Subgroups — #14545559

Order 2176.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763889	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763888	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763855	C₂	1	2	yes	maximal, complemented
4	#11763854	C₂	1	2	yes	maximal, complemented
5	#11763853	C₂	1	2	yes	maximal, complemented
6	#11763852	C₂	1	2	yes	maximal, complemented
7	#11763824	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587550	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₇⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#595	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763608	C₂	1	1	yes	maximal, characteristic
12	#41×D₁₇	C₂²	1	1	yes	characteristic
13	C₂×#63256	C₂²	1	1	yes	characteristic
14	#10587680	—	2	1	no	—
15	#10587680	—	2	1	no	—
16	#10587680	C₂²	1	2	yes	complemented
17	#10587680	C₂²	1	2	yes	complemented
18	#10587680	C₂²	1	2	yes	complemented
19	#10587678	—	2	2	no	—
20	#10587678	C₂²	1	2	yes	complemented
21	C₂×C₄×#595	C₂²	1	1	yes	characteristic
22	D₁₇⋊M₁₆	—	2	2	no	—
23	D₁₇⋊M₁₆	—	2	1	no	—
24	D₁₇⋊M₁₆	C₂²	1	1	yes	characteristic
25	D₁₇⋊M₁₆	C₂²	1	1	yes	characteristic
26	C₂×#63243	—	2	1	no	—
27	C₂×#63243	C₂²	1	1	yes	characteristic
28	C₁₇×#137	C₄	1	1	yes	characteristic, complemented
29	#10587635	C₄	1	1	yes	characteristic, complemented
30	#10587628	C₄	1	1	yes	characteristic, complemented
31	#10587605	C₂²	1	2	yes	—
32	#10587604	C₄	1	2	yes	complemented
33	#10587602	C₂²	1	2	yes	—
34	#10587600	C₄	1	2	yes	complemented
35	#10587594	C₄	1	1	yes	characteristic, complemented
36	M₁₆×D₁₇	C₂²	1	1	yes	characteristic
37	#10587550	C₂²	1	1	yes	characteristic, complemented
38	#10587550	C₂²	1	1	yes	characteristic, complemented
39	#10587550	C₂²	1	2	yes	complemented
40	#10587550	C₂²	1	2	yes	complemented
41	#10587548	C₂²	1	1	yes	characteristic, complemented
42	#10587548	C₂²	1	1	yes	characteristic, complemented
43	#10587536	C₂²	1	2	yes	complemented
44	#10587535	C₂²	1	2	yes	complemented
45	#10587532	C₂²	1	2	yes	complemented
46	C₈×#595	C₂²	1	2	yes	complemented
47	C₂²×#595	—	2	1	no	—
48	#41×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
50	#63256	—	2	1	no	—
51	#63256	—	2	1	no	—
52	#63256	C₂×C₄	1	2	yes	—
53	#63256	C₂³	1	2	yes	—
54	#63256	C₂³	1	1	yes	characteristic
55	Q₈×D₁₇	C₂×C₄	1	1	yes	characteristic
56	#63254	—	2	1	no	—
57	#63254	C₂³	1	1	yes	characteristic
58	D₄×D₁₇	—	2	1	no	—
59	D₄×D₁₇	C₂×C₄	1	1	yes	characteristic
60	#63252	—	2	1	no	—
61	#63252	C₂×C₄	1	1	yes	characteristic
62	C₂×D₆₈	—	2	1	no	—
63	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
67	#63247	—	2	1	no	—
68	#63247	—	2	2	no	—
69	#63247	D₄	1	2	yes	—
70	C₂×#3624	—	2	1	no	—
71	C₂×#3624	—	2	2	no	—
72	C₄×#595	—	2	1	no	—
73	C₄×#595	C₂³	1	1	yes	characteristic
74	C₄×#595	C₂³	1	1	yes	characteristic
75	#63243	—	2	1	no	—
76	#63243	—	2	2	no	—
77	#63243	—	2	2	no	—
78	#63243	—	2	2	no	—
79	#63243	D₄	1	1	yes	characteristic
80	#63243	D₄	1	1	yes	characteristic
81	#63243	C₂³	1	2	yes	—
82	#63243	C₂³	1	1	yes	characteristic
83	#63243	C₂³	1	1	yes	characteristic
84	#63242	—	2	1	no	—
85	#63242	—	2	2	no	—
86	#63242	—	2	2	no	—
87	#63242	—	2	2	no	—
88	#63242	C₂³	1	2	yes	—
89	Q₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
90	SD₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
91	M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
92	#63231	C₂×C₄	1	2	yes	complemented
93	#63230	C₂×C₄	1	2	yes	complemented
94	#63230	C₂×C₄	1	2	yes	complemented
95	#63228	C₂×C₄	1	2	yes	complemented
96	#63223	C₂×C₄	1	1	yes	characteristic
97	D₁₃₆	C₂×C₄	1	2	yes	complemented
98	#63219	C₂×C₄	1	2	yes	complemented
99	#63218	C₂×C₄	1	2	yes	—
100	C₈×D₁₇	C₂³	1	2	yes	—
101	C₂²×D₁₇	—	2	1	no	—
102	C₂²×D₁₇	—	2	1	no	—
103	C₂²×D₁₇	#41	1	1	yes	characteristic
104	C₂×#595	—	2	1	no	—
105	C₂×#595	—	2	1	no	—
106	C₂×#595	—	4	2	no	—
107	Q₈×C₁₇	—	2	1	no	—
108	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
109	Q₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	D₄×C₁₇	—	2	1	no	—
111	D₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²⋊D₁₇	—	2	1	no	—
116	C₂²⋊D₁₇	—	2	1	no	—
117	C₂²⋊D₁₇	—	2	1	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	D₆₈	—	2	2	no	—
121	D₆₈	—	2	2	no	—
122	D₆₈	—	2	1	no	—
123	D₆₈	—	2	1	no	—
124	D₆₈	C₂²×C₄	1	2	yes	—
125	D₆₈	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₁₇	—	2	1	no	—
127	C₄×D₁₇	—	2	1	no	—
128	C₄×D₁₇	—	2	2	no	—
129	C₄×D₁₇	—	2	1	no	—
130	C₄×D₁₇	—	2	2	no	—
131	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
133	C₄×D₁₇	#41	1	1	yes	characteristic
134	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
135	Dic₃₄	—	2	1	no	—
136	Dic₃₄	C₂²×C₄	1	1	yes	characteristic
137	#3624	—	2	1	no	—
138	#3624	—	2	1	no	—
139	#3624	—	2	2	no	—
140	#3624	C₂×D₄	1	2	yes	—
141	#3624	C₂×D₄	1	2	yes	—
142	C₈×C₁₇	C₂²×C₄	1	2	yes	—
143	#3622	C₂²×C₄	1	2	yes	—
144	#2956	—	17	1	no	maximal, Sylow
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
147	C₂×D₁₇	—	2	2	no	—
148	C₂×D₁₇	—	2	1	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₂×D₁₇	—	8	1	no	—
151	C₂×D₁₇	—	8	1	no	—
152	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₇	C₂×#41	1	1	yes	characteristic
154	#595	—	2	2	no	—
155	#595	—	4	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	2	no	—
158	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
159	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
160	Dic₁₇	—	2	1	no	—
161	Dic₁₇	C₄×D₄	1	1	yes	characteristic
162	Dic₁₇	C₂²×D₄	1	1	yes	characteristic
163	#575	—	17	1	no	—
164	#568	—	17	1	no	—
165	#568	—	17	1	no	—
166	#444	—	17	2	no	—
167	#444	—	17	2	no	—
168	#443	—	17	2	no	—
169	#443	—	17	2	no	—
170	#430	—	17	1	no	—
171	C₂×#104	—	17	1	no	—
172	C₂×#104	—	17	1	no	—
173	C₄×M₁₆	—	17	1	no	—
174	C₂×C₁₇	—	2	1	no	—
175	C₂×C₁₇	—	4	1	no	—
176	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
177	D₁₇	—	2	1	no	—
178	D₁₇	—	2	1	no	—
179	D₁₇	—	4	2	no	—
180	D₁₇	—	4	1	no	—
181	C₂×#41	—	17	1	no	—
182	C₂×#41	—	17	1	no	—
183	#137	—	17	1	no	—
184	#137	—	17	2	no	—
185	#137	#595	1	1	yes	characteristic, complemented
186	AGL(1,ℤ/8)	—	17	1	no	—
187	AGL(1,ℤ/8)	—	17	2	no	—
188	AGL(1,ℤ/8)	—	17	1	no	—
189	#135	—	17	2	no	—
190	#135	—	17	2	no	—
191	#131	—	17	2	no	—
192	#131	—	17	2	no	—
193	#131	—	17	2	no	—
194	#131	—	17	2	no	—
195	#131	—	34	2	no	—
196	#131	—	34	1	no	—
197	#131	—	34	1	no	—
198	C₂×M₁₆	—	17	1	no	—
199	C₂×M₁₆	—	17	1	no	—
200	C₂×M₁₆	—	17	1	no	—
201	C₂×M₁₆	—	17	1	no	—
202	C₂×M₁₆	—	34	1	no	—
203	C₂×M₁₆	—	34	1	no	—
204	C₂×M₁₆	—	34	2	no	—
205	C₂×C₄²	—	17	1	no	—
206	#108	—	17	2	no	—
207	#108	—	17	2	no	—
208	#104	—	17	1	no	—
209	#104	—	17	1	no	—
210	#104	—	17	1	no	—
211	#104	—	17	2	no	—
212	#104	—	17	1	no	—
213	#104	—	17	2	no	—
214	#97	—	17	2	no	—
215	C₄×C₈	—	17	2	no	—
216	C₁₇	#2956	1	1	yes	characteristic, complemented, Sylow
217	#41	—	17	2	no	—
218	#41	—	17	1	no	—
219	#41	—	17	2	no	—
220	#41	—	17	1	no	—
221	#41	—	17	1	no	—
222	#41	—	34	1	no	—
223	#41	—	34	1	no	—
224	#41	—	34	1	no	—
225	#41	—	34	1	no	—
226	#41	C₂×#595	1	1	yes	characteristic
227	C₂×Q₈	—	17	1	no	—
228	C₂×Q₈	C₂×#595	1	1	yes	characteristic
229	C₂×D₄	—	17	1	no	—
230	C₂×D₄	—	17	1	no	—
231	C₂×D₄	—	34	1	no	—
232	C₂×D₄	—	34	1	no	—
233	C₂²×C₄	—	17	1	no	—
234	C₂²×C₄	—	34	1	no	—
235	C₂²×C₄	—	34	1	no	—
236	C₂²×C₄	—	34	1	no	—
237	Q₁₆	—	17	2	no	—
238	Q₁₆	C₂×#595	1	2	yes	complemented
239	SD₁₆	—	17	2	no	—
240	SD₁₆	—	17	2	no	—
241	SD₁₆	—	17	2	no	—
242	SD₁₆	C₂×#595	1	2	yes	complemented
243	D₈	—	17	2	no	—
244	D₈	—	17	2	no	—
245	M₁₆	—	17	1	no	—
246	M₁₆	—	17	1	no	—
247	M₁₆	—	17	2	no	—
248	M₁₆	—	17	1	no	—
249	M₁₆	—	17	2	no	—
250	M₁₆	—	17	2	no	—
251	M₁₆	—	17	1	no	—
252	M₁₆	—	17	1	no	—
253	M₁₆	—	34	2	no	—
254	M₁₆	—	34	2	no	—
255	M₁₆	—	34	2	no	—
256	M₁₆	—	34	2	no	—
257	M₁₆	—	34	1	no	—
258	M₁₆	—	34	1	no	—
259	M₁₆	C₂×#595	1	1	yes	characteristic
260	C₂×C₈	—	17	2	no	—
261	C₂×C₈	—	17	2	no	—
262	C₂×C₈	—	34	2	no	—
263	C₂×C₈	—	34	2	no	—
264	C₂×C₈	—	34	2	no	—
265	C₂×C₈	—	34	2	no	—
266	C₂×C₈	—	34	1	no	—
267	C₂×C₈	—	34	1	no	—
268	C₄²	—	17	1	no	—
269	C₄²	—	17	1	no	—
270	C₄²	—	34	1	no	—
271	C₂³	—	17	1	no	—
272	C₂³	—	34	1	no	—
273	C₂³	—	34	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	17	1	no	—
276	Q₈	—	34	1	no	—
277	Q₈	C₂²×#595	1	1	yes	characteristic
278	Q₈	C₂²×#595	1	2	yes	—
279	D₄	—	2	1	no	—
280	D₄	—	17	2	no	—
281	D₄	—	17	1	no	—
282	D₄	—	34	1	no	—
283	D₄	—	34	2	no	—
284	D₄	—	34	2	no	—
285	D₄	—	34	1	no	—
286	D₄	—	34	1	no	—
287	D₄	—	34	1	no	—
288	D₄	—	34	1	no	—
289	D₄	C₂²×#595	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	17	1	no	—
293	C₂×C₄	—	17	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	2	no	—
299	C₂×C₄	—	34	2	no	—
300	C₂×C₄	—	34	1	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	34	1	no	—
305	C₂×C₄	—	68	2	no	—
306	C₂×C₄	C₂²×#595	1	1	yes	characteristic
307	C₈	—	17	2	no	—
308	C₈	—	17	2	no	—
309	C₈	—	17	2	no	—
310	C₈	—	34	1	no	—
311	C₈	—	34	1	no	—
312	C₈	—	34	2	no	—
313	C₈	C₂²×#595	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	17	1	no	—
316	C₂²	—	17	1	no	—
317	C₂²	—	34	2	no	—
318	C₂²	—	34	1	no	—
319	C₂²	—	68	1	no	—
320	C₂²	—	136	1	no	—
321	C₂²	—	136	1	no	—
322	C₂²	D₄×#595	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	17	1	no	—
326	C₄	—	17	1	no	—
327	C₄	—	34	2	no	—
328	C₄	—	34	1	no	—
329	C₄	—	68	1	no	—
330	C₄	D₄×#595	1	1	yes	characteristic
331	C₄	C₂³×#595	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	1	no	—
334	C₂	—	34	1	no	—
335	C₂	—	34	1	no	—
336	C₂	—	68	2	no	—
337	C₂	—	68	1	no	—
338	C₂	C₂×D₄×#595	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#595
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₄	C₂³×#595
Fitting	C₁₇×#137	C₄
socle	C₂×C₁₇	C₂×C₄×D₄
prime radical	#11763608	C₂
cosocle	C₄×D₁₇	C₂⁴
hypercenter	#137	#595
hypocenter	C₁₇	#2956
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #11763889	C₂	semidirect
2. #11763888	C₂	semidirect
3. #11763855	C₂	semidirect
4. #11763854	C₂	semidirect
5. #11763853	C₂	semidirect
6. #11763852	C₂	semidirect
7. #11763824	C₂	semidirect
8. C₂×#10587550	C₂	semidirect
9. D₁₇⋊#104	C₂	semidirect
10. M₁₆×#595	C₂	semidirect
16. #10587680	C₂²	semidirect
17. #10587680	C₂²	semidirect
18. #10587680	C₂²	semidirect
20. #10587678	C₂²	semidirect
28. C₁₇×#137	C₄	semidirect
29. #10587635	C₄	semidirect
30. #10587628	C₄	semidirect
32. #10587604	C₄	semidirect
34. #10587600	C₄	semidirect
35. #10587594	C₄	semidirect
37. #10587550	C₂²	semidirect
38. #10587550	C₂²	semidirect
39. #10587550	C₂²	semidirect
40. #10587550	C₂²	semidirect
41. #10587548	C₂²	semidirect
42. #10587548	C₂²	semidirect
43. #10587536	C₂²	semidirect
44. #10587535	C₂²	semidirect
45. #10587532	C₂²	semidirect
46. C₈×#595	C₂²	semidirect
89. Q₁₆×C₁₇	C₂×C₄	semidirect
90. SD₁₆×C₁₇	C₂×C₄	semidirect
92. #63231	C₂×C₄	semidirect
93. #63230	C₂×C₄	semidirect
94. #63230	C₂×C₄	semidirect
95. #63228	C₂×C₄	semidirect
97. D₁₃₆	C₂×C₄	semidirect
98. #63219	C₂×C₄	semidirect
185. #137	#595	semidirect
216. C₁₇	#2956	semidirect
238. Q₁₆	C₂×#595	semidirect
242. SD₁₆	C₂×#595	semidirect

Maximal subgroups

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

subgroup	index
#11763889	2
#11763888	2
#11763855	2
#11763854	2
#11763853	2
#11763852	2
#11763824	2
C₂×#10587550	2
D₁₇⋊#104	2
M₁₆×#595	2
#11763608	2
#2956	17