Subgroups — #15772138

Order 3328.

310 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
2	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
3	#13112778	C₂	1	2	yes	maximal, complemented
4	#13112762	C₂	1	2	yes	maximal
5	C₁₃×#2922	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
7	C₁₃×#433	—	2	1	no	—
8	C₁₃×#433	—	2	1	no	—
9	C₁₃×#431	—	2	1	no	—
10	C₁₃×#431	—	2	1	no	—
11	C₁₃×#424	C₄	1	1	yes	characteristic
12	C₁₃×#424	C₂²	1	1	yes	characteristic
13	C₁₃×#408	C₄	1	1	yes	characteristic
14	C₁₃×#408	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂²⋊#6090	C₂²	1	2	yes	—
18	C₂²⋊#6090	C₂²	1	2	yes	—
19	C₄⋊#6090	—	2	1	no	—
20	C₄⋊#6090	—	2	1	no	—
21	#11718053	—	2	1	no	—
22	C₄×#6090	—	2	1	no	—
23	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
30	C₄×D₄×C₁₃	—	2	1	no	—
31	C₄×D₄×C₁₃	—	2	1	no	—
32	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
33	C₄×D₄×C₁₃	C₈	1	1	yes	characteristic
34	C₁₃×#117	—	2	1	no	—
35	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	—	2	1	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₁₃×C₄⋊C₈	—	2	1	no	—
39	C₁₃×C₄⋊C₈	—	2	1	no	—
40	C₁₃×C₄⋊C₈	—	2	1	no	—
41	C₁₃×C₄⋊C₈	—	2	1	no	—
42	C₁₃×C₂²⋊C₈	—	2	1	no	—
43	C₁₃×C₂²⋊C₈	—	2	1	no	—
44	C₁₃×C₂²⋊C₈	—	2	1	no	—
45	C₁₃×C₂²⋊C₈	—	2	1	no	—
46	C₁₃×#97	—	2	1	no	—
47	C₄×C₈×C₁₃	—	2	1	no	—
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
52	C₂×#6090	—	2	2	no	—
53	C₂×#6090	—	2	2	no	—
54	#11464	—	13	1	no	maximal, Sylow
55	C₁₃×#41	—	4	1	no	—
56	C₁₃×#41	—	4	1	no	—
57	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂×D₄×C₁₃	—	2	1	no	—
59	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₁₃×M₁₆	—	4	1	no	—
67	C₁₃×M₁₆	—	4	1	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	4	1	no	—
71	C₂×C₈×C₁₃	—	4	1	no	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	—	2	1	no	—
77	C₁₃×C₄⋊C₄	—	2	1	no	—
78	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
80	C₁₃×C₂²⋊C₄	—	2	1	no	—
81	C₁₃×C₂²⋊C₄	—	2	1	no	—
82	C₁₃×C₂²⋊C₄	—	2	1	no	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
90	#6090	—	4	2	no	—
91	#6090	—	4	2	no	—
92	#2922	D₁₃	1	1	yes	characteristic
93	#2163	—	13	1	no	—
94	#2162	—	13	1	no	—
95	#1323	—	13	2	no	—
96	#1315	—	13	2	no	—
97	C₂³×C₁₃	—	2	1	no	—
98	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	Q₈×C₁₃	—	4	1	no	—
100	D₄×C₁₃	—	4	1	no	—
101	D₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₈×C₁₃	—	2	1	no	—
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	—	2	1	no	—
122	C₈×C₁₃	—	2	1	no	—
123	C₈×C₁₃	—	4	1	no	—
124	C₈×C₁₃	—	4	1	no	—
125	C₄×#41	Dic₁₃	1	1	yes	characteristic
126	#433	—	2	1	no	—
127	#433	—	2	1	no	—
128	#431	—	2	1	no	—
129	#431	—	2	1	no	—
130	#424	Dic₁₃	1	1	yes	characteristic
131	#424	C₂×D₁₃	1	1	yes	characteristic
132	#408	Dic₁₃	1	1	yes	characteristic
133	#408	C₂×D₁₃	1	1	yes	characteristic
134	C₄×M₁₆	Dic₁₃	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₄⋊C₁₆	—	26	1	no	—
137	C₄⋊C₁₆	—	26	1	no	—
138	C₂²⋊C₁₆	—	13	2	no	—
139	C₂²⋊C₁₆	—	13	2	no	—
140	#346	—	26	1	no	—
141	C₄×C₁₆	—	26	1	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	—	4	1	no	—
148	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	4	1	no	—
154	C₄×C₁₃	—	4	1	no	—
155	C₄×C₁₃	—	4	1	no	—
156	C₄×C₁₃	#350	1	2	yes	—
157	C₄×C₁₃	#410	1	1	yes	characteristic
158	C₄×C₁₃	#413	1	1	yes	characteristic
159	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
161	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₄×Q₈	#1055	1	1	yes	characteristic
165	C₄×Q₈	#1055	1	1	yes	characteristic
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	#1055	1	1	yes	characteristic
169	C₄×D₄	#1055	1	1	yes	characteristic
170	#117	—	2	1	no	—
171	#117	C₂×Dic₁₃	1	1	yes	characteristic
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂×C₁₆	—	26	2	no	—
175	C₂×C₁₆	—	26	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₂²⋊C₈	—	2	1	no	—
181	C₂²⋊C₈	—	2	1	no	—
182	C₂²⋊C₈	—	2	1	no	—
183	C₂²⋊C₈	—	2	1	no	—
184	#97	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂×C₁₃	—	4	1	no	—
191	C₂×C₁₃	—	8	1	no	—
192	C₂×C₁₃	#1458	1	1	yes	characteristic
193	C₂×C₁₃	C₂×#350	1	1	yes	characteristic
194	C₂×C₁₃	#2117	1	1	yes	characteristic
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₂×#1055	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	#6099	1	1	yes	characteristic
204	C₂²×C₄	#6099	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	C₂×#1055	1	1	yes	characteristic
219	C₄⋊C₄	C₂×#1055	1	1	yes	characteristic
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	C₂×#1055	1	1	yes	characteristic
227	C₄²	C₂×#1055	1	1	yes	characteristic
228	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
229	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
230	C₁₆	—	52	2	no	—
231	C₁₆	—	52	2	no	—
232	C₁₃	#11464	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	2	1	no	—
234	C₂³	C₂×#6099	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
253	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	#11718153	1	2	yes	—
276	C₄	#11718898	1	1	yes	characteristic
277	C₄	#11718901	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	#13113150	1	1	yes	characteristic
281	C₂	C₂×#11718153	1	1	yes	characteristic
282	C₂	#13116464	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2922	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2922	D₁₃
hypocenter	C₁₃	#11464
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

Complements

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

N	H ≅ G/N	type
1. #13116639	C₂	semidirect
2. #13116635	C₂	semidirect
3. #13112778	C₂	semidirect
232. C₁₃	#11464	semidirect

Maximal subgroups

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

subgroup	index
#13116639	2
#13116635	2
#13112778	2
#13112762	2
C₁₃×#2922	2
#11464	13