Subgroups — #15833762

Order 3328.

624 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	#13123912	C₂	1	2	yes	maximal, complemented
2	#13113701	C₂	1	4	yes	maximal, complemented
3	#13113666	C₂	1	4	yes	maximal, complemented
4	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113305	C₂	1	2	yes	maximal, complemented
6	#13113128	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#532	C₂²	1	2	yes	—
9	C₁₃×#500	C₄	1	1	yes	characteristic
10	C₁₃×#493	C₄	1	1	yes	characteristic
11	C₁₃×#486	C₄	1	2	yes	—
12	C₁₃×#466	C₄	1	2	yes	—
13	C₁₃×#466	C₄	1	2	yes	—
14	C₄×C₁₃×D₈	C₂²	1	2	yes	—
15	C₄×C₁₃×D₈	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718895	—	2	4	no	—
18	#11718673	C₂²	1	4	yes	—
19	#11718672	—	2	4	no	—
20	D₄×#1055	C₂²	1	4	yes	—
21	#11718587	C₂²	1	2	yes	—
22	#11718586	C₂²	1	1	yes	characteristic
23	#11718586	C₂²	1	1	yes	characteristic
24	D₄⋊#1055	C₂²	1	8	yes	—
25	#11718050	C₂²	1	2	yes	—
26	#11718049	C₂²	1	2	yes	—
27	#11718046	C₂²	1	2	yes	—
28	#11718046	C₂²	1	2	yes	—
29	C₂×C₁₃×#41	—	2	2	no	—
30	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₁₃×#127	C₂×C₄	1	2	yes	—
36	C₁₃×#124	—	2	2	no	—
37	C₁₃×#121	—	2	2	no	—
38	C₁₃×#121	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₁₃	—	2	2	no	—
40	C₄×D₄×C₁₃	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×#6099	—	2	4	no	—
43	C₂²×#1055	—	2	4	no	—
44	C₁₃×#107	C₂×C₄	1	2	yes	—
45	C₁₃×#107	C₂×C₄	1	2	yes	—
46	C₁₃×#102	C₂×C₄	1	4	yes	—
47	C₁₃×#102	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²⋊#1055	—	2	2	no	—
53	C₂²⋊#1055	—	2	1	no	—
54	C₂²⋊#1055	—	2	4	no	—
55	C₂²⋊#1055	—	2	4	no	—
56	C₂²⋊#1055	D₄	1	2	yes	—
57	C₄⋊#1055	—	2	1	no	—
58	C₄⋊#1055	D₄	1	2	yes	—
59	C₄⋊#1055	C₂³	1	4	yes	—
60	#88043	—	2	2	no	—
61	C₄×#1055	C₂³	1	2	yes	—
62	C₄×#1055	C₂³	1	2	yes	—
63	#17989	—	13	1	no	maximal, Sylow
64	C₁₃×#41	—	8	4	no	—
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	4	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
74	C₁₃×D₈	—	4	2	no	—
75	C₁₃×D₈	—	4	2	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
84	C₁₃×C₂²⋊C₄	—	2	4	no	—
85	C₁₃×C₂²⋊C₄	—	2	4	no	—
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	#41	1	1	yes	characteristic
89	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
90	#6099	—	8	4	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	—	8	4	no	—
94	C₂×#1055	C₂×D₄	1	4	yes	—
95	#3169	D₁₃	1	1	yes	characteristic
96	#2981	—	13	2	no	—
97	#1586	—	13	4	no	—
98	#1581	—	13	4	no	—
99	#1568	—	13	1	no	—
100	#1496	—	13	2	no	—
101	#1452	—	13	1	no	—
102	C₂³×C₁₃	—	2	4	no	—
103	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
104	Q₈×C₁₃	—	8	2	no	—
105	D₄×C₁₃	—	2	4	no	—
106	D₄×C₁₃	—	2	4	no	—
107	D₄×C₁₃	—	8	2	no	—
108	D₄×C₁₃	—	8	2	no	—
109	D₄×C₁₃	—	8	4	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	1	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₈×C₁₃	—	2	1	no	—
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	—	2	1	no	—
126	C₈×C₁₃	—	2	1	no	—
127	#1055	—	2	4	no	—
128	#1055	—	4	2	no	—
129	#1055	—	4	4	no	—
130	#532	C₂×D₁₃	1	2	yes	—
131	#500	Dic₁₃	1	1	yes	characteristic
132	#493	Dic₁₃	1	1	yes	characteristic
133	#486	Dic₁₃	1	2	yes	—
134	#466	Dic₁₃	1	2	yes	—
135	#466	Dic₁₃	1	2	yes	—
136	C₄×D₈	C₂×D₁₃	1	2	yes	—
137	C₄×D₈	C₂×D₁₃	1	2	yes	—
138	#436	—	13	4	no	—
139	#435	—	26	4	no	—
140	C₈×D₄	—	13	4	no	—
141	#432	—	13	2	no	—
142	#431	—	13	1	no	—
143	#431	—	13	1	no	—
144	#408	—	26	4	no	—
145	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
146	#335	—	13	2	no	—
147	#334	—	13	2	no	—
148	D₄⋊C₈	—	13	8	no	—
149	#322	—	13	2	no	—
150	#322	—	13	2	no	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	—	4	4	no	—
155	C₂²×C₁₃	—	4	4	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	#568	1	1	yes	characteristic
164	C₄×C₁₃	#568	1	1	yes	characteristic
165	C₄×C₁₃	#576	1	2	yes	—
166	C₂×#41	—	2	2	no	—
167	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
168	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
169	C₂×M₁₆	—	26	4	no	—
170	C₂²×C₈	—	26	4	no	—
171	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
173	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
174	#127	C₂×Dic₁₃	1	2	yes	—
175	#124	—	2	2	no	—
176	#121	—	2	2	no	—
177	#121	C₂×Dic₁₃	1	4	yes	—
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	C₂²×D₁₃	1	4	yes	—
180	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
181	#107	C₂×Dic₁₃	1	2	yes	—
182	#107	C₂×Dic₁₃	1	2	yes	—
183	C₄⋊C₈	—	13	2	no	—
184	C₄⋊C₈	—	13	4	no	—
185	C₄⋊C₈	—	26	1	no	—
186	#102	C₂×Dic₁₃	1	4	yes	—
187	#102	C₂×Dic₁₃	1	4	yes	—
188	C₂²⋊C₈	—	13	2	no	—
189	C₂²⋊C₈	—	26	4	no	—
190	C₂²⋊C₈	—	26	4	no	—
191	C₂²⋊C₈	—	26	2	no	—
192	C₂²⋊C₈	—	26	1	no	—
193	#97	—	26	2	no	—
194	C₄×C₈	—	13	2	no	—
195	C₄×C₈	—	13	2	no	—
196	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
197	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
200	C₂×C₁₃	—	4	1	no	—
201	C₂×C₁₃	—	8	4	no	—
202	C₂×C₁₃	#2932	1	1	yes	characteristic
203	C₂×C₁₃	#2948	1	1	yes	characteristic
204	C₂×C₁₃	#2957	1	1	yes	characteristic
205	#41	—	8	4	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
213	C₂²×C₄	#6129	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
215	D₈	—	4	2	no	—
216	D₈	—	4	2	no	—
217	M₁₆	—	104	4	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	13	4	no	—
221	C₂×C₈	—	26	4	no	—
222	C₂×C₈	—	26	2	no	—
223	C₂×C₈	—	104	4	no	—
224	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₄²	D₄×D₁₃	1	1	yes	characteristic
233	C₄²	#6129	1	1	yes	characteristic
234	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
235	C₄²	C₂³×D₁₃	1	1	yes	characteristic
236	C₁₃	#17989	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	2	4	no	—
238	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
239	Q₈	—	8	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	4	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
253	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×#6129	1	1	yes	characteristic
257	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	26	4	no	—
263	C₈	—	52	2	no	—
264	C₈	—	52	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	#11719462	1	2	yes	—
278	C₄	#11719524	1	1	yes	characteristic
279	C₄	#11719524	1	1	yes	characteristic
280	C₂	—	4	1	no	—
281	C₂	—	8	4	no	—
282	C₂	#13123554	1	1	yes	characteristic
283	C₂	#13123688	1	1	yes	characteristic
284	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#17989
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13123912	C₂	semidirect
2. #13113701	C₂	semidirect
3. #13113666	C₂	semidirect
4. #13113599	C₂	semidirect
5. #13113305	C₂	semidirect
6. #13113128	C₂	semidirect
236. C₁₃	#17989	semidirect

Maximal subgroups

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

subgroup	index
#13123912	2
#13113701	2
#13113666	2
#13113599	2
#13113305	2
#13113128	2
C₁₃×#3169	2
#17989	13