Subgroups — #15762900

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88123	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13112675	C₂	1	4	yes	maximal
4	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₂₆	—	2	1	no	—
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	#11718596	C₄	1	1	yes	characteristic
11	C₄×C₄⋊Dic₁₃	—	2	1	no	—
12	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	#11718590	—	2	1	no	—
14	#11718589	—	2	1	no	—
15	#11718589	C₄	1	2	yes	—
16	C₂×C₄⋊#1055	—	2	1	no	—
17	C₂×C₄×#1055	C₂²	1	2	yes	—
18	C₁₃×#336	—	2	1	no	—
19	#11718116	—	2	1	no	—
20	C₂²×Dic₂₆	—	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	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
34	C₂×#6101	—	2	2	no	—
35	C₂×C₄×Dic₁₃	—	2	2	no	—
36	C₂²×#1055	—	2	2	no	—
37	#88123	—	2	2	no	—
38	#88123	C₈	1	4	yes	complemented
39	C₄×Dic₂₆	—	4	4	no	—
40	C₁₃×C₄⋊C₈	—	2	2	no	—
41	C₁₃×C₄⋊C₈	Q₈	1	4	yes	complemented
42	C₄×C₈×C₁₃	—	4	2	no	—
43	#88071	—	2	2	no	—
44	#88071	—	2	2	no	—
45	C₄⋊#1055	—	2	4	no	—
46	C₄×#1055	—	2	4	no	—
47	#9864	—	13	1	no	maximal, Sylow
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	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	2	2	no	—
56	C₂²×Dic₁₃	—	2	2	no	—
57	C₂×Dic₂₆	—	2	4	no	—
58	C₂×Dic₂₆	—	4	2	no	—
59	C₂×Dic₂₆	—	4	2	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	4	2	no	—
62	C₂×C₈×C₁₃	—	4	2	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	4	1	no	—
67	C₄²×C₁₃	C₂×C₈	1	2	yes	—
68	C₄²×C₁₃	M₁₆	1	2	yes	—
69	C₄²×C₁₃	SD₁₆	1	2	yes	—
70	C₄²×C₁₃	Q₁₆	1	2	yes	—
71	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
72	C₄²×C₁₃	#41	1	2	yes	—
73	C₄⋊Dic₁₃	—	2	4	no	—
74	C₄⋊Dic₁₃	—	2	2	no	—
75	C₄⋊Dic₁₃	—	2	4	no	—
76	C₄⋊Dic₁₃	—	4	2	no	—
77	C₄⋊Dic₁₃	—	4	2	no	—
78	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
79	#6101	—	4	4	no	—
80	C₄×Dic₁₃	—	4	4	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	4	2	no	—
83	C₂×#1055	—	4	2	no	—
84	C₄×C₄⋊Q₈	—	13	1	no	—
85	C₄×C₄⋊C₈	—	13	1	no	—
86	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
87	#1279	—	13	4	no	—
88	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
98	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₁₃	#103	1	2	yes	—
101	C₂×C₄×C₁₃	#103	1	2	yes	—
102	C₂×C₄×C₁₃	#104	1	2	yes	—
103	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#131	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×Dic₁₃	—	2	4	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	—	4	2	no	—
116	C₂×Dic₁₃	—	4	2	no	—
117	C₂×Dic₁₃	—	4	2	no	—
118	C₂×Dic₁₃	—	4	2	no	—
119	Dic₂₆	—	4	8	no	—
120	C₈×C₁₃	—	4	4	no	—
121	#1055	—	4	4	no	—
122	C₂×C₄⋊Q₈	—	13	1	no	—
123	C₂×C₄×Q₈	—	26	1	no	—
124	C₂×C₄⋊C₈	—	26	1	no	—
125	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
126	C₂×C₄×C₈	—	2	1	no	—
127	C₂×C₄×C₈	—	13	2	no	—
128	#391	—	26	1	no	—
129	#389	—	13	2	no	—
130	#389	—	26	1	no	—
131	#384	—	13	1	no	—
132	C₄×C₄⋊C₄	—	13	2	no	—
133	C₄×C₄⋊C₄	—	26	1	no	—
134	C₄³	C₂×D₁₃	1	1	yes	characteristic
135	#336	—	2	1	no	—
136	#336	—	26	1	no	—
137	C₂²×C₁₃	#391	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₁₃	#408	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
141	C₂²×C₁₃	#418	1	1	yes	characteristic
142	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
143	C₂²×C₁₃	#421	1	1	yes	characteristic
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	4	1	no	—
149	C₄×C₁₃	Q₈⋊C₈	1	4	yes	—
150	C₄×C₁₃	C₈×Q₈	1	2	yes	—
151	C₄×C₁₃	#446	1	2	yes	—
152	C₄×C₁₃	#476	1	2	yes	—
153	C₄×C₁₃	#477	1	2	yes	—
154	Dic₁₃	—	4	4	no	—
155	Dic₁₃	—	8	2	no	—
156	C₂²×Q₈	—	26	1	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	26	2	no	—
159	C₄⋊Q₈	—	13	4	no	—
160	C₄⋊Q₈	—	26	2	no	—
161	C₄×Q₈	—	52	4	no	—
162	C₂×C₄⋊C₄	—	13	2	no	—
163	C₂×C₄⋊C₄	—	13	2	no	—
164	C₂×C₄⋊C₄	—	26	1	no	—
165	C₂×C₄⋊C₄	—	26	2	no	—
166	C₂×C₄⋊C₄	—	26	1	no	—
167	C₂×C₄⋊C₄	—	26	1	no	—
168	C₂×C₄²	—	26	2	no	—
169	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₄²	D₅₂	1	1	yes	characteristic
172	C₂×C₄²	D₅₂	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	26	4	no	—
178	C₄⋊C₈	Dic₂₆	1	4	yes	complemented
179	C₄×C₈	—	4	2	no	—
180	C₄×C₈	—	26	4	no	—
181	#95	—	26	2	no	—
182	#95	—	26	2	no	—
183	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
184	C₂×C₁₃	#1478	1	1	yes	characteristic
185	C₂×C₁₃	#1481	1	1	yes	characteristic
186	C₂×C₁₃	#1484	1	1	yes	characteristic
187	C₂×C₁₃	#1984	1	1	yes	characteristic
188	C₂×C₁₃	#1986	1	1	yes	characteristic
189	C₂×C₁₃	#1987	1	1	yes	characteristic
190	C₂×Q₈	—	26	4	no	—
191	C₂×Q₈	—	52	2	no	—
192	C₂×Q₈	—	52	2	no	—
193	C₂²×C₄	—	26	2	no	—
194	C₂²×C₄	—	26	2	no	—
195	C₂²×C₄	#6103	1	1	yes	characteristic
196	C₂²×C₄	#6103	1	1	yes	characteristic
197	C₂²×C₄	#6103	1	1	yes	characteristic
198	C₂²×C₄	#6103	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	26	4	no	—
206	C₂×C₈	—	52	2	no	—
207	C₂×C₈	—	52	2	no	—
208	C₄⋊C₄	—	13	4	no	—
209	C₄⋊C₄	—	26	2	no	—
210	C₄⋊C₄	—	26	4	no	—
211	C₄⋊C₄	—	26	4	no	—
212	C₄⋊C₄	—	52	4	no	—
213	C₄⋊C₄	—	52	2	no	—
214	C₄⋊C₄	—	52	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	52	4	no	—
220	C₄²	C₈×D₁₃	1	2	yes	—
221	C₄²	#6094	1	2	yes	—
222	C₄²	#6105	1	2	yes	—
223	C₄²	#6107	1	2	yes	—
224	C₄²	C₂×Dic₂₆	1	2	yes	—
225	C₄²	#6127	1	2	yes	—
226	C₁₃	#9864	1	1	yes	characteristic, complemented, Sylow
227	C₂³	C₂×#6103	1	1	yes	characteristic
228	Q₈	—	52	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	26	4	no	—
239	C₂×C₄	—	52	2	no	—
240	C₂×C₄	—	52	2	no	—
241	C₂×C₄	—	52	2	no	—
242	C₂×C₄	—	52	2	no	—
243	C₂×C₄	—	52	2	no	—
244	C₂×C₄	#88050	1	2	yes	—
245	C₂×C₄	#88050	1	2	yes	—
246	C₂×C₄	#88060	1	2	yes	—
247	C₂×C₄	#88060	1	2	yes	—
248	C₂×C₄	#88065	1	2	yes	—
249	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
250	C₂×C₄	#88123	1	1	yes	characteristic
251	C₂×C₄	#88129	1	1	yes	characteristic
252	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
253	C₂×C₄	C₂×#6094	1	1	yes	characteristic
254	C₂×C₄	#88155	1	2	yes	—
255	C₂×C₄	#88178	1	2	yes	—
256	C₂×C₄	C₂×#6105	1	1	yes	characteristic
257	C₂×C₄	C₂×#6107	1	1	yes	characteristic
258	C₂×C₄	#88203	1	2	yes	—
259	C₈	—	4	4	no	—
260	C₈	—	52	4	no	—
261	C₂²	#11718590	1	1	yes	characteristic
262	C₂²	C₂×#88050	1	1	yes	characteristic
263	C₂²	#11718662	1	1	yes	characteristic
264	C₂²	C₂×#88060	1	1	yes	characteristic
265	C₂²	#11718771	1	1	yes	characteristic
266	C₂²	C₂×#88065	1	1	yes	characteristic
267	C₂²	#11718800	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	52	4	no	—
274	C₄	—	104	2	no	—
275	C₄	#11718076	1	4	yes	—
276	C₄	C₈×Dic₂₆	1	2	yes	—
277	C₄	#11718340	1	2	yes	—
278	C₄	#11718669	1	2	yes	—
279	C₄	#11718682	1	2	yes	—
280	C₂	C₂×#11718076	1	1	yes	characteristic
281	C₂	#13113226	1	1	yes	characteristic
282	C₂	#13113238	1	1	yes	characteristic
283	C₂	#13113252	1	1	yes	characteristic
284	C₂	#13115772	1	1	yes	characteristic
285	C₂	#13115780	1	1	yes	characteristic
286	C₂	#13115784	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
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	#9864
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
38. #88123	C₈	semidirect
41. C₁₃×C₄⋊C₈	Q₈	semidirect
178. C₄⋊C₈	Dic₂₆	semidirect
226. C₁₃	#9864	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88123	2
C₄×C₄⋊#1055	2
#13112675	2
C₄×C₁₃×C₄⋊C₈	2
#9864	13