Subgroups — #13123998

Order 1664.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719184	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718501	C₂	1	2	yes	maximal, complemented
5	#11718500	C₂	1	2	yes	maximal, complemented
6	#11718391	C₂	1	2	yes	maximal, complemented
7	#11718386	C₂	1	2	yes	maximal, complemented
8	#11718349	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718340	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	#88184	C₄	1	1	yes	characteristic
14	#88183	C₄	1	2	yes	—
15	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
16	#88178	C₄	1	1	yes	characteristic
17	C₂²×#1055	C₂²	1	2	yes	—
18	C₂×#6094	C₂²	1	2	yes	—
19	C₂×C₈×D₁₃	C₂²	1	2	yes	—
20	#88130	C₄	1	2	yes	—
21	#88129	C₄	1	1	yes	characteristic
22	C₄×D₅₂	C₂²	1	1	yes	characteristic
23	#88126	C₂²	1	2	yes	—
24	#88124	C₄	1	1	yes	characteristic
25	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	#88060	C₂²	1	2	yes	—
33	#88060	C₂²	1	2	yes	—
34	#88055	C₂²	1	2	yes	—
35	#88054	C₂²	1	2	yes	—
36	#88054	C₂²	1	2	yes	—
37	C₈×Dic₁₃	C₂²	1	2	yes	—
38	C₄⋊#1055	C₂²	1	2	yes	—
39	C₄⋊#1055	C₂²	1	1	yes	characteristic
40	C₄⋊#1055	C₂²	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
45	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂³	1	2	yes	—
47	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
57	#6103	C₂×C₄	1	2	yes	—
58	#6103	C₂×C₄	1	2	yes	—
59	#6103	C₂×C₄	1	2	yes	—
60	#6103	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
63	#6101	C₂×C₄	1	2	yes	—
64	#6101	C₂×C₄	1	2	yes	—
65	#6101	C₂×C₄	1	2	yes	—
66	#6101	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₃	C₂³	1	2	yes	—
68	C₂×#1055	—	4	2	no	—
69	C₂×#1055	C₂³	1	2	yes	—
70	C₂×#1055	C₂³	1	2	yes	—
71	#6094	—	4	2	no	—
72	C₈×D₁₃	—	4	2	no	—
73	#2994	—	13	1	no	maximal, Sylow
74	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₁₃	—	2	4	no	—
87	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
90	D₅₂	—	2	2	no	—
91	C₄×D₁₃	—	2	2	no	—
92	C₄×D₁₃	—	2	2	no	—
93	Dic₂₆	—	2	2	no	—
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	—	2	1	no	—
98	#1055	—	2	2	no	—
99	#1055	—	2	2	no	—
100	#529	—	13	1	no	—
101	#446	—	13	1	no	—
102	C₈×Q₈	—	13	1	no	—
103	#436	—	13	1	no	—
104	#435	—	13	2	no	—
105	C₈×D₄	—	13	2	no	—
106	C₈×D₄	—	13	1	no	—
107	#433	—	13	2	no	—
108	#431	D₁₃	1	1	yes	characteristic, complemented
109	#424	—	13	2	no	—
110	C₂×C₄⋊C₈	—	13	1	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	#131	1	2	yes	—
114	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₁₃	—	2	2	no	—
116	C₂×D₁₃	—	2	2	no	—
117	C₂×D₁₃	—	2	2	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	#131	1	2	yes	—
123	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	2	2	no	—
128	C₂×M₁₆	—	13	2	no	—
129	C₂²×C₈	—	13	2	no	—
130	C₂²×C₈	—	13	2	no	—
131	#126	—	13	2	no	—
132	#125	—	13	1	no	—
133	#124	—	13	1	no	—
134	#123	—	13	2	no	—
135	C₂²⋊Q₈	—	13	1	no	—
136	#121	—	13	1	no	—
137	C₄×Q₈	—	13	1	no	—
138	C₄×D₄	—	13	2	no	—
139	C₄×D₄	—	13	1	no	—
140	#117	—	13	2	no	—
141	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
142	C₄⋊C₈	—	13	2	no	—
143	C₄⋊C₈	—	13	2	no	—
144	C₄⋊C₈	—	13	2	no	—
145	C₄⋊C₈	—	13	1	no	—
146	C₄⋊C₈	—	13	1	no	—
147	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
149	C₂²⋊C₈	—	13	2	no	—
150	C₂²⋊C₈	—	13	2	no	—
151	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
153	#97	—	13	2	no	—
154	C₄×C₈	—	13	2	no	—
155	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
156	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	#520	1	1	yes	characteristic
159	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
161	D₁₃	—	4	2	no	—
162	C₂×Q₈	—	13	1	no	—
163	C₂×D₄	—	13	2	no	—
164	C₂×D₄	—	13	1	no	—
165	C₂²×C₄	—	13	2	no	—
166	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
169	M₁₆	—	52	2	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	13	2	no	—
172	C₂×C₈	—	52	2	no	—
173	C₂×C₈	—	52	2	no	—
174	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
175	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₄⋊C₄	—	13	2	no	—
179	C₄⋊C₄	—	13	2	no	—
180	C₄⋊C₄	—	13	1	no	—
181	C₄⋊C₄	—	13	2	no	—
182	C₄⋊C₄	—	13	1	no	—
183	C₄⋊C₄	—	13	2	no	—
184	C₂²⋊C₄	—	13	2	no	—
185	C₂²⋊C₄	—	13	2	no	—
186	C₂²⋊C₄	—	13	2	no	—
187	C₂²⋊C₄	—	13	2	no	—
188	C₂²⋊C₄	—	13	2	no	—
189	C₄²	—	13	2	no	—
190	C₄²	C₄×D₁₃	1	1	yes	characteristic
191	C₄²	C₄×D₁₃	1	1	yes	characteristic
192	C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	C₁₃	#2994	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	13	2	no	—
196	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
197	Q₈	—	26	2	no	—
198	D₄	—	26	4	no	—
199	D₄	—	26	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	13	2	no	—
205	C₂×C₄	—	13	2	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	2	no	—
209	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	26	2	no	—
220	C₈	—	26	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	26	2	no	—
224	C₂²	—	26	2	no	—
225	C₂²	—	26	2	no	—
226	C₂²	#88155	1	2	yes	—
227	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	26	2	no	—
233	C₄	—	26	2	no	—
234	C₄	—	26	2	no	—
235	C₄	#88161	1	2	yes	—
236	C₄	#88256	1	1	yes	characteristic
237	C₄	#88257	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	52	2	no	—
240	C₂	#11719228	1	1	yes	characteristic
241	C₂	C₂×#88155	1	1	yes	characteristic
242	C₂	C₂×#88161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#431	C₂
prime radical	#88184	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#431	D₁₃
hypocenter	C₁₃	#2994
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. C₁₃×#431	C₂	semidirect
3. C₂×C₄⋊#1055	C₂	semidirect
4. #11718501	C₂	semidirect
5. #11718500	C₂	semidirect
6. #11718391	C₂	semidirect
7. #11718386	C₂	semidirect
8. #11718349	C₂	semidirect
9. C₈×D₅₂	C₂	semidirect
10. #11718340	C₂	semidirect
11. C₈×Dic₂₆	C₂	semidirect
108. #431	D₁₃	semidirect
194. C₁₃	#2994	semidirect

Maximal subgroups

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

subgroup	index
#11719184	2
C₁₃×#431	2
C₂×C₄⋊#1055	2
#11718501	2
#11718500	2
#11718391	2
#11718386	2
#11718349	2
C₈×D₅₂	2
#11718340	2
C₈×Dic₂₆	2
#2994	13