	C₁₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763359	C₂²	1	2	yes	—
9	#11763214	C₄	1	1	yes	characteristic
10	#11763186	C₄	1	1	yes	characteristic
11	#11763156	C₄	1	2	yes	—
12	#11763045	C₄	1	2	yes	—
13	#11763045	C₄	1	2	yes	—
14	C₄×D₁₃₆	C₂²	1	2	yes	—
15	C₄×D₁₃₆	C₂²	1	2	yes	—
16	#11762888	C₂²	1	4	yes	—
17	#11762879	—	2	4	no	—
18	C₈×D₆₈	C₂²	1	4	yes	—
19	#11762866	C₂²	1	2	yes	—
20	#11762774	—	2	4	no	—
21	#11762562	C₂²	1	2	yes	—
22	#11762560	C₂²	1	2	yes	—
23	#11762538	C₂²	1	8	yes	—
24	C₁₇×#431	C₂²	1	1	yes	characteristic
25	C₁₇×#431	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₁₇×#322	C₂²	1	2	yes	—
28	C₁₇×#322	C₂²	1	2	yes	—
29	C₂×#63252	—	2	2	no	—
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	#10587616	—	2	2	no	—
34	#10587616	C₂×C₄	1	4	yes	—
35	C₂×D₁₃₆	C₂×C₄	1	2	yes	—
36	C₂×D₁₃₆	C₂×C₄	1	2	yes	—
37	C₂×#63218	—	2	4	no	—
38	C₂×C₈×D₁₇	—	2	4	no	—
39	#10587561	—	2	2	no	—
40	#10587560	C₂×C₄	1	2	yes	—
41	C₄×D₆₈	—	2	2	no	—
42	C₄×D₆₈	C₂³	1	4	yes	—
43	#10587555	C₂×C₄	1	2	yes	—
44	C₁₇×C₄⋊C₈	—	2	1	no	—
45	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
46	C₁₇×C₂²⋊C₈	—	2	1	no	—
47	C₁₇×C₂²⋊C₈	D₄	1	2	yes	—
48	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₇	C₂³	1	2	yes	—
51	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₇	C₂³	1	2	yes	—
54	C₂²⋊#3622	—	2	2	no	—
55	#10587492	C₂×C₄	1	4	yes	—
56	#10587492	C₂×C₄	1	4	yes	—
57	#10587491	—	2	4	no	—
58	#10587491	—	2	4	no	—
59	#10587489	C₂×C₄	1	2	yes	—
60	#10587489	C₂×C₄	1	2	yes	—
61	C₄⋊#3622	C₂³	1	4	yes	—
62	#10587474	—	2	2	no	—
63	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₇	—	2	4	no	—
67	#63252	—	8	4	no	—
68	C₂×D₆₈	—	2	2	no	—
69	C₂×D₆₈	—	2	2	no	—
70	C₂×D₆₈	C₂²×C₄	1	4	yes	—
71	C₂×C₄×D₁₇	—	2	4	no	—
72	C₂×Dic₃₄	—	2	2	no	—
73	C₂×C₈×C₁₇	—	2	2	no	—
74	C₂×C₈×C₁₇	—	4	1	no	—
75	C₂×C₈×C₁₇	—	4	1	no	—
76	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂×D₄	1	4	yes	—
81	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₇	#41	1	1	yes	characteristic
84	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
85	#63227	—	2	4	no	—
86	#63227	—	2	4	no	—
87	C₄⋊Dic₁₇	—	2	2	no	—
88	C₄⋊Dic₁₇	C₂²×C₄	1	4	yes	—
89	C₂×#3622	—	2	4	no	—
90	D₁₃₆	—	4	2	no	—
91	D₁₃₆	—	4	2	no	—
92	#63218	—	8	4	no	—
93	C₈×D₁₇	—	8	4	no	—
94	#17989	—	17	1	no	maximal, Sylow
95	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
96	C₂²×D₁₇	—	2	4	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	2	no	—
100	C₂×C₄×C₁₇	—	4	1	no	—
101	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
107	C₂²⋊D₁₇	—	8	4	no	—
108	C₂×Dic₁₇	—	2	4	no	—
109	D₆₈	—	2	4	no	—
110	D₆₈	—	2	4	no	—
111	D₆₈	—	8	2	no	—
112	D₆₈	—	8	2	no	—
113	C₄×D₁₇	—	4	4	no	—
114	C₄×D₁₇	—	4	4	no	—
115	Dic₃₄	—	8	2	no	—
116	C₈×C₁₇	—	2	1	no	—
117	C₈×C₁₇	—	2	1	no	—
118	C₈×C₁₇	—	2	1	no	—
119	C₈×C₁₇	—	2	1	no	—
120	C₈×C₁₇	—	2	4	no	—
121	C₈×C₁₇	—	4	2	no	—
122	#3622	—	4	4	no	—
123	#3169	—	17	1	no	—
124	#2981	—	17	2	no	—
125	#1586	—	17	4	no	—
126	#1581	—	17	4	no	—
127	#1568	—	17	1	no	—
128	#1496	—	17	2	no	—
129	#1452	D₁₇	1	1	yes	characteristic, complemented
130	C₂²×C₁₇	—	2	1	no	—
131	C₂²×C₁₇	—	2	1	no	—
132	C₂²×C₁₇	—	2	1	no	—
133	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×D₁₇	—	4	4	no	—
135	C₂×D₁₇	—	4	4	no	—
136	C₂×D₁₇	—	4	4	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	—	2	1	no	—
141	C₄×C₁₇	#568	1	1	yes	characteristic
142	C₄×C₁₇	#568	1	1	yes	characteristic
143	C₄×C₁₇	#576	1	2	yes	—
144	Dic₁₇	—	4	4	no	—
145	#532	—	17	2	no	—
146	#500	—	17	1	no	—
147	#493	—	17	1	no	—
148	#486	—	17	2	no	—
149	#466	—	17	2	no	—
150	#466	—	17	2	no	—
151	C₄×D₈	—	17	2	no	—
152	C₄×D₈	—	17	2	no	—
153	#436	—	17	4	no	—
154	#435	—	34	4	no	—
155	C₈×D₄	—	17	4	no	—
156	#432	—	17	2	no	—
157	#431	C₂×D₁₇	1	1	yes	characteristic
158	#431	C₂×D₁₇	1	1	yes	characteristic
159	#408	—	34	4	no	—
160	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
161	#335	—	17	2	no	—
162	#334	—	17	2	no	—
163	D₄⋊C₈	—	17	8	no	—
164	#322	C₂×D₁₇	1	2	yes	—
165	#322	C₂×D₁₇	1	2	yes	—
166	C₂×C₁₇	—	4	1	no	—
167	C₂×C₁₇	#2932	1	1	yes	characteristic
168	C₂×C₁₇	#2948	1	1	yes	characteristic
169	C₂×C₁₇	#2957	1	1	yes	characteristic
170	D₁₇	—	8	4	no	—
171	C₂×#41	—	34	2	no	—
172	C₂×D₈	—	17	2	no	—
173	C₂×D₈	—	17	2	no	—
174	C₂×M₁₆	—	34	4	no	—
175	C₂²×C₈	—	34	4	no	—
176	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
178	C₄⋊Q₈	—	17	2	no	—
179	#127	—	17	2	no	—
180	#124	—	34	2	no	—
181	#121	—	17	4	no	—
182	#121	—	34	2	no	—
183	C₄×D₄	—	17	4	no	—
184	C₄×D₄	—	34	2	no	—
185	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
186	#107	—	17	2	no	—
187	#107	—	17	2	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	17	4	no	—
190	C₄⋊C₈	D₆₈	1	2	yes	—
191	#102	—	17	4	no	—
192	#102	—	17	4	no	—
193	C₂²⋊C₈	—	2	1	no	—
194	C₂²⋊C₈	—	34	2	no	—
195	C₂²⋊C₈	—	34	4	no	—
196	C₂²⋊C₈	—	34	4	no	—
197	C₂²⋊C₈	D₆₈	1	2	yes	—
198	#97	—	34	2	no	—
199	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
200	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₁₇	1	2	yes	—
202	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₁₇	1	2	yes	—
205	C₁₇	#17989	1	1	yes	characteristic, complemented, Sylow
206	#41	—	136	4	no	—
207	C₂×Q₈	—	34	2	no	—
208	C₂×D₄	—	17	4	no	—
209	C₂×D₄	—	34	4	no	—
210	C₂×D₄	—	34	2	no	—
211	C₂×D₄	—	34	2	no	—
212	C₂²×C₄	—	34	4	no	—
213	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
215	C₂²×C₄	#63252	1	1	yes	characteristic
216	D₈	—	68	2	no	—
217	D₈	—	68	2	no	—
218	M₁₆	—	136	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	34	4	no	—
223	C₂×C₈	—	136	4	no	—
224	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
228	C₂×C₈	C₂×D₆₈	1	4	yes	—
229	C₄⋊C₄	—	17	4	no	—
230	C₄⋊C₄	—	34	2	no	—
231	C₂²⋊C₄	—	34	4	no	—
232	C₂²⋊C₄	—	34	4	no	—
233	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
234	C₄²	C₂×D₆₈	1	1	yes	characteristic
235	C₄²	#63252	1	1	yes	characteristic
236	C₄²	C₂³×D₁₇	1	1	yes	characteristic
237	C₂³	—	34	4	no	—
238	C₂³	C₄×D₆₈	1	1	yes	characteristic
239	Q₈	—	136	2	no	—
240	D₄	—	34	4	no	—
241	D₄	—	34	4	no	—
242	D₄	—	136	2	no	—
243	D₄	—	136	2	no	—
244	D₄	—	136	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	34	4	no	—
250	C₂×C₄	—	68	4	no	—
251	C₂×C₄	—	68	4	no	—
252	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
253	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
254	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
257	C₂×C₄	C₂×#63252	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₈	—	68	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	68	4	no	—
269	C₂²	—	68	4	no	—
270	C₂²	—	68	4	no	—
271	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	68	4	no	—
277	C₄	#11763575	1	1	yes	characteristic
278	C₄	#11763575	1	1	yes	characteristic
279	C₄	#11763614	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	136	4	no	—
282	C₂	#14537503	1	1	yes	characteristic
283	C₂	#14537637	1	1	yes	characteristic
284	C₂	#14537702	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#1452	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	#14539363	C₂
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#1452	D₁₇
hypocenter	C₁₇	#17989
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #14537859	C₂	semidirect
3. #14527650	C₂	semidirect
4. #14527615	C₂	semidirect
5. #14527547	C₂	semidirect
6. #14527254	C₂	semidirect
7. C₁₇×#1452	C₂	semidirect
129. #1452	D₁₇	semidirect
205. C₁₇	#17989	semidirect

Maximal subgroups

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

subgroup	index
#14539363	2
#14537859	2
#14527650	2
#14527615	2
#14527547	2
#14527254	2
C₁₇×#1452	2
#17989	17