Subgroups — #16949409

Order 4352.

486 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762588	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762584	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#63227	C₂	1	1	yes	maximal, characteristic
5	#14526862	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526861	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₂²×#63214	C₂²	1	1	yes	characteristic
10	C₂²×#63214	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₁₇	—	2	2	no	—
12	C₂×C₁₆×D₁₇	—	2	2	no	—
13	C₂×C₁₆×D₁₇	C₄	1	4	yes	complemented
14	C₂×#10587491	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₄×#63227	C₄	1	1	yes	characteristic
17	#11762588	C₄	1	4	yes	complemented
18	#11762584	—	2	2	no	—
19	#11762564	C₄	1	1	yes	characteristic
20	#11762563	C₄	1	1	yes	characteristic
21	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₁₇×#346	—	2	2	no	—
25	C₂²×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#63227	C₈	1	2	yes	—
30	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
32	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₁₇	—	2	1	no	—
34	C₂×C₈×D₁₇	—	2	1	no	—
35	C₂×C₈×D₁₇	—	2	1	no	—
36	C₂×C₈×D₁₇	—	2	1	no	—
37	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₁₇	—	2	1	no	—
40	C₂×C₁₆×C₁₇	—	2	1	no	—
41	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
42	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
44	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
45	C₄×C₈×C₁₇	—	2	1	no	—
46	C₄×C₈×C₁₇	—	2	1	no	—
47	#10587502	C₈	1	2	yes	—
48	#10587491	C₈	1	4	yes	—
49	C₈×Dic₁₇	—	2	1	no	—
50	C₈×Dic₁₇	—	2	1	no	—
51	C₂×#63214	—	2	1	no	—
52	C₂×#63214	—	2	1	no	—
53	C₂×#63214	—	2	1	no	—
54	C₂×#63214	—	2	1	no	—
55	C₂×#63214	C₂×C₄	1	2	yes	—
56	C₂×#63214	C₂×C₄	1	2	yes	—
57	C₁₆×D₁₇	—	2	8	no	—
58	C₂³×D₁₇	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
62	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	—	2	1	no	—
67	C₂×C₄×D₁₇	C₄²	1	2	yes	—
68	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₁₇	—	2	1	no	—
70	C₂×C₈×C₁₇	—	2	1	no	—
71	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
77	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
78	C₄²×C₁₇	—	2	1	no	—
79	C₄²×C₁₇	—	2	1	no	—
80	#63227	—	2	4	no	—
81	C₄×Dic₁₇	—	2	1	no	—
82	C₄×Dic₁₇	—	2	1	no	—
83	C₂×#3622	—	2	1	no	—
84	C₂×#3622	—	2	1	no	—
85	C₂×#3622	—	2	1	no	—
86	C₂×#3622	—	2	1	no	—
87	C₂×#3622	C₄²	1	2	yes	—
88	C₂×#3622	C₂×C₈	1	2	yes	—
89	C₈×D₁₇	—	2	2	no	—
90	C₈×D₁₇	—	2	2	no	—
91	C₈×D₁₇	—	2	2	no	—
92	C₈×D₁₇	—	2	2	no	—
93	C₁₆×C₁₇	—	2	2	no	—
94	C₁₆×C₁₇	C₂²⋊C₄	1	4	yes	complemented
95	#63214	—	2	2	no	—
96	#63214	—	2	2	no	—
97	#12611	—	17	1	no	maximal, Sylow
98	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
99	C₂²×D₁₇	—	2	1	no	—
100	C₂²×D₁₇	—	2	1	no	—
101	C₂²×D₁₇	—	2	2	no	—
102	C₂²×D₁₇	C₄×C₈	1	2	yes	—
103	C₂²×D₁₇	M₃₂	1	4	yes	—
104	C₂×C₄×C₁₇	—	2	2	no	—
105	C₂×C₄×C₁₇	—	2	2	no	—
106	C₂×C₄×C₁₇	—	2	2	no	—
107	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
112	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
113	C₂×Dic₁₇	—	2	1	no	—
114	C₂×Dic₁₇	—	2	1	no	—
115	C₂×Dic₁₇	—	2	2	no	—
116	C₂×Dic₁₇	—	2	2	no	—
117	C₂×Dic₁₇	—	2	2	no	—
118	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
119	C₄×D₁₇	—	2	2	no	—
120	C₄×D₁₇	—	2	2	no	—
121	C₄×D₁₇	—	2	2	no	—
122	C₄×D₁₇	—	2	2	no	—
123	C₈×C₁₇	—	2	2	no	—
124	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₈×C₁₇	#117	1	1	yes	characteristic
126	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
127	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
128	#3622	—	2	2	no	—
129	#3622	—	2	2	no	—
130	C₂³×C₁₆	—	17	1	no	—
131	C₂×C₂²⋊C₁₆	—	17	1	no	—
132	C₂×#346	—	17	1	no	—
133	C₂×#346	D₁₇	1	1	yes	characteristic, complemented
134	C₈×C₂²⋊C₄	—	17	1	no	—
135	#1374	—	17	1	no	—
136	#1374	—	17	1	no	—
137	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
138	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
140	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
141	C₂²×C₁₇	#504	1	2	yes	—
142	C₂×D₁₇	—	2	2	no	—
143	C₂×D₁₇	—	2	4	no	—
144	C₂×D₁₇	—	2	4	no	—
145	C₂×D₁₇	—	2	2	no	—
146	C₂×D₁₇	#346	1	4	yes	complemented
147	C₄×C₁₇	—	4	2	no	—
148	C₄×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₁₇	#405	1	1	yes	characteristic
150	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
151	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
152	Dic₁₇	—	2	2	no	—
153	Dic₁₇	—	4	2	no	—
154	C₂³×C₈	—	17	1	no	—
155	C₂²×C₁₆	—	17	1	no	—
156	C₂²×C₁₆	—	17	1	no	—
157	C₂²×C₁₆	—	17	4	no	—
158	C₂²×C₁₆	—	34	2	no	—
159	C₂²×C₁₆	—	34	2	no	—
160	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
161	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
162	C₂×C₂²⋊C₈	—	17	1	no	—
163	C₂×C₄×C₈	—	17	1	no	—
164	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
165	C₄×C₂²⋊C₄	—	17	1	no	—
166	C₂²⋊C₁₆	—	17	4	no	—
167	#346	—	2	2	no	—
168	#346	—	34	2	no	—
169	#336	—	17	1	no	—
170	#336	—	17	1	no	—
171	C₂×C₁₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
173	C₂×C₁₇	#2108	1	1	yes	characteristic
174	C₂×C₁₇	#2168	1	2	yes	—
175	C₂×C₁₇	#2168	1	2	yes	—
176	D₁₇	—	2	4	no	—
177	C₂³×C₄	—	17	1	no	—
178	C₂²×C₈	—	17	1	no	—
179	C₂²×C₈	—	17	1	no	—
180	C₂²×C₈	—	17	2	no	—
181	C₂²×C₈	—	17	2	no	—
182	C₂²×C₈	—	34	1	no	—
183	C₂²×C₈	—	34	1	no	—
184	C₂²×C₈	—	34	1	no	—
185	C₂²×C₈	—	34	1	no	—
186	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
188	C₂×C₂²⋊C₄	—	17	2	no	—
189	C₂×C₄²	—	17	1	no	—
190	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	17	2	no	—
194	C₂×C₁₆	—	17	2	no	—
195	C₂×C₁₆	—	34	1	no	—
196	C₂×C₁₆	—	34	1	no	—
197	C₂×C₁₆	—	34	1	no	—
198	C₂×C₁₆	—	34	1	no	—
199	C₂×C₁₆	—	34	8	no	—
200	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
201	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
202	C₂×C₁₆	D₆₈	1	2	yes	—
203	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
204	C₂²⋊C₈	—	17	4	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	34	1	no	—
208	C₄×C₈	—	34	1	no	—
209	#95	—	17	2	no	—
210	C₁₇	#12611	1	1	yes	characteristic, complemented, Sylow
211	C₂⁴	—	17	1	no	—
212	C₂²×C₄	—	17	1	no	—
213	C₂²×C₄	—	17	2	no	—
214	C₂²×C₄	—	17	2	no	—
215	C₂²×C₄	—	17	2	no	—
216	C₂²×C₄	—	34	1	no	—
217	C₂²×C₄	—	34	1	no	—
218	C₂²×C₄	—	34	1	no	—
219	C₂²×C₄	—	34	1	no	—
220	C₂²×C₄	C₈×D₁₇	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	17	2	no	—
225	C₂×C₈	—	17	2	no	—
226	C₂×C₈	—	34	1	no	—
227	C₂×C₈	—	34	1	no	—
228	C₂×C₈	—	34	1	no	—
229	C₂×C₈	—	34	1	no	—
230	C₂×C₈	—	34	2	no	—
231	C₂×C₈	—	34	2	no	—
232	C₂×C₈	—	34	2	no	—
233	C₂×C₈	—	34	2	no	—
234	C₂×C₈	C₈×D₁₇	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
237	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
238	C₂×C₈	#63252	1	1	yes	characteristic
239	C₂×C₈	#63252	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
241	C₂²⋊C₄	—	34	4	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	34	1	no	—
245	C₄²	—	34	1	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	—	34	2	no	—
248	C₁₆	—	34	2	no	—
249	C₁₆	#63227	1	4	yes	complemented
250	C₂³	—	17	4	no	—
251	C₂³	—	17	2	no	—
252	C₂³	—	34	2	no	—
253	C₂³	—	34	1	no	—
254	C₂³	—	34	1	no	—
255	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	17	2	no	—
260	C₂×C₄	—	34	1	no	—
261	C₂×C₄	—	34	1	no	—
262	C₂×C₄	—	34	2	no	—
263	C₂×C₄	—	34	2	no	—
264	C₂×C₄	—	34	2	no	—
265	C₂×C₄	—	34	2	no	—
266	C₂×C₄	—	34	2	no	—
267	C₂×C₄	—	34	2	no	—
268	C₂×C₄	—	34	2	no	—
269	C₂×C₄	C₄²×D₁₇	1	1	yes	characteristic
270	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
271	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
272	C₂×C₄	#10587587	1	1	yes	characteristic
273	C₂×C₄	#10587587	1	1	yes	characteristic
274	C₂×C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	34	2	no	—
277	C₈	—	34	2	no	—
278	C₈	#10587558	1	1	yes	characteristic
279	C₈	C₄×D₆₈	1	1	yes	characteristic
280	C₈	C₂×#63227	1	1	yes	characteristic
281	C₈	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
282	C₂²	—	17	4	no	—
283	C₂²	—	34	2	no	—
284	C₂²	—	34	4	no	—
285	C₂²	—	34	4	no	—
286	C₂²	—	34	2	no	—
287	C₂²	C₄×C₈×D₁₇	1	1	yes	characteristic
288	C₂²	C₈×C₂²⋊D₁₇	1	1	yes	characteristic
289	C₂²	C₈×D₆₈	1	1	yes	characteristic
290	C₂²	M₃₂×D₁₇	1	2	yes	—
291	C₂²	#11763227	1	2	yes	—
292	C₄	—	4	2	no	—
293	C₄	—	34	2	no	—
294	C₄	—	68	2	no	—
295	C₄	C₄×#63227	1	1	yes	characteristic
296	C₄	#11762763	1	1	yes	characteristic
297	C₄	C₈×C₂²⋊D₁₇	1	1	yes	characteristic
298	C₄	C₈×D₆₈	1	1	yes	characteristic
299	C₂	—	34	4	no	—
300	C₂	C₈×#63227	1	1	yes	characteristic
301	C₂	D₁₇×#346	1	1	yes	characteristic
302	C₂	#14530382	1	1	yes	characteristic
303	C₂	#14530607	1	2	yes	—
304	C₂	#14530611	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₇
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₂×C₁₇×#346	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂³×D₁₇	C₂×C₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#12611
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. C₂×#11762584	C₂	semidirect
5. #14526862	C₂	semidirect
6. #14526861	C₂	semidirect
7. C₂×C₁₇×#346	C₂	semidirect
13. C₂×C₁₆×D₁₇	C₄	semidirect
17. #11762588	C₄	semidirect
94. C₁₆×C₁₇	C₂²⋊C₄	semidirect
133. C₂×#346	D₁₇	semidirect
146. C₂×D₁₇	#346	semidirect
210. C₁₇	#12611	semidirect
249. C₁₆	#63227	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₁₇	2
C₂×#11762588	2
C₂×#11762584	2
C₈×#63227	2
#14526862	2
#14526861	2
C₂×C₁₇×#346	2
#12611	17