Subgroups — #13123711

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88173	C₂	1	1	yes	maximal, characteristic
2	#11719279	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718687	C₂	1	2	yes	maximal, complemented
6	D₁₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718533	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718478	C₂	1	2	yes	maximal, complemented
9	#11718464	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718350	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6131	C₂²	1	2	yes	—
13	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	Q₈×Dic₁₃	C₂²	1	2	yes	—
16	C₂×#6106	C₂²	1	2	yes	—
17	#88173	C₄	1	8	yes	complemented
18	C₂×D₁₀₄	C₂²	1	1	yes	characteristic
19	C₂×#6094	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
21	#88147	—	2	2	no	—
22	#88147	C₂²	1	2	yes	—
23	C₄⋊C₄×D₁₃	—	2	2	no	—
24	C₄⋊C₄×D₁₃	C₂²	1	2	yes	—
25	C₄×D₅₂	—	2	2	no	—
26	C₄×D₅₂	C₂²	1	2	yes	—
27	#88126	—	2	2	no	—
28	#88126	C₂²	1	1	yes	characteristic
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#103	C₂²	1	2	yes	—
31	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	#88075	C₂²	1	2	yes	—
33	#88061	C₂²	1	2	yes	—
34	#88058	C₂²	1	1	yes	characteristic
35	#88055	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
37	#88049	C₂²	1	2	yes	—
38	#88047	C₂²	1	2	yes	—
39	#88043	C₂²	1	1	yes	characteristic
40	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
41	#6131	—	2	4	no	—
42	#6131	C₂×C₄	1	8	yes	complemented
43	C₂×D₅₂	—	2	2	no	—
44	C₂×D₅₂	C₂³	1	2	yes	—
45	C₂×C₄×D₁₃	—	2	2	no	—
46	C₂×C₄×D₁₃	—	2	2	no	—
47	C₂×C₄×D₁₃	—	2	2	no	—
48	C₂×C₄×D₁₃	—	2	2	no	—
49	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
50	C₁₃×Q₁₆	C₂×C₄	1	4	yes	complemented
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₄²×C₁₃	—	2	2	no	—
56	C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	#6106	C₂×C₄	1	8	yes	complemented
58	#6103	—	2	2	no	—
59	#6103	—	2	2	no	—
60	#6103	D₄	1	1	yes	characteristic
61	#6103	D₄	1	1	yes	characteristic
62	C₄⋊Dic₁₃	—	2	2	no	—
63	C₄⋊Dic₁₃	C₂³	1	2	yes	—
64	#6101	—	2	2	no	—
65	#6101	D₄	1	1	yes	characteristic
66	#6101	D₄	1	1	yes	characteristic
67	C₄×Dic₁₃	—	2	2	no	—
68	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	D₁₀₄	C₂×C₄	1	4	yes	complemented
72	#6094	—	2	2	no	—
73	C₈×D₁₃	C₂×C₄	1	4	yes	complemented
74	#2951	—	13	1	no	maximal, Sylow
75	C₂²×D₁₃	—	2	2	no	—
76	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₁₃	—	2	2	no	—
78	Q₈×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
88	D₅₂	—	2	8	no	—
89	D₅₂	—	2	2	no	—
90	D₅₂	C₂²×C₄	1	4	yes	—
91	C₄×D₁₃	—	2	8	no	—
92	C₄×D₁₃	—	8	2	no	—
93	C₄×D₁₃	—	8	2	no	—
94	C₄×D₁₃	—	8	2	no	—
95	C₄×D₁₃	C₂²×C₄	1	2	yes	—
96	C₄×D₁₃	#41	1	2	yes	—
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	C₂²×C₄	1	2	yes	—
99	#1055	—	2	1	no	—
100	#1055	C₂²×C₄	1	2	yes	—
101	C₂×#135	—	13	1	no	—
102	#520	—	13	2	no	—
103	#440	—	13	2	no	—
104	#440	—	13	2	no	—
105	C₄×Q₁₆	—	13	1	no	—
106	C₄×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
107	C₄×D₈	—	13	1	no	—
108	C₄×D₈	—	13	1	no	—
109	C₂×#107	—	13	1	no	—
110	#417	—	13	2	no	—
111	#405	—	13	1	no	—
112	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×D₁₃	—	2	1	no	—
114	C₂×D₁₃	—	2	1	no	—
115	C₂×D₁₃	—	2	4	no	—
116	C₂×D₁₃	—	4	2	no	—
117	C₂×D₁₃	—	4	2	no	—
118	C₂×D₁₃	C₄×D₄	1	2	yes	—
119	C₄×C₁₃	—	2	4	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	Dic₁₃	—	2	1	no	—
126	Dic₁₃	—	2	1	no	—
127	Dic₁₃	—	4	2	no	—
128	Dic₁₃	C₄×D₄	1	2	yes	—
129	C₂×#41	—	13	2	no	—
130	#135	—	13	8	no	—
131	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
132	C₂×SD₁₆	—	13	2	no	—
133	C₂×D₈	—	13	1	no	—
134	C₂×M₁₆	—	13	1	no	—
135	C₂²×C₈	—	13	1	no	—
136	C₄×Q₈	—	13	2	no	—
137	C₄×Q₈	C₂×D₁₃	1	2	yes	—
138	C₄×D₄	—	13	2	no	—
139	C₄×D₄	—	13	2	no	—
140	C₄×D₄	—	26	2	no	—
141	C₄×D₄	—	26	2	no	—
142	#117	—	13	1	no	—
143	#117	—	26	2	no	—
144	C₂×C₄⋊C₄	—	13	2	no	—
145	C₂×C₄⋊C₄	—	26	2	no	—
146	#107	—	13	1	no	—
147	#107	—	13	2	no	—
148	#107	C₂×D₁₃	1	1	yes	characteristic
149	#103	—	13	2	no	—
150	#103	C₂×D₁₃	1	2	yes	—
151	#102	—	13	2	no	—
152	#102	—	13	2	no	—
153	#97	—	13	1	no	—
154	#97	—	13	1	no	—
155	C₄×C₈	—	13	1	no	—
156	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
157	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂×C₁₃	#576	1	1	yes	characteristic
159	C₂×C₁₃	#578	1	1	yes	characteristic
160	D₁₃	—	2	2	no	—
161	D₁₃	—	4	4	no	—
162	#41	—	13	8	no	—
163	#41	—	26	4	no	—
164	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
165	C₂×D₄	—	13	2	no	—
166	C₂×D₄	—	26	2	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	26	2	no	—
169	C₂²×C₄	—	26	2	no	—
170	C₂²×C₄	—	26	2	no	—
171	C₂²×C₄	—	26	2	no	—
172	Q₁₆	C₄×D₁₃	1	4	yes	complemented
173	SD₁₆	—	13	8	no	—
174	D₈	—	13	4	no	—
175	M₁₆	—	26	2	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	4	no	—
178	C₂×C₈	—	13	1	no	—
179	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	13	1	no	—
183	C₄⋊C₄	—	13	1	no	—
184	C₄⋊C₄	—	13	2	no	—
185	C₄⋊C₄	—	26	2	no	—
186	C₄⋊C₄	—	26	2	no	—
187	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
188	C₂²⋊C₄	—	13	1	no	—
189	C₂²⋊C₄	—	13	1	no	—
190	C₂²⋊C₄	—	26	2	no	—
191	C₂²⋊C₄	—	26	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	13	1	no	—
194	C₄²	—	26	2	no	—
195	C₄²	C₂²×D₁₃	1	1	yes	characteristic
196	C₁₃	#2951	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	13	1	no	—
198	C₂³	—	26	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	C₂×C₄×D₁₃	1	4	yes	—
201	D₄	—	13	4	no	—
202	D₄	—	26	8	no	—
203	D₄	—	26	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	13	1	no	—
207	C₂×C₄	—	13	2	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	13	1	no	—
210	C₂×C₄	—	13	1	no	—
211	C₂×C₄	—	26	8	no	—
212	C₂×C₄	—	26	2	no	—
213	C₂×C₄	—	104	2	no	—
214	C₂×C₄	—	104	2	no	—
215	C₂×C₄	—	104	2	no	—
216	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
217	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	13	2	no	—
221	C₈	—	26	1	no	—
222	C₈	C₂×C₄×D₁₃	1	2	yes	—
223	C₂²	—	13	2	no	—
224	C₂²	—	26	1	no	—
225	C₂²	—	26	1	no	—
226	C₂²	—	26	4	no	—
227	C₂²	—	52	2	no	—
228	C₂²	—	52	2	no	—
229	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	4	no	—
233	C₄	—	4	2	no	—
234	C₄	—	13	2	no	—
235	C₄	—	26	1	no	—
236	C₄	—	26	1	no	—
237	C₄	—	52	2	no	—
238	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
239	C₄	#41×D₁₃	1	1	yes	characteristic
240	C₂	—	26	2	no	—
241	C₂	—	52	4	no	—
242	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic
243	C₂	#11719471	1	1	yes	characteristic
244	C₂	#11719474	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#88173	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#2951
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #11719279	C₂	semidirect
3. C₄×C₁₃×Q₁₆	C₂	semidirect
4. Q₁₆×Dic₁₃	C₂	semidirect
5. #11718687	C₂	semidirect
6. D₁₃×#107	C₂	semidirect
7. #11718533	C₂	semidirect
8. #11718478	C₂	semidirect
9. #11718464	C₂	semidirect
10. C₄×D₁₀₄	C₂	semidirect
11. #11718350	C₂	semidirect
17. #88173	C₄	semidirect
42. #6131	C₂×C₄	semidirect
50. C₁₃×Q₁₆	C₂×C₄	semidirect
57. #6106	C₂×C₄	semidirect
71. D₁₀₄	C₂×C₄	semidirect
73. C₈×D₁₃	C₂×C₄	semidirect
106. C₄×Q₁₆	D₁₃	semidirect
172. Q₁₆	C₄×D₁₃	semidirect
196. C₁₃	#2951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88173	2
#11719279	2
C₄×C₁₃×Q₁₆	2
Q₁₆×Dic₁₃	2
#11718687	2
D₁₃×#107	2
#11718533	2
#11718478	2
#11718464	2
C₄×D₁₀₄	2
#11718350	2
#2951	13