Subgroups — #13131596

Order 1664.

492 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719627	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719623	C₂	1	1	yes	maximal, characteristic
3	C₂×SD₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
4	#11719153	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719138	C₂	1	1	yes	maximal, characteristic
6	#11719106	C₂	1	4	yes	maximal, complemented
7	SD₁₆×#261	C₂	1	4	yes	maximal, complemented
8	#11719092	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719089	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×D₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊#261	—	2	2	no	—
13	Q₈×#261	—	2	2	no	—
14	Q₈×#261	C₂²	1	4	yes	complemented
15	D₄×#261	—	2	2	no	—
16	D₄×#261	C₂²	1	4	yes	—
17	#88237	—	2	2	no	—
18	#88237	C₂²	1	1	yes	characteristic
19	#88237	C₂²	1	1	yes	characteristic
20	#88237	C₂²	1	1	yes	characteristic
21	D₁₃⋊M₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×SD₁₆	C₄	1	1	yes	characteristic, complemented
23	C₂×#6106	C₄	1	1	yes	characteristic, complemented
24	C₂×#6105	C₄	1	1	yes	characteristic, complemented
25	SD₁₆×D₁₃	C₄	1	2	yes	complemented
26	SD₁₆×D₁₃	C₄	1	2	yes	complemented
27	SD₁₆×D₁₃	C₂²	1	2	yes	—
28	SD₁₆×D₁₃	C₂²	1	2	yes	—
29	C₂×#6095	C₄	1	1	yes	characteristic, complemented
30	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
31	#88117	C₂²	1	4	yes	complemented
32	#88115	C₂²	1	4	yes	—
33	#88102	C₂²	1	2	yes	complemented
34	#88101	C₂²	1	2	yes	complemented
35	#88100	C₂²	1	2	yes	complemented
36	C₈×#261	C₂²	1	2	yes	complemented
37	C₂³×D₁₃	—	2	1	no	—
38	C₂²×#261	—	2	4	no	—
39	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₁₃	—	2	1	no	—
42	Q₈×D₁₃	—	2	1	no	—
43	Q₈×D₁₃	—	2	1	no	—
44	Q₈×D₁₃	C₂×C₄	1	2	yes	complemented
45	Q₈×D₁₃	C₂³	1	2	yes	—
46	D₄×D₁₃	—	2	1	no	—
47	D₄×D₁₃	—	2	1	no	—
48	D₄×D₁₃	C₂×C₄	1	2	yes	—
49	D₄×D₁₃	C₂³	1	2	yes	—
50	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃	—	2	1	no	—
52	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₆	—	2	1	no	—
54	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic, complemented
55	C₂²⋊#261	—	2	1	no	—
56	C₂²⋊#261	—	2	1	no	—
57	C₂²⋊#261	—	2	4	no	—
58	C₂²⋊#261	D₄	1	2	yes	—
59	#6122	—	2	1	no	—
60	C₄⋊#261	—	2	4	no	—
61	C₄⋊#261	D₄	1	2	yes	—
62	C₄⋊#261	C₂³	1	2	yes	—
63	C₄⋊#261	C₂³	1	2	yes	—
64	C₄×#261	—	2	4	no	—
65	C₄×#261	—	2	1	no	—
66	C₄×#261	—	2	1	no	—
67	C₄×#261	C₂³	1	2	yes	—
68	#6118	—	2	1	no	—
69	D₁₃⋊C₈	C₂³	1	2	yes	—
70	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
71	C₁₃×SD₁₆	C₂×C₄	1	2	yes	complemented
72	C₂×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic, complemented
73	#6106	C₂×C₄	1	2	yes	complemented
74	#6106	C₂×C₄	1	2	yes	complemented
75	#6105	C₂×C₄	1	2	yes	complemented
76	#6105	C₂×C₄	1	2	yes	complemented
77	C₂×#1055	C₂×C₄	1	1	yes	characteristic, complemented
78	#6095	C₂×C₄	1	2	yes	complemented
79	#6095	C₂×C₄	1	2	yes	complemented
80	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
81	C₈×D₁₃	C₂×C₄	1	1	yes	characteristic, complemented
82	C₈×D₁₃	C₂³	1	1	yes	characteristic
83	C₈×D₁₃	C₂³	1	1	yes	characteristic
84	#2949	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	—	2	1	no	—
86	C₂²×D₁₃	—	2	1	no	—
87	C₂²×D₁₃	—	2	2	no	—
88	C₂²×D₁₃	—	2	2	no	—
89	C₂²×D₁₃	—	8	2	no	—
90	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×#261	—	2	2	no	—
92	C₂×#261	—	2	2	no	—
93	C₂×#261	—	8	4	no	—
94	C₂×#261	C₂×D₄	1	4	yes	—
95	Q₈×C₁₃	—	2	1	no	—
96	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
97	D₄×C₁₃	—	2	1	no	—
98	D₄×C₁₃	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²⋊D₁₃	—	2	2	no	—
102	C₂²⋊D₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	2	1	no	—
104	C₂×Dic₁₃	#41	1	1	yes	characteristic
105	D₅₂	—	2	1	no	—
106	D₅₂	C₂²×C₄	1	2	yes	—
107	C₄×D₁₃	—	2	2	no	—
108	C₄×D₁₃	—	2	2	no	—
109	C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄×D₁₃	#41	1	1	yes	characteristic
112	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
113	Dic₂₆	—	2	1	no	—
114	Dic₂₆	—	2	2	no	—
115	Dic₂₆	—	2	2	no	—
116	Dic₂₆	C₂²×C₄	1	2	yes	—
117	#1057	—	2	2	no	—
118	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
120	#1055	C₂²×C₄	1	1	yes	characteristic
121	#1055	C₂²×C₄	1	1	yes	characteristic
122	C₂²×SD₁₆	—	13	1	no	—
123	#519	—	13	1	no	—
124	#518	—	13	1	no	—
125	#440	—	13	4	no	—
126	C₄×SD₁₆	—	13	4	no	—
127	#427	—	13	1	no	—
128	#419	—	13	1	no	—
129	#418	—	13	1	no	—
130	#405	—	13	1	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	8	1	no	—
133	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
134	C₂×D₁₃	—	2	2	no	—
135	C₂×D₁₃	—	2	2	no	—
136	C₂×D₁₃	—	4	4	no	—
137	C₂×D₁₃	—	8	2	no	—
138	C₂×D₁₃	—	8	1	no	—
139	C₂×D₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂×D₁₃	C₂²×D₄	1	1	yes	characteristic
141	#261	—	2	4	no	—
142	#261	—	4	2	no	—
143	#261	—	4	2	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
147	Dic₁₃	—	2	2	no	—
148	Dic₁₃	C₄×D₄	1	1	yes	characteristic
149	Dic₁₃	C₂×#41	1	1	yes	characteristic
150	C₂²×Q₈	—	13	1	no	—
151	C₂²×D₄	—	13	1	no	—
152	C₂×SD₁₆	—	13	1	no	—
153	C₂×SD₁₆	—	13	1	no	—
154	C₂×SD₁₆	—	13	2	no	—
155	C₂×SD₁₆	—	13	2	no	—
156	C₂×SD₁₆	—	13	2	no	—
157	C₂×SD₁₆	—	13	2	no	—
158	C₂×SD₁₆	—	13	1	no	—
159	C₂×SD₁₆	#261	1	1	yes	characteristic, complemented
160	C₂×M₁₆	—	13	1	no	—
161	C₂²×C₈	—	13	1	no	—
162	C₄×Q₈	—	13	4	no	—
163	C₄×Q₈	—	26	2	no	—
164	C₄×D₄	—	13	4	no	—
165	C₄×D₄	—	26	2	no	—
166	#117	—	13	1	no	—
167	#117	—	13	1	no	—
168	#117	—	13	1	no	—
169	#117	—	26	2	no	—
170	C₂×C₂²⋊C₄	—	26	2	no	—
171	#107	—	13	2	no	—
172	#106	—	13	2	no	—
173	#103	—	13	4	no	—
174	#102	—	13	4	no	—
175	#97	—	13	2	no	—
176	C₄×C₈	—	13	2	no	—
177	C₂×C₁₃	—	2	1	no	—
178	C₂×C₁₃	—	4	2	no	—
179	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
180	D₁₃	—	2	1	no	—
181	D₁₃	—	4	2	no	—
182	D₁₃	#577	1	2	yes	—
183	C₂⁴	—	26	1	no	—
184	C₂×Q₈	—	13	2	no	—
185	C₂×Q₈	—	13	2	no	—
186	C₂×Q₈	—	13	1	no	—
187	C₂×Q₈	—	26	1	no	—
188	C₂×Q₈	—	26	1	no	—
189	C₂×Q₈	—	26	1	no	—
190	C₂×Q₈	C₂×#261	1	1	yes	characteristic, complemented
191	C₂×D₄	—	13	2	no	—
192	C₂×D₄	—	13	2	no	—
193	C₂×D₄	—	13	1	no	—
194	C₂×D₄	—	26	1	no	—
195	C₂×D₄	—	26	1	no	—
196	C₂×D₄	—	26	1	no	—
197	C₂×D₄	C₂×#261	1	1	yes	characteristic
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	26	1	no	—
200	C₂²×C₄	—	26	4	no	—
201	SD₁₆	—	13	2	no	—
202	SD₁₆	—	13	2	no	—
203	SD₁₆	—	13	2	no	—
204	SD₁₆	—	13	2	no	—
205	SD₁₆	—	13	2	no	—
206	SD₁₆	—	13	2	no	—
207	SD₁₆	C₂×#261	1	2	yes	complemented
208	SD₁₆	C₂×#261	1	2	yes	complemented
209	M₁₆	—	26	1	no	—
210	M₁₆	—	26	1	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	2	no	—
217	C₂×C₈	C₂×#261	1	1	yes	characteristic, complemented
218	C₄⋊C₄	—	13	2	no	—
219	C₄⋊C₄	—	13	2	no	—
220	C₄⋊C₄	—	13	2	no	—
221	C₄⋊C₄	—	26	4	no	—
222	C₂²⋊C₄	—	13	2	no	—
223	C₂²⋊C₄	—	26	4	no	—
224	C₂²⋊C₄	—	26	1	no	—
225	C₂²⋊C₄	—	26	1	no	—
226	C₄²	—	13	2	no	—
227	C₄²	—	26	1	no	—
228	C₄²	—	26	1	no	—
229	C₄²	—	26	4	no	—
230	C₁₃	#2949	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	2	1	no	—
232	C₂³	—	13	1	no	—
233	C₂³	—	26	2	no	—
234	C₂³	—	26	2	no	—
235	C₂³	—	26	1	no	—
236	C₂³	—	104	2	no	—
237	Q₈	—	2	1	no	—
238	Q₈	—	13	2	no	—
239	Q₈	—	26	2	no	—
240	Q₈	—	26	2	no	—
241	Q₈	—	26	1	no	—
242	Q₈	C₂²×#261	1	2	yes	—
243	D₄	—	2	1	no	—
244	D₄	—	13	2	no	—
245	D₄	—	26	2	no	—
246	D₄	—	26	2	no	—
247	D₄	—	26	1	no	—
248	D₄	C₂²×#261	1	2	yes	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	13	1	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	13	1	no	—
255	C₂×C₄	—	13	4	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	1	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	104	4	no	—
262	C₂×C₄	C₂²×#261	1	1	yes	characteristic
263	C₈	—	13	1	no	—
264	C₈	—	13	1	no	—
265	C₈	—	26	2	no	—
266	C₈	C₂²×#261	1	1	yes	characteristic
267	C₈	C₂²×#261	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	8	1	no	—
270	C₂²	—	13	1	no	—
271	C₂²	—	13	1	no	—
272	C₂²	—	26	2	no	—
273	C₂²	—	26	2	no	—
274	C₂²	—	52	4	no	—
275	C₂²	—	104	1	no	—
276	C₂²	—	104	2	no	—
277	C₂²	D₄×#261	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	13	1	no	—
280	C₄	—	13	1	no	—
281	C₄	—	26	4	no	—
282	C₄	—	26	2	no	—
283	C₄	—	52	2	no	—
284	C₄	—	52	2	no	—
285	C₄	D₄×#261	1	1	yes	characteristic
286	C₄	C₂³×#261	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	4	2	no	—
289	C₂	—	13	2	no	—
290	C₂	—	26	1	no	—
291	C₂	—	52	2	no	—
292	C₂	C₂×D₄×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₄	C₂³×#261
Fitting	C₂×C₁₃×SD₁₆	C₄
socle	C₂×C₁₃	C₂×C₄×D₄
prime radical	C₂×D₄×D₁₃	C₂²
cosocle	C₄×D₁₃	C₂⁴
hypercenter	C₂×SD₁₆	#261
hypocenter	C₁₃	#2949
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #11719627	C₂	semidirect
4. #11719153	C₂	semidirect
6. #11719106	C₂	semidirect
7. SD₁₆×#261	C₂	semidirect
8. #11719092	C₂	semidirect
9. #11719089	C₂	semidirect
14. Q₈×#261	C₂²	semidirect
22. C₂×C₁₃×SD₁₆	C₄	semidirect
23. C₂×#6106	C₄	semidirect
24. C₂×#6105	C₄	semidirect
25. SD₁₆×D₁₃	C₄	semidirect
26. SD₁₆×D₁₃	C₄	semidirect
29. C₂×#6095	C₄	semidirect
31. #88117	C₂²	semidirect
33. #88102	C₂²	semidirect
34. #88101	C₂²	semidirect
35. #88100	C₂²	semidirect
36. C₈×#261	C₂²	semidirect
39. C₂×Q₈×C₁₃	C₂×C₄	semidirect
44. Q₈×D₁₃	C₂×C₄	semidirect
54. C₂×Dic₂₆	C₂×C₄	semidirect
70. C₁₃×SD₁₆	C₂×C₄	semidirect
71. C₁₃×SD₁₆	C₂×C₄	semidirect
72. C₂×C₈×C₁₃	C₂×C₄	semidirect
73. #6106	C₂×C₄	semidirect
74. #6106	C₂×C₄	semidirect
75. #6105	C₂×C₄	semidirect
76. #6105	C₂×C₄	semidirect
77. C₂×#1055	C₂×C₄	semidirect
78. #6095	C₂×C₄	semidirect
79. #6095	C₂×C₄	semidirect
80. C₈×D₁₃	C₂×C₄	semidirect
81. C₈×D₁₃	C₂×C₄	semidirect
159. C₂×SD₁₆	#261	semidirect
190. C₂×Q₈	C₂×#261	semidirect
207. SD₁₆	C₂×#261	semidirect
208. SD₁₆	C₂×#261	semidirect
217. C₂×C₈	C₂×#261	semidirect
230. C₁₃	#2949	semidirect

Maximal subgroups

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

subgroup	index
#11719627	2
#11719623	2
C₂×SD₁₆×D₁₃	2
#11719153	2
#11719138	2
#11719106	2
SD₁₆×#261	2
#11719092	2
#11719089	2
#2949	13