Subgroups — #15861034

Order 3328.

994 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718786	C₂	1	1	yes	maximal, characteristic
2	#13123254	C₂	1	2	yes	maximal
3	C₂×#11718783	C₂	1	1	yes	maximal, characteristic
4	#13114744	C₂	1	4	yes	maximal, complemented
5	#108×Dic₁₃	C₂	1	4	yes	maximal, complemented
6	C₂×#11718150	C₂	1	2	yes	maximal
7	C₂×C₁₃×#430	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	1	yes	characteristic
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₂³×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#430	C₄	1	8	yes	complemented
14	C₂×C₁₃×#108	C₂²	1	4	yes	—
15	#11718786	C₄	1	8	yes	complemented
16	#11718783	—	2	4	no	—
17	#11718783	C₄	1	8	yes	—
18	M₁₆×Dic₁₃	C₂²	1	8	yes	—
19	C₂×#88059	C₂²	1	4	yes	—
20	#11718762	—	2	4	no	—
21	C₂×#88055	C₂²	1	2	yes	—
22	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
23	#11718150	C₂²	1	4	yes	—
24	#11718150	C₂²	1	4	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
31	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₃	—	2	2	no	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂×C₂²⋊Dic₁₃	Q₈	1	2	yes	—
36	#88180	C₂×C₄	1	4	yes	—
37	#88180	D₄	1	4	yes	—
38	C₂×C₄⋊Dic₁₃	Q₈	1	2	yes	—
39	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	4	no	—
42	C₂×#6099	C₂×C₄	1	8	yes	—
43	C₂×#6099	C₂³	1	4	yes	—
44	C₂²×#1055	—	2	2	no	—
45	C₂²×#1055	—	2	2	no	—
46	C₂²×#1055	—	2	2	no	—
47	C₂²×#1055	C₂×C₄	1	4	yes	—
48	C₂²×#1055	C₂³	1	1	yes	characteristic
49	C₂²×#1055	C₂³	1	1	yes	characteristic
50	C₁₃×#108	C₂×C₄	1	16	yes	—
51	#88059	C₂×C₄	1	16	yes	—
52	#88055	—	2	8	no	—
53	#88055	C₂×C₄	1	8	yes	—
54	C₈×Dic₁₃	—	2	8	no	—
55	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
56	#20459	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂×Q₈	1	4	yes	—
67	C₁₃×M₁₆	—	2	8	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	C₄²	1	16	yes	—
70	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
71	C₂×C₈×C₁₃	—	2	8	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
76	C₂²⋊Dic₁₃	C₄⋊C₄	1	8	yes	—
77	C₄⋊Dic₁₃	C₄⋊C₄	1	8	yes	—
78	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
79	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
80	#6099	—	2	8	no	—
81	#6099	—	2	4	no	—
82	#6099	C₂²×C₄	1	8	yes	—
83	C₂×#1055	—	2	8	no	—
84	C₂×#1055	—	2	8	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	2	1	no	—
87	C₂×#1055	—	2	1	no	—
88	C₂×#1055	—	2	1	no	—
89	C₂×#1055	—	2	1	no	—
90	C₂×#1055	C₂²×C₄	1	2	yes	—
91	C₂×#1055	C₂²×C₄	1	2	yes	—
92	C₂×#430	—	13	1	no	—
93	C₂×#430	D₁₃	1	1	yes	characteristic
94	#2873	—	13	2	no	—
95	C₂×#405	—	13	1	no	—
96	#1778	—	13	4	no	—
97	C₄×#108	—	13	4	no	—
98	C₂×#341	—	13	2	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
102	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
103	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₃	#117	1	4	yes	—
107	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
108	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
109	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₁₃	—	4	4	no	—
118	C₂×Dic₁₃	C₂×C₄⋊C₄	1	8	yes	—
119	C₈×C₁₃	—	2	8	no	—
120	C₈×C₁₃	C₂×C₄²	1	8	yes	—
121	#1055	—	2	2	no	—
122	#1055	—	2	2	no	—
123	#1055	—	2	8	no	—
124	C₂²×M₁₆	—	13	2	no	—
125	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
126	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
127	C₂³×C₈	—	13	1	no	—
128	C₂×#117	—	13	1	no	—
129	#430	—	13	8	no	—
130	#430	Dic₁₃	1	8	yes	complemented
131	C₂×#108	—	13	4	no	—
132	C₂×#108	C₂×D₁₃	1	4	yes	—
133	#405	—	13	8	no	—
134	#405	—	26	4	no	—
135	#405	—	26	4	no	—
136	C₄×M₁₆	—	13	8	no	—
137	C₂×#97	—	13	2	no	—
138	C₂×C₄×C₈	—	13	2	no	—
139	#341	—	13	4	no	—
140	#341	—	13	4	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	4	no	—
143	C₂²×C₁₃	—	4	2	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
146	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
150	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
151	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
152	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
154	Dic₁₃	—	2	8	no	—
155	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	13	8	no	—
159	C₂×M₁₆	—	13	4	no	—
160	C₂×M₁₆	—	26	2	no	—
161	C₂×M₁₆	—	26	4	no	—
162	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
163	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
164	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
165	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
166	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	13	4	no	—
169	C₂²×C₈	—	13	1	no	—
170	C₂²×C₈	—	13	1	no	—
171	C₂²×C₈	—	26	2	no	—
172	C₂²×C₈	—	26	2	no	—
173	C₂²×C₈	—	26	2	no	—
174	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
175	#117	—	13	4	no	—
176	#117	—	13	4	no	—
177	C₂×C₄⋊C₄	—	13	2	no	—
178	C₂×C₂²⋊C₄	—	13	2	no	—
179	C₂×C₄²	—	13	2	no	—
180	#108	—	13	16	no	—
181	#108	C₂×Dic₁₃	1	16	yes	—
182	#97	—	13	8	no	—
183	#97	—	26	8	no	—
184	C₄×C₈	—	13	8	no	—
185	C₄×C₈	—	26	8	no	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	2	4	no	—
189	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₁₃	#2916	1	2	yes	—
191	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	—	13	4	no	—
193	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
195	C₂²×C₄	D₄×D₁₃	1	2	yes	—
196	C₂²×C₄	#6129	1	2	yes	—
197	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	#6131	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
200	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
201	M₁₆	—	2	8	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	13	8	no	—
204	M₁₆	—	26	8	no	—
205	M₁₆	—	26	4	no	—
206	M₁₆	C₄×Dic₁₃	1	16	yes	—
207	M₁₆	C₂²×Dic₁₃	1	8	yes	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	26	8	no	—
213	C₂×C₈	—	26	8	no	—
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	—	26	1	no	—
216	C₂×C₈	—	26	1	no	—
217	C₂×C₈	—	26	1	no	—
218	C₂×C₈	—	26	1	no	—
219	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
220	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
221	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
222	C₄⋊C₄	—	13	8	no	—
223	C₂²⋊C₄	—	13	8	no	—
224	C₄²	—	13	4	no	—
225	C₄²	—	13	4	no	—
226	C₁₃	#20459	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	4	2	no	—
228	C₂³	#88141	1	2	yes	—
229	C₂³	C₄⋊C₄×D₁₃	1	2	yes	—
230	C₂³	Q₈×Dic₁₃	1	2	yes	—
231	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
232	C₂×C₄	—	13	8	no	—
233	C₂×C₄	—	52	4	no	—
234	C₂×C₄	#88141	1	2	yes	—
235	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
236	C₂×C₄	#88146	1	4	yes	—
237	C₂×C₄	#88147	1	4	yes	—
238	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
239	C₂×C₄	D₄×Dic₁₃	1	4	yes	—
240	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
241	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#6129	1	1	yes	characteristic
245	C₂×C₄	C₂×#6131	1	1	yes	characteristic
246	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
247	C₈	—	2	8	no	—
248	C₈	—	26	2	no	—
249	C₈	—	26	2	no	—
250	C₈	—	26	8	no	—
251	C₈	C₂×C₄×Dic₁₃	1	8	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₄⋊C₄×Dic₁₃	1	4	yes	—
257	C₂²	C₂×#88141	1	1	yes	characteristic
258	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
259	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
260	C₄	—	26	8	no	—
261	C₄	C₄⋊C₄×Dic₁₃	1	4	yes	—
262	C₄	C₂×#88146	1	1	yes	characteristic
263	C₄	C₂×#88147	1	1	yes	characteristic
264	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
265	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	4	no	—
269	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
270	C₂	#13123446	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#430	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#430	D₁₃
hypocenter	C₁₃	#20459
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
4. #13114744	C₂	semidirect
5. #108×Dic₁₃	C₂	semidirect
13. C₁₃×#430	C₄	semidirect
15. #11718786	C₄	semidirect
130. #430	Dic₁₃	semidirect
226. C₁₃	#20459	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718786	2
#13123254	2
C₂×#11718783	2
#13114744	2
#108×Dic₁₃	2
C₂×#11718150	2
C₂×C₁₃×#430	2
#20459	13