Subgroups — #15781474

Order 3328.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
2	#13114715	C₂	1	1	yes	maximal, characteristic
3	#13112952	C₂	1	2	yes	maximal
4	#13112937	C₂	1	2	yes	maximal
5	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	—	2	1	no	—
7	C₁₃×#500	—	2	1	no	—
8	C₁₃×#499	—	2	1	no	—
9	C₁₃×#499	C₄	1	2	yes	complemented
10	C₁₃×#498	C₄	1	2	yes	complemented
11	C₁₃×#427	—	2	1	no	—
12	C₁₃×#427	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88059	C₂²	1	1	yes	characteristic
15	#11718767	C₂²	1	1	yes	characteristic
16	#11718762	—	2	1	no	—
17	C₂×#88052	C₂²	1	2	yes	—
18	#11718115	—	2	1	no	—
19	#11718055	—	2	1	no	—
20	#11718055	—	2	1	no	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×#125	—	2	2	no	—
25	C₁₃×C₂²⋊Q₈	—	4	2	no	—
26	C₄×Q₈×C₁₃	—	4	2	no	—
27	C₁₃×#117	—	2	2	no	—
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	#88180	—	2	1	no	—
30	C₂×#6099	—	2	1	no	—
31	C₁₃×#107	—	2	2	no	—
32	C₁₃×#107	—	2	2	no	—
33	C₁₃×#106	—	2	2	no	—
34	C₁₃×#106	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
39	#88059	—	2	1	no	—
40	#88059	Q₈	1	2	yes	complemented
41	#88058	—	2	1	no	—
42	#88057	Q₈	1	2	yes	complemented
43	#88055	—	4	1	no	—
44	C₈×Dic₁₃	—	4	1	no	—
45	#88052	—	2	2	no	—
46	#88052	—	2	2	no	—
47	C₂×#6090	—	2	2	no	—
48	#13271	—	13	1	no	maximal, Sylow
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
57	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
63	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₄⋊C₄	—	4	4	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₁₃×C₂²⋊C₄	—	4	2	no	—
68	C₄²×C₁₃	—	2	1	no	—
69	C₄²×C₁₃	—	4	2	no	—
70	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²⋊Dic₁₃	—	4	1	no	—
73	C₄⋊Dic₁₃	—	2	2	no	—
74	C₄×Dic₁₃	—	4	1	no	—
75	#6099	—	2	2	no	—
76	#6099	—	4	1	no	—
77	C₂×#1055	—	4	1	no	—
78	#6090	—	4	2	no	—
79	#3160	D₁₃	1	1	yes	characteristic
80	C₂×#364	—	13	1	no	—
81	#1772	—	13	1	no	—
82	#1389	—	13	2	no	—
83	#1384	—	13	2	no	—
84	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
85	Q₈×C₁₃	—	8	4	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	—	4	4	no	—
90	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×Dic₁₃	—	4	2	no	—
97	C₈×C₁₃	#103	1	1	yes	characteristic
98	C₈×C₁₃	#103	1	1	yes	characteristic
99	C₈×C₁₃	#103	1	1	yes	characteristic
100	C₈×C₁₃	#103	1	1	yes	characteristic
101	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
102	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
103	C₈×C₁₃	#124	1	1	yes	characteristic
104	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
105	#1055	—	4	2	no	—
106	#533	—	2	1	no	—
107	C₂×M₃₂	—	13	2	no	—
108	#500	—	2	1	no	—
109	#499	—	2	1	no	—
110	#499	Dic₁₃	1	2	yes	complemented
111	#498	Dic₁₃	1	2	yes	complemented
112	C₂×#108	—	13	1	no	—
113	#427	—	2	1	no	—
114	#427	—	13	1	no	—
115	#427	C₂×D₁₃	1	2	yes	—
116	#405	—	26	1	no	—
117	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
118	#364	—	26	1	no	—
119	#364	—	26	1	no	—
120	#339	—	26	1	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	#418	1	1	yes	characteristic
123	C₂²×C₁₃	#476	1	1	yes	characteristic
124	C₂²×C₁₃	#477	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	4	1	no	—
127	C₄×C₁₃	—	8	4	no	—
128	C₄×C₁₃	#391	1	1	yes	characteristic
129	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
130	C₄×C₁₃	#476	1	1	yes	characteristic
131	C₄×C₁₃	#477	1	1	yes	characteristic
132	Dic₁₃	—	8	2	no	—
133	C₂×M₁₆	—	26	1	no	—
134	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	2	no	—
137	#125	—	2	2	no	—
138	C₂²⋊Q₈	—	4	2	no	—
139	C₄×Q₈	—	4	2	no	—
140	#117	—	2	2	no	—
141	#117	—	26	1	no	—
142	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
143	M₃₂	—	26	2	no	—
144	M₃₂	—	26	2	no	—
145	C₂×C₁₆	—	26	2	no	—
146	#108	—	13	2	no	—
147	#108	—	26	1	no	—
148	#107	—	2	2	no	—
149	#107	—	2	2	no	—
150	#107	—	26	1	no	—
151	#106	—	2	2	no	—
152	#106	—	13	2	no	—
153	#106	C₂×Dic₁₃	1	4	yes	—
154	#97	—	52	1	no	—
155	C₄×C₈	—	52	1	no	—
156	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
157	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₂×C₁₃	—	2	1	no	—
161	C₂×C₁₃	—	2	1	no	—
162	C₂×C₁₃	—	2	1	no	—
163	C₂×C₁₃	#1987	1	1	yes	characteristic
164	C₂×Q₈	—	4	2	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
167	M₁₆	—	26	2	no	—
168	M₁₆	—	52	1	no	—
169	C₂×C₈	—	52	1	no	—
170	C₂×C₈	#6106	1	1	yes	characteristic
171	C₂×C₈	#6106	1	1	yes	characteristic
172	C₂×C₈	#6107	1	1	yes	characteristic
173	C₂×C₈	#6107	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
177	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
178	C₂×C₈	#6131	1	1	yes	characteristic
179	C₂×C₈	#6131	1	1	yes	characteristic
180	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
181	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	26	2	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	52	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	4	2	no	—
190	C₄²	—	52	1	no	—
191	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
192	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₁₆	—	52	2	no	—
194	C₁₃	#13271	1	1	yes	characteristic, complemented, Sylow
195	C₂³	#88186	1	1	yes	characteristic
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	#88186	1	1	yes	characteristic
203	C₂×C₄	C₂×#6106	1	1	yes	characteristic
204	C₂×C₄	C₂×#6107	1	1	yes	characteristic
205	C₂×C₄	#88200	1	1	yes	characteristic
206	C₂×C₄	#88200	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
208	C₈	—	52	2	no	—
209	C₈	#88075	1	1	yes	characteristic
210	C₈	#88075	1	1	yes	characteristic
211	C₈	#88075	1	1	yes	characteristic
212	C₈	#88075	1	1	yes	characteristic
213	C₈	#88198	1	1	yes	characteristic
214	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
215	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
216	C₈	#88201	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#11718745	1	1	yes	characteristic
219	C₂²	#11718752	1	1	yes	characteristic
220	C₂²	#11718876	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	104	2	no	—
225	C₄	#11718745	1	1	yes	characteristic
226	C₄	#11718752	1	1	yes	characteristic
227	C₄	C₂×#88075	1	1	yes	characteristic
228	C₄	#11718891	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#13115785	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11718745
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂×C₁₃	#1987
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#13271
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
9. C₁₃×#499	C₄	semidirect
10. C₁₃×#498	C₄	semidirect
40. #88059	Q₈	semidirect
42. #88057	Q₈	semidirect
110. #499	Dic₁₃	semidirect
111. #498	Dic₁₃	semidirect
194. C₁₃	#13271	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718055	2
#13114715	2
#13112952	2
#13112937	2
C₁₃×#3160	2
#13271	13