Subgroups — #13117697

Order 1664.

532 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88126	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#6103	C₂	1	2	yes	maximal
4	#11718597	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718594	C₂	1	2	yes	maximal, complemented
6	#11718329	C₂	1	2	yes	maximal
7	#11718325	C₂	1	1	yes	maximal, characteristic
8	D₁₃×#95	C₂	1	1	yes	maximal, characteristic
9	#11718316	C₂	1	2	yes	maximal, complemented
10	#11718312	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718310	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#6103	C₂²	1	2	yes	—
17	C₂×#6103	C₂²	1	1	yes	characteristic
18	C₂×#6103	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×#6101	C₂²	1	2	yes	—
21	C₂×#6101	C₂²	1	1	yes	characteristic
22	C₂×#6101	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	#88126	C₄	1	8	yes	complemented
26	C₁₃×#95	C₂²	1	1	yes	characteristic
27	C₁₃×#95	C₂²	1	1	yes	characteristic
28	C₁₃×#95	C₂²	1	2	yes	—
29	#88071	C₂²	1	1	yes	characteristic
30	#88071	C₂²	1	2	yes	—
31	#88071	C₂²	1	1	yes	characteristic
32	#88071	C₂²	1	2	yes	—
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	2	yes	—
36	C₂³×D₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×Dic₁₃	C₂³	1	2	yes	—
43	C₂²×Dic₁₃	C₂³	1	2	yes	—
44	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₃	—	2	4	no	—
48	C₂×C₄×D₁₃	—	2	4	no	—
49	C₂×C₄×D₁₃	—	2	4	no	—
50	C₂×C₄×D₁₃	—	4	2	no	—
51	C₂×C₄×D₁₃	C₂×C₄	1	4	yes	—
52	C₄²×C₁₃	—	2	4	no	—
53	C₄²×C₁₃	C₂×C₄	1	4	yes	complemented
54	#6103	—	2	4	no	—
55	#6103	C₂×C₄	1	4	yes	—
56	#6103	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₁₃	—	4	2	no	—
58	#6101	—	2	4	no	—
59	#6101	C₂×C₄	1	4	yes	complemented
60	#6101	C₂×C₄	1	4	yes	complemented
61	C₄×Dic₁₃	—	4	2	no	—
62	C₄×Dic₁₃	C₂×C₄	1	4	yes	complemented
63	#2343	—	13	1	no	maximal, Sylow
64	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×D₁₃	—	2	2	no	—
66	C₂²×D₁₃	—	2	1	no	—
67	C₂²×D₁₃	—	2	1	no	—
68	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×D₁₃	#41	1	4	yes	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₃	#41	1	4	yes	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
95	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
96	C₄×D₁₃	—	2	8	no	—
97	C₄×D₁₃	—	4	8	no	—
98	C₂×#117	—	13	1	no	—
99	#387	—	13	2	no	—
100	#387	—	13	2	no	—
101	#385	—	13	2	no	—
102	#383	—	13	1	no	—
103	#382	—	13	1	no	—
104	#381	—	13	1	no	—
105	C₄×C₄⋊C₄	—	13	1	no	—
106	C₄×C₂²⋊C₄	—	13	2	no	—
107	#376	D₁₃	1	1	yes	characteristic, complemented
108	C₂×#95	—	13	1	no	—
109	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×#41	1	2	yes	—
111	C₂²×C₁₃	C₂×#41	1	2	yes	—
112	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×D₁₃	—	2	2	no	—
115	C₂×D₁₃	—	2	2	no	—
116	C₂×D₁₃	—	2	4	no	—
117	C₂×D₁₃	—	2	4	no	—
118	C₂×D₁₃	#117	1	4	yes	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	4	no	—
123	C₄×C₁₃	—	4	2	no	—
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	4	2	no	—
128	Dic₁₃	—	4	2	no	—
129	C₂³×C₄	—	13	1	no	—
130	C₂³×C₄	—	13	2	no	—
131	#117	—	13	8	no	—
132	C₂×C₄⋊C₄	—	13	1	no	—
133	C₂×C₄⋊C₄	—	13	1	no	—
134	C₂×C₄⋊C₄	—	13	2	no	—
135	C₂×C₄⋊C₄	—	13	2	no	—
136	C₂×C₂²⋊C₄	—	13	1	no	—
137	C₂×C₂²⋊C₄	—	13	1	no	—
138	C₂×C₂²⋊C₄	—	13	2	no	—
139	C₂×C₄²	—	13	1	no	—
140	C₂×C₄²	—	13	2	no	—
141	C₂×C₄²	C₂×D₁₃	1	2	yes	—
142	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
143	#95	—	13	1	no	—
144	#95	—	13	2	no	—
145	#95	—	13	2	no	—
146	#95	—	13	2	no	—
147	#95	—	13	1	no	—
148	#95	—	13	2	no	—
149	#95	—	13	2	no	—
150	#95	C₂×D₁₃	1	1	yes	characteristic
151	#95	C₂×D₁₃	1	1	yes	characteristic
152	#95	C₂×D₁₃	1	2	yes	—
153	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
154	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
155	C₂×C₁₃	#520	1	1	yes	characteristic
156	C₂×C₁₃	#551	1	2	yes	—
157	C₂×C₁₃	#552	1	2	yes	—
158	D₁₃	—	2	4	no	—
159	C₂⁴	—	13	1	no	—
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	1	no	—
163	C₂²×C₄	—	13	4	no	—
164	C₂²×C₄	—	13	2	no	—
165	C₂²×C₄	—	13	2	no	—
166	C₂²×C₄	—	26	4	no	—
167	C₂²×C₄	—	26	4	no	—
168	C₂²×C₄	—	26	4	no	—
169	C₂²×C₄	—	52	2	no	—
170	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
171	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
175	C₄⋊C₄	—	13	4	no	—
176	C₄⋊C₄	—	13	4	no	—
177	C₄⋊C₄	—	26	4	no	—
178	C₄⋊C₄	—	52	2	no	—
179	C₂²⋊C₄	—	13	4	no	—
180	C₂²⋊C₄	—	13	4	no	—
181	C₂²⋊C₄	—	26	4	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	13	4	no	—
184	C₄²	—	52	2	no	—
185	C₄²	C₄×D₁₃	1	4	yes	complemented
186	C₁₃	#2343	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	13	2	no	—
188	C₂³	—	13	4	no	—
189	C₂³	—	26	2	no	—
190	C₂³	—	26	1	no	—
191	C₂³	—	26	1	no	—
192	C₂³	C₂³×D₁₃	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	13	2	no	—
203	C₂×C₄	—	13	2	no	—
204	C₂×C₄	—	13	2	no	—
205	C₂×C₄	—	26	8	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	1	no	—
209	C₂×C₄	—	26	1	no	—
210	C₂×C₄	—	26	2	no	—
211	C₂×C₄	—	26	2	no	—
212	C₂×C₄	—	26	2	no	—
213	C₂×C₄	—	26	2	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	52	8	no	—
217	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
220	C₂×C₄	#6127	1	4	yes	—
221	C₂²	—	13	4	no	—
222	C₂²	—	26	2	no	—
223	C₂²	—	26	2	no	—
224	C₂²	—	26	4	no	—
225	C₂²	—	26	4	no	—
226	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
227	C₂²	C₂×#6127	1	2	yes	—
228	C₂²	#41×D₁₃	1	2	yes	—
229	C₂²	#88256	1	1	yes	characteristic
230	C₂²	#88257	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	—	26	2	no	—
237	C₄	—	26	2	no	—
238	C₄	—	26	2	no	—
239	C₄	—	52	2	no	—
240	C₄	—	52	2	no	—
241	C₂	—	26	4	no	—
242	C₂	C₄×#6127	1	1	yes	characteristic
243	C₂	D₁₃×#117	1	1	yes	characteristic
244	C₂	#11719228	1	1	yes	characteristic
245	C₂	#11719232	1	2	yes	—
246	C₂	#11719234	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#376	C₂
socle, cosocle	C₂³×C₁₃	C₂⁴
prime radical	C₂³×D₁₃	C₂³
hypercenter	#376	D₁₃
hypocenter	C₁₃	#2343
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #376

Complements

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

N	H ≅ G/N	type
2. C₁₃×#376	C₂	semidirect
4. #11718597	C₂	semidirect
5. #11718594	C₂	semidirect
9. #11718316	C₂	semidirect
10. #11718312	C₂	semidirect
11. #11718310	C₂	semidirect
25. #88126	C₄	semidirect
53. C₄²×C₁₃	C₂×C₄	semidirect
59. #6101	C₂×C₄	semidirect
60. #6101	C₂×C₄	semidirect
62. C₄×Dic₁₃	C₂×C₄	semidirect
107. #376	D₁₃	semidirect
185. C₄²	C₄×D₁₃	semidirect
186. C₁₃	#2343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88126	2
C₁₃×#376	2
C₄×#6103	2
#11718597	2
#11718594	2
#11718329	2
#11718325	2
D₁₃×#95	2
#11718316	2
#11718312	2
#11718310	2
#2343	13