Subgroups — #15779264

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718093	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112914	C₂	1	1	yes	maximal, characteristic
4	#13112913	C₂	1	2	yes	maximal
5	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	2	yes	—
13	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	#11718093	—	2	4	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×C₁₆	—	2	2	no	—
28	C₂×C₁₃×C₁₆	—	2	2	no	—
29	C₂×C₁₃×C₁₆	—	2	2	no	—
30	C₄×C₈×C₁₃	—	2	2	no	—
31	C₈×Dic₁₃	—	2	2	no	—
32	C₈×Dic₁₃	—	2	2	no	—
33	C₈×Dic₁₃	—	2	2	no	—
34	C₈×Dic₁₃	D₄	1	2	yes	—
35	C₈×Dic₁₃	Q₈	1	2	yes	—
36	C₂×#6090	—	2	2	no	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	—	2	2	no	—
39	C₄×#1055	—	2	2	no	—
40	#12879	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	1	no	—
50	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
51	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	#41	1	2	yes	—
54	C₂×C₈×C₁₃	#41	1	2	yes	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄×Dic₁₃	—	2	2	no	—
58	C₄×Dic₁₃	—	2	1	no	—
59	C₄×Dic₁₃	—	2	1	no	—
60	C₄×Dic₁₃	—	2	2	no	—
61	C₄×Dic₁₃	—	2	2	no	—
62	C₄×Dic₁₃	C₁₆	1	4	yes	complemented
63	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
64	C₄×Dic₁₃	M₁₆	1	2	yes	—
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	—	2	1	no	—
67	C₂×#1055	—	2	1	no	—
68	C₂×#1055	—	2	1	no	—
69	C₂×#1055	C₄⋊C₄	1	2	yes	—
70	C₂×#1055	M₁₆	1	2	yes	—
71	C₁₃×C₁₆	—	4	2	no	—
72	#6090	—	4	2	no	—
73	C₂×C₄⋊C₁₆	—	13	2	no	—
74	C₄²×C₈	—	13	1	no	—
75	#1374	—	13	2	no	—
76	#1374	—	13	1	no	—
77	#1374	D₁₃	1	1	yes	characteristic
78	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	1	no	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	—	2	1	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	1	no	—
97	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
98	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
99	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
100	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
101	C₂×Dic₁₃	M₃₂	1	2	yes	—
102	C₂×Dic₁₃	M₃₂	1	2	yes	—
103	C₈×C₁₃	—	2	2	no	—
104	C₈×C₁₃	#124	1	2	yes	—
105	C₈×C₁₃	#125	1	2	yes	—
106	#1055	—	2	2	no	—
107	#1055	—	2	2	no	—
108	C₂²×C₁₆	—	13	2	no	—
109	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
110	C₂×C₄×C₈	—	13	1	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₂×C₄×C₈	—	13	1	no	—
113	C₂×C₄×C₈	—	13	1	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	C₄³	—	13	1	no	—
117	C₄⋊C₁₆	—	26	4	no	—
118	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₁₃	#431	1	1	yes	characteristic
120	C₂²×C₁₃	#431	1	1	yes	characteristic
121	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
122	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
124	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	#382	1	1	yes	characteristic
128	C₄×C₁₃	#423	1	1	yes	characteristic
129	C₄×C₁₃	#432	1	1	yes	characteristic
130	C₄×C₁₃	#432	1	1	yes	characteristic
131	Dic₁₃	—	2	2	no	—
132	Dic₁₃	—	2	2	no	—
133	Dic₁₃	—	2	2	no	—
134	Dic₁₃	C₄⋊C₁₆	1	4	yes	—
135	C₂²×C₈	—	13	1	no	—
136	C₂²×C₈	—	13	1	no	—
137	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
139	C₂×C₄²	—	13	1	no	—
140	C₂×C₄²	—	13	1	no	—
141	C₂×C₄²	—	13	1	no	—
142	C₂×C₄²	—	13	1	no	—
143	C₂×C₄²	—	13	1	no	—
144	C₂×C₄²	—	13	1	no	—
145	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	26	2	no	—
150	C₂×C₁₆	—	26	2	no	—
151	C₂×C₁₆	—	26	2	no	—
152	C₄×C₈	—	2	2	no	—
153	C₄×C₈	—	13	2	no	—
154	C₄×C₈	—	13	2	no	—
155	C₄×C₈	—	26	2	no	—
156	C₄×C₈	—	26	2	no	—
157	C₄×C₈	—	26	2	no	—
158	C₄×C₈	—	26	2	no	—
159	C₂×C₁₃	#1831	1	1	yes	characteristic
160	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
161	C₂×C₁₃	#2150	1	1	yes	characteristic
162	C₂×C₁₃	#2162	1	1	yes	characteristic
163	C₂×C₁₃	#2162	1	1	yes	characteristic
164	C₂×C₁₃	#2163	1	1	yes	characteristic
165	C₂×C₁₃	#2163	1	1	yes	characteristic
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	13	1	no	—
169	C₂²×C₄	—	13	1	no	—
170	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	2	no	—
177	C₂×C₈	—	26	1	no	—
178	C₂×C₈	—	26	1	no	—
179	C₂×C₈	—	26	1	no	—
180	C₂×C₈	—	26	1	no	—
181	C₂×C₈	#6094	1	2	yes	—
182	C₂×C₈	#6127	1	2	yes	—
183	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
184	C₂×C₈	#6129	1	2	yes	—
185	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	13	2	no	—
189	C₄²	—	13	2	no	—
190	C₄²	—	13	4	no	—
191	C₄²	—	26	2	no	—
192	C₄²	—	26	2	no	—
193	C₄²	—	26	1	no	—
194	C₄²	—	26	1	no	—
195	C₄²	—	26	2	no	—
196	C₁₆	—	4	2	no	—
197	C₁₆	—	52	2	no	—
198	C₁₃	#12879	1	1	yes	characteristic, complemented, Sylow
199	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	13	2	no	—
205	C₂×C₄	—	13	2	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	13	2	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	13	2	no	—
210	C₂×C₄	—	26	1	no	—
211	C₂×C₄	—	26	1	no	—
212	C₂×C₄	—	26	1	no	—
213	C₂×C₄	—	26	1	no	—
214	C₂×C₄	—	26	1	no	—
215	C₂×C₄	—	26	1	no	—
216	C₂×C₄	C₁₆×D₁₃	1	2	yes	—
217	C₂×C₄	#88038	1	2	yes	—
218	C₂×C₄	#88126	1	1	yes	characteristic
219	C₂×C₄	#88131	1	1	yes	characteristic
220	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
221	C₂×C₄	C₂×#6094	1	1	yes	characteristic
222	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
223	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	26	2	no	—
226	C₈	—	26	2	no	—
227	C₈	#88139	1	2	yes	—
228	C₈	#88143	1	2	yes	—
229	C₂²	#11718346	1	1	yes	characteristic
230	C₂²	#11718379	1	1	yes	characteristic
231	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
232	C₂²	C₂×C₁₆×D₁₃	1	1	yes	characteristic
233	C₂²	C₂×#88038	1	1	yes	characteristic
234	C₂²	D₁₃×M₃₂	1	1	yes	characteristic
235	C₂²	D₁₃×M₃₂	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	13	4	no	—
239	C₄	—	26	2	no	—
240	C₄	—	26	2	no	—
241	C₄	—	26	2	no	—
242	C₄	#11718311	1	1	yes	characteristic
243	C₄	#11718366	1	1	yes	characteristic
244	C₄	#11718380	1	1	yes	characteristic
245	C₄	#11718498	1	1	yes	characteristic
246	C₂	#13114951	1	1	yes	characteristic
247	C₂	D₁₃×C₄⋊C₁₆	1	1	yes	characteristic
248	C₂	#13116590	1	1	yes	characteristic
249	C₂	#13116632	1	1	yes	characteristic
250	C₂	#13116633	1	1	yes	characteristic
251	C₂	#13116636	1	1	yes	characteristic
252	C₂	#13116637	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₁₃×#1374	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1374	D₁₃
hypocenter	C₁₃	#12879
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
62. C₄×Dic₁₃	C₁₆	semidirect
198. C₁₃	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718093	2
C₄×C₈×Dic₁₃	2
#13112914	2
#13112913	2
C₁₃×#1374	2
#12879	13