Subgroups — #17192383

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14541797	C₂	1	1	yes	maximal, characteristic
2	#14531053	C₂	1	2	yes	maximal, complemented
3	#14531041	C₂	1	1	yes	maximal, characteristic
4	#14530846	C₂	1	2	yes	maximal, complemented
5	#14530831	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530826	C₂	1	2	yes	maximal, complemented
7	#14530824	C₂	1	1	yes	maximal, characteristic
8	#14530693	C₂	1	1	yes	maximal, characteristic, complemented
9	#14530646	C₂	1	2	yes	maximal, complemented
10	C₄×#10587500	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#3395	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763544	—	2	1	no	—
13	C₂×#10587500	C₂²	1	1	yes	characteristic
14	C₂×#10587500	C₂²	1	2	yes	—
15	C₂×#10587499	C₂²	1	2	yes	—
16	C₂×#10587499	C₂²	1	2	yes	—
17	C₂×#10587499	C₂²	1	2	yes	—
18	#11763193	C₂²	1	2	yes	—
19	#11763186	C₂²	1	1	yes	characteristic
20	#11763025	—	2	2	no	—
21	#11763010	—	2	1	no	—
22	C₄×Dic₆₈	C₂²	1	1	yes	characteristic
23	C₄×D₁₃₆	C₂²	1	2	yes	—
24	#11762634	C₂²	1	1	yes	characteristic
25	C₄⋊#63214	C₂²	1	1	yes	characteristic
26	C₄⋊#63214	C₂²	1	2	yes	—
27	#11762613	C₂²	1	1	yes	characteristic
28	#11762613	C₂²	1	2	yes	—
29	#11762612	C₂²	1	1	yes	characteristic
30	#11762609	C₂²	1	2	yes	—
31	#11762585	C₂²	1	2	yes	—
32	C₄×#63214	C₂²	1	1	yes	characteristic
33	C₁₇×#557	—	2	1	no	—
34	C₁₇×#494	C₂²	1	2	yes	—
35	C₁₇×#494	C₂²	1	1	yes	characteristic
36	C₁₇×#462	—	2	1	no	—
37	C₁₇×#462	—	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	—	2	1	no	—
42	C₂×Q₁₆×C₁₇	—	2	2	no	—
43	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
44	C₂×Q₁₆×C₁₇	C₂³	1	1	yes	characteristic
45	C₁₇×C₄⋊Q₈	—	2	1	no	—
46	C₁₇×C₄⋊Q₈	—	2	2	no	—
47	C₁₇×#125	—	4	1	no	—
48	C₁₇×#125	—	4	2	no	—
49	C₄×Q₈×C₁₇	—	2	1	no	—
50	C₄×Q₈×C₁₇	—	2	2	no	—
51	C₄×Q₈×C₁₇	—	4	1	no	—
52	C₄×Q₈×C₁₇	D₄	1	1	yes	characteristic
53	#10587633	—	4	1	no	—
54	C₂×Dic₆₈	C₂³	1	1	yes	characteristic
55	C₂×D₁₃₆	—	2	1	no	—
56	C₂×D₁₃₆	C₂³	1	2	yes	—
57	C₂×#63219	—	2	2	no	—
58	#10587580	—	4	2	no	—
59	#10587579	—	4	1	no	—
60	#10587561	—	2	2	no	—
61	#10587560	—	2	1	no	—
62	C₄×D₆₈	—	2	2	no	—
63	C₄×Dic₃₄	—	2	1	no	—
64	C₁₇×#107	C₂³	1	1	yes	characteristic
65	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
66	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
67	C₁₇×#103	—	2	1	no	—
68	C₁₇×#103	—	2	2	no	—
69	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₇	C₂³	1	2	yes	—
71	#10587500	—	2	2	no	—
72	#10587500	—	4	2	no	—
73	#10587499	—	2	4	no	—
74	#10587499	—	4	2	no	—
75	#10587499	—	4	2	no	—
76	#10587492	—	2	2	no	—
77	#10587489	C₂³	1	1	yes	characteristic
78	#10587487	—	2	1	no	—
79	C₂×#63214	C₂³	1	1	yes	characteristic
80	C₂×#63214	C₂³	1	2	yes	—
81	C₂×#63214	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₁₇	—	2	2	no	—
83	C₂×Q₈×C₁₇	—	2	1	no	—
84	C₂×Q₈×C₁₇	D₈	1	1	yes	characteristic
85	C₂×D₆₈	—	2	2	no	—
86	C₂×D₆₈	—	4	1	no	—
87	C₂×D₆₈	—	4	1	no	—
88	C₂×C₄×D₁₇	—	4	2	no	—
89	C₂×Dic₃₄	—	2	1	no	—
90	Q₁₆×C₁₇	—	2	2	no	—
91	Q₁₆×C₁₇	—	2	1	no	—
92	Q₁₆×C₁₇	—	2	1	no	—
93	Q₁₆×C₁₇	—	2	2	no	—
94	Q₁₆×C₁₇	—	8	2	no	—
95	Q₁₆×C₁₇	—	8	1	no	—
96	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₁₇	—	2	1	no	—
100	C₄⋊C₄×C₁₇	—	4	2	no	—
101	C₄⋊C₄×C₁₇	—	4	1	no	—
102	C₄⋊C₄×C₁₇	—	4	2	no	—
103	C₄⋊C₄×C₁₇	—	4	1	no	—
104	C₄⋊C₄×C₁₇	—	4	2	no	—
105	C₄⋊C₄×C₁₇	—	4	1	no	—
106	C₄⋊C₄×C₁₇	D₈	1	1	yes	characteristic
107	C₄⋊C₄×C₁₇	D₈	1	2	yes	—
108	C₄²×C₁₇	—	4	2	no	—
109	C₄²×C₁₇	—	4	1	no	—
110	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₇	C₂×D₄	1	2	yes	—
112	#63227	—	4	2	no	—
113	#63227	—	4	2	no	—
114	#63227	—	4	2	no	—
115	C₄⋊Dic₁₇	—	2	1	no	—
116	#63225	—	4	1	no	—
117	C₄×Dic₁₇	—	4	1	no	—
118	Dic₆₈	—	2	1	no	—
119	Dic₆₈	—	2	1	no	—
120	D₁₃₆	—	2	2	no	—
121	D₁₃₆	—	2	2	no	—
122	D₁₃₆	—	8	1	no	—
123	#63219	—	8	2	no	—
124	#63214	—	2	1	no	—
125	#63214	—	2	2	no	—
126	#63214	#41	1	2	yes	—
127	#33967	—	17	1	no	maximal, Sylow
128	C₂²×D₁₇	—	4	2	no	—
129	Q₈×C₁₇	—	2	2	no	—
130	Q₈×C₁₇	—	4	2	no	—
131	Q₈×C₁₇	—	4	1	no	—
132	Q₈×C₁₇	—	4	1	no	—
133	Q₈×C₁₇	—	4	2	no	—
134	C₂×C₄×C₁₇	—	4	1	no	—
135	C₂×C₄×C₁₇	—	4	2	no	—
136	C₂×C₄×C₁₇	—	4	1	no	—
137	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×Dic₁₇	—	4	1	no	—
143	C₂×Dic₁₇	—	4	1	no	—
144	D₆₈	—	4	2	no	—
145	D₆₈	—	4	2	no	—
146	D₆₈	—	8	2	no	—
147	D₆₈	—	16	1	no	—
148	C₄×D₁₇	—	16	2	no	—
149	Dic₃₄	—	4	1	no	—
150	Dic₃₄	—	4	1	no	—
151	C₈×C₁₇	—	2	2	no	—
152	C₈×C₁₇	—	2	1	no	—
153	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
154	C₈×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
155	#3396	—	17	1	no	—
156	#3395	D₁₇	1	1	yes	characteristic, complemented
157	#2251	—	17	2	no	—
158	#2248	—	17	1	no	—
159	#2211	—	17	2	no	—
160	#2209	—	17	1	no	—
161	#2208	—	17	2	no	—
162	#2207	—	17	1	no	—
163	#2183	—	17	1	no	—
164	#2175	—	17	2	no	—
165	C₄×Q₃₂	—	17	1	no	—
166	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₁₇	—	8	2	no	—
168	C₂×D₁₇	—	8	2	no	—
169	C₂×D₁₇	—	8	2	no	—
170	C₄×C₁₇	—	2	2	no	—
171	C₄×C₁₇	—	2	2	no	—
172	C₄×C₁₇	—	2	1	no	—
173	C₄×C₁₇	—	8	2	no	—
174	C₄×C₁₇	—	8	1	no	—
175	C₄×C₁₇	—	8	1	no	—
176	C₄×C₁₇	#510	1	2	yes	—
177	C₄×C₁₇	#550	1	1	yes	characteristic
178	C₄×C₁₇	#576	1	1	yes	characteristic
179	Dic₁₇	—	8	1	no	—
180	Dic₁₇	—	8	1	no	—
181	#559	—	34	1	no	—
182	#557	—	2	1	no	—
183	C₂×Q₃₂	—	17	1	no	—
184	C₂×Q₃₂	—	17	2	no	—
185	C₂×SD₃₂	—	17	2	no	—
186	C₂×SD₃₂	—	17	2	no	—
187	C₂×SD₃₂	—	17	2	no	—
188	#495	—	17	2	no	—
189	#494	C₂×D₁₇	1	2	yes	—
190	#494	C₂×D₁₇	1	1	yes	characteristic
191	#493	—	17	1	no	—
192	#463	—	34	2	no	—
193	#462	—	2	1	no	—
194	#462	—	2	2	no	—
195	#460	—	34	1	no	—
196	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
197	C₄×Q₁₆	—	17	1	no	—
198	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
199	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
200	C₄×D₈	—	17	2	no	—
201	#366	—	17	1	no	—
202	C₄⋊C₁₆	—	17	1	no	—
203	C₄⋊C₁₆	—	17	2	no	—
204	#358	—	17	1	no	—
205	#358	—	17	2	no	—
206	#358	—	17	1	no	—
207	#357	—	17	2	no	—
208	#346	—	17	2	no	—
209	C₄×C₁₆	—	17	1	no	—
210	C₂×C₁₇	#3391	1	1	yes	characteristic
211	C₂×C₁₇	C₂×#510	1	1	yes	characteristic
212	C₂×C₁₇	#3416	1	1	yes	characteristic
213	D₁₇	—	16	2	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	—	17	1	no	—
217	C₂×Q₁₆	C₂²×D₁₇	1	2	yes	—
218	C₂×Q₁₆	C₂²×D₁₇	1	1	yes	characteristic
219	C₂×SD₁₆	—	34	2	no	—
220	C₂×D₈	—	17	2	no	—
221	C₂×D₈	—	34	1	no	—
222	C₄⋊Q₈	—	2	2	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	#127	—	34	1	no	—
225	#125	—	4	1	no	—
226	#125	—	4	2	no	—
227	#124	—	34	2	no	—
228	#124	—	68	1	no	—
229	#123	—	68	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	4	1	no	—
233	C₄×Q₈	—	34	1	no	—
234	C₄×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
235	C₄×D₄	—	34	2	no	—
236	C₄×D₄	—	68	1	no	—
237	Q₃₂	—	34	2	no	—
238	Q₃₂	—	68	2	no	—
239	SD₃₂	—	34	4	no	—
240	SD₃₂	—	68	2	no	—
241	SD₃₂	—	68	2	no	—
242	C₂×C₁₆	—	17	1	no	—
243	C₂×C₁₆	—	17	1	no	—
244	C₂×C₁₆	—	17	2	no	—
245	#107	—	17	1	no	—
246	#107	C₂²×D₁₇	1	1	yes	characteristic
247	C₄⋊C₈	C₂²⋊D₁₇	1	1	yes	characteristic
248	C₄⋊C₈	C₂²⋊D₁₇	1	2	yes	—
249	#103	—	2	2	no	—
250	#103	—	2	1	no	—
251	#103	—	34	1	no	—
252	#102	—	34	2	no	—
253	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₁₇	1	2	yes	—
255	C₁₇	#33967	1	1	yes	characteristic, complemented, Sylow
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	34	1	no	—
259	C₂×Q₈	#63228	1	1	yes	characteristic
260	C₂×D₄	—	34	2	no	—
261	C₂×D₄	—	68	1	no	—
262	C₂×D₄	—	68	1	no	—
263	C₂²×C₄	—	68	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	8	1	no	—
269	Q₁₆	—	8	2	no	—
270	Q₁₆	—	34	1	no	—
271	Q₁₆	—	34	1	no	—
272	SD₁₆	—	136	2	no	—
273	D₈	—	34	2	no	—
274	D₈	—	34	2	no	—
275	D₈	—	136	1	no	—
276	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
277	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
278	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	34	1	no	—
287	C₄⋊C₄	—	68	1	no	—
288	C₄⋊C₄	#63228	1	1	yes	characteristic
289	C₄⋊C₄	#63228	1	2	yes	—
290	C₂²⋊C₄	—	68	2	no	—
291	C₂²⋊C₄	—	68	2	no	—
292	C₂²⋊C₄	—	68	2	no	—
293	C₄²	—	4	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	68	1	no	—
296	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
297	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
298	C₁₆	—	17	2	no	—
299	C₁₆	—	34	1	no	—
300	C₁₆	—	34	2	no	—
301	C₂³	—	68	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	68	1	no	—
308	Q₈	—	68	1	no	—
309	D₄	—	272	1	no	—
310	D₄	—	68	2	no	—
311	D₄	—	68	2	no	—
312	D₄	—	136	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	272	2	no	—
317	C₂×C₄	—	68	1	no	—
318	C₂×C₄	—	68	1	no	—
319	C₂×C₄	C₂×#63228	1	2	yes	—
320	C₂×C₄	C₂×#63228	1	2	yes	—
321	C₂×C₄	C₂×#63228	1	1	yes	characteristic
322	C₂×C₄	C₂×#63228	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	#41×D₁₇	1	1	yes	characteristic
327	C₈	#10587696	1	1	yes	characteristic
328	C₂²	—	136	2	no	—
329	C₂²	—	136	2	no	—
330	C₂²	—	136	2	no	—
331	C₂²	C₂²×#63228	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	136	1	no	—
339	C₄	—	136	1	no	—
340	C₄	#11763250	1	2	yes	—
341	C₄	#11763478	1	1	yes	characteristic
342	C₄	#11763612	1	1	yes	characteristic
343	C₂	—	272	2	no	—
344	C₂	#14541756	1	1	yes	characteristic
345	C₂	C₂×#11763250	1	1	yes	characteristic
346	C₂	#14541928	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#3395	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	#11763186	C₂²
hypercenter	#3395	D₁₇
hypocenter	C₁₇	#33967
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

Complements

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

N	H ≅ G/N	type
2. #14531053	C₂	semidirect
4. #14530846	C₂	semidirect
5. #14530831	C₂	semidirect
6. #14530826	C₂	semidirect
8. #14530693	C₂	semidirect
9. #14530646	C₂	semidirect
10. C₄×#10587500	C₂	semidirect
11. C₁₇×#3395	C₂	semidirect
156. #3395	D₁₇	semidirect
255. C₁₇	#33967	semidirect

Maximal subgroups

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

subgroup	index
#14541797	2
#14531053	2
#14531041	2
#14530846	2
#14530831	2
#14530826	2
#14530824	2
#14530693	2
#14530646	2
C₄×#10587500	2
C₁₇×#3395	2
#33967	17