Subgroups — #17978625

Order 4352.

642 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	#14545564	C₂	1	2	yes	maximal, complemented
2	#14545187	C₂	1	4	yes	maximal, complemented
3	#14545185	C₂	1	4	yes	maximal, complemented
4	D₁₇⋊#368	C₂	1	1	yes	maximal, characteristic, complemented
5	#14545100	C₂	1	2	yes	maximal, complemented
6	#14544992	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11763242	C₂	1	1	yes	maximal, characteristic
8	#11763889	—	2	2	no	—
9	#11763865	C₂²	1	1	yes	characteristic
10	#11763855	—	2	4	no	—
11	#11763853	C₂²	1	8	yes	complemented
12	D₁₇⋊#108	C₂²	1	2	yes	—
13	#11763808	—	2	2	no	—
14	#11763772	C₂²	1	1	yes	characteristic
15	#11763738	C₂²	1	2	yes	complemented
16	#11763737	C₂²	1	2	yes	complemented
17	#11763724	C₂²	1	8	yes	complemented
18	#11763696	C₂²	1	2	yes	complemented
19	C₁₆×#595	C₂²	1	2	yes	complemented
20	C₂×#10587605	C₂²	1	2	yes	—
21	#11763242	C₄	1	2	yes	complemented
22	#11763242	C₄	1	2	yes	complemented
23	#11763242	C₂²	1	2	yes	—
24	#11763242	C₂²	1	2	yes	—
25	C₂×#10587499	C₄	1	2	yes	complemented
26	C₂×D₂₇₂	C₄	1	1	yes	characteristic, complemented
27	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
28	C₂×C₁₇×Q₃₂	C₄	1	1	yes	characteristic, complemented
29	C₂×#63256	—	2	2	no	—
30	#10587680	—	2	8	no	—
31	#10587680	—	4	4	no	—
32	#10587677	D₄	1	1	yes	characteristic
33	D₁₇⋊M₁₆	—	2	2	no	—
34	D₁₇⋊M₁₆	—	4	4	no	—
35	D₁₇⋊M₁₆	D₄	1	1	yes	characteristic
36	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
37	C₂×#63230	—	2	2	no	—
38	#10587605	—	2	2	no	—
39	#10587605	—	2	2	no	—
40	#10587605	C₂×C₄	1	4	yes	—
41	#10587605	C₂³	1	4	yes	—
42	C₂×D₁₃₆	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
44	#10587550	—	2	8	no	—
45	#10587538	—	2	1	no	—
46	#10587536	C₂³	1	4	yes	—
47	#10587535	—	2	2	no	—
48	#10587532	D₄	1	2	yes	—
49	C₈×#595	C₂³	1	2	yes	—
50	#10587530	—	2	1	no	—
51	#10587529	C₂³	1	2	yes	—
52	C₁₇×Q₃₂	C₂×C₄	1	2	yes	—
53	C₁₇×Q₃₂	C₂×C₄	1	2	yes	—
54	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
55	#10587499	C₂×C₄	1	4	yes	—
56	#10587499	C₂×C₄	1	4	yes	—
57	C₂×#63214	C₂×C₄	1	1	yes	characteristic
58	D₂₇₂	C₂×C₄	1	2	yes	—
59	D₂₇₂	C₂×C₄	1	2	yes	—
60	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
61	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
62	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
63	C₁₆×D₁₇	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₁₇	—	2	2	no	—
65	#63256	—	2	4	no	—
66	#63256	—	2	4	no	—
67	#63256	—	4	2	no	—
68	#63256	—	4	2	no	—
69	C₂×D₆₈	—	2	2	no	—
70	C₂×D₆₈	—	4	2	no	—
71	C₂×C₄×D₁₇	—	4	2	no	—
72	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
73	C₂²⋊#595	—	2	1	no	—
74	#63247	—	4	2	no	—
75	#63247	D₈	1	2	yes	—
76	C₄⋊#595	—	2	1	no	—
77	C₄×#595	C₂×D₄	1	2	yes	—
78	#63243	—	2	4	no	—
79	#63243	—	4	8	no	—
80	#63243	D₈	1	2	yes	—
81	#63242	—	4	2	no	—
82	#63242	—	4	8	no	—
83	#63242	C₂×D₄	1	2	yes	—
84	Q₁₆×C₁₇	—	2	2	no	—
85	Q₁₆×C₁₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	#63230	—	2	4	no	—
88	#63230	—	2	4	no	—
89	C₂×#3622	#41	1	1	yes	characteristic
90	D₁₃₆	—	2	2	no	—
91	D₁₃₆	C₂²×C₄	1	4	yes	—
92	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
94	C₈×D₁₇	#41	1	1	yes	characteristic
95	C₈×D₁₇	C₂⁴	1	1	yes	characteristic
96	#63216	—	2	2	no	—
97	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
99	#63214	C₂²×C₄	1	1	yes	characteristic
100	#63214	C₂²×C₄	1	1	yes	characteristic
101	#33465	—	17	1	no	maximal, Sylow
102	C₂²×D₁₇	—	4	2	no	—
103	C₂²×D₁₇	#135	1	1	yes	characteristic
104	C₂×#595	—	2	2	no	—
105	Q₈×C₁₇	—	2	4	no	—
106	Q₈×C₁₇	—	4	2	no	—
107	C₂×C₄×C₁₇	—	4	2	no	—
108	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
109	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
110	D₆₈	—	2	4	no	—
111	D₆₈	—	4	4	no	—
112	D₆₈	—	4	4	no	—
113	D₆₈	—	4	2	no	—
114	C₄×D₁₇	—	4	4	no	—
115	C₄×D₁₇	—	4	4	no	—
116	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
117	C₄×D₁₇	C₂×D₈	1	1	yes	characteristic
118	C₄×D₁₇	#135	1	1	yes	characteristic
119	C₄×D₁₇	C₂²×D₄	1	1	yes	characteristic
120	#3624	—	4	4	no	—
121	#3624	C₂×D₈	1	4	yes	—
122	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
123	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
124	#3622	C₄×D₄	1	1	yes	characteristic
125	#3622	C₂×#41	1	1	yes	characteristic
126	C₂×#508	—	17	1	no	—
127	#2957	—	17	2	no	—
128	#2179	—	17	4	no	—
129	#2178	—	17	4	no	—
130	C₂×#368	—	17	1	no	—
131	#2143	—	17	2	no	—
132	#2108	—	17	1	no	—
133	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
134	C₂×D₁₇	—	4	1	no	—
135	C₂×D₁₇	—	4	4	no	—
136	C₂×D₁₇	—	16	2	no	—
137	C₂×D₁₇	C₄×D₈	1	1	yes	characteristic
138	C₂×D₁₇	C₂×#135	1	1	yes	characteristic
139	#595	—	4	2	no	—
140	C₄×C₁₇	—	4	4	no	—
141	C₄×C₁₇	C₄×D₈	1	1	yes	characteristic
142	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
143	Dic₁₇	C₄×D₈	1	1	yes	characteristic
144	Dic₁₇	C₂²×D₈	1	1	yes	characteristic
145	C₂×#135	—	17	2	no	—
146	#568	—	34	2	no	—
147	#508	—	17	2	no	—
148	#508	—	17	2	no	—
149	#508	—	17	2	no	—
150	#508	—	17	2	no	—
151	C₂×Q₃₂	#595	1	1	yes	characteristic, complemented
152	C₂×SD₃₂	—	17	2	no	—
153	C₂×D₁₆	—	17	1	no	—
154	C₂×M₃₂	—	17	1	no	—
155	C₂²×C₁₆	—	17	1	no	—
156	#444	—	34	4	no	—
157	#443	—	17	8	no	—
158	C₂×#108	—	17	2	no	—
159	#421	—	34	2	no	—
160	#405	—	17	1	no	—
161	#368	—	17	2	no	—
162	#368	—	17	2	no	—
163	#359	—	17	8	no	—
164	#346	—	17	2	no	—
165	C₄×C₁₆	—	17	2	no	—
166	C₂×C₁₇	—	2	1	no	—
167	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
168	D₁₇	—	2	1	no	—
169	D₁₇	—	2	1	no	—
170	D₁₇	—	8	4	no	—
171	C₂×#41	—	34	2	no	—
172	#135	—	17	4	no	—
173	#135	—	17	4	no	—
174	#135	—	34	2	no	—
175	#135	—	34	2	no	—
176	C₂×Q₁₆	C₂×#595	1	2	yes	—
177	C₂×SD₁₆	—	34	2	no	—
178	C₂×D₈	—	17	2	no	—
179	#131	—	34	8	no	—
180	#131	—	68	4	no	—
181	C₂×M₁₆	—	17	1	no	—
182	C₂×M₁₆	—	34	2	no	—
183	C₂×M₁₆	—	68	4	no	—
184	C₂²×C₈	—	17	1	no	—
185	#117	—	17	1	no	—
186	Q₃₂	C₂×#595	1	2	yes	—
187	Q₃₂	C₂×#595	1	2	yes	—
188	SD₃₂	—	17	4	no	—
189	SD₃₂	—	17	4	no	—
190	D₁₆	—	17	2	no	—
191	D₁₆	—	17	2	no	—
192	M₃₂	—	34	1	no	—
193	M₃₂	—	34	1	no	—
194	C₂×C₁₆	—	17	1	no	—
195	C₂×C₁₆	—	17	1	no	—
196	C₂×C₁₆	—	17	1	no	—
197	C₂×C₁₆	—	17	1	no	—
198	C₂×C₁₆	—	17	1	no	—
199	C₂×C₁₆	—	17	2	no	—
200	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
201	#108	—	17	4	no	—
202	#108	—	34	2	no	—
203	#104	—	34	8	no	—
204	#97	—	17	2	no	—
205	C₄×C₈	—	17	2	no	—
206	C₁₇	#33465	1	1	yes	characteristic, complemented, Sylow
207	#41	—	34	4	no	—
208	#41	—	34	4	no	—
209	#41	—	68	2	no	—
210	#41	—	68	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	34	2	no	—
213	C₂×D₄	—	68	2	no	—
214	C₂²×C₄	—	17	1	no	—
215	C₂²×C₄	—	68	2	no	—
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×#595	1	4	yes	—
218	SD₁₆	—	34	4	no	—
219	SD₁₆	—	34	4	no	—
220	D₈	—	17	4	no	—
221	D₈	—	34	2	no	—
222	M₁₆	—	17	2	no	—
223	M₁₆	—	17	2	no	—
224	M₁₆	—	34	4	no	—
225	M₁₆	—	68	8	no	—
226	M₁₆	—	68	2	no	—
227	C₂×C₈	—	17	2	no	—
228	C₂×C₈	—	17	1	no	—
229	C₂×C₈	—	17	1	no	—
230	C₂×C₈	—	17	1	no	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	—	17	1	no	—
233	C₂×C₈	—	68	8	no	—
234	C₂×C₈	—	68	2	no	—
235	C₂×C₈	C₂²×#595	1	1	yes	characteristic
236	C₄⋊C₄	—	34	1	no	—
237	C₂²⋊C₄	—	34	1	no	—
238	C₄²	—	17	2	no	—
239	C₁₆	—	17	1	no	—
240	C₁₆	—	17	1	no	—
241	C₁₆	—	34	2	no	—
242	C₁₆	C₂²×#595	1	1	yes	characteristic
243	C₁₆	C₂²×#595	1	1	yes	characteristic
244	C₂³	—	17	1	no	—
245	C₂³	—	68	2	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	34	4	no	—
249	D₄	—	68	4	no	—
250	D₄	—	68	4	no	—
251	D₄	—	68	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	17	1	no	—
254	C₂×C₄	—	17	1	no	—
255	C₂×C₄	—	17	1	no	—
256	C₂×C₄	—	17	1	no	—
257	C₂×C₄	—	17	1	no	—
258	C₂×C₄	—	34	2	no	—
259	C₂×C₄	—	68	4	no	—
260	C₂×C₄	—	68	4	no	—
261	C₂×C₄	D₄×#595	1	1	yes	characteristic
262	C₈	—	17	4	no	—
263	C₈	—	17	1	no	—
264	C₈	—	17	1	no	—
265	C₈	—	68	4	no	—
266	C₈	D₄×#595	1	1	yes	characteristic
267	C₈	C₂³×#595	1	1	yes	characteristic
268	C₂²	—	272	2	no	—
269	C₂²	—	17	1	no	—
270	C₂²	—	17	1	no	—
271	C₂²	—	68	1	no	—
272	C₂²	—	68	4	no	—
273	C₂²	D₈×#595	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	17	1	no	—
276	C₄	—	17	1	no	—
277	C₄	—	68	2	no	—
278	C₄	D₈×#595	1	1	yes	characteristic
279	C₄	C₂×D₄×#595	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	34	1	no	—
282	C₂	—	34	1	no	—
283	C₂	—	136	4	no	—
284	C₂	C₂×D₈×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#595
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₈	C₂³×#595
Fitting	C₂×C₁₇×Q₃₂	C₄
socle	C₂×C₁₇	C₂×C₄×D₈
prime radical	C₂×#11763242	C₂
cosocle	C₈×D₁₇	C₂⁴
hypercenter	C₂×Q₃₂	#595
hypocenter	C₁₇	#33465
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #14545564	C₂	semidirect
2. #14545187	C₂	semidirect
3. #14545185	C₂	semidirect
4. D₁₇⋊#368	C₂	semidirect
5. #14545100	C₂	semidirect
6. #14544992	C₂	semidirect
11. #11763853	C₂²	semidirect
15. #11763738	C₂²	semidirect
16. #11763737	C₂²	semidirect
17. #11763724	C₂²	semidirect
18. #11763696	C₂²	semidirect
19. C₁₆×#595	C₂²	semidirect
21. #11763242	C₄	semidirect
22. #11763242	C₄	semidirect
25. C₂×#10587499	C₄	semidirect
26. C₂×D₂₇₂	C₄	semidirect
28. C₂×C₁₇×Q₃₂	C₄	semidirect
151. C₂×Q₃₂	#595	semidirect
206. C₁₇	#33465	semidirect

Maximal subgroups

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

subgroup	index
#14545564	2
#14545187	2
#14545185	2
D₁₇⋊#368	2
#14545100	2
#14544992	2
C₂×#11763242	2
#33465	17