Subgroups — #15763035

Order 3328.

538 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123234	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13112720	C₂	1	4	yes	maximal
4	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	1	no	—
6	C₂×#88123	—	2	1	no	—
7	C₂×#88123	C₄	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄³×C₁₃	C₂²	1	1	yes	characteristic
12	#11718596	—	2	1	no	—
13	#11718596	—	2	1	no	—
14	#11718596	C₄	1	2	yes	—
15	#11718596	C₂²	1	2	yes	—
16	C₂×C₄⋊#1055	C₂²	1	2	yes	—
17	C₂×C₄×#1055	—	2	1	no	—
18	#11718116	—	2	1	no	—
19	C₂²×Dic₂₆	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	C₂×C₄⋊Dic₁₃	—	2	2	no	—
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
34	C₂²×#1055	—	2	2	no	—
35	#88123	—	4	4	no	—
36	#88123	—	4	2	no	—
37	#88123	—	4	2	no	—
38	#88123	—	4	4	no	—
39	C₁₃×C₄⋊C₈	—	2	4	no	—
40	C₁₃×C₄⋊C₈	—	4	2	no	—
41	C₁₃×C₄⋊C₈	—	4	2	no	—
42	C₄⋊#1055	—	2	2	no	—
43	C₄⋊#1055	Q₈	1	4	yes	complemented
44	C₄×#1055	—	4	2	no	—
45	#9896	—	13	1	no	maximal, Sylow
46	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₁₃	—	2	4	no	—
54	C₂×Dic₂₆	—	4	4	no	—
55	C₂×Dic₂₆	—	4	4	no	—
56	C₂×Dic₂₆	—	4	4	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	—	4	2	no	—
59	C₂×C₈×C₁₃	—	4	2	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	D₈	1	2	yes	—
64	C₄²×C₁₃	SD₁₆	1	2	yes	—
65	C₄²×C₁₃	SD₁₆	1	2	yes	—
66	C₄²×C₁₃	SD₁₆	1	2	yes	—
67	C₄²×C₁₃	Q₁₆	1	2	yes	—
68	C₄²×C₁₃	Q₁₆	1	2	yes	—
69	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
70	C₄²×C₁₃	#41	1	2	yes	—
71	C₄⋊Dic₁₃	—	4	4	no	—
72	C₄⋊Dic₁₃	—	4	4	no	—
73	C₄⋊Dic₁₃	—	4	4	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	C₄⋊Dic₁₃	—	4	4	no	—
77	C₄⋊Dic₁₃	—	4	4	no	—
78	C₂×#1055	—	2	4	no	—
79	C₂×#1055	—	4	2	no	—
80	C₂×#1055	—	4	2	no	—
81	#2868	—	13	1	no	—
82	#1842	D₁₃	1	1	yes	characteristic
83	C₄×C₄⋊C₈	—	13	1	no	—
84	#1296	—	13	4	no	—
85	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	#100	1	2	yes	—
93	C₂×C₄×C₁₃	#101	1	2	yes	—
94	C₂×C₄×C₁₃	#102	1	2	yes	—
95	C₂×C₄×C₁₃	#102	1	2	yes	—
96	C₂×C₄×C₁₃	#103	1	2	yes	—
97	C₂×C₄×C₁₃	#103	1	2	yes	—
98	C₂×C₄×C₁₃	#103	1	2	yes	—
99	C₂×C₄×C₁₃	#103	1	2	yes	—
100	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×Dic₁₃	—	4	4	no	—
112	C₂×Dic₁₃	—	4	4	no	—
113	C₂×Dic₁₃	—	4	4	no	—
114	Dic₂₆	—	8	8	no	—
115	C₈×C₁₃	—	8	2	no	—
116	#1055	—	4	4	no	—
117	C₂×C₄⋊Q₈	—	13	2	no	—
118	C₂×C₄⋊Q₈	—	26	1	no	—
119	C₂×C₄⋊Q₈	—	26	1	no	—
120	C₂×C₄⋊C₈	—	2	1	no	—
121	C₂×C₄⋊C₈	—	2	1	no	—
122	C₂×C₄⋊C₈	—	13	2	no	—
123	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
124	C₂×C₄×C₈	—	26	1	no	—
125	#384	—	13	2	no	—
126	#384	—	13	2	no	—
127	#384	—	26	1	no	—
128	#384	—	26	1	no	—
129	C₄³	C₂×D₁₃	1	1	yes	characteristic
130	#336	—	26	1	no	—
131	C₂²×C₁₃	#391	1	1	yes	characteristic
132	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
137	C₂²×C₁₃	#418	1	1	yes	characteristic
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	#332	1	4	yes	—
142	C₄×C₁₃	#333	1	4	yes	—
143	C₄×C₁₃	#476	1	2	yes	—
144	C₄×C₁₃	#477	1	2	yes	—
145	C₄×C₁₃	#486	1	1	yes	characteristic
146	C₄×C₁₃	#487	1	1	yes	characteristic
147	C₄×C₁₃	#498	1	1	yes	characteristic
148	C₄×C₁₃	#500	1	1	yes	characteristic
149	Dic₁₃	—	8	4	no	—
150	C₂²×Q₈	—	26	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	26	2	no	—
153	C₄⋊Q₈	—	52	4	no	—
154	C₄⋊Q₈	—	52	4	no	—
155	C₄⋊Q₈	—	52	2	no	—
156	C₄⋊Q₈	—	52	2	no	—
157	C₂×C₄⋊C₄	—	13	4	no	—
158	C₂×C₄⋊C₄	—	26	2	no	—
159	C₂×C₄⋊C₄	—	26	2	no	—
160	C₂×C₄⋊C₄	—	26	2	no	—
161	C₂×C₄⋊C₄	—	26	2	no	—
162	C₂×C₄⋊C₄	—	26	2	no	—
163	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
165	C₂×C₄²	D₅₂	1	1	yes	characteristic
166	C₂×C₄²	D₅₂	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	4	2	no	—
172	C₄⋊C₈	—	4	2	no	—
173	C₄⋊C₈	—	13	4	no	—
174	C₄⋊C₈	—	26	2	no	—
175	C₄×C₈	—	52	2	no	—
176	C₂×C₁₃	C₂×#332	1	1	yes	characteristic
177	C₂×C₁₃	C₂×#333	1	1	yes	characteristic
178	C₂×C₁₃	#1545	1	1	yes	characteristic
179	C₂×C₁₃	#1549	1	1	yes	characteristic
180	C₂×C₁₃	#1987	1	1	yes	characteristic
181	C₂×C₁₃	#1990	1	1	yes	characteristic
182	C₂×C₁₃	#1997	1	1	yes	characteristic
183	C₂×Q₈	—	52	4	no	—
184	C₂×Q₈	—	52	4	no	—
185	C₂×Q₈	—	52	4	no	—
186	C₂²×C₄	—	26	4	no	—
187	C₂²×C₄	#6103	1	1	yes	characteristic
188	C₂²×C₄	#6103	1	1	yes	characteristic
189	C₂²×C₄	#6103	1	1	yes	characteristic
190	C₂²×C₄	#6103	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	26	4	no	—
198	C₂×C₈	—	52	2	no	—
199	C₂×C₈	—	52	2	no	—
200	C₄⋊C₄	—	52	4	no	—
201	C₄⋊C₄	—	52	4	no	—
202	C₄⋊C₄	—	52	4	no	—
203	C₄⋊C₄	—	52	2	no	—
204	C₄⋊C₄	—	52	2	no	—
205	C₄⋊C₄	—	52	4	no	—
206	C₄⋊C₄	—	52	4	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	#6095	1	2	yes	—
211	C₄²	Dic₅₂	1	2	yes	—
212	C₄²	#6104	1	2	yes	—
213	C₄²	#6105	1	2	yes	—
214	C₄²	#6106	1	2	yes	—
215	C₄²	#6107	1	2	yes	—
216	C₄²	#6129	1	2	yes	—
217	C₄²	Q₈×D₁₃	1	2	yes	—
218	C₁₃	#9896	1	1	yes	characteristic, complemented, Sylow
219	C₂³	C₂×#6103	1	1	yes	characteristic
220	Q₈	—	104	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	52	4	no	—
228	C₂×C₄	—	52	4	no	—
229	C₂×C₄	—	52	4	no	—
230	C₂×C₄	#88049	1	2	yes	—
231	C₂×C₄	#88049	1	2	yes	—
232	C₂×C₄	#88050	1	2	yes	—
233	C₂×C₄	#88050	1	2	yes	—
234	C₂×C₄	#88056	1	2	yes	—
235	C₂×C₄	#88056	1	2	yes	—
236	C₂×C₄	#88063	1	2	yes	—
237	C₂×C₄	#88064	1	2	yes	—
238	C₂×C₄	#88141	1	2	yes	—
239	C₂×C₄	#88151	1	2	yes	—
240	C₂×C₄	C₂×#6095	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
242	C₂×C₄	#88162	1	2	yes	—
243	C₂×C₄	C₂×#6104	1	1	yes	characteristic
244	C₂×C₄	C₂×#6105	1	1	yes	characteristic
245	C₂×C₄	#88190	1	1	yes	characteristic
246	C₂×C₄	C₂×#6106	1	1	yes	characteristic
247	C₂×C₄	C₂×#6107	1	1	yes	characteristic
248	C₂×C₄	#88198	1	1	yes	characteristic
249	C₈	—	8	2	no	—
250	C₈	—	52	4	no	—
251	C₂²	C₂×#88049	1	1	yes	characteristic
252	C₂²	C₂×#88050	1	1	yes	characteristic
253	C₂²	#11718635	1	1	yes	characteristic
254	C₂²	C₂×#88056	1	1	yes	characteristic
255	C₂²	C₂×#88063	1	1	yes	characteristic
256	C₂²	#11718792	1	1	yes	characteristic
257	C₂²	C₂×#88064	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	104	4	no	—
262	C₄	#11718073	1	4	yes	—
263	C₄	#11718074	1	4	yes	—
264	C₄	#11718503	1	2	yes	—
265	C₄	#11718511	1	2	yes	—
266	C₄	#11718731	1	1	yes	characteristic
267	C₄	#11718739	1	1	yes	characteristic
268	C₄	#11718740	1	1	yes	characteristic
269	C₄	#11718742	1	1	yes	characteristic
270	C₂	C₂×#11718074	1	1	yes	characteristic
271	C₂	C₂×#11718073	1	1	yes	characteristic
272	C₂	#13113505	1	1	yes	characteristic
273	C₂	#13113522	1	1	yes	characteristic
274	C₂	#13115783	1	1	yes	characteristic
275	C₂	#13115796	1	1	yes	characteristic
276	C₂	#13115824	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1842	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1842	D₁₃
hypocenter	C₁₃	#9896
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
43. C₄⋊#1055	Q₈	semidirect
218. C₁₃	#9896	semidirect

Maximal subgroups

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

subgroup	index
#13123234	2
C₄×C₄⋊#1055	2
#13112720	2
C₁₃×#1842	2
#9896	13