Subgroups — #15861708

Order 3328.

1090 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₄×#88180	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₁₃	C₂	1	4	yes	maximal, complemented
4	C₄⋊C₈×Dic₁₃	C₂	1	4	yes	maximal, complemented
5	#13114610	C₂	1	4	yes	maximal
6	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₄	1	2	yes	—
9	C₂³×#1055	C₂²	1	2	yes	—
10	C₁₃×#431	C₄	1	8	yes	complemented
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₄×C₂²⋊Dic₁₃	C₄	1	2	yes	—
15	C₄×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₈×Dic₁₃	C₂²	1	8	yes	—
17	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
18	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	#11718593	C₄	1	2	yes	—
20	C₄²×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₄⋊#1055	C₄	1	8	yes	complemented
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	C₂×C₄⋊#1055	C₂²	1	2	yes	—
24	#11718142	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
36	#88180	C₈	1	16	yes	complemented
37	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
41	C₂²×#1055	—	4	4	no	—
42	C₂²×#1055	C₂×C₄	1	8	yes	—
43	C₂²×#1055	C₂³	1	4	yes	—
44	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
45	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
47	#88071	C₂×C₄	1	4	yes	—
48	#88071	C₂×C₄	1	4	yes	—
49	C₈×Dic₁₃	—	4	8	no	—
50	C₄⋊#1055	C₂×C₄	1	8	yes	—
51	C₄⋊#1055	C₂×C₄	1	8	yes	—
52	#20543	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	4	2	no	—
57	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
58	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
72	C₄²×C₁₃	C₄²	1	4	yes	—
73	C₄²×C₁₃	C₄²	1	4	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²⋊Dic₁₃	C₂×C₈	1	16	yes	—
79	C₄⋊Dic₁₃	C₂×C₈	1	16	yes	complemented
80	C₄×Dic₁₃	—	2	4	no	—
81	C₄×Dic₁₃	—	2	4	no	—
82	C₄×Dic₁₃	—	4	4	no	—
83	C₄×Dic₁₃	C₂×C₈	1	16	yes	complemented
84	C₂×#1055	—	2	8	no	—
85	C₂×#1055	—	4	8	no	—
86	C₂×#1055	C₂²×C₄	1	8	yes	—
87	C₂×#431	D₁₃	1	1	yes	characteristic
88	C₂²×C₄⋊C₈	—	13	1	no	—
89	C₄×#117	—	13	1	no	—
90	C₈×C₄⋊C₄	—	13	4	no	—
91	C₄×C₄⋊C₈	—	13	4	no	—
92	C₈×C₂²⋊C₄	—	13	4	no	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
96	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
97	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	C₄×C₈	1	16	yes	complemented
106	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
116	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
117	C₂×Dic₁₃	—	2	8	no	—
118	C₂×Dic₁₃	—	2	8	no	—
119	C₂×Dic₁₃	C₂²×C₈	1	16	yes	—
120	C₈×C₁₃	—	2	8	no	—
121	#1055	—	2	8	no	—
122	C₂³×C₈	—	13	2	no	—
123	C₂×#117	—	13	2	no	—
124	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
125	#431	Dic₁₃	1	8	yes	complemented
126	C₂×C₄⋊C₈	—	13	8	no	—
127	C₂×C₄⋊C₈	—	13	2	no	—
128	C₂×C₄⋊C₈	—	13	2	no	—
129	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
130	C₂×C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
131	C₂×C₄×C₈	—	13	8	no	—
132	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
133	C₄×C₄⋊C₄	—	13	2	no	—
134	C₄×C₄⋊C₄	—	13	2	no	—
135	C₄×C₂²⋊C₄	—	13	2	no	—
136	C₄×C₂²⋊C₄	—	13	2	no	—
137	#376	—	13	2	no	—
138	C₄³	—	13	2	no	—
139	#336	—	13	8	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	8	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
144	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
145	C₂²×C₁₃	#520	1	2	yes	—
146	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
147	C₂²×C₁₃	#568	1	2	yes	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
152	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
153	C₄×C₁₃	#518	1	1	yes	characteristic
154	C₄×C₁₃	#519	1	1	yes	characteristic
155	C₄×C₁₃	#520	1	2	yes	—
156	Dic₁₃	—	2	16	no	—
157	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
158	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
159	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
160	C₂²×C₈	—	13	4	no	—
161	C₂²×C₈	—	13	8	no	—
162	C₂²×C₈	—	52	4	no	—
163	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
164	#117	—	13	16	no	—
165	C₂×C₄⋊C₄	—	13	4	no	—
166	C₂×C₂²⋊C₄	—	13	4	no	—
167	C₂×C₄²	—	13	4	no	—
168	C₂×C₄²	—	13	4	no	—
169	C₂×C₄²	—	13	4	no	—
170	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₁₃	1	4	yes	—
173	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
174	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
176	C₄⋊C₈	—	13	8	no	—
177	C₄⋊C₈	—	13	8	no	—
178	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
179	C₂²⋊C₈	C₂×Dic₁₃	1	8	yes	—
180	C₄×C₈	—	52	8	no	—
181	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
182	#95	—	13	4	no	—
183	#95	—	13	4	no	—
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	#2274	1	1	yes	characteristic
186	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
187	C₂×C₁₃	#2873	1	1	yes	characteristic
188	C₂×C₁₃	#2988	1	4	yes	—
189	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	13	8	no	—
194	C₂²×C₄	C₈×D₁₃	1	8	yes	—
195	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
196	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
197	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
204	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	13	8	no	—
208	C₂×C₈	—	26	8	no	—
209	C₂×C₈	—	52	8	no	—
210	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
211	C₄⋊C₄	—	13	16	no	—
212	C₂²⋊C₄	—	13	16	no	—
213	C₄²	—	13	16	no	—
214	C₄²	—	26	4	no	—
215	C₄²	—	26	4	no	—
216	C₄²	—	52	4	no	—
217	C₄²	C₄×Dic₁₃	1	4	yes	—
218	C₄²	C₄×Dic₁₃	1	4	yes	—
219	C₄²	C₂×C₄×D₁₃	1	2	yes	—
220	C₄²	C₂×C₄×D₁₃	1	2	yes	—
221	C₄²	C₂²×Dic₁₃	1	2	yes	—
222	C₄²	C₂²×Dic₁₃	1	2	yes	—
223	C₁₃	#20543	1	1	yes	characteristic, complemented, Sylow
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×C₈×D₁₃	1	4	yes	—
227	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
228	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	13	16	no	—
237	C₂×C₄	—	26	8	no	—
238	C₂×C₄	—	26	8	no	—
239	C₂×C₄	C₈×Dic₁₃	1	16	yes	complemented
240	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
241	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
242	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
247	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₄	#88256	1	2	yes	—
249	C₂×C₄	#88257	1	2	yes	—
250	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₈	—	26	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×C₈×Dic₁₃	1	4	yes	—
257	C₂²	#11719228	1	2	yes	—
258	C₂²	C₂²×C₈×D₁₃	1	2	yes	—
259	C₂²	#11719450	1	2	yes	—
260	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	26	16	no	—
265	C₄	C₂×C₈×Dic₁₃	1	4	yes	—
266	C₄	C₂×C₈×Dic₁₃	1	4	yes	—
267	C₄	#11719221	1	1	yes	characteristic
268	C₄	#11719226	1	1	yes	characteristic
269	C₄	#11719532	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	#13117192	1	1	yes	characteristic
272	C₂	C₂²×C₈×Dic₁₃	1	1	yes	characteristic
273	C₂	#13123254	1	1	yes	characteristic
274	C₂	#13123944	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₂×C₁₃×#431	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#431	D₁₃
hypocenter	C₁₃	#20543
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. C₈×C₄⋊Dic₁₃	C₂	semidirect
4. C₄⋊C₈×Dic₁₃	C₂	semidirect
10. C₁₃×#431	C₄	semidirect
21. C₂×C₄⋊#1055	C₄	semidirect
36. #88180	C₈	semidirect
79. C₄⋊Dic₁₃	C₂×C₈	semidirect
83. C₄×Dic₁₃	C₂×C₈	semidirect
105. C₂×C₄×C₁₃	C₄×C₈	semidirect
125. #431	Dic₁₃	semidirect
223. C₁₃	#20543	semidirect
239. C₂×C₄	C₈×Dic₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#1055	2
C₄×#88180	2
C₈×C₄⋊Dic₁₃	2
C₄⋊C₈×Dic₁₃	2
#13114610	2
C₂×C₁₃×#431	2
#20543	13