Subgroups — #13119427

Order 1664.

440 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	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#11718649	C₂	1	1	yes	maximal, characteristic
3	#11718649	C₂	1	1	yes	maximal, characteristic
4	#11718638	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718634	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718601	C₂	1	2	yes	maximal, complemented
7	#11718601	C₂	1	2	yes	maximal, complemented
8	#11718600	C₂	1	1	yes	maximal, characteristic
9	C₄×#6101	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718336	C₂	1	2	yes	maximal, complemented
11	#11718336	C₂	1	2	yes	maximal, complemented
12	C₂²×D₅₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
17	C₂×#6103	C₂²	1	2	yes	—
18	C₂×#6103	C₂²	1	2	yes	—
19	C₂×#6103	C₂²	1	2	yes	—
20	C₂×#6103	C₂²	1	2	yes	—
21	C₂×#6103	C₂²	1	2	yes	—
22	C₂×#6103	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×#6101	C₂²	1	1	yes	characteristic
25	C₂×#6101	C₂²	1	1	yes	characteristic
26	C₂×#6101	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#95	C₂²	1	2	yes	—
30	#88071	C₂²	1	1	yes	characteristic
31	#88071	C₂²	1	2	yes	—
32	#88071	C₂²	1	2	yes	—
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	1	yes	characteristic
35	C₂³×D₁₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂³	1	2	yes	—
44	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂×D₅₂	—	2	2	no	—
47	C₂×D₅₂	—	2	2	no	—
48	C₂×D₅₂	—	2	2	no	—
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₄²×C₁₃	—	2	2	no	—
52	C₄²×C₁₃	—	4	2	no	—
53	#6103	—	2	4	no	—
54	#6103	—	2	4	no	—
55	#6103	—	4	2	no	—
56	#6103	—	4	2	no	—
57	#6103	—	4	2	no	—
58	#6103	—	4	2	no	—
59	C₄⋊Dic₁₃	—	2	2	no	—
60	#6101	—	2	2	no	—
61	#6101	—	2	2	no	—
62	#6101	—	4	2	no	—
63	C₄×Dic₁₃	—	2	2	no	—
64	C₄×Dic₁₃	—	2	2	no	—
65	#2513	—	13	1	no	maximal, Sylow
66	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂²×D₁₃	—	2	2	no	—
68	C₂²×D₁₃	—	2	2	no	—
69	C₂²×D₁₃	—	2	2	no	—
70	C₂²×D₁₃	—	2	2	no	—
71	C₂²×D₁₃	—	2	2	no	—
72	C₂²×D₁₃	—	2	2	no	—
73	C₂²×D₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	#41	1	2	yes	—
86	C₂×C₄×C₁₃	#41	1	2	yes	—
87	C₂×C₄×C₁₃	#41	1	2	yes	—
88	C₂×C₄×C₁₃	#41	1	2	yes	—
89	C₂×Dic₁₃	—	2	1	no	—
90	C₂×Dic₁₃	—	2	1	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	1	no	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	—	2	1	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	2	no	—
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	—	2	1	no	—
103	C₂×Dic₁₃	#41	1	2	yes	—
104	D₅₂	—	4	4	no	—
105	#397	—	13	2	no	—
106	#397	—	13	2	no	—
107	#390	—	13	1	no	—
108	#390	—	13	1	no	—
109	#390	—	13	1	no	—
110	#389	—	13	1	no	—
111	#388	—	13	2	no	—
112	#388	—	13	2	no	—
113	#382	—	13	1	no	—
114	C₄×C₄⋊C₄	—	13	1	no	—
115	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
116	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×D₁₃	—	4	2	no	—
124	C₂×D₁₃	—	4	2	no	—
125	C₂×D₁₃	—	4	2	no	—
126	C₂×D₁₃	—	4	2	no	—
127	C₂×D₁₃	—	4	2	no	—
128	C₂×D₁₃	—	4	2	no	—
129	C₂×D₁₃	—	4	2	no	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	4	2	no	—
132	C₄×C₁₃	—	4	1	no	—
133	C₄×C₁₃	—	4	2	no	—
134	C₄×C₁₃	#126	1	4	yes	—
135	Dic₁₃	—	2	2	no	—
136	Dic₁₃	—	4	1	no	—
137	Dic₁₃	—	4	1	no	—
138	Dic₁₃	—	4	1	no	—
139	Dic₁₃	—	4	2	no	—
140	C₂²×D₄	—	13	1	no	—
141	C₂×C₄⋊C₄	—	13	1	no	—
142	C₂×C₄⋊C₄	—	13	1	no	—
143	C₂×C₄⋊C₄	—	13	1	no	—
144	C₂×C₄⋊C₄	—	13	2	no	—
145	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	—	13	2	no	—
148	C₂×C₂²⋊C₄	—	13	2	no	—
149	C₂×C₂²⋊C₄	—	13	2	no	—
150	C₂×C₂²⋊C₄	—	13	2	no	—
151	C₂×C₂²⋊C₄	—	13	2	no	—
152	C₂×C₂²⋊C₄	—	13	2	no	—
153	C₂×C₄²	—	13	1	no	—
154	C₂×C₄²	—	13	1	no	—
155	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
156	C₂×C₄²	C₂×D₁₃	1	2	yes	—
157	#95	—	13	1	no	—
158	#95	—	13	1	no	—
159	#95	—	13	2	no	—
160	#95	—	13	2	no	—
161	#95	—	13	2	no	—
162	#95	C₂×D₁₃	1	2	yes	—
163	C₂×C₁₃	C₂×#126	1	1	yes	characteristic
164	C₂×C₁₃	#529	1	1	yes	characteristic
165	C₂×C₁₃	#540	1	1	yes	characteristic
166	C₂×C₁₃	#553	1	1	yes	characteristic
167	C₂×C₁₃	#555	1	1	yes	characteristic
168	C₂×C₁₃	#559	1	1	yes	characteristic
169	C₂×C₁₃	#559	1	1	yes	characteristic
170	D₁₃	—	8	2	no	—
171	C₂⁴	—	13	2	no	—
172	C₂×D₄	—	26	2	no	—
173	C₂×D₄	—	26	2	no	—
174	C₂×D₄	—	26	2	no	—
175	C₂²×C₄	—	13	1	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	—	13	1	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
181	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
183	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	26	2	no	—
188	C₄⋊C₄	—	26	2	no	—
189	C₄⋊C₄	—	26	2	no	—
190	C₄⋊C₄	—	52	2	no	—
191	C₂²⋊C₄	—	26	4	no	—
192	C₂²⋊C₄	—	26	4	no	—
193	C₂²⋊C₄	—	52	2	no	—
194	C₂²⋊C₄	—	52	2	no	—
195	C₂²⋊C₄	—	52	2	no	—
196	C₂²⋊C₄	—	52	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	26	2	no	—
200	C₄²	—	26	2	no	—
201	C₁₃	#2513	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	26	2	no	—
203	C₂³	—	26	2	no	—
204	C₂³	—	26	2	no	—
205	C₂³	—	26	2	no	—
206	C₂³	—	26	2	no	—
207	C₂³	—	26	2	no	—
208	C₂³	—	26	2	no	—
209	C₂³	C₂³×D₁₃	1	1	yes	characteristic
210	D₄	—	52	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	13	2	no	—
223	C₂×C₄	—	26	1	no	—
224	C₂×C₄	—	26	1	no	—
225	C₂×C₄	—	26	1	no	—
226	C₂×C₄	—	26	1	no	—
227	C₂×C₄	—	26	1	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	26	1	no	—
230	C₂×C₄	—	26	1	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	26	1	no	—
233	C₂×C₄	—	26	1	no	—
234	C₂×C₄	—	26	1	no	—
235	C₂×C₄	—	26	2	no	—
236	C₂×C₄	—	26	1	no	—
237	C₂×C₄	#6127	1	2	yes	—
238	C₂×C₄	#6127	1	2	yes	—
239	C₂×C₄	#6127	1	2	yes	—
240	C₂×C₄	#6131	1	2	yes	—
241	C₂²	—	52	2	no	—
242	C₂²	—	52	2	no	—
243	C₂²	—	52	2	no	—
244	C₂²	—	52	2	no	—
245	C₂²	—	52	2	no	—
246	C₂²	—	52	2	no	—
247	C₂²	—	52	2	no	—
248	C₂²	C₂×#6127	1	1	yes	characteristic
249	C₂²	C₂×#6127	1	1	yes	characteristic
250	C₂²	C₂×#6127	1	1	yes	characteristic
251	C₂²	#88251	1	1	yes	characteristic
252	C₂²	C₂×#6131	1	1	yes	characteristic
253	C₂²	#41×D₁₃	1	1	yes	characteristic
254	C₂²	#88256	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	26	2	no	—
260	C₄	—	52	1	no	—
261	C₄	—	52	1	no	—
262	C₄	—	52	1	no	—
263	C₄	—	52	2	no	—
264	C₄	#88130	1	4	yes	—
265	C₂	—	104	2	no	—
266	C₂	C₂×#88130	1	1	yes	characteristic
267	C₂	#11719219	1	1	yes	characteristic
268	C₂	#11719237	1	1	yes	characteristic
269	C₂	#11719264	1	1	yes	characteristic
270	C₂	#11719267	1	1	yes	characteristic
271	C₂	#11719285	1	1	yes	characteristic
272	C₂	#11719290	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#2513
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₄×C₁₃×C₄⋊C₄	C₂	semidirect
4. #11718638	C₂	semidirect
5. #11718634	C₂	semidirect
6. #11718601	C₂	semidirect
7. #11718601	C₂	semidirect
9. C₄×#6101	C₂	semidirect
10. #11718336	C₂	semidirect
11. #11718336	C₂	semidirect
115. C₄×C₄⋊C₄	D₁₃	semidirect
201. C₁₃	#2513	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₃×C₄⋊C₄	2
#11718649	2
#11718638	2
#11718634	2
#11718601	2
#11718600	2
C₄×#6101	2
#11718336	2
#2513	13