Subgroups — #15873251

Order 3328.

978 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	C₂×#11718586	C₂	1	2	yes	maximal
2	#13115555	C₂	1	8	yes	maximal, complemented
3	#13114816	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
6	C₂²×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
10	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₁₃×#95	C₄	1	2	yes	—
12	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
13	#11718586	—	2	8	no	—
14	C₂×C₄⋊#1055	C₂²	1	4	yes	—
15	C₂×C₄⋊#1055	C₂²	1	8	yes	—
16	C₂×C₄×#1055	C₂²	1	4	yes	—
17	#11718116	C₂²	1	8	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
24	C₂×C₁₃×C₄⋊C₄	C₈	1	8	yes	complemented
25	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₃	—	4	4	no	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	D₄	1	4	yes	—
31	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
34	C₂²×#1055	C₂³	1	8	yes	—
35	C₁₃×#95	C₂×C₄	1	8	yes	—
36	C₂²⋊#1055	—	2	8	no	—
37	C₄⋊#1055	—	2	8	no	—
38	C₄⋊#1055	—	4	8	no	—
39	C₄×#1055	—	2	8	no	—
40	#21951	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	—	4	1	no	—
48	C₂²×C₄×C₁₃	—	4	1	no	—
49	C₂²×C₄×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
54	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	8	no	—
65	C₁₃×C₄⋊C₄	C₂×C₈	1	16	yes	—
66	C₄²×C₁₃	—	2	8	no	—
67	C₄²×C₁₃	—	4	4	no	—
68	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
69	C₄²×C₁₃	C₂×D₄	1	4	yes	—
70	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
71	C₂×#1055	—	2	8	no	—
72	C₂×#1055	—	2	8	no	—
73	C₂×#1055	—	2	8	no	—
74	C₂×#431	—	13	2	no	—
75	C₂×C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
76	#1939	—	13	8	no	—
77	#1797	—	13	4	no	—
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	—	2	1	no	—
82	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
84	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
85	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	4	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	4	8	no	—
99	C₂×C₄×C₁₃	—	4	8	no	—
100	C₂×C₄×C₁₃	C₄⋊C₈	1	16	yes	complemented
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
105	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
110	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
111	#1055	—	4	8	no	—
112	C₂²×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
113	C₂²×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
114	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
115	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
116	#431	—	26	8	no	—
117	C₂×C₄⋊C₈	—	13	4	no	—
118	C₂×C₄⋊C₈	—	13	8	no	—
119	C₂×C₂²⋊C₈	—	13	4	no	—
120	C₂×C₄×C₈	—	13	4	no	—
121	C₄×C₄⋊C₄	Dic₁₃	1	4	yes	—
122	C₄×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
123	C₂×#95	Dic₁₃	1	2	yes	—
124	#336	—	13	8	no	—
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
132	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₁₃	#518	1	1	yes	characteristic
134	C₂²×C₁₃	#519	1	1	yes	characteristic
135	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
136	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
137	C₂²×C₁₃	#568	1	1	yes	characteristic
138	C₂²×C₁₃	#568	1	1	yes	characteristic
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	—	2	8	no	—
142	C₄×C₁₃	—	4	4	no	—
143	C₄×C₁₃	C₂×C₄⋊C₈	1	8	yes	—
144	C₄×C₁₃	#537	1	2	yes	—
145	C₄×C₁₃	#544	1	2	yes	—
146	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
147	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
148	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
149	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
151	C₂²×C₈	—	13	8	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	#1055	1	8	yes	complemented
154	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
155	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
156	C₂×C₄²	—	4	4	no	—
157	C₂×C₄²	Dic₂₆	1	4	yes	—
158	C₂×C₄²	D₅₂	1	4	yes	—
159	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
160	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
161	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
162	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
163	C₄⋊C₈	—	26	8	no	—
164	C₄⋊C₈	—	52	8	no	—
165	C₂²⋊C₈	—	26	8	no	—
166	C₄×C₈	—	26	8	no	—
167	#95	C₂×Dic₁₃	1	8	yes	—
168	C₂×C₁₃	—	2	4	no	—
169	C₂×C₁₃	#2316	1	1	yes	characteristic
170	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
171	C₂×C₁₃	#2905	1	1	yes	characteristic
172	C₂×C₁₃	#2972	1	1	yes	characteristic
173	C₂×C₁₃	#2975	1	1	yes	characteristic
174	C₂×C₁₃	#2982	1	1	yes	characteristic
175	C₂×C₁₃	#2983	1	1	yes	characteristic
176	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	C₂×#1055	1	8	yes	—
187	C₂²×C₄	C₂×#1055	1	4	yes	—
188	C₂²×C₄	#6099	1	4	yes	—
189	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
190	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
191	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
192	C₂²×C₄	C₂×D₅₂	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
195	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
196	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
198	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
199	C₂×C₈	—	26	8	no	—
200	C₂×C₈	—	26	8	no	—
201	C₂×C₈	—	26	8	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	C₂×#1055	1	16	yes	—
204	C₄²	—	2	8	no	—
205	C₄²	—	4	4	no	—
206	C₄²	C₄⋊Dic₁₃	1	8	yes	—
207	C₄²	C₂×Dic₂₆	1	4	yes	—
208	C₄²	C₂×D₅₂	1	4	yes	—
209	C₁₃	#21951	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	C₂²×#1055	1	2	yes	—
215	C₂³	C₂×#6099	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
217	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	C₄⋊#1055	1	16	yes	complemented
233	C₂×C₄	C₂²×#1055	1	8	yes	—
234	C₂×C₄	C₂²×#1055	1	4	yes	—
235	C₂×C₄	C₂×#6099	1	4	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
239	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
241	C₂×C₄	#88251	1	2	yes	—
242	C₂×C₄	#88254	1	2	yes	—
243	C₈	—	52	8	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	C₂×C₄⋊#1055	1	4	yes	—
251	C₂²	C₂³×#1055	1	1	yes	characteristic
252	C₂²	C₂²×#6099	1	1	yes	characteristic
253	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
254	C₂²	#11719503	1	1	yes	characteristic
255	C₂²	#11719517	1	1	yes	characteristic
256	C₂²	#11719524	1	1	yes	characteristic
257	C₂²	#11719524	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	8	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	4	no	—
262	C₄	C₂×C₄⋊#1055	1	8	yes	—
263	C₄	#11719205	1	2	yes	—
264	C₄	#11719215	1	2	yes	—
265	C₂	—	2	4	no	—
266	C₂	#13117454	1	1	yes	characteristic
267	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic
268	C₂	#13123400	1	1	yes	characteristic
269	C₂	#13123860	1	1	yes	characteristic
270	C₂	#13123877	1	1	yes	characteristic
271	C₂	#13123917	1	1	yes	characteristic
272	C₂	#13123921	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₄×C₁₃×C₄⋊C₄	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#21951
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #13115555	C₂	semidirect
24. C₂×C₁₃×C₄⋊C₄	C₈	semidirect
100. C₂×C₄×C₁₃	C₄⋊C₈	semidirect
153. C₂×C₄⋊C₄	#1055	semidirect
209. C₁₃	#21951	semidirect
232. C₂×C₄	C₄⋊#1055	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
#13115555	2
#13114816	2
C₂×C₄×C₁₃×C₄⋊C₄	2
#21951	13