Subgroups — #14537685

Order 2176.

456 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763607	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587587	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762929	C₂	1	4	yes	maximal, complemented
4	#11762924	C₂	1	4	yes	maximal, complemented
5	#11762857	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762826	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762825	C₂	1	1	yes	maximal, characteristic, complemented
8	#11762764	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×#63252	C₂²	1	1	yes	characteristic
12	C₁₇×#131	—	2	1	no	—
13	C₁₇×#131	—	2	1	no	—
14	C₁₇×#131	C₂²	1	2	yes	complemented
15	C₁₇×#131	C₂²	1	2	yes	complemented
16	C₂×M₁₆×C₁₇	—	2	1	no	—
17	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	—	2	1	no	—
19	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	#10587637	C₄	1	2	yes	complemented
21	#10587636	C₂²	1	4	yes	—
22	#10587635	C₄	1	2	yes	complemented
23	#10587628	C₄	1	2	yes	complemented
24	#10587618	C₄	1	2	yes	complemented
25	#10587612	C₂²	1	1	yes	characteristic
26	C₂×#63223	C₂²	1	1	yes	characteristic
27	C₂×#63223	C₂²	1	1	yes	characteristic
28	#10587587	—	2	1	no	—
29	#10587587	—	2	1	no	—
30	#10587587	C₂²	1	2	yes	complemented
31	#10587587	C₂²	1	2	yes	complemented
32	C₂×#63218	—	2	1	no	—
33	C₂×C₈×D₁₇	—	2	1	no	—
34	#10587508	C₂²	1	4	yes	complemented
35	#10587496	C₂²	1	4	yes	complemented
36	#10587493	C₂²	1	4	yes	complemented
37	#10587486	C₂²	1	2	yes	complemented
38	C₈×Dic₁₇	C₂²	1	2	yes	complemented
39	#41×C₁₇	—	2	1	no	—
40	#41×C₁₇	—	2	1	no	—
41	#41×C₁₇	C₂×C₄	1	2	yes	—
42	#41×C₁₇	C₂³	1	2	yes	—
43	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄×C₁₇	—	2	1	no	—
45	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₇	—	2	1	no	—
49	#63252	—	2	1	no	—
50	#63252	—	2	1	no	—
51	#63252	C₂×C₄	1	2	yes	—
52	#63252	C₂³	1	2	yes	—
53	C₂×D₆₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₇	—	2	1	no	—
55	C₂×Dic₃₄	C₂×C₄	1	1	yes	characteristic
56	M₁₆×C₁₇	—	2	1	no	—
57	M₁₆×C₁₇	—	2	2	no	—
58	M₁₆×C₁₇	—	2	2	no	—
59	M₁₆×C₁₇	C₂³	1	2	yes	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	2	no	—
62	C₂×C₈×C₁₇	—	2	2	no	—
63	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₁₇	—	2	1	no	—
70	#63231	C₂×C₄	1	4	yes	—
71	#63230	C₂×C₄	1	4	yes	—
72	#63229	C₂×C₄	1	4	yes	—
73	#63228	C₂×C₄	1	4	yes	—
74	C₄⋊Dic₁₇	—	2	1	no	—
75	C₄×Dic₁₇	C₂³	1	2	yes	—
76	#63223	—	2	1	no	—
77	#63223	C₂×C₄	1	2	yes	—
78	#63223	C₂×C₄	1	2	yes	—
79	#63223	C₂³	1	2	yes	—
80	C₂×#3622	—	2	1	no	—
81	C₂×#3622	C₂³	1	2	yes	—
82	#63218	—	2	2	no	—
83	#63218	—	2	2	no	—
84	C₈×D₁₇	—	2	2	no	—
85	C₈×D₁₇	—	2	2	no	—
86	C₂³×C₁₇	—	2	1	no	—
87	C₂³×C₁₇	#41	1	1	yes	characteristic
88	C₂²×D₁₇	—	2	1	no	—
89	Q₈×C₁₇	—	2	1	no	—
90	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
91	D₄×C₁₇	—	2	2	no	—
92	D₄×C₁₇	—	2	2	no	—
93	D₄×C₁₇	—	2	1	no	—
94	D₄×C₁₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	2	no	—
97	C₂×C₄×C₁₇	—	2	2	no	—
98	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₇	—	2	2	no	—
105	C₂²⋊D₁₇	—	2	2	no	—
106	C₂×Dic₁₇	—	2	1	no	—
107	C₂×Dic₁₇	—	2	2	no	—
108	D₆₈	—	2	1	no	—
109	D₆₈	C₂²×C₄	1	2	yes	—
110	C₄×D₁₇	—	2	2	no	—
111	C₄×D₁₇	—	2	2	no	—
112	Dic₃₄	—	2	1	no	—
113	Dic₃₄	C₂²×C₄	1	2	yes	—
114	C₈×C₁₇	—	2	2	no	—
115	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
118	C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
119	#3622	—	2	2	no	—
120	#3622	C₂²×C₄	1	4	yes	—
121	#2955	—	17	1	no	maximal, Sylow
122	C₂²×C₁₇	—	2	2	no	—
123	C₂²×C₁₇	—	4	1	no	—
124	C₂²×C₁₇	—	4	2	no	—
125	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
126	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
127	C₂²×C₁₇	C₂×#41	1	1	yes	characteristic
128	C₂×D₁₇	—	2	2	no	—
129	C₂×D₁₇	—	4	2	no	—
130	C₄×C₁₇	—	2	2	no	—
131	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
133	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
134	C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
135	Dic₁₇	—	2	2	no	—
136	Dic₁₇	—	4	2	no	—
137	#575	—	17	1	no	—
138	C₂×#131	—	17	1	no	—
139	C₂×#131	D₁₇	1	1	yes	characteristic, complemented
140	#444	—	17	4	no	—
141	#443	—	17	4	no	—
142	#430	—	17	1	no	—
143	#421	—	17	1	no	—
144	#421	—	17	1	no	—
145	#405	—	17	1	no	—
146	C₂×C₁₇	—	2	1	no	—
147	C₂×C₁₇	—	2	1	no	—
148	C₂×C₁₇	—	2	1	no	—
149	C₂×C₁₇	—	4	2	no	—
150	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
151	D₁₇	—	4	2	no	—
152	C₂×#41	—	17	1	no	—
153	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
154	#137	—	17	2	no	—
155	#137	—	17	2	no	—
156	AGL(1,ℤ/8)	—	17	2	no	—
157	AGL(1,ℤ/8)	—	17	2	no	—
158	#135	—	17	4	no	—
159	#131	—	2	1	no	—
160	#131	—	2	1	no	—
161	#131	—	17	2	no	—
162	#131	—	17	2	no	—
163	#131	—	34	1	no	—
164	#131	—	34	1	no	—
165	#131	C₂×D₁₇	1	2	yes	complemented
166	#131	C₂×D₁₇	1	2	yes	complemented
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	17	1	no	—
169	C₂×M₁₆	—	17	1	no	—
170	C₂×M₁₆	—	34	1	no	—
171	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	34	1	no	—
174	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
175	#117	—	17	1	no	—
176	#108	—	17	4	no	—
177	#104	—	17	4	no	—
178	#104	—	17	4	no	—
179	#97	—	17	2	no	—
180	C₄×C₈	—	17	2	no	—
181	C₁₇	#2955	1	1	yes	characteristic, complemented, Sylow
182	#41	—	2	1	no	—
183	#41	—	2	1	no	—
184	#41	—	17	2	no	—
185	#41	—	17	2	no	—
186	#41	—	34	1	no	—
187	#41	—	34	1	no	—
188	#41	C₄×D₁₇	1	2	yes	—
189	#41	C₂²×D₁₇	1	2	yes	—
190	C₂×Q₈	—	17	1	no	—
191	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	17	1	no	—
194	C₂×D₄	—	34	1	no	—
195	C₂×D₄	C₄×D₁₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	34	1	no	—
198	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
199	Q₁₆	—	17	4	no	—
200	SD₁₆	—	17	4	no	—
201	SD₁₆	—	17	4	no	—
202	D₈	—	17	4	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	17	2	no	—
207	M₁₆	—	17	2	no	—
208	M₁₆	—	17	2	no	—
209	M₁₆	—	34	1	no	—
210	M₁₆	—	34	2	no	—
211	M₁₆	—	34	2	no	—
212	M₁₆	C₂²×D₁₇	1	2	yes	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	17	2	no	—
217	C₂×C₈	—	34	1	no	—
218	C₂×C₈	—	34	2	no	—
219	C₂×C₈	—	34	2	no	—
220	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
222	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
223	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
226	C₄⋊C₄	—	34	1	no	—
227	C₂²⋊C₄	—	34	1	no	—
228	C₄²	—	17	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	34	1	no	—
231	C₂³	#63252	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	17	2	no	—
234	Q₈	—	34	1	no	—
235	Q₈	C₂×C₄×D₁₇	1	2	yes	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	17	2	no	—
240	D₄	—	34	2	no	—
241	D₄	—	34	2	no	—
242	D₄	—	34	1	no	—
243	D₄	C₂×C₄×D₁₇	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	34	1	no	—
248	C₂×C₄	—	34	2	no	—
249	C₂×C₄	—	34	2	no	—
250	C₂×C₄	—	34	2	no	—
251	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
253	C₂×C₄	#63252	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
256	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	17	4	no	—
259	C₈	—	34	2	no	—
260	C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
261	C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
262	C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
263	C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	34	2	no	—
268	C₂²	—	68	2	no	—
269	C₂²	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
270	C₂²	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
271	C₂²	C₂×#63252	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	34	2	no	—
274	C₄	—	68	2	no	—
275	C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
276	C₄	C₄×C₂²⋊D₁₇	1	1	yes	characteristic
277	C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	4	2	no	—
283	C₂	—	68	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₁₇
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂×C₁₇×#131	C₂
socle	C₂×C₁₇	C₂×C₄×D₄
prime radical	#11763607	C₂
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂×#131	D₁₇
hypocenter	C₁₇	#2955
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#10587587	C₂	semidirect
3. #11762929	C₂	semidirect
4. #11762924	C₂	semidirect
5. #11762857	C₂	semidirect
6. #11762826	C₂	semidirect
7. #11762825	C₂	semidirect
8. #11762764	C₂	semidirect
9. C₂×C₁₇×#131	C₂	semidirect
14. C₁₇×#131	C₂²	semidirect
15. C₁₇×#131	C₂²	semidirect
20. #10587637	C₄	semidirect
22. #10587635	C₄	semidirect
23. #10587628	C₄	semidirect
24. #10587618	C₄	semidirect
30. #10587587	C₂²	semidirect
31. #10587587	C₂²	semidirect
34. #10587508	C₂²	semidirect
35. #10587496	C₂²	semidirect
36. #10587493	C₂²	semidirect
37. #10587486	C₂²	semidirect
38. C₈×Dic₁₇	C₂²	semidirect
139. C₂×#131	D₁₇	semidirect
165. #131	C₂×D₁₇	semidirect
166. #131	C₂×D₁₇	semidirect
181. C₁₇	#2955	semidirect

Maximal subgroups

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

subgroup	index
#11763607	2
C₂×#10587587	2
#11762929	2
#11762924	2
#11762857	2
#11762826	2
#11762825	2
#11762764	2
C₂×C₁₇×#131	2
#2955	17