Subgroups — #17192074

Order 4352.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541773	C₂	1	1	yes	maximal, characteristic
2	#14531023	C₂	1	2	yes	maximal
3	#14531013	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530928	C₂	1	1	yes	maximal, characteristic
5	#14530923	C₂	1	2	yes	maximal, complemented
6	#14530846	C₂	1	2	yes	maximal
7	#14530834	C₂	1	1	yes	maximal, characteristic, complemented
8	#14530693	C₂	1	1	yes	maximal, characteristic, complemented
9	#14530653	C₂	1	2	yes	maximal, complemented
10	C₄×#10587499	C₂	1	1	yes	maximal, characteristic
11	Q₈×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763532	—	2	1	no	—
13	C₂×#10587500	C₂²	1	2	yes	—
14	C₂×#10587499	C₂²	1	1	yes	characteristic
15	#11763193	C₂²	1	2	yes	—
16	#11763189	C₂²	1	1	yes	characteristic
17	#11763032	—	2	2	no	—
18	#11763015	—	2	1	no	—
19	C₄×Dic₆₈	C₂²	1	2	yes	—
20	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
21	#11762636	C₂²	1	1	yes	characteristic
22	#11762636	C₂²	1	2	yes	—
23	C₄⋊#63214	C₂²	1	1	yes	characteristic
24	C₄⋊#63214	C₂²	1	2	yes	—
25	#11762613	C₂²	1	1	yes	characteristic
26	#11762612	C₂²	1	2	yes	—
27	#11762612	C₂²	1	2	yes	—
28	#11762612	C₂²	1	2	yes	—
29	#11762609	C₂²	1	2	yes	—
30	#11762609	C₂²	1	1	yes	characteristic
31	#11762585	C₂²	1	2	yes	—
32	C₄×#63214	C₂²	1	1	yes	characteristic
33	Q₈²×C₁₇	—	2	1	no	—
34	C₁₇×#500	C₂²	1	2	yes	—
35	C₁₇×#500	C₂²	1	1	yes	characteristic
36	C₁₇×#477	—	2	1	no	—
37	C₁₇×#477	—	2	2	no	—
38	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₁₇×C₄⋊Q₈	—	2	2	no	—
43	C₁₇×C₄⋊Q₈	—	2	1	no	—
44	C₁₇×C₄⋊Q₈	—	4	1	no	—
45	C₁₇×C₄⋊Q₈	—	4	2	no	—
46	C₄×Q₈×C₁₇	—	2	2	no	—
47	C₄×Q₈×C₁₇	—	2	1	no	—
48	C₄×Q₈×C₁₇	—	4	1	no	—
49	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
50	#10587632	—	4	1	no	—
51	C₂×Dic₆₈	—	2	1	no	—
52	C₂×Dic₆₈	C₂³	1	2	yes	—
53	C₂×D₁₃₆	C₂³	1	1	yes	characteristic
54	C₂×#63219	—	2	2	no	—
55	#10587584	—	4	2	no	—
56	#10587578	—	4	1	no	—
57	#10587561	—	2	2	no	—
58	C₄×D₆₈	—	2	1	no	—
59	#10587555	—	2	1	no	—
60	C₄×Dic₃₄	—	2	2	no	—
61	C₁₇×#107	—	2	2	no	—
62	C₁₇×#107	—	2	1	no	—
63	C₁₇×#107	C₂³	1	1	yes	characteristic
64	C₁₇×#107	C₂³	1	2	yes	—
65	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
67	C₁₇×#103	—	2	2	no	—
68	C₁₇×#103	—	2	1	no	—
69	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₇	C₂³	1	2	yes	—
71	#10587500	—	2	4	no	—
72	#10587499	—	2	2	no	—
73	#10587492	—	2	1	no	—
74	#10587489	C₂³	1	1	yes	characteristic
75	#10587487	—	2	2	no	—
76	C₂×#63214	C₂³	1	1	yes	characteristic
77	C₂×#63214	C₂³	1	2	yes	—
78	C₂×#63214	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₇	—	2	1	no	—
80	C₂×Q₈×C₁₇	—	4	2	no	—
81	C₂×Q₈×C₁₇	—	4	1	no	—
82	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
83	C₂×D₆₈	—	2	1	no	—
84	C₂×C₄×D₁₇	—	4	1	no	—
85	C₂×Dic₃₄	—	2	2	no	—
86	Q₁₆×C₁₇	—	2	1	no	—
87	Q₁₆×C₁₇	#41	1	2	yes	—
88	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₁₇	—	2	1	no	—
92	C₄⋊C₄×C₁₇	—	2	2	no	—
93	C₄⋊C₄×C₁₇	—	4	2	no	—
94	C₄⋊C₄×C₁₇	—	4	1	no	—
95	C₄⋊C₄×C₁₇	—	4	2	no	—
96	C₄⋊C₄×C₁₇	—	4	1	no	—
97	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
98	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
99	C₄²×C₁₇	—	4	2	no	—
100	C₄²×C₁₇	—	4	1	no	—
101	C₄²×C₁₇	C₂×D₄	1	2	yes	—
102	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
103	#63227	—	4	1	no	—
104	#63227	—	4	2	no	—
105	#63227	—	4	2	no	—
106	C₄⋊Dic₁₇	—	2	1	no	—
107	C₄⋊Dic₁₇	—	4	1	no	—
108	C₄⋊Dic₁₇	—	4	1	no	—
109	#63225	—	4	2	no	—
110	C₄×Dic₁₇	—	4	2	no	—
111	Dic₆₈	—	2	2	no	—
112	Dic₆₈	—	2	2	no	—
113	Dic₆₈	—	8	1	no	—
114	D₁₃₆	—	2	1	no	—
115	D₁₃₆	—	2	1	no	—
116	#63219	—	8	2	no	—
117	#63214	—	2	1	no	—
118	#63214	—	2	2	no	—
119	#63214	—	2	1	no	—
120	#33936	—	17	1	no	maximal, Sylow
121	C₂²×D₁₇	—	4	1	no	—
122	Q₈×C₁₇	—	2	2	no	—
123	Q₈×C₁₇	—	2	2	no	—
124	Q₈×C₁₇	—	4	1	no	—
125	Q₈×C₁₇	—	16	2	no	—
126	Q₈×C₁₇	—	16	1	no	—
127	C₂×C₄×C₁₇	—	4	1	no	—
128	C₂×C₄×C₁₇	—	4	2	no	—
129	C₂×C₄×C₁₇	—	4	1	no	—
130	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₁₇	—	4	2	no	—
136	C₂×Dic₁₇	—	4	1	no	—
137	D₆₈	—	4	1	no	—
138	D₆₈	—	4	1	no	—
139	C₄×D₁₇	—	8	2	no	—
140	Dic₃₄	—	4	2	no	—
141	Dic₃₄	—	4	2	no	—
142	C₈×C₁₇	—	2	2	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
145	C₈×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
146	#3393	—	17	1	no	—
147	Q₈×Q₁₆	D₁₇	1	1	yes	characteristic, complemented
148	#2243	—	17	2	no	—
149	#2241	—	17	1	no	—
150	#2226	—	17	1	no	—
151	#2225	—	17	2	no	—
152	#2211	—	17	2	no	—
153	#2209	—	17	1	no	—
154	#2183	—	17	1	no	—
155	#2176	—	17	2	no	—
156	C₄×SD₃₂	—	17	1	no	—
157	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₁₇	—	8	1	no	—
159	C₂×D₁₇	—	8	1	no	—
160	C₂×D₁₇	—	8	1	no	—
161	C₄×C₁₇	—	2	2	no	—
162	C₄×C₁₇	—	2	2	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	4	2	no	—
165	C₄×C₁₇	—	8	2	no	—
166	C₄×C₁₇	—	8	1	no	—
167	C₄×C₁₇	#510	1	2	yes	—
168	C₄×C₁₇	#548	1	1	yes	characteristic
169	C₄×C₁₇	#577	1	1	yes	characteristic
170	Dic₁₇	—	8	2	no	—
171	Dic₁₇	—	8	1	no	—
172	Q₈²	—	2	1	no	—
173	#556	—	34	1	no	—
174	C₂×Q₃₂	—	17	2	no	—
175	C₂×SD₃₂	—	17	1	no	—
176	#500	C₂×D₁₇	1	2	yes	—
177	#500	C₂×D₁₇	1	1	yes	characteristic
178	#495	—	17	2	no	—
179	#494	—	17	1	no	—
180	#477	—	2	1	no	—
181	#477	—	2	2	no	—
182	#464	—	34	2	no	—
183	#461	—	34	1	no	—
184	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
185	C₄×Q₁₆	—	17	2	no	—
186	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
187	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
188	C₄×D₈	—	17	1	no	—
189	#367	—	17	1	no	—
190	#367	—	17	2	no	—
191	C₄⋊C₁₆	—	17	1	no	—
192	C₄⋊C₁₆	—	17	2	no	—
193	#358	—	17	1	no	—
194	#358	—	17	2	no	—
195	#358	—	17	2	no	—
196	#358	—	17	2	no	—
197	#357	—	17	1	no	—
198	#357	—	17	2	no	—
199	#346	—	17	2	no	—
200	C₄×C₁₆	—	17	1	no	—
201	C₂×C₁₇	#3358	1	1	yes	characteristic
202	C₂×C₁₇	C₂×#510	1	1	yes	characteristic
203	C₂×C₁₇	#3416	1	1	yes	characteristic
204	D₁₇	—	16	1	no	—
205	C₂×Q₁₆	—	17	2	no	—
206	C₂×Q₁₆	—	34	1	no	—
207	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
208	C₂×SD₁₆	—	34	2	no	—
209	C₂×D₈	—	17	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	—	4	1	no	—
213	C₄⋊Q₈	—	4	2	no	—
214	C₄⋊Q₈	—	34	1	no	—
215	#126	—	68	2	no	—
216	#124	—	34	2	no	—
217	C₂²⋊Q₈	—	68	1	no	—
218	C₄×Q₈	—	2	1	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×Q₈	—	34	2	no	—
222	C₄×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
223	C₄×D₄	—	34	1	no	—
224	#117	—	68	1	no	—
225	Q₃₂	—	34	4	no	—
226	SD₃₂	—	34	2	no	—
227	C₂×C₁₆	—	17	1	no	—
228	C₂×C₁₆	—	17	1	no	—
229	C₂×C₁₆	—	17	2	no	—
230	#107	—	2	1	no	—
231	#107	—	2	2	no	—
232	#107	—	17	1	no	—
233	#107	C₂²×D₁₇	1	1	yes	characteristic
234	#107	C₂²×D₁₇	1	2	yes	—
235	C₄⋊C₈	C₂²⋊D₁₇	1	1	yes	characteristic
236	C₄⋊C₈	C₂²⋊D₁₇	1	2	yes	—
237	#103	—	2	1	no	—
238	#103	—	2	2	no	—
239	#103	—	34	2	no	—
240	#102	—	34	1	no	—
241	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₇	1	2	yes	—
243	C₁₇	#33936	1	1	yes	characteristic, complemented, Sylow
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	34	2	no	—
248	C₂×Q₈	#63228	1	1	yes	characteristic
249	C₂×D₄	—	34	1	no	—
250	C₂²×C₄	—	68	1	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	34	2	no	—
253	Q₁₆	—	34	2	no	—
254	Q₁₆	—	136	1	no	—
255	Q₁₆	#63256	1	2	yes	—
256	SD₁₆	—	136	2	no	—
257	D₈	—	34	1	no	—
258	D₈	—	34	1	no	—
259	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
261	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	—	68	1	no	—
270	C₄⋊C₄	—	68	2	no	—
271	C₄⋊C₄	—	68	1	no	—
272	C₄⋊C₄	#63228	1	1	yes	characteristic
273	C₄⋊C₄	#63228	1	2	yes	—
274	C₂²⋊C₄	—	68	2	no	—
275	C₂²⋊C₄	—	68	1	no	—
276	C₂²⋊C₄	—	68	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	68	2	no	—
280	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
282	C₁₆	—	34	1	no	—
283	C₁₆	—	34	2	no	—
284	C₁₆	—	34	1	no	—
285	C₂³	—	68	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	16	2	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	68	2	no	—
292	Q₈	—	68	2	no	—
293	D₄	—	68	1	no	—
294	D₄	—	68	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	68	1	no	—
299	C₂×C₄	—	68	2	no	—
300	C₂×C₄	—	136	2	no	—
301	C₂×C₄	C₂×#63228	1	2	yes	—
302	C₂×C₄	C₂×#63228	1	2	yes	—
303	C₂×C₄	C₂×#63228	1	1	yes	characteristic
304	C₂×C₄	C₂×#63228	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×#63256	1	1	yes	characteristic
309	C₈	#10587694	1	1	yes	characteristic
310	C₂²	—	136	1	no	—
311	C₂²	—	136	1	no	—
312	C₂²	—	136	1	no	—
313	C₂²	C₂²×#63228	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	136	1	no	—
321	C₄	—	136	2	no	—
322	C₄	#11763250	1	2	yes	—
323	C₄	#11763470	1	1	yes	characteristic
324	C₄	#11763617	1	1	yes	characteristic
325	C₂	—	272	1	no	—
326	C₂	#14541419	1	1	yes	characteristic
327	C₂	C₂×#11763250	1	1	yes	characteristic
328	C₂	#14541931	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	Q₈×Q₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂×D₁₃₆	C₂³
hypercenter	Q₈×Q₁₆	D₁₇
hypocenter	C₁₇	#33936
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #14531013	C₂	semidirect
5. #14530923	C₂	semidirect
7. #14530834	C₂	semidirect
8. #14530693	C₂	semidirect
9. #14530653	C₂	semidirect
11. Q₈×Q₁₆×C₁₇	C₂	semidirect
147. Q₈×Q₁₆	D₁₇	semidirect
243. C₁₇	#33936	semidirect

Maximal subgroups

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

subgroup	index
#14541773	2
#14531023	2
#14531013	2
#14530928	2
#14530923	2
#14530846	2
#14530834	2
#14530693	2
#14530653	2
C₄×#10587499	2
Q₈×Q₁₆×C₁₇	2
#33936	17