Subgroups — #15778205

Order 3328.

428 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123638	C₂	1	1	yes	maximal, characteristic
2	#13116573	C₂	1	1	yes	maximal, characteristic
3	#13116565	C₂	1	1	yes	maximal, characteristic
4	#13114720	C₂	1	1	yes	maximal, characteristic, complemented
5	#13112951	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112938	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88158	C₂²	1	1	yes	characteristic
9	C₄×#6127	—	2	1	no	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
13	#11718774	—	2	1	no	—
14	C₂×#88059	C₂²	1	1	yes	characteristic
15	#11718767	C₂²	1	1	yes	characteristic
16	#11718767	C₂²	1	1	yes	characteristic
17	#11718762	—	2	1	no	—
18	#11718377	C₄	1	2	yes	complemented
19	#11718372	C₄	1	2	yes	complemented
20	#11718371	—	2	2	no	—
21	C₁₃×#347	—	2	2	no	—
22	#11718115	—	2	1	no	—
23	#11718110	C₄	1	4	yes	complemented
24	#11718102	C₄	1	4	yes	complemented
25	C₂×#6127	—	2	1	no	—
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₄×C₂²⋊D₁₃	—	4	2	no	—
30	#88180	—	2	1	no	—
31	#88180	—	2	1	no	—
32	C₂×#6099	—	2	1	no	—
33	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
34	#88158	—	2	1	no	—
35	#88158	—	2	1	no	—
36	#88158	C₂×C₄	1	2	yes	—
37	#88158	C₂×C₄	1	2	yes	—
38	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
39	C₂×#6095	—	2	1	no	—
40	C₄×D₅₂	—	2	2	no	—
41	#88126	—	4	1	no	—
42	C₄²×D₁₃	—	4	1	no	—
43	C₄×Dic₂₆	—	2	2	no	—
44	C₁₃×M₃₂	—	2	1	no	—
45	C₁₃×M₃₂	—	2	1	no	—
46	C₁₃×M₃₂	—	2	1	no	—
47	C₁₃×M₃₂	C₂×C₄	1	2	yes	complemented
48	C₂×C₁₃×C₁₆	—	2	1	no	—
49	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
50	C₁₃×#97	C₂×C₄	1	2	yes	—
51	C₁₃×#97	D₄	1	2	yes	—
52	#88061	—	2	2	no	—
53	#88059	—	2	1	no	—
54	#88059	C₂×C₄	1	2	yes	complemented
55	#88058	—	2	1	no	—
56	#88058	—	2	1	no	—
57	#88057	C₂×C₄	1	2	yes	—
58	#88057	C₂×C₄	1	2	yes	complemented
59	#88056	—	2	2	no	—
60	#88055	—	4	1	no	—
61	C₈×Dic₁₃	—	4	1	no	—
62	#12649	—	13	1	no	maximal, Sylow
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₃	—	4	1	no	—
66	#6127	—	2	2	no	—
67	#6127	—	2	2	no	—
68	#6127	—	4	1	no	—
69	#6127	—	4	1	no	—
70	C₂×D₅₂	—	2	1	no	—
71	C₂×C₄×D₁₃	—	4	1	no	—
72	C₂×C₄×D₁₃	—	4	2	no	—
73	C₂×Dic₂₆	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
85	C₂²⋊Dic₁₃	—	4	1	no	—
86	C₂²⋊Dic₁₃	—	4	1	no	—
87	#6103	—	4	2	no	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	2	2	no	—
90	#6101	—	4	2	no	—
91	C₄×Dic₁₃	—	4	2	no	—
92	C₄×Dic₁₃	—	4	1	no	—
93	C₄×Dic₁₃	—	4	1	no	—
94	#6099	—	2	2	no	—
95	#6099	—	4	1	no	—
96	C₂×#1055	—	4	1	no	—
97	Dic₅₂	—	2	1	no	—
98	Dic₅₂	C₄²	1	2	yes	—
99	D₁₀₄	—	2	1	no	—
100	D₁₀₄	C₄²	1	2	yes	—
101	#6095	—	2	2	no	—
102	#6095	—	2	2	no	—
103	C₁₃×C₁₆	—	2	2	no	—
104	C₁₃×C₁₆	—	2	2	no	—
105	#2943	—	13	1	no	—
106	#2145	—	13	1	no	—
107	#2143	—	13	1	no	—
108	C₂×#347	D₁₃	1	1	yes	characteristic, complemented
109	#1773	—	13	1	no	—
110	#1389	—	13	1	no	—
111	#1384	—	13	1	no	—
112	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂²×D₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	8	1	no	—
116	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
122	C₂²⋊D₁₃	—	4	2	no	—
123	C₂²⋊D₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	1	no	—
125	C₂×Dic₁₃	—	4	2	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	D₅₂	—	2	2	no	—
128	D₅₂	—	4	1	no	—
129	C₄×D₁₃	—	4	2	no	—
130	C₄×D₁₃	—	4	2	no	—
131	C₄×D₁₃	—	8	4	no	—
132	Dic₂₆	—	2	2	no	—
133	Dic₂₆	—	4	1	no	—
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
136	C₈×C₁₃	#117	1	1	yes	characteristic
137	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
138	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
139	#1055	—	4	2	no	—
140	C₂×#135	—	13	1	no	—
141	C₄×#41	—	26	1	no	—
142	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
143	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
144	#442	—	13	2	no	—
145	#441	—	13	2	no	—
146	#440	—	26	2	no	—
147	C₂×#108	—	13	1	no	—
148	#427	—	13	1	no	—
149	#427	—	13	1	no	—
150	#416	—	26	1	no	—
151	#405	—	26	1	no	—
152	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
153	#360	—	13	4	no	—
154	#359	—	13	4	no	—
155	#347	—	2	2	no	—
156	#339	—	26	1	no	—
157	C₂²×C₁₃	—	4	1	no	—
158	C₂²×C₁₃	#418	1	1	yes	characteristic
159	C₂²×C₁₃	#440	1	1	yes	characteristic
160	C₂²×C₁₃	#442	1	1	yes	characteristic
161	C₂×D₁₃	—	4	2	no	—
162	C₂×D₁₃	—	8	2	no	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₄×C₁₃	#417	1	1	yes	characteristic
166	C₄×C₁₃	#440	1	1	yes	characteristic
167	C₄×C₁₃	#442	1	1	yes	characteristic
168	Dic₁₃	—	4	2	no	—
169	Dic₁₃	—	8	2	no	—
170	Dic₁₃	—	8	2	no	—
171	C₂×#41	—	26	1	no	—
172	#135	—	13	2	no	—
173	#135	—	13	2	no	—
174	#135	—	26	1	no	—
175	#135	—	26	1	no	—
176	C₂×Q₁₆	—	13	1	no	—
177	C₂×SD₁₆	—	26	1	no	—
178	C₂×D₈	—	13	1	no	—
179	C₂×M₁₆	—	26	1	no	—
180	C₂²×C₈	D₅₂	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₄×Q₈	—	26	2	no	—
183	C₄×D₄	—	26	2	no	—
184	C₄×D₄	—	52	2	no	—
185	#117	—	26	1	no	—
186	#117	—	26	1	no	—
187	#117	—	52	1	no	—
188	C₂×C₄²	—	52	1	no	—
189	C₂×C₄²	D₅₂	1	1	yes	characteristic
190	M₃₂	—	2	1	no	—
191	M₃₂	—	2	1	no	—
192	M₃₂	—	2	1	no	—
193	M₃₂	C₄×D₁₃	1	2	yes	complemented
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	C₄×D₁₃	1	2	yes	complemented
196	#108	—	13	2	no	—
197	#108	—	26	1	no	—
198	#107	—	26	1	no	—
199	#107	—	26	1	no	—
200	#106	—	13	2	no	—
201	#106	—	13	2	no	—
202	#103	—	26	2	no	—
203	#102	—	26	2	no	—
204	#97	—	52	1	no	—
205	#97	C₄×D₁₃	1	2	yes	—
206	#97	C₂²⋊D₁₃	1	2	yes	—
207	C₄×C₈	—	52	1	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	#1762	1	1	yes	characteristic
212	D₁₃	—	8	2	no	—
213	#41	—	26	2	no	—
214	#41	—	26	2	no	—
215	#41	—	52	1	no	—
216	#41	—	52	1	no	—
217	C₂×Q₈	—	26	1	no	—
218	C₂×D₄	—	26	1	no	—
219	C₂×D₄	—	52	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	52	2	no	—
222	C₂²×C₄	—	52	1	no	—
223	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
224	Q₁₆	—	13	2	no	—
225	Q₁₆	—	26	1	no	—
226	SD₁₆	—	26	2	no	—
227	SD₁₆	—	26	2	no	—
228	D₈	—	13	2	no	—
229	D₈	—	26	1	no	—
230	M₁₆	—	26	2	no	—
231	M₁₆	—	52	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	52	1	no	—
235	C₂×C₈	#6103	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
239	C₂×C₈	#6127	1	1	yes	characteristic
240	C₂×C₈	#6127	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
242	C₄⋊C₄	—	26	2	no	—
243	C₄⋊C₄	—	26	2	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₂²⋊C₄	—	52	2	no	—
246	C₂²⋊C₄	—	52	1	no	—
247	C₂²⋊C₄	—	52	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	52	2	no	—
250	C₄²	—	52	1	no	—
251	C₄²	—	52	1	no	—
252	C₄²	#6103	1	2	yes	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₃	#12649	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	52	1	no	—
257	C₂³	#88162	1	1	yes	characteristic
258	Q₈	—	26	2	no	—
259	Q₈	—	52	1	no	—
260	D₄	—	26	2	no	—
261	D₄	—	52	2	no	—
262	D₄	—	52	2	no	—
263	D₄	—	52	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	—	52	1	no	—
267	C₂×C₄	—	52	2	no	—
268	C₂×C₄	—	52	2	no	—
269	C₂×C₄	—	52	2	no	—
270	C₂×C₄	—	52	2	no	—
271	C₂×C₄	—	104	4	no	—
272	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
273	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
274	C₂×C₄	#88162	1	1	yes	characteristic
275	C₂×C₄	#88162	1	1	yes	characteristic
276	C₂×C₄	#88163	1	1	yes	characteristic
277	C₂×C₄	C₂×#6103	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	52	2	no	—
280	C₈	C₄²×D₁₃	1	1	yes	characteristic
281	C₈	#88126	1	1	yes	characteristic
282	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
283	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
284	C₂²	—	4	1	no	—
285	C₂²	—	52	2	no	—
286	C₂²	—	104	2	no	—
287	C₂²	#11718371	1	1	yes	characteristic
288	C₂²	#11718372	1	1	yes	characteristic
289	C₂²	#11718792	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	52	2	no	—
292	C₄	—	104	2	no	—
293	C₄	—	104	2	no	—
294	C₄	#11718371	1	1	yes	characteristic
295	C₄	#11718372	1	1	yes	characteristic
296	C₄	C₄×#6103	1	1	yes	characteristic
297	C₄	#11718794	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	104	2	no	—
302	C₂	#13114662	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718372
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#347	C₂
socle	C₂×C₁₃	#1762
prime radical	C₂×#88158	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#347	D₁₃
hypocenter	C₁₃	#12649
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #13114720	C₂	semidirect
5. #13112951	C₂	semidirect
6. #13112938	C₂	semidirect
7. C₂×C₁₃×#347	C₂	semidirect
18. #11718377	C₄	semidirect
19. #11718372	C₄	semidirect
23. #11718110	C₄	semidirect
24. #11718102	C₄	semidirect
47. C₁₃×M₃₂	C₂×C₄	semidirect
49. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
54. #88059	C₂×C₄	semidirect
58. #88057	C₂×C₄	semidirect
108. C₂×#347	D₁₃	semidirect
193. M₃₂	C₄×D₁₃	semidirect
195. C₂×C₁₆	C₄×D₁₃	semidirect
255. C₁₃	#12649	semidirect

Maximal subgroups

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

subgroup	index
#13123638	2
#13116573	2
#13116565	2
#13114720	2
#13112951	2
#13112938	2
C₂×C₁₃×#347	2
#12649	13