Subgroups — #16933486

Order 4352.

506 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14529704	C₂	1	2	yes	maximal, complemented
2	C₄×#10587489	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527279	C₂	1	2	yes	maximal, complemented
4	C₈×#63223	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#3169	C₂	1	1	yes	maximal, characteristic
6	#11762862	C₂²	1	1	yes	characteristic
7	C₂×#10587489	—	2	1	no	—
8	C₂×#10587508	—	2	4	no	—
9	C₄×#63223	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	#11762572	—	2	1	no	—
12	#11762543	—	2	4	no	—
13	#11762531	—	2	1	no	—
14	C₈×#3622	—	2	1	no	—
15	C₁₇×#532	C₂²	1	2	yes	—
16	C₁₇×#500	C₄	1	1	yes	characteristic, complemented
17	C₁₇×#493	C₄	1	1	yes	characteristic, complemented
18	C₁₇×#486	C₄	1	2	yes	complemented
19	C₁₇×#466	—	2	1	no	—
20	C₁₇×#466	—	2	1	no	—
21	C₄×D₈×C₁₇	—	2	1	no	—
22	C₄×D₈×C₁₇	—	2	1	no	—
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₂×#41×C₁₇	—	2	2	no	—
25	C₂×D₈×C₁₇	—	2	1	no	—
26	C₂×D₈×C₁₇	—	2	1	no	—
27	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₁₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₁₇×#127	C₂×C₄	1	2	yes	—
31	C₁₇×#124	—	2	2	no	—
32	C₁₇×#121	—	2	2	no	—
33	C₁₇×#121	—	2	2	no	—
34	C₄×D₄×C₁₇	—	2	2	no	—
35	C₄×D₄×C₁₇	—	2	2	no	—
36	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₇	—	2	2	no	—
38	C₂×C₄×Dic₁₇	—	2	2	no	—
39	C₂×#63223	D₄	1	2	yes	—
40	C₁₇×#107	—	2	1	no	—
41	C₁₇×#107	—	2	1	no	—
42	C₁₇×#102	—	2	2	no	—
43	C₁₇×#102	—	2	2	no	—
44	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
48	#10587508	—	4	8	no	—
49	#10587502	—	2	2	no	—
50	C₂²⋊#3622	D₄	1	2	yes	—
51	#10587489	—	2	4	no	—
52	C₄⋊#3622	—	2	1	no	—
53	#10587474	—	2	1	no	—
54	C₄×#3622	—	2	1	no	—
55	C₄×#3622	—	2	1	no	—
56	#41×C₁₇	—	4	4	no	—
57	#41×C₁₇	—	4	4	no	—
58	C₂×Q₈×C₁₇	—	2	2	no	—
59	C₂×D₄×C₁₇	—	2	2	no	—
60	C₂×D₄×C₁₇	—	2	2	no	—
61	C₂×D₄×C₁₇	—	2	2	no	—
62	C₂×D₄×C₁₇	—	4	2	no	—
63	C₂²×C₄×C₁₇	—	4	2	no	—
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₁₇	—	4	2	no	—
68	D₈×C₁₇	—	4	2	no	—
69	D₈×C₁₇	—	4	2	no	—
70	C₂×C₈×C₁₇	—	2	2	no	—
71	C₂×C₈×C₁₇	—	4	1	no	—
72	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	—	2	2	no	—
77	C₄⋊C₄×C₁₇	—	2	2	no	—
78	C₂²⋊C₄×C₁₇	—	4	2	no	—
79	C₂²⋊C₄×C₁₇	—	4	2	no	—
80	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₇	#41	1	1	yes	characteristic
83	C₄²×C₁₇	#41	1	1	yes	characteristic
84	C₄⋊Dic₁₇	—	2	4	no	—
85	C₄⋊Dic₁₇	—	4	2	no	—
86	C₄×Dic₁₇	—	4	4	no	—
87	#63223	—	2	2	no	—
88	#63223	D₈	1	4	yes	complemented
89	C₂×#3622	—	2	2	no	—
90	C₂×#3622	—	2	2	no	—
91	#9719	—	17	1	no	maximal, Sylow
92	C₂³×C₁₇	—	4	2	no	—
93	C₂³×C₁₇	#127	1	1	yes	characteristic
94	Q₈×C₁₇	—	4	4	no	—
95	D₄×C₁₇	—	4	2	no	—
96	D₄×C₁₇	—	4	2	no	—
97	D₄×C₁₇	—	4	4	no	—
98	D₄×C₁₇	—	8	2	no	—
99	D₄×C₁₇	—	8	2	no	—
100	D₄×C₁₇	—	8	2	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	4	1	no	—
104	C₂×C₄×C₁₇	—	4	2	no	—
105	C₂×C₄×C₁₇	—	8	4	no	—
106	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₇	#135	1	2	yes	—
114	C₂×Dic₁₇	—	4	4	no	—
115	C₂×Dic₁₇	—	8	2	no	—
116	C₂×Dic₁₇	—	8	2	no	—
117	C₈×C₁₇	—	2	1	no	—
118	C₈×C₁₇	—	2	1	no	—
119	C₈×C₁₇	—	2	1	no	—
120	C₈×C₁₇	#104	1	2	yes	complemented
121	#3622	—	2	4	no	—
122	#3622	—	4	2	no	—
123	#3169	D₁₇	1	1	yes	characteristic
124	#1980	—	17	2	no	—
125	C₄×#107	—	17	1	no	—
126	#1502	—	17	2	no	—
127	C₈×M₁₆	—	17	1	no	—
128	C₂²×C₁₇	—	2	1	no	—
129	C₂²×C₁₇	—	2	1	no	—
130	C₂²×C₁₇	—	8	2	no	—
131	C₂²×C₁₇	—	8	2	no	—
132	C₂²×C₁₇	—	8	2	no	—
133	C₂²×C₁₇	#390	1	1	yes	characteristic
134	C₂²×C₁₇	#493	1	1	yes	characteristic
135	C₂²×C₁₇	#495	1	1	yes	characteristic
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	8	2	no	—
140	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
141	C₄×C₁₇	#421	1	1	yes	characteristic
142	C₄×C₁₇	C₄×D₈	1	2	yes	—
143	C₄×C₁₇	#466	1	2	yes	—
144	Dic₁₇	—	8	4	no	—
145	#532	C₂×D₁₇	1	2	yes	—
146	#500	Dic₁₇	1	1	yes	characteristic, complemented
147	#493	Dic₁₇	1	1	yes	characteristic, complemented
148	#486	Dic₁₇	1	2	yes	complemented
149	#466	—	2	1	no	—
150	#466	—	2	1	no	—
151	C₄×D₈	—	2	1	no	—
152	C₄×D₈	—	2	1	no	—
153	#431	—	17	1	no	—
154	C₂×#107	—	34	1	no	—
155	C₂×#104	—	34	4	no	—
156	C₄×M₁₆	—	17	1	no	—
157	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
158	C₄×C₄⋊C₄	—	17	2	no	—
159	#340	—	34	1	no	—
160	#327	—	34	4	no	—
161	#322	—	34	1	no	—
162	C₈²	—	34	1	no	—
163	C₂×C₁₇	—	2	1	no	—
164	C₂×C₁₇	—	2	1	no	—
165	C₂×C₁₇	—	16	2	no	—
166	C₂×C₁₇	#1970	1	1	yes	characteristic
167	C₂×C₁₇	#1976	1	1	yes	characteristic
168	C₂×C₁₇	#1980	1	1	yes	characteristic
169	C₂×#41	—	2	2	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×M₁₆	—	17	2	no	—
173	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
175	C₄⋊Q₈	C₂×Dic₁₇	1	2	yes	—
176	#127	C₂×Dic₁₇	1	2	yes	—
177	#124	—	2	2	no	—
178	#121	—	2	2	no	—
179	#121	—	2	2	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	34	2	no	—
183	C₂×C₄²	—	34	2	no	—
184	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
185	#107	—	2	1	no	—
186	#107	—	2	1	no	—
187	#107	—	34	4	no	—
188	C₄⋊C₈	—	34	1	no	—
189	#104	—	68	8	no	—
190	#102	—	2	2	no	—
191	#102	—	2	2	no	—
192	C₂²⋊C₈	—	17	2	no	—
193	#97	—	34	1	no	—
194	C₄×C₈	—	34	1	no	—
195	C₄×C₈	—	34	1	no	—
196	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
197	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
198	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
200	#95	—	34	2	no	—
201	C₁₇	#9719	1	1	yes	characteristic, complemented, Sylow
202	#41	—	4	4	no	—
203	#41	—	4	4	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	68	2	no	—
211	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
214	D₈	—	4	2	no	—
215	D₈	—	4	2	no	—
216	M₁₆	—	17	4	no	—
217	M₁₆	—	34	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	34	2	no	—
221	C₂×C₈	—	34	2	no	—
222	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	34	4	no	—
229	C₄⋊C₄	—	68	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	68	4	no	—
233	C₄²	#63254	1	1	yes	characteristic
234	C₄²	#63254	1	1	yes	characteristic
235	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
236	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
237	C₂³	—	4	2	no	—
238	C₂³	#10587626	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	68	4	no	—
252	C₂×C₄	—	136	2	no	—
253	C₂×C₄	—	136	2	no	—
254	C₂×C₄	D₈×D₁₇	1	2	yes	—
255	C₂×C₄	#10587598	1	2	yes	—
256	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
258	C₂×C₄	#10587622	1	1	yes	characteristic
259	C₂×C₄	#10587624	1	1	yes	characteristic
260	C₂×C₄	#10587624	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	34	4	no	—
266	C₈	—	68	2	no	—
267	C₈	#10587508	1	2	yes	complemented
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	#11762712	1	1	yes	characteristic
274	C₂²	#11763184	1	1	yes	characteristic
275	C₂²	#11763192	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	8	2	no	—
280	C₄	—	136	4	no	—
281	C₄	C₂×#10587508	1	1	yes	characteristic
282	C₄	#11762826	1	1	yes	characteristic
283	C₄	D₈×Dic₁₇	1	2	yes	—
284	C₄	#11763042	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	16	2	no	—
288	C₂	#14529651	1	1	yes	characteristic
289	C₂	#14529683	1	1	yes	characteristic
290	C₂	#14529704	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587624
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting, prime radical	C₁₇×#3169	C₂
socle	C₂²×C₁₇	#390
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#3169	D₁₇
hypocenter	C₁₇	#9719
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #14529704	C₂	semidirect
2. C₄×#10587489	C₂	semidirect
3. #14527279	C₂	semidirect
4. C₈×#63223	C₂	semidirect
16. C₁₇×#500	C₄	semidirect
17. C₁₇×#493	C₄	semidirect
18. C₁₇×#486	C₄	semidirect
88. #63223	D₈	semidirect
120. C₈×C₁₇	#104	semidirect
146. #500	Dic₁₇	semidirect
147. #493	Dic₁₇	semidirect
148. #486	Dic₁₇	semidirect
201. C₁₇	#9719	semidirect
267. C₈	#10587508	semidirect

Maximal subgroups

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

subgroup	index
#14529704	2
C₄×#10587489	2
#14527279	2
C₈×#63223	2
C₁₇×#3169	2
#9719	17