Subgroups — #15756043

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115008	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113261	C₂	1	2	yes	maximal
4	#13112733	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	—	2	1	no	—
7	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₁₃	—	2	1	no	—
9	C₈×D₄×C₁₃	—	2	1	no	—
10	C₈×D₄×C₁₃	C₄	1	2	yes	—
11	C₈×D₄×C₁₃	C₄	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88059	—	2	1	no	—
16	C₂×#88059	—	2	1	no	—
17	#11718586	C₂²	1	2	yes	—
18	#11718585	C₂²	1	2	yes	—
19	D₄⋊#1055	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	C₂²×D₄×C₁₃	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	—	2	1	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	4	2	no	—
28	C₂²×C₈×C₁₃	—	4	2	no	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₄×D₄×C₁₃	—	2	1	no	—
32	C₄×D₄×C₁₃	—	2	1	no	—
33	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×D₄×C₁₃	D₄	1	2	yes	—
35	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
36	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×#6099	—	2	2	no	—
39	C₁₃×C₄⋊C₈	—	2	1	no	—
40	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₁₃×C₂²⋊C₈	—	2	2	no	—
42	C₁₃×C₂²⋊C₈	—	2	2	no	—
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²⋊#1055	—	2	2	no	—
48	#88059	—	4	2	no	—
49	#88059	—	4	2	no	—
50	C₄⋊#1055	—	2	2	no	—
51	C₄⋊#1055	—	2	2	no	—
52	C₄⋊#1055	—	2	2	no	—
53	C₄×#1055	—	2	2	no	—
54	#8658	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	—	2	1	no	—
56	C₂×D₄×C₁₃	—	2	4	no	—
57	C₂×D₄×C₁₃	—	2	1	no	—
58	C₂×D₄×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	4	1	no	—
73	C₂×C₈×C₁₃	—	4	1	no	—
74	C₂×C₈×C₁₃	—	4	4	no	—
75	C₂×C₈×C₁₃	—	4	4	no	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
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	D₈	1	2	yes	—
81	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
82	C₁₃×C₄⋊C₄	—	2	1	no	—
83	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
84	C₁₃×C₂²⋊C₄	—	2	2	no	—
85	C₁₃×C₂²⋊C₄	—	2	2	no	—
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	#41	1	1	yes	characteristic
90	#6099	—	4	2	no	—
91	#6099	—	4	2	no	—
92	C₂×#1055	—	4	2	no	—
93	C₂×#1055	—	4	2	no	—
94	C₂×C₈×D₄	D₁₃	1	1	yes	characteristic
95	#1848	—	13	1	no	—
96	#1564	—	13	1	no	—
97	#1486	—	13	2	no	—
98	#1301	—	13	2	no	—
99	C₂³×C₁₃	—	2	4	no	—
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	—	4	1	no	—
103	C₂³×C₁₃	—	4	1	no	—
104	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
105	D₄×C₁₃	—	2	4	no	—
106	D₄×C₁₃	—	2	2	no	—
107	D₄×C₁₃	#108	1	4	yes	complemented
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	4	2	no	—
130	C₈×C₁₃	#102	1	2	yes	—
131	C₈×C₁₃	#102	1	2	yes	—
132	#1055	—	8	2	no	—
133	#1055	—	8	2	no	—
134	C₂³×C₈	—	2	1	no	—
135	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
136	C₈×D₄	—	2	1	no	—
137	C₈×D₄	—	2	1	no	—
138	C₈×D₄	Dic₁₃	1	2	yes	—
139	C₈×D₄	Dic₁₃	1	2	yes	—
140	#431	—	13	2	no	—
141	C₂×#108	—	26	1	no	—
142	C₂×#108	—	26	1	no	—
143	#423	—	13	2	no	—
144	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
145	C₂×C₂²⋊C₈	—	2	1	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#335	—	26	1	no	—
148	#334	—	26	1	no	—
149	D₄⋊C₈	—	26	4	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	—	2	4	no	—
153	C₂²×C₁₃	—	4	4	no	—
154	C₂²×C₁₃	—	4	4	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	—	4	2	no	—
157	C₂²×C₁₃	#380	1	1	yes	characteristic
158	C₂²×C₁₃	#474	1	1	yes	characteristic
159	C₂²×C₁₃	#476	1	1	yes	characteristic
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
165	C₄×C₁₃	#417	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#108	1	2	yes	—
167	C₄×C₁₃	#459	1	1	yes	characteristic
168	C₄×C₁₃	#461	1	1	yes	characteristic
169	C₂²×D₄	Dic₂₆	1	1	yes	characteristic
170	C₂³×C₄	—	2	1	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	4	2	no	—
176	C₂²×C₈	—	4	2	no	—
177	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	D₅₂	1	2	yes	—
182	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
183	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	—	2	1	no	—
185	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
186	#108	—	52	2	no	—
187	#108	—	52	2	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	26	2	no	—
190	C₄⋊C₈	—	26	2	no	—
191	C₄⋊C₈	—	26	2	no	—
192	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
193	C₂²⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	26	2	no	—
196	C₄×C₈	—	26	2	no	—
197	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
198	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	4	4	no	—
204	C₂×C₁₃	#1802	1	1	yes	characteristic
205	C₂×C₁₃	#1810	1	1	yes	characteristic
206	C₂×C₁₃	#1814	1	1	yes	characteristic
207	C₂⁴	—	2	1	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	C₄⋊Dic₁₃	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
219	C₂²×C₄	#6127	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
221	M₁₆	—	52	2	no	—
222	M₁₆	—	52	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	4	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	52	2	no	—
231	C₂×C₈	—	52	2	no	—
232	C₂×C₈	#6104	1	2	yes	—
233	C₂×C₈	#6105	1	2	yes	—
234	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂×D₅₂	1	1	yes	characteristic
243	C₄²	#6127	1	1	yes	characteristic
244	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
246	C₁₃	#8658	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	2	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	#88178	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	2	no	—
255	D₄	#88059	1	4	yes	complemented
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	#88178	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
267	C₂×C₄	#88180	1	1	yes	characteristic
268	C₂×C₄	#88184	1	1	yes	characteristic
269	C₂×C₄	#88184	1	1	yes	characteristic
270	C₂×C₄	C₂×#6104	1	1	yes	characteristic
271	C₂×C₄	C₂×#6105	1	1	yes	characteristic
272	C₂×C₄	#88203	1	1	yes	characteristic
273	C₂×C₄	#88205	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	4	2	no	—
278	C₈	—	104	2	no	—
279	C₈	—	104	2	no	—
280	C₈	#88072	1	2	yes	—
281	C₈	#88072	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#11718668	1	1	yes	characteristic
290	C₂²	#11718669	1	1	yes	characteristic
291	C₂²	#11718873	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	4	2	no	—
296	C₄	#11718676	1	1	yes	characteristic
297	C₄	#11718680	1	1	yes	characteristic
298	C₄	C₂×#88059	1	2	yes	—
299	C₄	C₂×#88072	1	1	yes	characteristic
300	C₄	#11718893	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	4	no	—
304	C₂	#13114836	1	1	yes	characteristic
305	C₂	#13114874	1	1	yes	characteristic
306	C₂	#13114890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈×D₄×C₁₃	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂²×D₄×C₁₃	Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈×D₄	D₁₃
hypocenter	C₁₃	#8658
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #13115008	C₂	semidirect
2. #13113584	C₂	semidirect
4. #13112733	C₂	semidirect
107. D₄×C₁₃	#108	semidirect
246. C₁₃	#8658	semidirect
255. D₄	#88059	semidirect

Maximal subgroups

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

subgroup	index
#13115008	2
#13113584	2
#13113261	2
#13112733	2
C₂×C₈×D₄×C₁₃	2
#8658	13