Subgroups — #13123706

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×#88166	C₂	1	1	yes	maximal, characteristic
2	#11719251	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	D₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718679	C₂	1	2	yes	maximal, complemented
6	D₁₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718534	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718432	C₂	1	2	yes	maximal, complemented
9	#11718418	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718350	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6129	C₂²	1	2	yes	—
13	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₂²	1	2	yes	—
15	D₄×Dic₁₃	C₂²	1	2	yes	—
16	C₂×#6105	C₂²	1	2	yes	—
17	C₄×C₂²⋊D₁₃	—	2	2	no	—
18	C₂×#6101	—	2	2	no	—
19	#88166	C₄	1	8	yes	complemented
20	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
21	C₂×#6094	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₃	C₂²	1	2	yes	—
24	#88141	C₂²	1	2	yes	—
25	#88135	—	2	2	no	—
26	#88131	—	2	2	no	—
27	#88126	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₆	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#102	C₂²	1	2	yes	—
31	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	#88072	C₂²	1	2	yes	—
33	#88058	C₂²	1	1	yes	characteristic
34	#88056	C₂²	1	2	yes	—
35	#88055	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
37	#88050	C₂²	1	2	yes	—
38	#88047	C₂²	1	2	yes	—
39	#88043	C₂²	1	1	yes	characteristic
40	C₂×D₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂×C₂²⋊D₁₃	—	2	2	no	—
43	C₂²×Dic₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	—	2	2	no	—
45	#6129	—	2	4	no	—
46	#6129	C₂×C₄	1	8	yes	complemented
47	C₂×C₄×D₁₃	—	2	2	no	—
48	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₂₆	C₂³	1	2	yes	—
50	C₁₃×D₈	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	C₂³	1	2	yes	—
54	C₁₃×C₂²⋊C₄	—	2	2	no	—
55	C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₁₃	—	2	2	no	—
57	#6105	C₂×C₄	1	8	yes	complemented
58	#6103	D₄	1	1	yes	characteristic
59	#6103	D₄	1	1	yes	characteristic
60	C₄⋊Dic₁₃	C₂³	1	2	yes	—
61	#6101	—	2	2	no	—
62	#6101	—	2	2	no	—
63	#6101	—	2	2	no	—
64	#6101	D₄	1	1	yes	characteristic
65	#6101	D₄	1	1	yes	characteristic
66	C₄×Dic₁₃	—	2	2	no	—
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	Dic₅₂	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	C₂³×C₁₃	—	2	2	no	—
76	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
77	D₄×C₁₃	—	2	2	no	—
78	D₄×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	8	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	C₂²⋊D₁₃	—	2	8	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	C₂×Dic₁₃	—	2	8	no	—
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	8	2	no	—
89	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄×D₁₃	—	8	2	no	—
93	C₄×D₁₃	C₂²×C₄	1	2	yes	—
94	C₄×D₁₃	#41	1	2	yes	—
95	Dic₂₆	—	2	2	no	—
96	Dic₂₆	C₂²×C₄	1	4	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	—	13	1	no	—
107	C₄×D₈	—	13	1	no	—
108	C₄×D₈	D₁₃	1	1	yes	characteristic, complemented
109	C₂×#107	—	13	1	no	—
110	#417	—	13	2	no	—
111	#405	—	13	1	no	—
112	C₂²×C₁₃	—	2	4	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₁₃	—	2	1	no	—
117	C₂×D₁₃	—	2	1	no	—
118	C₂×D₁₃	C₄×D₄	1	2	yes	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	4	2	no	—
122	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	Dic₁₃	—	2	4	no	—
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	—	13	1	no	—
132	C₂×SD₁₆	—	13	2	no	—
133	C₂×D₈	C₂×D₁₃	1	1	yes	characteristic
134	C₂×M₁₆	—	13	1	no	—
135	C₂²×C₈	—	13	1	no	—
136	C₄×Q₈	—	13	2	no	—
137	C₄×Q₈	—	13	2	no	—
138	C₄×D₄	—	13	2	no	—
139	C₄×D₄	—	26	2	no	—
140	C₄×D₄	—	26	2	no	—
141	C₄×D₄	C₂×D₁₃	1	2	yes	—
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	—	13	2	no	—
151	#102	—	13	2	no	—
152	#102	C₂×D₁₃	1	2	yes	—
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	—	4	4	no	—
158	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂×C₁₃	#576	1	1	yes	characteristic
160	C₂×C₁₃	#578	1	1	yes	characteristic
161	D₁₃	—	2	2	no	—
162	#41	—	13	8	no	—
163	#41	—	26	4	no	—
164	C₂×Q₈	—	13	2	no	—
165	C₂×D₄	—	26	2	no	—
166	C₂×D₄	C₂²×D₁₃	1	2	yes	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	13	1	no	—
169	C₂²×C₄	—	26	2	no	—
170	C₂²×C₄	—	26	2	no	—
171	C₂²×C₄	—	26	2	no	—
172	Q₁₆	—	13	4	no	—
173	SD₁₆	—	13	8	no	—
174	D₈	C₄×D₁₃	1	4	yes	complemented
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	—	13	1	no	—
182	C₄⋊C₄	—	13	1	no	—
183	C₄⋊C₄	—	13	2	no	—
184	C₄⋊C₄	—	26	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	—	2	2	no	—
189	C₂²⋊C₄	—	13	1	no	—
190	C₂²⋊C₄	—	13	1	no	—
191	C₂²⋊C₄	—	26	2	no	—
192	C₄²	—	13	1	no	—
193	C₄²	—	26	2	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	—	2	2	no	—
198	C₂³	—	13	1	no	—
199	Q₈	—	13	4	no	—
200	Q₈	—	26	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	26	8	no	—
203	D₄	C₂×C₄×D₁₃	1	4	yes	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	8	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	—	26	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	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	13	2	no	—
227	C₂²	—	26	1	no	—
228	C₂²	—	26	1	no	—
229	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	13	2	no	—
234	C₄	—	26	1	no	—
235	C₄	—	26	1	no	—
236	C₄	—	26	4	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	—	4	4	no	—
241	C₂	—	26	2	no	—
242	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic
243	C₂	#11719462	1	1	yes	characteristic
244	C₂	#11719476	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×D₈	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#88166	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×D₈	D₁₃
hypocenter	C₁₃	#2951
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #11719251	C₂	semidirect
3. C₄×C₁₃×D₈	C₂	semidirect
4. D₈×Dic₁₃	C₂	semidirect
5. #11718679	C₂	semidirect
6. D₁₃×#107	C₂	semidirect
7. #11718534	C₂	semidirect
8. #11718432	C₂	semidirect
9. #11718418	C₂	semidirect
10. C₄×Dic₅₂	C₂	semidirect
11. #11718350	C₂	semidirect
19. #88166	C₄	semidirect
46. #6129	C₂×C₄	semidirect
50. C₁₃×D₈	C₂×C₄	semidirect
57. #6105	C₂×C₄	semidirect
71. Dic₅₂	C₂×C₄	semidirect
73. C₈×D₁₃	C₂×C₄	semidirect
108. C₄×D₈	D₁₃	semidirect
174. D₈	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×#88166	2
#11719251	2
C₄×C₁₃×D₈	2
D₈×Dic₁₃	2
#11718679	2
D₁₃×#107	2
#11718534	2
#11718432	2
#11718418	2
C₄×Dic₅₂	2
#11718350	2
#2951	13