Subgroups — #14541363

Order 2176.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763530	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763459	C₂	1	1	yes	maximal, characteristic
3	#11763213	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763143	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763142	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763132	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763130	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763087	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763052	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762885	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762851	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#468	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63254	—	2	1	no	—
17	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₇×#121	C₂²	1	1	yes	characteristic
21	#10587632	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
23	#10587624	C₂²	1	1	yes	characteristic
24	#10587621	—	2	1	no	—
25	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
27	#10587612	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
29	C₂×#63218	C₂²	1	1	yes	characteristic
30	#10587584	—	2	1	no	—
31	#10587578	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	#10587574	C₂²	1	1	yes	characteristic
34	#10587574	C₂²	1	1	yes	characteristic
35	#10587573	—	2	1	no	—
36	#10587571	—	2	1	no	—
37	#10587569	—	2	1	no	—
38	#10587565	—	2	1	no	—
39	#10587564	—	2	1	no	—
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#103	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	C₂²⋊#3622	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587488	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
65	C₂²×Dic₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	#63254	—	4	4	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
70	C₂×Dic₃₄	—	2	1	no	—
71	C₂×Dic₃₄	—	2	1	no	—
72	SD₁₆×C₁₇	—	2	2	no	—
73	M₁₆×C₁₇	—	2	2	no	—
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	D₄	1	1	yes	characteristic
88	C₄⋊Dic₁₇	—	2	1	no	—
89	C₄⋊Dic₁₇	—	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	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	D₄	1	1	yes	characteristic
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
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	C₂³×C₁₇	—	2	1	no	—
107	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	D₄×C₁₇	—	2	1	no	—
112	D₄×C₁₇	—	2	1	no	—
113	D₄×C₁₇	—	4	2	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
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	—	2	1	no	—
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₇	—	2	2	no	—
122	C₂²⋊D₁₇	—	4	2	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	4	2	no	—
128	C₂×Dic₁₇	—	4	2	no	—
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	C₄×D₁₇	—	4	2	no	—
135	Dic₃₄	—	4	2	no	—
136	Dic₃₄	—	4	2	no	—
137	C₈×C₁₇	—	2	1	no	—
138	C₈×C₁₇	#41	1	2	yes	complemented
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3353	—	17	1	no	maximal, Sylow
142	C₂²×C₁₇	—	2	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	4	1	no	—
146	C₂²×C₁₇	—	4	1	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
157	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₁₇	—	2	1	no	—
159	Dic₁₇	—	2	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	—	4	1	no	—
164	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
165	#556	—	17	1	no	—
166	#547	—	17	1	no	—
167	#500	—	17	1	no	—
168	#484	—	17	1	no	—
169	#484	—	17	1	no	—
170	#482	—	17	1	no	—
171	#482	—	17	1	no	—
172	#475	—	17	1	no	—
173	#474	—	17	1	no	—
174	#468	—	17	1	no	—
175	#468	D₁₇	1	1	yes	characteristic, complemented
176	C₄×SD₁₆	—	17	1	no	—
177	#436	—	17	1	no	—
178	#428	—	17	1	no	—
179	#428	—	17	1	no	—
180	C₂×C₁₇	—	4	1	no	—
181	C₂×C₁₇	—	8	1	no	—
182	C₂×C₁₇	#547	1	1	yes	characteristic
183	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×C₁₇	#578	1	1	yes	characteristic
185	D₁₇	—	4	1	no	—
186	C₂×#41	—	34	1	no	—
187	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
188	C₂×M₁₆	—	17	1	no	—
189	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
190	C₄⋊Q₈	—	17	1	no	—
191	C₄⋊Q₈	—	17	1	no	—
192	#126	—	34	1	no	—
193	#126	—	34	1	no	—
194	#124	—	34	1	no	—
195	#123	—	34	1	no	—
196	#123	—	34	1	no	—
197	C₂²⋊Q₈	—	17	1	no	—
198	C₂²⋊Q₈	—	34	1	no	—
199	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
200	#121	—	17	1	no	—
201	#121	C₂×D₁₇	1	1	yes	characteristic
202	C₄×Q₈	—	17	1	no	—
203	C₄×Q₈	—	34	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	17	1	no	—
206	#117	—	17	1	no	—
207	#117	—	17	1	no	—
208	C₂×C₄⋊C₄	—	17	1	no	—
209	C₂×C₄⋊C₄	—	17	1	no	—
210	#107	—	17	1	no	—
211	#107	—	17	1	no	—
212	#107	—	17	1	no	—
213	#107	—	17	1	no	—
214	#107	—	17	1	no	—
215	#107	C₂×D₁₇	1	1	yes	characteristic
216	#106	—	17	1	no	—
217	#106	—	17	1	no	—
218	#106	—	17	1	no	—
219	C₄⋊C₈	—	17	1	no	—
220	#103	—	17	1	no	—
221	#103	—	17	1	no	—
222	#103	C₂×D₁₇	1	1	yes	characteristic
223	#102	—	17	1	no	—
224	#102	—	17	1	no	—
225	#102	C₂×D₁₇	1	1	yes	characteristic
226	C₂²⋊C₈	—	17	1	no	—
227	C₂²⋊C₈	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₁₇	#3353	1	1	yes	characteristic, complemented, Sylow
230	#41	—	68	4	no	—
231	C₂×Q₈	—	34	1	no	—
232	C₂×Q₈	—	34	1	no	—
233	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	17	1	no	—
236	C₂×D₄	—	34	1	no	—
237	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
238	C₂²×C₄	—	17	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	C₂²×C₄	—	34	1	no	—
241	C₂²×C₄	—	34	1	no	—
242	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
243	SD₁₆	—	2	2	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	34	2	no	—
246	C₂×C₈	—	17	1	no	—
247	C₂×C₈	—	17	1	no	—
248	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	—	17	1	no	—
253	C₄⋊C₄	—	17	1	no	—
254	C₄⋊C₄	—	17	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	17	1	no	—
267	C₂²⋊C₄	—	17	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₄²	—	17	1	no	—
274	C₄²	—	34	1	no	—
275	C₄²	—	34	1	no	—
276	C₄²	—	34	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	17	1	no	—
279	C₂³	D₄×D₁₇	1	1	yes	characteristic
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	68	2	no	—
283	Q₈	—	68	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	34	2	no	—
288	D₄	—	68	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	17	1	no	—
295	C₂×C₄	—	17	1	no	—
296	C₂×C₄	—	17	1	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	—	34	1	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	—	68	2	no	—
304	C₂×C₄	—	68	2	no	—
305	C₂×C₄	—	68	2	no	—
306	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	34	1	no	—
310	C₈	—	34	1	no	—
311	C₈	#63254	1	2	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	34	1	no	—
319	C₂²	—	34	1	no	—
320	C₂²	—	34	1	no	—
321	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	17	2	no	—
327	C₄	—	34	1	no	—
328	C₄	—	34	1	no	—
329	C₄	—	68	1	no	—
330	C₄	—	68	1	no	—
331	C₄	—	68	1	no	—
332	C₄	—	68	1	no	—
333	C₄	C₂×#63254	1	1	yes	characteristic
334	C₄	#10587697	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	8	1	no	—
337	C₂	—	68	1	no	—
338	C₂	#11763459	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₁₇	1	1	yes	characteristic
340	C₂	#11763625	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×#468	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763459	C₂
hypercenter	#468	D₁₇
hypocenter	C₁₇	#3353
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #11763530	C₂	semidirect
3. #11763213	C₂	semidirect
4. #11763143	C₂	semidirect
5. #11763142	C₂	semidirect
6. #11763132	C₂	semidirect
7. #11763130	C₂	semidirect
8. #11763092	C₂	semidirect
9. #11763087	C₂	semidirect
10. #11763052	C₂	semidirect
11. SD₁₆×Dic₁₇	C₂	semidirect
12. #11762885	C₂	semidirect
13. #11762851	C₂	semidirect
14. #11762850	C₂	semidirect
15. C₁₇×#468	C₂	semidirect
138. C₈×C₁₇	#41	semidirect
175. #468	D₁₇	semidirect
229. C₁₇	#3353	semidirect
311. C₈	#63254	semidirect

Maximal subgroups

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

subgroup	index
#11763530	2
#11763459	2
#11763213	2
#11763143	2
#11763142	2
#11763132	2
#11763130	2
#11763092	2
#11763087	2
#11763052	2
SD₁₆×Dic₁₇	2
#11762885	2
#11762851	2
#11762850	2
C₁₇×#468	2
#3353	17