Subgroups — #15763058

Order 3328.

430 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	#13123199	C₂	1	1	yes	maximal, characteristic
2	#13114953	C₂	1	1	yes	maximal, characteristic
3	#13112720	C₂	1	2	yes	maximal
4	#13112720	C₂	1	2	yes	maximal
5	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#88124	C₄	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	—	2	1	no	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718597	C₄	1	2	yes	—
11	#11718596	C₂²	1	1	yes	characteristic
12	#11718596	C₂²	1	1	yes	characteristic
13	#11718589	—	2	1	no	—
14	#11718589	—	2	1	no	—
15	#11718589	—	2	1	no	—
16	#11718589	—	2	1	no	—
17	C₂×C₄⋊#1055	C₂²	1	2	yes	—
18	C₁₃×#336	—	2	1	no	—
19	#11718116	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	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	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	—	2	1	no	—
30	C₂×C₄⋊Dic₁₃	—	2	1	no	—
31	C₂×C₄⋊Dic₁₃	—	2	1	no	—
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
35	C₂×#6101	—	2	2	no	—
36	C₂×#6101	—	2	2	no	—
37	C₂²×#1055	—	2	2	no	—
38	#88124	—	4	4	no	—
39	C₁₃×C₄⋊C₈	—	2	2	no	—
40	C₁₃×C₄⋊C₈	Q₈	1	4	yes	—
41	C₄×C₈×C₁₃	—	4	2	no	—
42	#88071	—	2	2	no	—
43	#88071	—	2	2	no	—
44	C₄⋊#1055	—	2	4	no	—
45	#9901	—	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	2	no	—
54	C₂²×Dic₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	2	4	no	—
56	C₂×C₈×C₁₃	—	4	2	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	—	2	1	no	—
65	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
66	C₄²×C₁₃	#41	1	2	yes	—
67	C₄⋊Dic₁₃	—	4	2	no	—
68	C₄⋊Dic₁₃	—	4	2	no	—
69	C₄⋊Dic₁₃	—	4	2	no	—
70	C₄⋊Dic₁₃	—	4	1	no	—
71	C₄⋊Dic₁₃	—	4	1	no	—
72	C₄⋊Dic₁₃	—	4	2	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	#6101	—	4	4	no	—
75	#6101	—	4	4	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	—	4	2	no	—
78	C₂×#1055	—	4	2	no	—
79	#2859	—	13	1	no	—
80	#1831	—	13	1	no	—
81	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
82	#1296	—	13	2	no	—
83	#1296	—	13	2	no	—
84	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	2	1	no	—
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	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	#100	1	2	yes	—
98	C₂×C₄×C₁₃	#101	1	2	yes	—
99	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#135	1	2	yes	—
104	C₂×C₄×C₁₃	#135	1	2	yes	—
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	—	4	2	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₈×C₁₃	—	4	4	no	—
115	#1055	—	8	2	no	—
116	C₂×#125	—	13	2	no	—
117	C₂×C₄⋊C₈	—	13	2	no	—
118	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
119	C₂×C₄×C₈	—	2	1	no	—
120	#389	—	26	1	no	—
121	#389	—	26	1	no	—
122	#389	—	26	1	no	—
123	#389	—	26	1	no	—
124	#384	—	13	1	no	—
125	#384	—	13	1	no	—
126	#382	—	13	2	no	—
127	C₄³	C₂×D₁₃	1	1	yes	characteristic
128	#336	—	2	1	no	—
129	#336	—	26	1	no	—
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	#416	1	1	yes	characteristic
135	C₂²×C₁₃	#416	1	1	yes	characteristic
136	C₂²×C₁₃	#416	1	1	yes	characteristic
137	C₂²×C₁₃	#417	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	—	4	1	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	#478	1	2	yes	—
144	C₄×C₁₃	#479	1	2	yes	—
145	C₄×C₁₃	#488	1	2	yes	—
146	C₄×C₁₃	#499	1	2	yes	—
147	Dic₁₃	—	8	2	no	—
148	Dic₁₃	—	8	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	26	2	no	—
151	#125	—	52	4	no	—
152	C₂×C₄⋊C₄	—	13	2	no	—
153	C₂×C₄⋊C₄	—	13	2	no	—
154	C₂×C₄⋊C₄	—	26	2	no	—
155	C₂×C₄⋊C₄	—	26	2	no	—
156	C₂×C₄⋊C₄	—	26	1	no	—
157	C₂×C₄⋊C₄	—	26	1	no	—
158	C₂×C₄⋊C₄	—	26	1	no	—
159	C₂×C₄⋊C₄	—	26	1	no	—
160	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
162	C₂×C₄²	D₅₂	1	1	yes	characteristic
163	C₂×C₄²	D₅₂	1	1	yes	characteristic
164	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	26	4	no	—
169	C₄⋊C₈	Dic₂₆	1	4	yes	—
170	C₄×C₈	—	4	2	no	—
171	#95	—	26	2	no	—
172	#95	—	26	2	no	—
173	C₂×C₁₃	#1543	1	1	yes	characteristic
174	C₂×C₁₃	#1544	1	1	yes	characteristic
175	C₂×C₁₃	#1551	1	1	yes	characteristic
176	C₂×C₁₃	#1552	1	1	yes	characteristic
177	C₂×C₁₃	#1989	1	1	yes	characteristic
178	C₂×C₁₃	#1991	1	1	yes	characteristic
179	C₂×C₁₃	#1997	1	1	yes	characteristic
180	C₂²×C₄	—	26	2	no	—
181	C₂²×C₄	—	26	2	no	—
182	C₂²×C₄	#6103	1	1	yes	characteristic
183	C₂²×C₄	#6103	1	1	yes	characteristic
184	C₂²×C₄	#6103	1	1	yes	characteristic
185	C₂²×C₄	#6103	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	52	2	no	—
193	C₂×C₈	—	52	2	no	—
194	C₂×C₈	—	52	2	no	—
195	C₄⋊C₄	—	52	2	no	—
196	C₄⋊C₄	—	52	2	no	—
197	C₄⋊C₄	—	52	4	no	—
198	C₄⋊C₄	—	52	2	no	—
199	C₄⋊C₄	—	52	2	no	—
200	C₄⋊C₄	—	52	1	no	—
201	C₄⋊C₄	—	52	1	no	—
202	C₄⋊C₄	—	52	2	no	—
203	C₄⋊C₄	—	52	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₂×Dic₂₆	1	2	yes	—
212	C₄²	#6127	1	2	yes	—
213	C₁₃	#9901	1	1	yes	characteristic, complemented, Sylow
214	C₂³	C₂×#6103	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	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	2	no	—
228	C₂×C₄	—	52	2	no	—
229	C₂×C₄	—	52	2	no	—
230	C₂×C₄	—	52	2	no	—
231	C₂×C₄	—	52	2	no	—
232	C₂×C₄	—	52	2	no	—
233	C₂×C₄	#88063	1	2	yes	—
234	C₂×C₄	#88064	1	2	yes	—
235	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
236	C₂×C₄	#88123	1	1	yes	characteristic
237	C₂×C₄	#88129	1	1	yes	characteristic
238	C₂×C₄	#88158	1	2	yes	—
239	C₂×C₄	#88158	1	2	yes	—
240	C₂×C₄	#88178	1	2	yes	—
241	C₂×C₄	#88186	1	1	yes	characteristic
242	C₂×C₄	#88196	1	1	yes	characteristic
243	C₂×C₄	#88204	1	2	yes	—
244	C₈	—	4	4	no	—
245	C₈	—	104	2	no	—
246	C₂²	#11718590	1	1	yes	characteristic
247	C₂²	#11718627	1	1	yes	characteristic
248	C₂²	#11718663	1	1	yes	characteristic
249	C₂²	#11718774	1	1	yes	characteristic
250	C₂²	#11718774	1	1	yes	characteristic
251	C₂²	C₂×#88063	1	1	yes	characteristic
252	C₂²	C₂×#88064	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	104	2	no	—
259	C₄	—	104	2	no	—
260	C₄	#11718344	1	2	yes	—
261	C₄	#11718358	1	2	yes	—
262	C₄	#11718670	1	2	yes	—
263	C₄	#11718683	1	2	yes	—
264	C₂	#13113496	1	1	yes	characteristic
265	C₂	#13113500	1	1	yes	characteristic
266	C₂	#13113529	1	1	yes	characteristic
267	C₂	#13113534	1	1	yes	characteristic
268	C₂	#13115793	1	1	yes	characteristic
269	C₂	#13115801	1	1	yes	characteristic
270	C₂	#13115825	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#9901
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
213. C₁₃	#9901	semidirect

Maximal subgroups

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

subgroup	index
#13123199	2
#13114953	2
#13112720	2
C₄×C₁₃×C₄⋊C₈	2
#9901	13