Subgroups — #16950813

Order 4352.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763227	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762629	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530543	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11762592	C₂	1	1	yes	maximal, characteristic
5	#14528623	C₂	1	1	yes	maximal, characteristic
6	#14526861	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587587	C₂²	1	1	yes	characteristic
9	#11763227	—	2	2	no	—
10	#11763227	—	2	2	no	—
11	C₂×#10587483	C₂²	1	1	yes	characteristic
12	C₂×#10587469	—	2	1	no	—
13	C₂²×#63214	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₁₇	—	2	1	no	—
15	C₂×#10587496	C₄	1	1	yes	characteristic
16	#11762803	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
18	#11762629	—	2	2	no	—
19	C₄⋊#63214	—	2	1	no	—
20	C₄⋊#63214	—	2	1	no	—
21	#11762592	—	2	2	no	—
22	#11762573	C₄	1	1	yes	characteristic
23	#11762568	C₄	1	1	yes	characteristic
24	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
26	C₁₇×#405	C₂²	1	1	yes	characteristic
27	C₂×#63252	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×#63223	C₂×C₄	1	1	yes	characteristic
33	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
34	#10587587	—	2	1	no	—
35	#10587587	—	2	1	no	—
36	#10587587	D₄	1	2	yes	—
37	#10587587	Q₈	1	2	yes	—
38	C₂×#63218	—	2	1	no	—
39	C₂×C₈×D₁₇	—	2	1	no	—
40	C₁₇×M₃₂	—	2	1	no	—
41	C₁₇×M₃₂	D₄	1	2	yes	—
42	C₂×C₁₆×C₁₇	—	2	1	no	—
43	C₂×C₁₆×C₁₇	—	2	1	no	—
44	C₂×C₁₆×C₁₇	—	2	1	no	—
45	C₂×C₁₆×C₁₇	—	2	1	no	—
46	C₁₇×#97	—	2	1	no	—
47	C₄×C₈×C₁₇	—	2	1	no	—
48	#10587496	—	2	2	no	—
49	C₈×Dic₁₇	—	2	1	no	—
50	C₈×Dic₁₇	—	2	1	no	—
51	#10587483	—	2	1	no	—
52	#10587483	—	2	1	no	—
53	C₂×#63214	—	2	1	no	—
54	C₂×#63214	—	2	1	no	—
55	C₂×#63214	—	2	1	no	—
56	C₂×#63214	D₄	1	1	yes	characteristic
57	C₂×#63214	D₄	1	1	yes	characteristic
58	#10587481	C₈	1	1	yes	characteristic
59	#10587481	C₈	1	1	yes	characteristic
60	#10587480	C₈	1	1	yes	characteristic
61	#10587480	C₈	1	1	yes	characteristic
62	#10587469	—	2	4	no	—
63	C₁₆×D₁₇	—	2	4	no	—
64	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	#63252	—	2	1	no	—
68	#63252	—	2	1	no	—
69	#63252	C₄⋊C₄	1	2	yes	—
70	#63252	M₁₆	1	2	yes	—
71	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
74	M₁₆×C₁₇	—	2	1	no	—
75	M₁₆×C₁₇	—	2	1	no	—
76	C₂×C₈×C₁₇	—	2	1	no	—
77	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₇	—	2	1	no	—
86	C₄²×C₁₇	—	2	1	no	—
87	C₄×Dic₁₇	—	2	1	no	—
88	C₄×Dic₁₇	—	2	1	no	—
89	#63223	—	2	1	no	—
90	#63223	C₄⋊C₄	1	2	yes	—
91	C₂×#3622	—	2	1	no	—
92	C₂×#3622	—	2	1	no	—
93	C₂×#3622	—	2	1	no	—
94	C₂×#3622	C₂×C₈	1	1	yes	characteristic
95	C₂×#3622	C₂×C₈	1	1	yes	characteristic
96	#63218	—	2	2	no	—
97	#63218	—	2	2	no	—
98	C₈×D₁₇	—	2	2	no	—
99	C₈×D₁₇	—	2	2	no	—
100	C₁₆×C₁₇	—	2	2	no	—
101	C₁₆×C₁₇	—	4	1	no	—
102	#63214	—	2	1	no	—
103	#63214	—	2	1	no	—
104	#63214	—	4	1	no	—
105	#12917	—	17	1	no	maximal, Sylow
106	C₂³×C₁₇	#131	1	1	yes	characteristic
107	C₂²×D₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
116	C₂²⋊D₁₇	—	2	2	no	—
117	C₂²⋊D₁₇	—	2	2	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	—	2	2	no	—
120	C₂×Dic₁₇	—	4	1	no	—
121	C₂×Dic₁₇	—	4	1	no	—
122	D₆₈	—	2	1	no	—
123	D₆₈	C₄⋊C₈	1	2	yes	—
124	C₄×D₁₇	—	2	2	no	—
125	C₄×D₁₇	—	2	2	no	—
126	Dic₃₄	—	2	1	no	—
127	Dic₃₄	C₄⋊C₈	1	2	yes	—
128	C₈×C₁₇	—	4	1	no	—
129	C₈×C₁₇	#120	1	1	yes	characteristic
130	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₁₇	#123	1	1	yes	characteristic
133	#3622	—	2	2	no	—
134	#3622	—	2	1	no	—
135	#3622	—	2	1	no	—
136	C₂×#504	—	17	1	no	—
137	C₂×#364	—	17	1	no	—
138	#2151	—	17	1	no	—
139	C₂×#350	—	17	1	no	—
140	#1764	—	17	1	no	—
141	#1379	D₁₇	1	1	yes	characteristic, complemented
142	#1374	—	17	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	#424	1	1	yes	characteristic
146	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
147	C₂²×C₁₇	#435	1	1	yes	characteristic
148	C₂×D₁₇	—	2	2	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₂×D₁₇	—	4	1	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	#385	1	1	yes	characteristic
154	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
155	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
156	C₄×C₁₇	#435	1	1	yes	characteristic
157	Dic₁₇	—	2	2	no	—
158	Dic₁₇	—	4	2	no	—
159	C₂×#131	—	17	1	no	—
160	#504	—	34	2	no	—
161	#504	—	34	2	no	—
162	C₂×M₃₂	—	17	1	no	—
163	C₂×M₃₂	—	34	1	no	—
164	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
165	C₂²×C₁₆	—	17	1	no	—
166	C₂²×C₁₆	—	34	1	no	—
167	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
168	C₂×#104	—	17	1	no	—
169	#416	—	17	1	no	—
170	#405	C₂×D₁₇	1	1	yes	characteristic
171	C₂×C₄×C₈	—	17	1	no	—
172	#364	—	34	2	no	—
173	C₄⋊C₁₆	—	34	1	no	—
174	C₄⋊C₁₆	—	34	1	no	—
175	#350	—	34	2	no	—
176	#341	—	17	1	no	—
177	#339	—	17	1	no	—
178	C₂×C₁₇	—	2	1	no	—
179	C₂×C₁₇	—	2	1	no	—
180	C₂×C₁₇	#1850	1	1	yes	characteristic
181	C₂×C₁₇	#2170	1	1	yes	characteristic
182	C₂×C₁₇	#2171	1	1	yes	characteristic
183	D₁₇	—	4	2	no	—
184	C₂×#41	—	17	1	no	—
185	#131	—	17	2	no	—
186	#131	—	17	2	no	—
187	#131	—	34	1	no	—
188	#131	—	34	1	no	—
189	C₂×M₁₆	—	17	1	no	—
190	C₂×M₁₆	—	34	1	no	—
191	C₂×M₁₆	C₄×D₁₇	1	1	yes	characteristic
192	C₂²×C₈	—	17	1	no	—
193	C₂²×C₈	—	34	1	no	—
194	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
195	#117	C₄×D₁₇	1	1	yes	characteristic
196	C₂×C₄²	—	17	1	no	—
197	M₃₂	—	2	1	no	—
198	M₃₂	—	34	4	no	—
199	M₃₂	—	34	1	no	—
200	M₃₂	—	34	1	no	—
201	M₃₂	D₆₈	1	2	yes	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	17	1	no	—
207	C₂×C₁₆	—	17	1	no	—
208	C₂×C₁₆	—	34	4	no	—
209	C₂×C₁₆	—	34	1	no	—
210	C₂×C₁₆	—	34	1	no	—
211	C₂×C₁₆	—	34	1	no	—
212	#104	—	34	2	no	—
213	#103	—	17	1	no	—
214	#103	—	17	1	no	—
215	#102	—	17	1	no	—
216	#102	—	17	1	no	—
217	#97	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	34	1	no	—
220	C₄×C₈	—	34	1	no	—
221	C₁₇	#12917	1	1	yes	characteristic, complemented, Sylow
222	#41	—	17	2	no	—
223	#41	—	17	2	no	—
224	#41	—	34	1	no	—
225	#41	—	34	1	no	—
226	C₂×Q₈	—	17	1	no	—
227	C₂×D₄	—	17	1	no	—
228	C₂×D₄	—	34	1	no	—
229	C₂²×C₄	—	34	1	no	—
230	C₂²×C₄	—	34	1	no	—
231	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	17	2	no	—
235	M₁₆	—	34	1	no	—
236	M₁₆	—	34	2	no	—
237	M₁₆	—	34	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₂×C₈	—	34	1	no	—
242	C₂×C₈	—	34	1	no	—
243	C₂×C₈	—	34	1	no	—
244	C₂×C₈	—	34	2	no	—
245	C₂×C₈	—	34	2	no	—
246	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
247	C₂×C₈	#63252	1	1	yes	characteristic
248	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
249	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
250	C₂×C₈	#63254	1	1	yes	characteristic
251	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
252	C₄⋊C₄	C₈×D₁₇	1	1	yes	characteristic
253	C₄⋊C₄	C₈×D₁₇	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	34	1	no	—
257	C₄²	—	34	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	4	1	no	—
260	C₁₆	—	34	1	no	—
261	C₁₆	—	34	1	no	—
262	C₁₆	—	68	1	no	—
263	C₂³	—	34	1	no	—
264	C₂³	#10587593	1	1	yes	characteristic
265	Q₈	—	17	2	no	—
266	Q₈	—	34	1	no	—
267	D₄	—	17	2	no	—
268	D₄	—	34	2	no	—
269	D₄	—	34	2	no	—
270	D₄	—	34	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	34	1	no	—
274	C₂×C₄	—	34	2	no	—
275	C₂×C₄	—	34	2	no	—
276	C₂×C₄	—	34	2	no	—
277	C₂×C₄	—	68	1	no	—
278	C₂×C₄	—	68	1	no	—
279	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
280	C₂×C₄	#10587567	1	1	yes	characteristic
281	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
283	C₂×C₄	#10587587	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
285	C₈	—	4	1	no	—
286	C₈	—	34	1	no	—
287	C₈	—	34	1	no	—
288	C₈	—	34	2	no	—
289	C₈	#10587568	1	1	yes	characteristic
290	C₈	#10587569	1	1	yes	characteristic
291	C₈	#10587582	1	1	yes	characteristic
292	C₈	#10587583	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	34	2	no	—
296	C₂²	—	68	1	no	—
297	C₂²	—	68	1	no	—
298	C₂²	#11762834	1	1	yes	characteristic
299	C₂²	#11762872	1	1	yes	characteristic
300	C₂²	#11762882	1	1	yes	characteristic
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	34	2	no	—
304	C₄	—	68	2	no	—
305	C₄	#11762683	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₁₇	1	1	yes	characteristic
307	C₄	C₈×D₆₈	1	1	yes	characteristic
308	C₄	#11762883	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	68	2	no	—
312	C₂	#14528963	1	1	yes	characteristic
313	C₂	#14530618	1	1	yes	characteristic
314	C₂	#14530623	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1379	C₂
socle	C₂²×C₁₇	C₈×D₄
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1379	D₁₇
hypocenter	C₁₇	#12917
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
2. C₂×#11762629	C₂	semidirect
3. #14530543	C₂	semidirect
6. #14526861	C₂	semidirect
7. C₁₇×#1379	C₂	semidirect
141. #1379	D₁₇	semidirect
221. C₁₇	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763227	2
C₂×#11762629	2
#14530543	2
C₂×#11762592	2
#14528623	2
#14526861	2
C₁₇×#1379	2
#12917	17