Subgroups — #15756363

Order 3328.

522 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113150	C₂	1	2	yes	maximal
2	#13112669	C₂	1	2	yes	maximal, complemented
3	#13112665	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
6	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
9	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
10	C₁₃×#376	C₄	1	1	yes	characteristic
11	C₁₃×#376	C₂²	1	1	yes	characteristic
12	C₄³×C₁₃	C₄	1	1	yes	characteristic
13	C₄³×C₁₃	C₂²	1	1	yes	characteristic
14	#11718586	C₂²	1	4	yes	—
15	#11718120	—	2	2	no	—
16	#11718117	—	2	2	no	—
17	C₂³×C₄×C₁₃	—	2	1	no	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₁₃×#117	—	2	2	no	—
20	C₁₃×#117	—	2	2	no	—
21	C₁₃×#117	—	2	2	no	—
22	C₁₃×#117	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
24	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
25	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
27	C₂×C₄²×C₁₃	—	2	1	no	—
28	C₂×C₄²×C₁₃	—	2	2	no	—
29	C₂×C₄²×C₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₈	1	4	yes	complemented
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#95	—	2	2	no	—
39	C₁₃×#95	—	2	2	no	—
40	C₂²⋊#1055	—	2	4	no	—
41	C₄⋊#1055	—	2	4	no	—
42	C₄×#1055	—	2	4	no	—
43	#8730	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	4	4	no	—
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	2	4	no	—
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₂²⋊C₄	—	2	4	no	—
64	C₁₃×C₂²⋊C₄	—	2	4	no	—
65	C₄²×C₁₃	—	2	4	no	—
66	C₄²×C₁₃	—	2	1	no	—
67	C₄²×C₁₃	—	2	1	no	—
68	C₄²×C₁₃	—	2	1	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	—	4	4	no	—
73	C₄²×C₁₃	—	4	4	no	—
74	C₄²×C₁₃	C₂×C₈	1	2	yes	—
75	C₄²×C₁₃	C₂×C₈	1	2	yes	—
76	C₄²×C₁₃	M₁₆	1	2	yes	—
77	C₄²×C₁₃	M₁₆	1	2	yes	—
78	C₄²×C₁₃	#41	1	2	yes	—
79	C₄²×C₁₃	#41	1	2	yes	—
80	C₂×#1055	—	4	4	no	—
81	C₄×#117	D₁₃	1	1	yes	characteristic
82	#1458	—	13	2	no	—
83	#1276	—	13	2	no	—
84	#1274	—	13	2	no	—
85	C₂³×C₁₃	—	2	1	no	—
86	C₂³×C₁₃	—	2	1	no	—
87	C₂³×C₁₃	—	4	2	no	—
88	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
89	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	8	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#123	1	4	yes	—
112	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
116	#1055	—	8	4	no	—
117	C₂×#117	Dic₁₃	1	1	yes	characteristic
118	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
119	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
120	#431	—	13	4	no	—
121	C₄×C₄⋊C₄	—	2	2	no	—
122	C₄×C₂²⋊C₄	—	2	2	no	—
123	#376	Dic₁₃	1	1	yes	characteristic
124	#376	C₂×D₁₃	1	1	yes	characteristic
125	C₄³	Dic₁₃	1	1	yes	characteristic
126	C₄³	C₂×D₁₃	1	1	yes	characteristic
127	#324	—	26	2	no	—
128	#323	—	26	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	4	2	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	#381	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂²×C₁₃	#407	1	1	yes	characteristic
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₄×C₁₃	—	4	2	no	—
142	C₄×C₁₃	—	4	2	no	—
143	C₄×C₁₃	—	4	2	no	—
144	C₄×C₁₃	#431	1	1	yes	characteristic
145	C₄×C₁₃	#431	1	1	yes	characteristic
146	C₄×C₁₃	#432	1	1	yes	characteristic
147	C₄×C₁₃	#432	1	1	yes	characteristic
148	C₂³×C₄	—	2	1	no	—
149	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
150	#117	—	2	2	no	—
151	#117	—	2	2	no	—
152	#117	—	2	2	no	—
153	#117	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	2	yes	—
156	C₂×C₂²⋊C₄	—	2	1	no	—
157	C₂×C₂²⋊C₄	C₂²⋊D₁₃	1	2	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	—	2	2	no	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	#1055	1	4	yes	complemented
163	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	C₄⋊C₈	—	26	4	no	—
170	C₂²⋊C₈	—	26	4	no	—
171	C₄×C₈	—	26	4	no	—
172	#95	—	2	2	no	—
173	#95	—	2	2	no	—
174	C₂×C₁₃	—	2	1	no	—
175	C₂×C₁₃	—	2	1	no	—
176	C₂×C₁₃	—	4	2	no	—
177	C₂×C₁₃	#1817	1	1	yes	characteristic
178	C₂×C₁₃	#1827	1	2	yes	—
179	C₂⁴	C₂²⋊Dic₁₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	C₂×#1055	1	2	yes	—
191	C₂²×C₄	#6099	1	2	yes	—
192	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
196	C₂×C₈	—	52	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	4	4	no	—
209	C₄²	—	4	4	no	—
210	C₄²	C₂×#1055	1	2	yes	—
211	C₄²	C₂×#1055	1	2	yes	—
212	C₄²	#6099	1	2	yes	—
213	C₄²	#6099	1	2	yes	—
214	C₄²	#6127	1	2	yes	—
215	C₄²	#6127	1	2	yes	—
216	C₁₃	#8730	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂²⋊#1055	1	2	yes	—
221	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂²⋊#1055	1	2	yes	—
241	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
242	C₂×C₄	C₂×#6099	1	1	yes	characteristic
243	C₂×C₄	C₂×#6099	1	1	yes	characteristic
244	C₂×C₄	C₂×#6099	1	1	yes	characteristic
245	C₂×C₄	#88180	1	1	yes	characteristic
246	C₂×C₄	#88180	1	1	yes	characteristic
247	C₂×C₄	#88183	1	4	yes	—
248	C₈	—	104	4	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
255	C₂²	#11718868	1	1	yes	characteristic
256	C₂²	#11718872	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	#11718586	1	1	yes	characteristic
265	C₄	#11718586	1	1	yes	characteristic
266	C₄	#11718587	1	1	yes	characteristic
267	C₄	#11718587	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	4	2	no	—
271	C₂	#13114904	1	1	yes	characteristic
272	C₂	#13114940	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88180
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#117	C₂
socle	C₂²×C₁₃	#381
prime radical	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#8730
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #13112669	C₂	semidirect
3. #13112665	C₂	semidirect
31. C₂×C₄²×C₁₃	C₈	semidirect
162. C₂×C₄²	#1055	semidirect
216. C₁₃	#8730	semidirect

Maximal subgroups

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

subgroup	index
#13113150	2
#13112669	2
#13112665	2
C₄×C₁₃×#117	2
#8730	13