Subgroups — #16778942

Order 3328.

666 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊#431	C₂	1	2	yes	maximal
2	#13129662	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#6094	C₂	1	1	yes	maximal, characteristic
4	C₂²×#6094	C₄	1	2	yes	—
5	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
6	#11719125	—	2	2	no	—
7	D₁₃⋊(C₂²⋊C₈)	—	2	2	no	—
8	#11719119	C₂²	1	4	yes	—
9	#11719051	—	2	2	no	—
10	D₁₃⋊(C₄⋊C₈)	—	2	2	no	—
11	#11719049	C₄	1	4	yes	—
12	C₄×D₁₃⋊C₈	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
14	C₂×#88055	C₂²	1	2	yes	—
15	C₂×C₄×#1055	C₄	1	1	yes	characteristic
16	C₄×#6094	—	2	2	no	—
17	C₄×#6094	—	2	2	no	—
18	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
21	C₂×D₁₃⋊C₈	—	2	4	no	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	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	C₂³	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	2	yes	—
28	C₂×#6094	—	2	4	no	—
29	C₂×#6094	—	2	4	no	—
30	C₂×#6094	C₈	1	8	yes	complemented
31	C₄²×D₁₃	C₂×C₄	1	2	yes	—
32	C₄²×D₁₃	C₂×C₄	1	2	yes	—
33	C₄²×D₁₃	D₄	1	2	yes	—
34	C₄²×D₁₃	D₄	1	2	yes	—
35	C₂²⋊#1057	—	2	4	no	—
36	#88112	—	2	4	no	—
37	#88111	—	2	4	no	—
38	C₄⋊#1057	—	2	4	no	—
39	C₄×#1057	—	2	4	no	—
40	C₄×#1057	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₁₃	—	2	1	no	—
42	C₄×C₈×C₁₃	—	2	1	no	—
43	#88055	—	2	4	no	—
44	C₄×#1055	—	2	1	no	—
45	C₄×#1055	—	2	1	no	—
46	#7860	—	13	1	no	maximal, Sylow
47	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×D₁₃	—	4	2	no	—
55	C₂×C₄×D₁₃	—	4	2	no	—
56	C₂×C₄×D₁₃	C₄²	1	2	yes	—
57	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
63	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
64	C₂×#1057	—	2	8	no	—
65	D₁₃⋊C₈	—	4	8	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
69	C₄²×C₁₃	C₄²	1	1	yes	characteristic
70	C₄²×C₁₃	C₄²	1	1	yes	characteristic
71	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
74	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
75	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
76	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
77	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
78	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	C₂×C₈	1	4	yes	complemented
82	#6094	—	2	16	no	—
83	C₂×#431	—	13	2	no	—
84	C₂×C₄×M₁₆	—	13	1	no	—
85	#1275	—	13	4	no	—
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×D₁₃	—	4	1	no	—
88	C₂²×D₁₃	—	4	1	no	—
89	C₂²×D₁₃	#95	1	2	yes	—
90	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
101	C₂×Dic₁₃	—	4	1	no	—
102	C₂×Dic₁₃	—	4	1	no	—
103	C₂×Dic₁₃	#95	1	2	yes	—
104	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
105	C₂×Dic₁₃	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₄×D₁₃	—	2	4	no	—
108	C₄×D₁₃	—	2	4	no	—
109	C₄×D₁₃	—	4	2	no	—
110	C₄×D₁₃	—	4	2	no	—
111	C₄×D₁₃	C₄×C₈	1	4	yes	—
112	C₄×D₁₃	#97	1	4	yes	—
113	C₄×D₁₃	C₂²⋊C₈	1	8	yes	—
114	#1057	—	4	8	no	—
115	C₈×C₁₃	—	2	4	no	—
116	#1055	—	2	4	no	—
117	C₂²×M₁₆	—	13	2	no	—
118	C₂²×C₄²	—	13	1	no	—
119	#431	—	13	4	no	—
120	#431	—	13	4	no	—
121	#431	—	26	2	no	—
122	#431	—	26	2	no	—
123	C₂×C₄⋊C₈	—	26	2	no	—
124	C₂×C₂²⋊C₈	—	26	2	no	—
125	C₄×M₁₆	—	26	2	no	—
126	C₄×M₁₆	—	26	2	no	—
127	C₂×#97	—	13	2	no	—
128	C₂×C₄×C₈	—	13	4	no	—
129	C₂×C₄×C₈	—	13	1	no	—
130	C₂×C₄×C₈	#261	1	1	yes	characteristic
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
135	C₂×D₁₃	—	2	2	no	—
136	C₂×D₁₃	—	2	2	no	—
137	C₂×D₁₃	—	4	4	no	—
138	C₂×D₁₃	#336	1	4	yes	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	H(3,ℤ/4)	1	2	yes	—
142	C₄×C₁₃	#338	1	2	yes	—
143	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
144	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
145	C₄×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
146	Dic₁₃	—	2	2	no	—
147	Dic₁₃	—	2	2	no	—
148	Dic₁₃	#336	1	4	yes	—
149	C₂³×C₄	—	13	1	no	—
150	C₂³×C₄	—	13	1	no	—
151	C₂³×C₄	—	13	1	no	—
152	C₂×M₁₆	—	13	8	no	—
153	C₂×M₁₆	—	26	4	no	—
154	C₂×M₁₆	—	26	4	no	—
155	C₂²×C₈	—	13	2	no	—
156	C₂²×C₈	—	26	4	no	—
157	C₂²×C₈	C₂×#261	1	2	yes	—
158	C₂×C₄²	—	13	1	no	—
159	C₂×C₄²	—	13	1	no	—
160	C₂×C₄²	—	13	1	no	—
161	C₂×C₄²	—	13	2	no	—
162	C₂×C₄²	—	13	2	no	—
163	C₂×C₄²	—	13	2	no	—
164	C₂×C₄²	—	13	2	no	—
165	C₂×C₄²	C₂×#261	1	1	yes	characteristic
166	C₄⋊C₈	—	26	4	no	—
167	C₄⋊C₈	—	26	4	no	—
168	C₂²⋊C₈	—	26	4	no	—
169	C₂²⋊C₈	—	26	4	no	—
170	#97	—	26	4	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	—	13	8	no	—
174	C₄×C₈	—	26	4	no	—
175	C₄×C₈	—	26	1	no	—
176	C₄×C₈	—	26	1	no	—
177	C₂×C₁₃	—	2	2	no	—
178	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
179	C₂×C₁₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#338	1	1	yes	characteristic
181	D₁₃	—	2	4	no	—
182	C₂⁴	—	13	1	no	—
183	C₂²×C₄	—	13	1	no	—
184	C₂²×C₄	—	13	4	no	—
185	C₂²×C₄	—	13	4	no	—
186	C₂²×C₄	—	13	2	no	—
187	C₂²×C₄	—	13	2	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	—	52	2	no	—
195	C₂²×C₄	—	52	2	no	—
196	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
198	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
199	M₁₆	—	26	16	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	13	4	no	—
203	C₂×C₈	—	26	2	no	—
204	C₂×C₈	—	26	2	no	—
205	C₂×C₈	—	26	8	no	—
206	C₂×C₈	—	52	8	no	—
207	C₂×C₈	D₁₃⋊C₈	1	4	yes	complemented
208	C₄²	—	13	2	no	—
209	C₄²	—	13	2	no	—
210	C₄²	—	13	2	no	—
211	C₄²	—	13	2	no	—
212	C₄²	—	13	2	no	—
213	C₄²	—	13	2	no	—
214	C₄²	C₄×#261	1	1	yes	characteristic
215	C₄²	C₄×#261	1	1	yes	characteristic
216	C₄²	C₂²⋊#261	1	1	yes	characteristic
217	C₄²	C₂²⋊#261	1	1	yes	characteristic
218	C₁₃	#7860	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	13	2	no	—
220	C₂³	—	13	4	no	—
221	C₂³	—	52	1	no	—
222	C₂³	—	52	1	no	—
223	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	13	4	no	—
227	C₂×C₄	—	13	2	no	—
228	C₂×C₄	—	13	4	no	—
229	C₂×C₄	—	13	4	no	—
230	C₂×C₄	—	13	8	no	—
231	C₂×C₄	—	13	2	no	—
232	C₂×C₄	—	13	2	no	—
233	C₂×C₄	—	26	4	no	—
234	C₂×C₄	—	26	4	no	—
235	C₂×C₄	—	52	2	no	—
236	C₂×C₄	—	52	2	no	—
237	C₂×C₄	—	52	1	no	—
238	C₂×C₄	—	52	1	no	—
239	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
240	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
242	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
243	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
244	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
245	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
246	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	—	26	4	no	—
249	C₈	—	52	8	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	13	4	no	—
253	C₂²	—	26	2	no	—
254	C₂²	—	26	2	no	—
255	C₂²	—	52	4	no	—
256	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
257	C₂²	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	13	4	no	—
261	C₄	—	26	2	no	—
262	C₄	—	26	2	no	—
263	C₄	#11718267	1	2	yes	—
264	C₄	#11718269	1	2	yes	—
265	C₄	C₄×D₁₃⋊C₈	1	1	yes	characteristic
266	C₄	#11719047	1	1	yes	characteristic
267	C₄	D₁₃⋊(C₂²⋊C₈)	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	26	4	no	—
270	C₂	#13130497	1	1	yes	characteristic
271	C₂	C₂×#11718267	1	1	yes	characteristic
272	C₂	C₂×#11718269	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×#95
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#7860
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13129662	C₂	semidirect
30. C₂×#6094	C₈	semidirect
68. C₂×C₈×C₁₃	C₂×C₈	semidirect
81. C₂×#1055	C₂×C₈	semidirect
207. C₂×C₈	D₁₃⋊C₈	semidirect
218. C₁₃	#7860	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊#431	2
#13129662	2
C₂×C₄×#6094	2
#7860	13