Subgroups — #14538579

Order 2176.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763365	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₁₇	C₂	1	1	yes	maximal, characteristic
3	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763098	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763088	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763087	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762831	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762812	C₂	1	1	yes	maximal, characteristic
14	#11762812	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₇	—	2	1	no	—
18	C₁₇×#125	C₂²	1	1	yes	characteristic, complemented
19	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
20	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₇	—	2	1	no	—
22	C₂×#63218	C₂²	1	1	yes	characteristic
23	C₂×#63218	C₂²	1	1	yes	characteristic
24	#10587582	—	2	1	no	—
25	#10587582	—	2	1	no	—
26	#10587579	C₂²	1	1	yes	characteristic
27	#10587578	C₂²	1	1	yes	characteristic
28	#10587578	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
30	#10587574	C₂²	1	1	yes	characteristic, complemented
31	#10587574	C₂²	1	1	yes	characteristic, complemented
32	#10587573	—	2	1	no	—
33	#10587573	—	2	1	no	—
34	#10587567	—	2	1	no	—
35	C₂²⋊C₄×D₁₇	—	2	1	no	—
36	C₄×D₆₈	C₂²	1	1	yes	characteristic
37	C₄²×D₁₇	C₂²	1	1	yes	characteristic
38	#10587556	C₂²	1	1	yes	characteristic, complemented
39	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
40	C₁₇×#106	C₂²	1	1	yes	characteristic, complemented
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×#102	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic, complemented
49	#10587488	C₂²	1	1	yes	characteristic, complemented
50	#10587485	C₂²	1	1	yes	characteristic, complemented
51	#10587485	C₂²	1	1	yes	characteristic, complemented
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic, complemented
55	#10587479	C₂²	1	1	yes	characteristic, complemented
56	#10587478	C₂²	1	1	yes	characteristic, complemented
57	#10587478	C₂²	1	1	yes	characteristic, complemented
58	C₄⋊#3622	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂²×Dic₁₇	—	2	1	no	—
64	D₄×D₁₇	—	2	2	no	—
65	D₄×D₁₇	Q₈	1	4	yes	complemented
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	4	4	no	—
71	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	1	no	—
75	C₂×Dic₃₄	—	2	1	no	—
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	D₄	1	1	yes	characteristic
101	C₄×Dic₁₇	D₄	1	1	yes	characteristic
102	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	#63218	—	2	2	no	—
106	#63218	—	2	2	no	—
107	C₂³×C₁₇	—	2	1	no	—
108	C₂²×D₁₇	—	2	1	no	—
109	C₂²×D₁₇	—	2	4	no	—
110	C₂²×D₁₇	—	4	2	no	—
111	C₂²×D₁₇	—	4	2	no	—
112	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	C₂×Q₈	1	2	yes	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	4	2	no	—
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₇	—	2	4	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	4	2	no	—
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	D₆₈	—	2	1	no	—
131	D₆₈	C₂×Q₈	1	2	yes	—
132	C₄×D₁₇	—	2	2	no	—
133	C₄×D₁₇	—	4	1	no	—
134	C₄×D₁₇	—	4	2	no	—
135	C₄×D₁₇	—	4	2	no	—
136	C₄×D₁₇	—	4	2	no	—
137	C₄×D₁₇	C₂×Q₈	1	2	yes	—
138	C₄×D₁₇	#41	1	2	yes	—
139	Dic₃₄	—	4	2	no	—
140	Dic₃₄	—	4	2	no	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3077	—	17	1	no	maximal, Sylow
146	C₂²×C₁₇	—	2	2	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	—	4	1	no	—
149	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	—	2	1	no	—
152	C₂×D₁₇	—	2	2	no	—
153	C₂×D₁₇	—	4	8	no	—
154	C₂×D₁₇	—	4	1	no	—
155	C₂×D₁₇	—	4	1	no	—
156	C₂×D₁₇	C₂²⋊Q₈	1	2	yes	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
163	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	C₂²⋊Q₈	1	2	yes	—
169	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
170	#533	—	17	1	no	—
171	C₂×C₄×D₄	—	17	1	no	—
172	#478	—	17	1	no	—
173	#478	—	17	1	no	—
174	#478	—	17	1	no	—
175	#478	D₁₇	1	1	yes	characteristic, complemented
176	#476	—	17	1	no	—
177	#476	—	17	1	no	—
178	#474	—	17	1	no	—
179	#474	—	17	1	no	—
180	#428	—	17	1	no	—
181	#428	—	17	1	no	—
182	#423	—	17	1	no	—
183	#418	—	17	1	no	—
184	#418	—	17	1	no	—
185	C₂×C₁₇	—	4	2	no	—
186	C₂×C₁₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂×C₁₇	#575	1	1	yes	characteristic
189	D₁₇	—	2	2	no	—
190	D₁₇	—	4	2	no	—
191	C₂²×D₄	—	17	1	no	—
192	C₂³×C₄	—	34	1	no	—
193	C₂×M₁₆	—	17	1	no	—
194	C₂×M₁₆	—	17	1	no	—
195	C₄⋊Q₈	—	17	1	no	—
196	C₄⋊Q₈	—	17	1	no	—
197	#125	—	17	1	no	—
198	#125	C₂×D₁₇	1	1	yes	characteristic, complemented
199	C₂²⋊Q₈	—	34	1	no	—
200	C₂²⋊Q₈	—	34	1	no	—
201	C₄×Q₈	—	34	1	no	—
202	C₄×Q₈	—	34	1	no	—
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	17	1	no	—
206	C₄×D₄	—	34	1	no	—
207	C₄×D₄	—	34	1	no	—
208	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
209	#117	—	17	1	no	—
210	#117	—	17	1	no	—
211	C₂×C₄⋊C₄	—	17	1	no	—
212	C₂×C₂²⋊C₄	—	34	1	no	—
213	C₂×C₄²	—	17	1	no	—
214	#107	—	17	1	no	—
215	#107	—	17	1	no	—
216	#107	—	17	1	no	—
217	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
218	#106	—	17	1	no	—
219	#106	—	17	1	no	—
220	#106	—	17	1	no	—
221	#106	C₂×D₁₇	1	1	yes	characteristic, complemented
222	C₄⋊C₈	—	17	1	no	—
223	C₄⋊C₈	—	17	1	no	—
224	C₄⋊C₈	—	17	1	no	—
225	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
226	#102	—	17	1	no	—
227	#102	—	17	1	no	—
228	#102	—	17	1	no	—
229	#102	—	17	1	no	—
230	#102	—	17	1	no	—
231	#102	—	17	1	no	—
232	#102	C₂×D₁₇	1	1	yes	characteristic
233	#102	C₂×D₁₇	1	1	yes	characteristic
234	C₁₇	#3077	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	—	34	1	no	—
236	C₂×Q₈	—	34	1	no	—
237	C₂×Q₈	—	34	1	no	—
238	C₂×D₄	—	17	4	no	—
239	C₂×D₄	—	17	1	no	—
240	C₂×D₄	—	34	2	no	—
241	C₂×D₄	—	34	1	no	—
242	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	17	1	no	—
245	C₂²×C₄	—	17	1	no	—
246	C₂²×C₄	—	17	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	C₂²×C₄	—	34	1	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	—	68	4	no	—
252	M₁₆	—	34	2	no	—
253	M₁₆	—	34	2	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	17	1	no	—
261	C₄⋊C₄	—	17	1	no	—
262	C₄⋊C₄	—	17	1	no	—
263	C₄⋊C₄	—	34	1	no	—
264	C₄⋊C₄	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₂²⋊C₄	—	34	1	no	—
277	C₄²	—	17	1	no	—
278	C₄²	—	17	1	no	—
279	C₄²	—	17	1	no	—
280	C₄²	—	34	1	no	—
281	C₄²	—	34	1	no	—
282	C₄²	C₂²×D₁₇	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	17	1	no	—
285	C₂³	—	34	4	no	—
286	C₂³	—	34	1	no	—
287	C₂³	—	68	2	no	—
288	C₂³	—	68	2	no	—
289	Q₈	—	68	2	no	—
290	Q₈	—	68	2	no	—
291	D₄	—	2	1	no	—
292	D₄	—	17	2	no	—
293	D₄	—	34	4	no	—
294	D₄	—	34	1	no	—
295	D₄	Q₈×D₁₇	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	17	1	no	—
301	C₂×C₄	—	17	1	no	—
302	C₂×C₄	—	17	2	no	—
303	C₂×C₄	—	17	2	no	—
304	C₂×C₄	—	17	1	no	—
305	C₂×C₄	—	34	2	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	68	1	no	—
310	C₂×C₄	—	68	2	no	—
311	C₂×C₄	—	68	2	no	—
312	C₂×C₄	—	68	2	no	—
313	C₂×C₄	—	68	2	no	—
314	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
315	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	34	1	no	—
320	C₈	—	34	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	17	2	no	—
325	C₂²	—	34	1	no	—
326	C₂²	—	34	1	no	—
327	C₂²	—	34	2	no	—
328	C₂²	—	68	8	no	—
329	C₂²	—	68	1	no	—
330	C₂²	—	68	1	no	—
331	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	17	2	no	—
338	C₄	—	17	2	no	—
339	C₄	—	34	1	no	—
340	C₄	—	68	1	no	—
341	C₄	—	68	1	no	—
342	C₄	—	68	1	no	—
343	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
344	C₄	#41×D₁₇	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	34	2	no	—
347	C₂	—	68	2	no	—
348	C₂	C₂²⋊Q₈×D₁₇	1	1	yes	characteristic
349	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
350	C₂	#11763606	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#478	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₄×D₁₇	C₂²
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3077
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #11763365	C₂	semidirect
3. #11763110	C₂	semidirect
4. #11763109	C₂	semidirect
5. #11763106	C₂	semidirect
6. #11763098	C₂	semidirect
7. #11763097	C₂	semidirect
8. #11763088	C₂	semidirect
9. #11763087	C₂	semidirect
10. #11762850	C₂	semidirect
11. #11762848	C₂	semidirect
12. #11762831	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
18. C₁₇×#125	C₂²	semidirect
30. #10587574	C₂²	semidirect
31. #10587574	C₂²	semidirect
38. #10587556	C₂²	semidirect
39. C₁₇×#107	C₂²	semidirect
40. C₁₇×#106	C₂²	semidirect
41. C₁₇×C₄⋊C₈	C₂²	semidirect
48. #10587489	C₂²	semidirect
49. #10587488	C₂²	semidirect
50. #10587485	C₂²	semidirect
51. #10587485	C₂²	semidirect
54. #10587479	C₂²	semidirect
55. #10587479	C₂²	semidirect
56. #10587478	C₂²	semidirect
57. #10587478	C₂²	semidirect
58. C₄⋊#3622	C₂²	semidirect
65. D₄×D₁₇	Q₈	semidirect
175. #478	D₁₇	semidirect
198. #125	C₂×D₁₇	semidirect
217. #107	C₂×D₁₇	semidirect
221. #106	C₂×D₁₇	semidirect
225. C₄⋊C₈	C₂×D₁₇	semidirect
234. C₁₇	#3077	semidirect

Maximal subgroups

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

subgroup	index
#11763365	2
C₄×D₄×D₁₇	2
#11763110	2
#11763109	2
#11763106	2
#11763098	2
#11763097	2
#11763088	2
#11763087	2
#11762850	2
#11762848	2
#11762831	2
#11762812	2
C₁₇×#478	2
#3077	17