Subgroups — #13126579

Order 1664.

482 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	C₂²×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88158	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719257	C₂	1	2	yes	maximal, complemented
4	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718777	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718776	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718506	C₂	1	2	yes	maximal, complemented
8	#11718399	C₂	1	2	yes	maximal, complemented
9	#11718394	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718357	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6127	C₂²	1	2	yes	—
13	C₂²×Dic₂₆	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
17	#88193	—	2	2	no	—
18	#88184	C₂²	1	2	yes	—
19	#88178	C₂²	1	2	yes	—
20	C₂×Dic₅₂	—	2	4	no	—
21	C₂×Dic₅₂	C₂²	1	2	yes	—
22	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
24	#88158	—	2	4	no	—
25	C₂×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₂×#6095	C₂²	1	2	yes	—
27	C₂×#6095	C₂²	1	2	yes	—
28	#88150	—	2	2	no	—
29	#88139	—	2	2	no	—
30	#88135	—	2	2	no	—
31	#88129	C₂²	1	2	yes	—
32	C₄×Dic₂₆	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
37	#88061	C₂²	1	2	yes	—
38	#88058	C₂²	1	1	yes	characteristic
39	#88056	C₂²	1	2	yes	—
40	#88056	C₂²	1	2	yes	—
41	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₃	—	2	2	no	—
45	C₂×C₂²⋊D₁₃	—	2	2	no	—
46	C₂²×Dic₁₃	—	2	2	no	—
47	#6127	—	2	4	no	—
48	#6127	—	4	2	no	—
49	C₂×D₅₂	C₂³	1	2	yes	—
50	C₂×C₄×D₁₃	—	2	2	no	—
51	C₂×Dic₂₆	—	2	4	no	—
52	C₂×Dic₂₆	—	2	2	no	—
53	C₂×Dic₂₆	—	4	2	no	—
54	C₂×Dic₂₆	C₂³	1	2	yes	—
55	C₂×Dic₂₆	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
64	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
65	C₄²×C₁₃	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₁₃	—	2	2	no	—
67	#6103	—	2	2	no	—
68	#6103	—	2	2	no	—
69	#6103	—	2	2	no	—
70	C₄⋊Dic₁₃	C₂³	1	2	yes	—
71	#6101	—	2	2	no	—
72	#6101	—	2	2	no	—
73	C₄×Dic₁₃	—	2	2	no	—
74	Dic₅₂	—	2	4	no	—
75	Dic₅₂	—	2	2	no	—
76	Dic₅₂	D₄	1	4	yes	complemented
77	D₁₀₄	—	2	2	no	—
78	#6095	—	2	4	no	—
79	#6095	—	4	2	no	—
80	#3285	—	13	1	no	maximal, Sylow
81	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×D₁₃	—	2	2	no	—
84	D₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₁₃	—	4	4	no	—
95	C₂²⋊D₁₃	—	4	4	no	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	2	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	4	4	no	—
100	D₅₂	—	2	2	no	—
101	D₅₂	—	2	2	no	—
102	C₄×D₁₃	—	4	4	no	—
103	Dic₂₆	—	2	2	no	—
104	Dic₂₆	—	2	2	no	—
105	Dic₂₆	—	2	2	no	—
106	Dic₂₆	—	4	4	no	—
107	Dic₂₆	C₂×D₄	1	4	yes	—
108	C₈×C₁₃	—	2	1	no	—
109	C₈×C₁₃	—	2	1	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	C₂×D₄	1	2	yes	—
112	C₂×#135	—	13	1	no	—
113	C₂²×Q₁₆	—	13	1	no	—
114	#548	—	13	2	no	—
115	#495	—	13	1	no	—
116	#467	—	13	1	no	—
117	#466	—	13	1	no	—
118	#464	—	13	2	no	—
119	#452	—	13	2	no	—
120	#448	—	13	2	no	—
121	C₄×Q₁₆	—	13	1	no	—
122	C₈×D₄	D₁₃	1	1	yes	characteristic, complemented
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	#135	1	2	yes	complemented
129	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×D₁₃	—	4	2	no	—
131	C₂×D₁₃	—	4	2	no	—
132	C₂×D₁₃	—	4	2	no	—
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	Dic₁₃	—	2	4	no	—
140	Dic₁₃	—	4	2	no	—
141	Dic₁₃	—	4	2	no	—
142	C₂×#41	—	13	2	no	—
143	C₂²×Q₈	—	13	2	no	—
144	#135	—	26	4	no	—
145	C₂×Q₁₆	—	13	1	no	—
146	C₂×Q₁₆	—	13	2	no	—
147	C₂×Q₁₆	—	13	1	no	—
148	C₂×Q₁₆	—	26	4	no	—
149	C₂×SD₁₆	—	13	2	no	—
150	C₂×SD₁₆	—	13	2	no	—
151	C₂×D₈	—	13	1	no	—
152	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
153	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
154	#124	—	13	2	no	—
155	#124	—	26	2	no	—
156	C₂²⋊Q₈	—	13	2	no	—
157	C₂²⋊Q₈	—	26	2	no	—
158	#121	—	13	2	no	—
159	#121	—	26	2	no	—
160	C₄×Q₈	—	13	2	no	—
161	C₄×D₄	—	26	2	no	—
162	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
163	#107	—	13	1	no	—
164	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
165	#103	—	13	2	no	—
166	#103	—	13	2	no	—
167	#102	—	13	2	no	—
168	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
169	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
170	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
171	C₂×C₁₃	—	2	2	no	—
172	C₂×C₁₃	—	4	1	no	—
173	C₂×C₁₃	D₄²	1	1	yes	characteristic
174	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
175	C₂×C₁₃	#578	1	1	yes	characteristic
176	D₁₃	—	8	2	no	—
177	#41	—	26	4	no	—
178	#41	—	52	2	no	—
179	C₂×Q₈	—	13	2	no	—
180	C₂×Q₈	—	13	2	no	—
181	C₂×Q₈	—	26	4	no	—
182	C₂×Q₈	—	26	2	no	—
183	C₂×Q₈	—	52	2	no	—
184	C₂×D₄	—	13	2	no	—
185	C₂×D₄	—	26	2	no	—
186	C₂×D₄	—	26	2	no	—
187	C₂×D₄	D₅₂	1	1	yes	characteristic
188	C₂²×C₄	—	26	2	no	—
189	C₂²×C₄	—	26	2	no	—
190	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
192	Q₁₆	—	13	4	no	—
193	Q₁₆	—	26	4	no	—
194	Q₁₆	—	26	2	no	—
195	SD₁₆	—	26	4	no	—
196	SD₁₆	—	52	2	no	—
197	D₈	—	26	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
201	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	13	2	no	—
205	C₄⋊C₄	—	26	2	no	—
206	C₄⋊C₄	—	26	2	no	—
207	C₄⋊C₄	D₅₂	1	1	yes	characteristic
208	C₂²⋊C₄	—	26	2	no	—
209	C₂²⋊C₄	—	26	2	no	—
210	C₂²⋊C₄	—	26	2	no	—
211	C₂²⋊C₄	—	26	2	no	—
212	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
213	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
214	C₄²	—	26	2	no	—
215	C₄²	C₂²×D₁₃	1	1	yes	characteristic
216	C₁₃	#3285	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	26	2	no	—
218	C₂³	C₂×D₅₂	1	1	yes	characteristic
219	C₂³	C₂×D₅₂	1	1	yes	characteristic
220	Q₈	—	13	4	no	—
221	Q₈	—	26	2	no	—
222	Q₈	—	26	2	no	—
223	Q₈	—	26	2	no	—
224	Q₈	—	52	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	26	2	no	—
227	D₄	—	26	2	no	—
228	D₄	—	52	4	no	—
229	D₄	—	52	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	26	2	no	—
235	C₂×C₄	—	26	2	no	—
236	C₂×C₄	—	26	2	no	—
237	C₂×C₄	—	52	4	no	—
238	C₂×C₄	—	52	4	no	—
239	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
240	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
241	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
242	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₁₃	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	52	2	no	—
254	C₂²	—	52	2	no	—
255	C₂²	—	52	2	no	—
256	C₂²	#88158	1	2	yes	complemented
257	C₂²	C₂²×D₅₂	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	26	4	no	—
263	C₄	—	52	2	no	—
264	C₄	—	52	2	no	—
265	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
266	C₄	#88256	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	104	2	no	—
270	C₂	D₄×D₅₂	1	1	yes	characteristic
271	C₂	C₂×#88158	1	1	yes	characteristic
272	C₂	#11719458	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₈×D₄×C₁₃	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	C₈×D₄	D₁₃
hypocenter	C₁₃	#3285
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₅₂	C₂	semidirect
2. C₂×#88158	C₂	semidirect
3. #11719257	C₂	semidirect
4. C₈×D₄×C₁₃	C₂	semidirect
5. #11718777	C₂	semidirect
6. #11718776	C₂	semidirect
7. #11718506	C₂	semidirect
8. #11718399	C₂	semidirect
9. #11718394	C₂	semidirect
10. C₄×Dic₅₂	C₂	semidirect
11. #11718357	C₂	semidirect
76. Dic₅₂	D₄	semidirect
122. C₈×D₄	D₁₃	semidirect
128. C₂²×C₁₃	#135	semidirect
216. C₁₃	#3285	semidirect
256. C₂²	#88158	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₂	2
C₂×#88158	2
#11719257	2
C₈×D₄×C₁₃	2
#11718777	2
#11718776	2
#11718506	2
#11718399	2
#11718394	2
C₄×Dic₅₂	2
#11718357	2
#3285	13