Subgroups — #15756578

Order 3328.

422 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	#13114987	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112671	C₂	1	1	yes	maximal, characteristic
4	#13112670	C₂	1	2	yes	maximal
5	C₁₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88054	—	2	2	no	—
9	C₂×#88054	—	2	2	no	—
10	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
27	C₂²×#1055	—	2	2	no	—
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₁₃×C₄⋊C₈	—	2	2	no	—
31	C₁₃×C₄⋊C₈	—	2	2	no	—
32	C₄×C₈×C₁₃	—	2	2	no	—
33	#88054	—	4	4	no	—
34	#88054	—	4	4	no	—
35	C₈×Dic₁₃	—	2	2	no	—
36	C₈×Dic₁₃	—	2	2	no	—
37	C₈×Dic₁₃	—	2	2	no	—
38	C₈×Dic₁₃	C₈	1	4	yes	complemented
39	C₄⋊#1055	—	2	2	no	—
40	C₄⋊#1055	—	2	2	no	—
41	C₄×#1055	—	2	2	no	—
42	#8784	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	—	2	1	no	—
51	C₂×C₈×C₁₃	—	2	1	no	—
52	C₂×C₈×C₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	—	4	2	no	—
55	C₂×C₈×C₁₃	—	4	2	no	—
56	C₂×C₈×C₁₃	—	4	2	no	—
57	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
59	C₄²×C₁₃	#41	1	2	yes	—
60	C₄²×C₁₃	#41	1	2	yes	—
61	C₄×Dic₁₃	—	2	1	no	—
62	C₄×Dic₁₃	—	2	1	no	—
63	C₄×Dic₁₃	—	2	2	no	—
64	C₄×Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
66	C₄×Dic₁₃	M₁₆	1	2	yes	—
67	C₄×Dic₁₃	D₈	1	2	yes	—
68	C₄×Dic₁₃	SD₁₆	1	4	yes	—
69	C₄×Dic₁₃	Q₁₆	1	2	yes	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	—	2	1	no	—
73	C₂×#1055	—	2	1	no	—
74	C₂×#1055	—	4	2	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	C₂×C₈	1	2	yes	—
78	C₂×#1055	M₁₆	1	2	yes	—
79	#1842	—	13	2	no	—
80	C₄²×C₈	—	13	1	no	—
81	#1277	—	13	2	no	—
82	#1277	—	13	1	no	—
83	#1277	D₁₃	1	1	yes	characteristic
84	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#124	1	2	yes	—
90	C₂×C₄×C₁₃	#125	1	2	yes	—
91	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
103	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
104	C₂×Dic₁₃	#106	1	2	yes	—
105	C₂×Dic₁₃	#106	1	2	yes	—
106	C₂×Dic₁₃	#107	1	2	yes	—
107	C₂×Dic₁₃	#107	1	2	yes	—
108	C₂×Dic₁₃	#108	1	2	yes	—
109	C₂×Dic₁₃	#108	1	2	yes	—
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	—	8	2	no	—
113	#1055	—	2	2	no	—
114	#1055	—	2	2	no	—
115	#1055	—	8	2	no	—
116	C₂×C₄⋊C₈	—	13	2	no	—
117	C₂×C₄⋊C₈	—	26	2	no	—
118	C₂×C₄⋊C₈	—	26	2	no	—
119	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₂×C₄×C₈	—	13	1	no	—
122	C₂×C₄×C₈	—	13	1	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₂×C₄×C₈	—	13	1	no	—
125	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
126	C₄³	—	13	1	no	—
127	C₂²×C₁₃	#382	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₁₃	#423	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
132	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	#431	1	1	yes	characteristic
136	C₄×C₁₃	#431	1	1	yes	characteristic
137	C₄×C₁₃	#432	1	1	yes	characteristic
138	C₄×C₁₃	#432	1	1	yes	characteristic
139	C₄×C₁₃	#486	1	2	yes	—
140	C₄×C₁₃	#487	1	2	yes	—
141	Dic₁₃	—	2	2	no	—
142	Dic₁₃	—	2	2	no	—
143	Dic₁₃	#334	1	4	yes	—
144	Dic₁₃	#335	1	4	yes	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	1	no	—
148	C₂²×C₈	—	26	2	no	—
149	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
150	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
151	C₂×C₄²	—	13	1	no	—
152	C₂×C₄²	—	13	1	no	—
153	C₂×C₄²	—	13	1	no	—
154	C₂×C₄²	—	13	1	no	—
155	C₂×C₄²	—	13	1	no	—
156	C₂×C₄²	—	13	1	no	—
157	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	26	2	no	—
161	C₄⋊C₈	—	26	2	no	—
162	C₄⋊C₈	—	52	4	no	—
163	C₄⋊C₈	—	52	4	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	13	4	no	—
166	C₄×C₈	—	26	2	no	—
167	C₄×C₈	—	26	2	no	—
168	C₄×C₈	—	26	2	no	—
169	C₄×C₈	—	26	2	no	—
170	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
172	C₂×C₁₃	#1566	1	1	yes	characteristic
173	C₂×C₁₃	#1567	1	1	yes	characteristic
174	C₂×C₁₃	#1831	1	1	yes	characteristic
175	C₂×C₁₃	#1834	1	1	yes	characteristic
176	C₂×C₁₃	#1837	1	1	yes	characteristic
177	C₂²×C₄	—	13	1	no	—
178	C₂²×C₄	—	13	1	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	—	13	1	no	—
181	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
183	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	13	2	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	26	1	no	—
194	C₂×C₈	—	26	1	no	—
195	C₂×C₈	—	26	1	no	—
196	C₂×C₈	—	26	1	no	—
197	C₂×C₈	—	52	2	no	—
198	C₂×C₈	—	52	2	no	—
199	C₂×C₈	—	52	2	no	—
200	C₂×C₈	C₈×D₁₃	1	2	yes	—
201	C₂×C₈	#6094	1	2	yes	—
202	C₄²	—	13	2	no	—
203	C₄²	—	13	2	no	—
204	C₄²	—	13	2	no	—
205	C₄²	—	13	4	no	—
206	C₄²	—	13	2	no	—
207	C₄²	—	26	2	no	—
208	C₄²	—	26	2	no	—
209	C₄²	—	26	1	no	—
210	C₄²	—	26	1	no	—
211	C₄²	#6127	1	2	yes	—
212	C₄²	#6129	1	2	yes	—
213	C₁₃	#8784	1	1	yes	characteristic, complemented, Sylow
214	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	13	2	no	—
218	C₂×C₄	—	13	2	no	—
219	C₂×C₄	—	13	2	no	—
220	C₂×C₄	—	13	2	no	—
221	C₂×C₄	—	13	2	no	—
222	C₂×C₄	—	13	2	no	—
223	C₂×C₄	—	13	2	no	—
224	C₂×C₄	—	13	2	no	—
225	C₂×C₄	—	26	1	no	—
226	C₂×C₄	—	26	1	no	—
227	C₂×C₄	—	26	1	no	—
228	C₂×C₄	—	26	1	no	—
229	C₂×C₄	#88126	1	1	yes	characteristic
230	C₂×C₄	#88131	1	1	yes	characteristic
231	C₂×C₄	#88139	1	2	yes	—
232	C₂×C₄	#88143	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×#6094	1	1	yes	characteristic
235	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
236	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
237	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
238	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
239	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	8	2	no	—
243	C₈	—	26	2	no	—
244	C₈	—	26	2	no	—
245	C₈	—	104	2	no	—
246	C₂²	#11718311	1	1	yes	characteristic
247	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
248	C₂²	#11718498	1	1	yes	characteristic
249	C₂²	D₁₃×#106	1	1	yes	characteristic
250	C₂²	D₁₃×#107	1	1	yes	characteristic
251	C₂²	D₁₃×#108	1	1	yes	characteristic
252	C₂²	D₁₃×#108	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	13	4	no	—
255	C₄	—	13	4	no	—
256	C₄	—	26	2	no	—
257	C₄	—	26	2	no	—
258	C₄	#11718346	1	1	yes	characteristic
259	C₄	#11718366	1	1	yes	characteristic
260	C₄	#11718379	1	1	yes	characteristic
261	C₄	#11718380	1	1	yes	characteristic
262	C₄	#11718421	1	2	yes	—
263	C₄	#11718453	1	2	yes	—
264	C₂	D₁₃×#334	1	1	yes	characteristic
265	C₂	D₁₃×#335	1	1	yes	characteristic
266	C₂	#13113590	1	1	yes	characteristic
267	C₂	#13113593	1	1	yes	characteristic
268	C₂	#13114951	1	1	yes	characteristic
269	C₂	#13114962	1	1	yes	characteristic
270	C₂	#13114972	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1277	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1277	D₁₃
hypocenter	C₁₃	#8784
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
38. C₈×Dic₁₃	C₈	semidirect
213. C₁₃	#8784	semidirect

Maximal subgroups

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

subgroup	index
#13114987	2
C₄×C₈×Dic₁₃	2
#13112671	2
#13112670	2
C₁₃×#1277	2
#8784	13