Subgroups — #14540571

Order 2176.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763626	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763616	C₂	1	2	yes	maximal, complemented
3	#11763614	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763193	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763084	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763079	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763073	C₂	1	2	yes	maximal, complemented
8	#11762988	C₂	1	2	yes	maximal, complemented
9	#11762983	C₂	1	2	yes	maximal, complemented
10	#11762927	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#10587697	C₂²	1	2	yes	complemented
13	#10587696	C₂²	1	2	yes	—
14	C₁₇×#135	C₂²	1	1	yes	characteristic
15	C₁₇×#131	C₂²	1	1	yes	characteristic
16	C₁₇×#131	C₂²	1	1	yes	characteristic, complemented
17	#10587637	C₂²	1	2	yes	—
18	#10587635	C₂²	1	2	yes	complemented
19	#10587604	—	2	2	no	—
20	#10587602	—	2	2	no	—
21	SD₁₆×D₁₇	—	2	2	no	—
22	#10587597	—	2	2	no	—
23	#10587595	—	2	2	no	—
24	#10587595	C₂²	1	2	yes	complemented
25	#10587594	—	2	2	no	—
26	#10587594	C₂²	1	2	yes	—
27	C₂×Dic₆₈	—	2	1	no	—
28	C₂×Dic₆₈	C₂²	1	1	yes	characteristic, complemented
29	#10587590	—	2	1	no	—
30	#10587590	—	2	1	no	—
31	#10587590	C₂²	1	1	yes	characteristic
32	C₂×D₁₃₆	—	2	1	no	—
33	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
34	C₂×#63219	C₂²	1	2	yes	complemented
35	#10587587	C₂²	1	2	yes	complemented
36	#10587561	C₂²	1	2	yes	complemented
37	C₁₇×#108	C₂²	1	1	yes	characteristic, complemented
38	C₁₇×#104	C₂²	1	1	yes	characteristic
39	C₁₇×#104	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
41	#10587495	C₂²	1	2	yes	complemented
42	#10587494	C₂²	1	2	yes	complemented
43	#10587490	C₂²	1	1	yes	characteristic, complemented
44	#10587476	C₂²	1	2	yes	complemented
45	#41×C₁₇	C₂³	1	1	yes	characteristic
46	#41×C₁₇	C₂³	1	1	yes	characteristic
47	#63256	—	2	2	no	—
48	Q₈×D₁₇	—	2	2	no	—
49	#63254	—	2	2	no	—
50	#63254	—	4	2	no	—
51	D₄×D₁₇	—	2	2	no	—
52	D₄×D₁₇	—	4	2	no	—
53	#63252	—	2	2	no	—
54	#63252	—	2	2	no	—
55	#63252	C₂³	1	2	yes	—
56	C₂×D₆₈	—	2	2	no	—
57	C₂×D₆₈	C₂³	1	2	yes	—
58	C₂×Dic₃₄	—	2	2	no	—
59	C₂×Dic₃₄	C₂³	1	2	yes	—
60	Q₁₆×C₁₇	D₄	1	1	yes	characteristic
61	SD₁₆×C₁₇	D₄	1	1	yes	characteristic
62	SD₁₆×C₁₇	D₄	1	1	yes	characteristic
63	D₈×C₁₇	D₄	1	1	yes	characteristic
64	M₁₆×C₁₇	—	2	1	no	—
65	M₁₆×C₁₇	—	2	1	no	—
66	M₁₆×C₁₇	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₇	—	2	1	no	—
69	C₂×C₈×C₁₇	—	2	1	no	—
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₄²×C₁₇	C₂³	1	1	yes	characteristic
73	#63231	—	2	2	no	—
74	#63230	—	2	2	no	—
75	#63229	—	2	2	no	—
76	#63228	—	2	2	no	—
77	#63227	—	4	2	no	—
78	#63223	C₂³	1	2	yes	—
79	Dic₆₈	—	2	2	no	—
80	Dic₆₈	—	2	1	no	—
81	Dic₆₈	—	2	1	no	—
82	Dic₆₈	D₄	1	1	yes	characteristic
83	D₁₃₆	—	2	2	no	—
84	D₁₃₆	—	2	1	no	—
85	D₁₃₆	—	2	1	no	—
86	D₁₃₆	D₄	1	1	yes	characteristic
87	#63219	—	2	2	no	—
88	#63219	—	2	2	no	—
89	#63219	—	2	2	no	—
90	#63219	—	2	2	no	—
91	#63219	D₄	1	2	yes	—
92	#63218	—	2	2	no	—
93	C₈×D₁₇	—	2	2	no	—
94	C₂²×D₁₇	—	2	2	no	—
95	C₂²×D₁₇	—	4	2	no	—
96	Q₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₁₇	C₂×D₄	1	1	yes	characteristic
98	D₄×C₁₇	—	2	1	no	—
99	D₄×C₁₇	—	2	1	no	—
100	D₄×C₁₇	C₂×D₄	1	1	yes	characteristic
101	D₄×C₁₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₁₇	—	2	2	no	—
107	C₂²⋊D₁₇	—	4	2	no	—
108	C₂²⋊D₁₇	—	4	2	no	—
109	C₂×Dic₁₇	—	2	2	no	—
110	C₂×Dic₁₇	—	4	2	no	—
111	D₆₈	—	2	2	no	—
112	D₆₈	—	2	2	no	—
113	D₆₈	—	4	2	no	—
114	D₆₈	C₂×D₄	1	2	yes	—
115	C₄×D₁₇	—	2	2	no	—
116	C₄×D₁₇	—	4	2	no	—
117	C₄×D₁₇	—	4	2	no	—
118	Dic₃₄	—	2	2	no	—
119	Dic₃₄	—	2	2	no	—
120	Dic₃₄	—	4	2	no	—
121	Dic₃₄	C₂×D₄	1	2	yes	—
122	C₈×C₁₇	—	2	1	no	—
123	C₈×C₁₇	—	2	1	no	—
124	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
128	#3622	—	2	2	no	—
129	#3290	—	17	1	no	maximal, Sylow
130	C₂²×C₁₇	—	2	1	no	—
131	C₂²×C₁₇	—	2	1	no	—
132	C₂²×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×D₁₇	—	2	2	no	—
134	C₂×D₁₇	—	4	2	no	—
135	C₂×D₁₇	—	8	2	no	—
136	C₂×D₁₇	—	16	2	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	4	1	no	—
140	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
142	Dic₁₇	—	2	2	no	—
143	Dic₁₇	—	4	2	no	—
144	#578	—	17	1	no	—
145	#577	—	17	2	no	—
146	#576	—	17	1	no	—
147	#495	—	17	1	no	—
148	#473	—	17	1	no	—
149	#472	—	17	1	no	—
150	#471	—	17	2	no	—
151	#455	—	17	2	no	—
152	#454	—	17	2	no	—
153	#443	—	17	1	no	—
154	#443	D₁₇	1	1	yes	characteristic, complemented
155	C₂×C₁₇	—	2	1	no	—
156	C₂×C₁₇	—	4	1	no	—
157	C₂×C₁₇	—	4	1	no	—
158	C₂×C₁₇	D₄²	1	1	yes	characteristic
159	D₁₇	—	4	2	no	—
160	D₁₇	—	8	2	no	—
161	2₋¹⁺⁴	—	17	2	no	—
162	H(5,𝔽₂)	—	17	2	no	—
163	#137	—	17	2	no	—
164	#137	—	17	2	no	—
165	#137	—	34	2	no	—
166	#137	—	34	2	no	—
167	AGL(1,ℤ/8)	—	17	2	no	—
168	AGL(1,ℤ/8)	—	17	2	no	—
169	AGL(1,ℤ/8)	—	34	2	no	—
170	AGL(1,ℤ/8)	—	34	2	no	—
171	#135	—	17	1	no	—
172	#135	—	34	1	no	—
173	#135	—	34	1	no	—
174	#135	—	34	2	no	—
175	#135	C₂×D₁₇	1	1	yes	characteristic
176	C₂×Q₁₆	—	17	1	no	—
177	C₂×Q₁₆	—	34	1	no	—
178	C₂×SD₁₆	—	17	2	no	—
179	C₂×SD₁₆	—	34	2	no	—
180	C₂×D₈	—	17	1	no	—
181	C₂×D₈	—	34	1	no	—
182	#131	—	17	2	no	—
183	#131	C₂×D₁₇	1	1	yes	characteristic
184	#131	C₂×D₁₇	1	1	yes	characteristic, complemented
185	#124	—	17	2	no	—
186	#108	—	17	1	no	—
187	#108	C₂×D₁₇	1	1	yes	characteristic, complemented
188	#104	—	17	2	no	—
189	#104	C₂×D₁₇	1	1	yes	characteristic
190	#104	C₂×D₁₇	1	1	yes	characteristic, complemented
191	#101	—	17	2	no	—
192	#100	—	17	2	no	—
193	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic, complemented
194	C₁₇	#3290	1	1	yes	characteristic, complemented, Sylow
195	#41	—	17	2	no	—
196	#41	—	34	2	no	—
197	#41	—	34	2	no	—
198	#41	—	34	2	no	—
199	#41	—	34	2	no	—
200	#41	—	68	2	no	—
201	#41	C₂²×D₁₇	1	1	yes	characteristic
202	#41	C₂²×D₁₇	1	1	yes	characteristic
203	C₂×Q₈	—	17	2	no	—
204	C₂×Q₈	—	34	2	no	—
205	C₂×Q₈	—	34	2	no	—
206	C₂×D₄	—	17	2	no	—
207	C₂×D₄	—	34	2	no	—
208	C₂×D₄	—	34	2	no	—
209	C₂×D₄	—	68	2	no	—
210	Q₁₆	—	17	1	no	—
211	Q₁₆	—	34	2	no	—
212	Q₁₆	—	34	2	no	—
213	Q₁₆	—	34	1	no	—
214	Q₁₆	—	34	1	no	—
215	Q₁₆	D₆₈	1	1	yes	characteristic
216	SD₁₆	—	17	2	no	—
217	SD₁₆	—	34	2	no	—
218	SD₁₆	—	34	2	no	—
219	SD₁₆	—	34	2	no	—
220	SD₁₆	—	34	2	no	—
221	SD₁₆	—	34	2	no	—
222	SD₁₆	—	34	2	no	—
223	SD₁₆	D₆₈	1	1	yes	characteristic
224	SD₁₆	D₆₈	1	1	yes	characteristic
225	D₈	—	17	1	no	—
226	D₈	—	34	2	no	—
227	D₈	—	34	2	no	—
228	D₈	—	34	1	no	—
229	D₈	—	34	1	no	—
230	D₈	D₆₈	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	17	2	no	—
234	M₁₆	—	34	2	no	—
235	M₁₆	C₂²×D₁₇	1	1	yes	characteristic
236	M₁₆	C₂²×D₁₇	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	34	2	no	—
240	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
242	C₂²⋊C₄	—	68	2	no	—
243	C₄²	C₂²×D₁₇	1	1	yes	characteristic
244	C₂³	—	34	2	no	—
245	C₂³	—	68	2	no	—
246	Q₈	—	17	2	no	—
247	Q₈	—	34	2	no	—
248	Q₈	—	34	2	no	—
249	Q₈	—	68	2	no	—
250	Q₈	C₂×D₆₈	1	1	yes	characteristic
251	Q₈	C₂×D₆₈	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	17	2	no	—
255	D₄	—	34	2	no	—
256	D₄	—	34	2	no	—
257	D₄	—	34	2	no	—
258	D₄	—	68	2	no	—
259	D₄	—	68	2	no	—
260	D₄	—	68	2	no	—
261	D₄	C₂×D₆₈	1	1	yes	characteristic
262	D₄	C₂×D₆₈	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	34	2	no	—
267	C₂×C₄	—	34	2	no	—
268	C₂×C₄	—	68	2	no	—
269	C₂×C₄	—	68	2	no	—
270	C₂×C₄	—	68	2	no	—
271	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	34	2	no	—
275	C₈	C₂×D₆₈	1	1	yes	characteristic
276	C₈	C₂×D₆₈	1	1	yes	characteristic
277	C₈	D₄×D₁₇	1	1	yes	characteristic
278	C₈	D₄×D₁₇	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	272	2	no	—
282	C₂²	—	34	2	no	—
283	C₂²	—	68	2	no	—
284	C₂²	—	136	2	no	—
285	C₂²	#10587696	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	34	2	no	—
290	C₄	—	68	2	no	—
291	C₄	C₂²×D₆₈	1	1	yes	characteristic
292	C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	68	2	no	—
297	C₂	—	136	2	no	—
298	C₂	D₄×D₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#443	C₂
socle	C₂×C₁₇	D₄²
hypercenter	#443	D₁₇
hypocenter	C₁₇	#3290
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #11763626	C₂	semidirect
2. #11763616	C₂	semidirect
3. #11763614	C₂	semidirect
4. #11763193	C₂	semidirect
5. #11763084	C₂	semidirect
6. #11763079	C₂	semidirect
7. #11763073	C₂	semidirect
8. #11762988	C₂	semidirect
9. #11762983	C₂	semidirect
10. #11762927	C₂	semidirect
11. C₁₇×#443	C₂	semidirect
12. #10587697	C₂²	semidirect
16. C₁₇×#131	C₂²	semidirect
18. #10587635	C₂²	semidirect
24. #10587595	C₂²	semidirect
28. C₂×Dic₆₈	C₂²	semidirect
34. C₂×#63219	C₂²	semidirect
35. #10587587	C₂²	semidirect
36. #10587561	C₂²	semidirect
37. C₁₇×#108	C₂²	semidirect
39. C₁₇×#104	C₂²	semidirect
40. C₄×C₈×C₁₇	C₂²	semidirect
41. #10587495	C₂²	semidirect
42. #10587494	C₂²	semidirect
43. #10587490	C₂²	semidirect
44. #10587476	C₂²	semidirect
154. #443	D₁₇	semidirect
184. #131	C₂×D₁₇	semidirect
187. #108	C₂×D₁₇	semidirect
190. #104	C₂×D₁₇	semidirect
193. C₄×C₈	C₂×D₁₇	semidirect
194. C₁₇	#3290	semidirect

Maximal subgroups

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

subgroup	index
#11763626	2
#11763616	2
#11763614	2
#11763193	2
#11763084	2
#11763079	2
#11763073	2
#11762988	2
#11762983	2
#11762927	2
C₁₇×#443	2
#3290	17