Subgroups — #13123913

Order 1664.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719253	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718790	C₂	1	2	yes	maximal, complemented
4	#11718788	C₂	1	2	yes	maximal, complemented
5	D₄×#1055	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718496	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718495	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718391	C₂	1	2	yes	maximal, complemented
9	#11718388	C₂	1	2	yes	maximal, complemented
10	#11718349	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718346	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6127	C₂²	1	2	yes	—
13	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
14	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	#88194	C₄	1	1	yes	characteristic
16	#88192	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
18	C₂²×#1055	C₂²	1	2	yes	—
19	C₂×#6094	C₂²	1	2	yes	—
20	C₂×C₈×D₁₃	C₂²	1	2	yes	—
21	#88149	C₄	1	1	yes	characteristic
22	#88142	C₄	1	1	yes	characteristic
23	#88139	C₄	1	2	yes	—
24	#88138	C₄	1	2	yes	—
25	C₄×D₅₂	C₂²	1	1	yes	characteristic
26	C₄²×D₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
30	C₂²⋊#1055	C₂²	1	2	yes	—
31	#88060	C₂²	1	2	yes	—
32	#88060	C₂²	1	2	yes	—
33	#88060	C₂²	1	2	yes	—
34	#88060	C₂²	1	1	yes	characteristic
35	#88060	C₂²	1	1	yes	characteristic
36	#88055	C₂²	1	2	yes	—
37	#88054	C₂²	1	1	yes	characteristic
38	#88054	C₂²	1	1	yes	characteristic
39	C₄⋊#1055	C₂²	1	1	yes	characteristic
40	C₄×#1055	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	#6127	—	4	2	no	—
47	#6127	—	4	2	no	—
48	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
49	C₂×D₅₂	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₃	C₂³	1	2	yes	—
54	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
55	C₁₃×M₁₆	—	4	2	no	—
56	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂³	1	2	yes	—
59	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
60	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
61	C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
63	#6103	C₂×C₄	1	2	yes	—
64	#6103	C₂×C₄	1	2	yes	—
65	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
66	#6101	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
68	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₂×#1055	—	4	2	no	—
71	C₂×#1055	C₂³	1	1	yes	characteristic
72	C₂×#1055	C₂³	1	2	yes	—
73	C₂×#1055	C₂³	1	1	yes	characteristic
74	#6094	—	4	2	no	—
75	C₈×D₁₃	—	4	2	no	—
76	#2981	—	13	1	no	maximal, Sylow
77	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
80	D₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₁₃	—	4	2	no	—
88	C₂²⋊D₁₃	—	4	2	no	—
89	C₂²⋊D₁₃	—	4	2	no	—
90	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
94	D₅₂	—	2	2	no	—
95	D₅₂	—	4	2	no	—
96	C₄×D₁₃	—	2	2	no	—
97	C₄×D₁₃	—	2	2	no	—
98	C₄×D₁₃	—	2	2	no	—
99	C₄×D₁₃	—	2	1	no	—
100	C₄×D₁₃	—	2	1	no	—
101	C₄×D₁₃	—	4	1	no	—
102	Dic₂₆	—	4	2	no	—
103	C₈×C₁₃	—	2	2	no	—
104	C₈×C₁₃	—	2	1	no	—
105	C₈×C₁₃	—	2	1	no	—
106	#1055	—	2	1	no	—
107	#1055	—	2	2	no	—
108	#1055	—	2	1	no	—
109	#532	—	13	1	no	—
110	#436	—	13	1	no	—
111	#436	D₁₃	1	1	yes	characteristic, complemented
112	#435	—	13	2	no	—
113	#435	—	13	2	no	—
114	C₈×D₄	—	13	1	no	—
115	C₈×D₄	—	13	1	no	—
116	#432	—	13	1	no	—
117	#431	—	13	1	no	—
118	#408	—	13	2	no	—
119	#408	—	13	2	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₁₃	—	2	1	no	—
125	C₂×D₁₃	—	2	2	no	—
126	C₂×D₁₃	—	2	1	no	—
127	C₂×D₁₃	—	2	2	no	—
128	C₂×D₁₃	—	2	1	no	—
129	C₂×D₁₃	—	2	2	no	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	#131	1	2	yes	—
133	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
134	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
135	Dic₁₃	—	2	1	no	—
136	Dic₁₃	—	2	2	no	—
137	Dic₁₃	—	2	1	no	—
138	Dic₁₃	—	2	1	no	—
139	C₂×#41	—	13	2	no	—
140	C₂×M₁₆	—	13	2	no	—
141	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
142	C₂²×C₈	—	13	2	no	—
143	C₂²×C₈	—	13	2	no	—
144	C₄⋊Q₈	—	13	1	no	—
145	#127	—	13	1	no	—
146	#124	—	13	2	no	—
147	#121	—	13	1	no	—
148	#121	—	13	1	no	—
149	#121	—	13	2	no	—
150	C₄×D₄	—	13	2	no	—
151	C₄×D₄	—	13	1	no	—
152	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
153	C₂×C₄²	—	13	1	no	—
154	C₄⋊C₈	—	13	1	no	—
155	C₄⋊C₈	—	13	1	no	—
156	C₄⋊C₈	—	13	1	no	—
157	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
158	C₂²⋊C₈	—	13	2	no	—
159	C₂²⋊C₈	—	13	1	no	—
160	C₂²⋊C₈	—	13	1	no	—
161	C₂²⋊C₈	—	13	2	no	—
162	C₂²⋊C₈	—	13	2	no	—
163	C₂²⋊C₈	—	13	2	no	—
164	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
165	#97	—	13	2	no	—
166	C₄×C₈	—	13	1	no	—
167	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
168	C₂×C₁₃	—	4	2	no	—
169	C₂×C₁₃	#518	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
171	C₂×C₁₃	#568	1	1	yes	characteristic
172	D₁₃	—	4	1	no	—
173	D₁₃	—	4	2	no	—
174	#41	—	52	2	no	—
175	#41	—	52	2	no	—
176	C₂×Q₈	—	13	2	no	—
177	C₂×D₄	—	13	2	no	—
178	C₂×D₄	—	13	2	no	—
179	C₂×D₄	—	13	2	no	—
180	C₂×D₄	—	13	1	no	—
181	C₂×D₄	—	13	2	no	—
182	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
183	C₂²×C₄	—	13	1	no	—
184	C₂²×C₄	—	13	1	no	—
185	C₂²×C₄	—	13	2	no	—
186	C₂²×C₄	—	13	1	no	—
187	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	52	2	no	—
190	C₂×C₈	—	13	1	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	52	2	no	—
194	C₂×C₈	—	52	2	no	—
195	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
196	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
197	C₂×C₈	C₂²×D₁₃	1	2	yes	—
198	C₄⋊C₄	—	13	2	no	—
199	C₄⋊C₄	—	13	1	no	—
200	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
201	C₂²⋊C₄	—	13	2	no	—
202	C₂²⋊C₄	—	13	2	no	—
203	C₂²⋊C₄	—	13	2	no	—
204	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
205	C₄²	—	13	1	no	—
206	C₄²	—	13	1	no	—
207	C₄²	—	13	1	no	—
208	C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	C₁₃	#2981	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	13	2	no	—
211	C₂³	—	13	1	no	—
212	C₂³	C₂×C₄×D₁₃	1	2	yes	—
213	Q₈	—	52	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	26	2	no	—
216	D₄	—	52	2	no	—
217	D₄	—	52	2	no	—
218	D₄	—	52	2	no	—
219	D₄	—	52	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	13	1	no	—
223	C₂×C₄	—	13	1	no	—
224	C₂×C₄	—	13	1	no	—
225	C₂×C₄	—	13	2	no	—
226	C₂×C₄	—	26	2	no	—
227	C₂×C₄	—	26	2	no	—
228	C₂×C₄	—	26	1	no	—
229	C₂×C₄	—	26	2	no	—
230	C₂×C₄	—	26	1	no	—
231	C₂×C₄	—	52	1	no	—
232	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	26	1	no	—
240	C₈	—	26	1	no	—
241	C₈	—	26	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	26	1	no	—
246	C₂²	—	26	2	no	—
247	C₂²	—	26	1	no	—
248	C₂²	—	26	2	no	—
249	C₂²	—	26	1	no	—
250	C₂²	—	26	2	no	—
251	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	26	1	no	—
255	C₄	—	26	1	no	—
256	C₄	—	26	1	no	—
257	C₄	—	26	2	no	—
258	C₄	#88161	1	2	yes	—
259	C₄	#88251	1	1	yes	characteristic
260	C₄	#88256	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	52	1	no	—
263	C₂	—	52	2	no	—
264	C₂	#11719250	1	1	yes	characteristic
265	C₂	C₂×#88161	1	1	yes	characteristic
266	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#436	C₂
prime radical	#11719253	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#436	D₁₃
hypocenter	C₁₃	#2981
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₁₃×#436	C₂	semidirect
3. #11718790	C₂	semidirect
4. #11718788	C₂	semidirect
5. D₄×#1055	C₂	semidirect
6. #11718496	C₂	semidirect
7. #11718495	C₂	semidirect
8. #11718391	C₂	semidirect
9. #11718388	C₂	semidirect
10. #11718349	C₂	semidirect
11. #11718346	C₂	semidirect
111. #436	D₁₃	semidirect
209. C₁₃	#2981	semidirect

Maximal subgroups

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

subgroup	index
#11719253	2
C₁₃×#436	2
#11718790	2
#11718788	2
D₄×#1055	2
#11718496	2
#11718495	2
#11718391	2
#11718388	2
#11718349	2
#11718346	2
#2981	13