Subgroups — #13125475

Order 1664.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719239	C₂	1	2	yes	maximal, complemented
2	C₁₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718795	C₂	1	2	yes	maximal, complemented
4	#11718777	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718776	C₂	1	1	yes	maximal, characteristic
6	#11718375	C₂	1	2	yes	maximal, complemented
7	#11718371	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718357	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718343	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#117	C₂²	1	1	yes	characteristic
15	#88184	C₂²	1	2	yes	—
16	#88178	C₂²	1	2	yes	—
17	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
18	C₂×D₁₀₄	C₂²	1	1	yes	characteristic
19	C₂×#6095	C₂²	1	2	yes	—
20	#88152	—	2	2	no	—
21	#88150	—	2	2	no	—
22	#88139	—	2	2	no	—
23	#88137	—	2	2	no	—
24	#88129	C₂²	1	2	yes	—
25	C₄×D₅₂	C₂²	1	2	yes	—
26	#88123	C₂²	1	2	yes	—
27	C₄×Dic₂₆	C₂²	1	2	yes	—
28	C₁₃×#97	C₂²	1	1	yes	characteristic
29	C₁₃×#97	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	#88061	C₂²	1	2	yes	—
33	#88061	C₂²	1	2	yes	—
34	#88058	C₂²	1	1	yes	characteristic
35	#88058	C₂²	1	1	yes	characteristic
36	#88058	C₂²	1	2	yes	—
37	#88056	C₂²	1	2	yes	—
38	#88056	C₂²	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₃	—	2	2	no	—
41	C₂×D₅₂	C₂³	1	2	yes	—
42	C₂×C₄×D₁₃	—	2	2	no	—
43	C₂×Dic₂₆	—	2	2	no	—
44	C₂×Dic₂₆	C₂³	1	2	yes	—
45	C₁₃×M₁₆	—	2	2	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	C₂³	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	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
53	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
54	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₁₃	—	2	2	no	—
58	#6103	—	2	2	no	—
59	#6103	—	2	2	no	—
60	#6103	—	2	2	no	—
61	#6103	—	2	2	no	—
62	C₄⋊Dic₁₃	—	2	2	no	—
63	C₄⋊Dic₁₃	C₂³	1	2	yes	—
64	C₄⋊Dic₁₃	C₂³	1	2	yes	—
65	#6101	—	2	2	no	—
66	#6101	—	2	2	no	—
67	C₄×Dic₁₃	—	2	2	no	—
68	Dic₅₂	—	2	2	no	—
69	D₁₀₄	—	2	2	no	—
70	#6095	—	2	4	no	—
71	#3175	—	13	1	no	maximal, Sylow
72	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×D₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₁₃	—	8	2	no	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	D₅₂	—	2	2	no	—
87	D₅₂	—	2	2	no	—
88	C₄×D₁₃	—	8	2	no	—
89	Dic₂₆	—	2	2	no	—
90	Dic₂₆	—	2	2	no	—
91	Dic₂₆	—	8	2	no	—
92	C₈×C₁₃	—	2	1	no	—
93	C₈×C₁₃	—	2	1	no	—
94	C₈×C₁₃	#41	1	2	yes	—
95	C₈×C₁₃	#41	1	2	yes	—
96	#542	—	13	2	no	—
97	#500	—	13	1	no	—
98	#495	—	13	1	no	—
99	#489	—	13	2	no	—
100	#468	—	13	2	no	—
101	#467	—	13	1	no	—
102	#466	—	13	1	no	—
103	#440	—	13	2	no	—
104	C₄×Q₁₆	—	13	1	no	—
105	C₄×D₈	—	13	1	no	—
106	#405	D₁₃	1	1	yes	characteristic, complemented
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×D₁₃	—	4	2	no	—
112	C₂×D₁₃	—	4	2	no	—
113	C₂×D₁₃	—	4	2	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
120	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
121	Dic₁₃	—	4	2	no	—
122	Dic₁₃	—	4	2	no	—
123	Dic₁₃	—	4	2	no	—
124	C₂×Q₁₆	—	13	1	no	—
125	C₂×SD₁₆	—	13	2	no	—
126	C₂×D₈	—	13	1	no	—
127	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
129	C₄⋊Q₈	—	13	2	no	—
130	#126	—	26	2	no	—
131	#124	—	13	2	no	—
132	#124	—	26	2	no	—
133	#123	—	26	2	no	—
134	C₂²⋊Q₈	—	13	2	no	—
135	C₂²⋊Q₈	—	26	2	no	—
136	#121	—	13	2	no	—
137	C₄×Q₈	—	13	2	no	—
138	C₄×D₄	—	13	2	no	—
139	#117	C₂×D₁₃	1	1	yes	characteristic
140	#107	—	13	1	no	—
141	#107	—	13	1	no	—
142	#107	—	13	2	no	—
143	#103	—	13	2	no	—
144	#103	—	13	2	no	—
145	#102	—	13	2	no	—
146	#102	—	13	2	no	—
147	#97	C₂×D₁₃	1	1	yes	characteristic
148	#97	C₂×D₁₃	1	1	yes	characteristic
149	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
150	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂×C₁₃	—	4	1	no	—
152	C₂×C₁₃	#532	1	1	yes	characteristic
153	C₂×C₁₃	#576	1	1	yes	characteristic
154	C₂×C₁₃	#578	1	1	yes	characteristic
155	D₁₃	—	8	2	no	—
156	C₂×Q₈	—	13	2	no	—
157	C₂×Q₈	—	26	2	no	—
158	C₂×D₄	—	13	2	no	—
159	C₂×D₄	—	26	2	no	—
160	C₂²×C₄	—	26	2	no	—
161	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
162	Q₁₆	—	26	2	no	—
163	SD₁₆	—	26	4	no	—
164	D₈	—	26	2	no	—
165	M₁₆	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₄⋊C₄	—	13	2	no	—
172	C₄⋊C₄	—	13	2	no	—
173	C₄⋊C₄	—	26	2	no	—
174	C₄⋊C₄	—	26	2	no	—
175	C₄⋊C₄	—	26	2	no	—
176	C₄⋊C₄	D₅₂	1	1	yes	characteristic
177	C₄⋊C₄	D₅₂	1	1	yes	characteristic
178	C₂²⋊C₄	—	26	2	no	—
179	C₂²⋊C₄	—	26	2	no	—
180	C₂²⋊C₄	—	26	2	no	—
181	C₂²⋊C₄	—	26	2	no	—
182	C₂²⋊C₄	—	26	2	no	—
183	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
184	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
185	C₄²	—	26	2	no	—
186	C₄²	C₂²×D₁₃	1	1	yes	characteristic
187	C₄²	C₂²×D₁₃	1	1	yes	characteristic
188	C₁₃	#3175	1	1	yes	characteristic, complemented, Sylow
189	C₂³	—	26	2	no	—
190	C₂³	C₂×D₅₂	1	1	yes	characteristic
191	Q₈	—	26	2	no	—
192	Q₈	—	26	2	no	—
193	Q₈	—	104	2	no	—
194	D₄	—	26	2	no	—
195	D₄	—	26	2	no	—
196	D₄	—	104	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	26	2	no	—
200	C₂×C₄	—	26	2	no	—
201	C₂×C₄	—	26	2	no	—
202	C₂×C₄	—	104	2	no	—
203	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
204	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
205	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
206	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
207	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#6127	1	2	yes	—
212	C₈	#6127	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	52	2	no	—
217	C₂²	—	52	2	no	—
218	C₂²	—	52	2	no	—
219	C₂²	C₂²×D₅₂	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	52	2	no	—
226	C₄	—	52	2	no	—
227	C₄	—	52	2	no	—
228	C₄	C₂×#6127	1	1	yes	characteristic
229	C₄	C₂×#6127	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	104	2	no	—
232	C₂	#11719182	1	1	yes	characteristic
233	C₂	#11719457	1	1	yes	characteristic
234	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	C₁₃×#405	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	#11718776	C₂
hypercenter	#405	D₁₃
hypocenter	C₁₃	#3175
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #11719239	C₂	semidirect
2. C₁₃×#405	C₂	semidirect
3. #11718795	C₂	semidirect
4. #11718777	C₂	semidirect
6. #11718375	C₂	semidirect
7. #11718371	C₂	semidirect
8. C₄×Dic₅₂	C₂	semidirect
9. #11718357	C₂	semidirect
10. C₄×D₁₀₄	C₂	semidirect
11. #11718343	C₂	semidirect
106. #405	D₁₃	semidirect
188. C₁₃	#3175	semidirect

Maximal subgroups

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

subgroup	index
#11719239	2
C₁₃×#405	2
#11718795	2
#11718777	2
#11718776	2
#11718375	2
#11718371	2
C₄×Dic₅₂	2
#11718357	2
C₄×D₁₀₄	2
#11718343	2
#3175	13