Subgroups — #15761850

Order 3328.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13113230	C₂	1	2	yes	maximal
4	#13113159	C₂	1	2	yes	maximal
5	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
8	C₁₃×#431	C₂²	1	1	yes	characteristic
9	C₁₃×#431	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88058	C₄	1	1	yes	characteristic
12	C₂×#88056	C₄	1	2	yes	—
13	#11718586	C₂²	1	2	yes	—
14	C₄×Dic₅₂	—	2	1	no	—
15	C₄×Dic₅₂	—	2	1	no	—
16	C₄×Dic₅₂	—	2	1	no	—
17	C₄×Dic₅₂	—	2	1	no	—
18	C₁₃×#322	—	2	1	no	—
19	C₈²×C₁₃	—	2	1	no	—
20	#11718076	—	2	4	no	—
21	#11718061	—	2	4	no	—
22	#11718050	—	2	1	no	—
23	#11718049	—	2	1	no	—
24	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
29	C₂×#6101	—	2	2	no	—
30	C₂×C₄×Dic₁₃	—	2	2	no	—
31	C₂×Dic₅₂	—	2	1	no	—
32	C₂×Dic₅₂	—	2	1	no	—
33	C₂×Dic₅₂	—	2	2	no	—
34	C₂×Dic₅₂	C₈	1	4	yes	complemented
35	C₄×Dic₂₆	—	2	2	no	—
36	C₄×Dic₂₆	—	2	2	no	—
37	C₄×Dic₂₆	—	2	2	no	—
38	C₄×Dic₂₆	—	2	2	no	—
39	C₁₃×C₄⋊C₈	—	2	1	no	—
40	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
41	C₁₃×C₂²⋊C₈	—	2	1	no	—
42	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₂²⋊#1055	—	2	2	no	—
50	#88058	—	2	1	no	—
51	#88058	—	2	1	no	—
52	#88056	—	2	2	no	—
53	#88056	—	2	2	no	—
54	C₄⋊#1055	—	2	2	no	—
55	C₄×#1055	—	2	2	no	—
56	#9669	—	13	1	no	maximal, Sylow
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	2	2	no	—
61	C₂²×Dic₁₃	—	2	2	no	—
62	C₂×Dic₂₆	—	2	2	no	—
63	C₂×Dic₂₆	—	2	2	no	—
64	C₂×Dic₂₆	—	2	2	no	—
65	C₂×Dic₂₆	—	2	2	no	—
66	C₂×Dic₂₆	—	2	2	no	—
67	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	#41	1	1	yes	characteristic
82	C₄⋊Dic₁₃	—	2	2	no	—
83	C₄⋊Dic₁₃	—	2	2	no	—
84	#6101	—	4	2	no	—
85	#6101	—	4	2	no	—
86	C₄×Dic₁₃	—	4	2	no	—
87	C₄×Dic₁₃	—	4	2	no	—
88	C₂×#1055	—	4	2	no	—
89	Dic₅₂	—	2	4	no	—
90	Dic₅₂	—	2	4	no	—
91	C₂×C₄×Q₁₆	—	13	1	no	—
92	#1564	—	13	1	no	—
93	#1479	—	13	2	no	—
94	#1461	—	13	2	no	—
95	#1450	D₁₃	1	1	yes	characteristic
96	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	4	1	no	—
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
109	C₂×Dic₁₃	—	2	4	no	—
110	C₂×Dic₁₃	—	4	2	no	—
111	C₂×Dic₁₃	—	4	2	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	—	8	2	no	—
114	Dic₂₆	—	2	4	no	—
115	Dic₂₆	—	2	4	no	—
116	Dic₂₆	—	4	2	no	—
117	Dic₂₆	—	4	2	no	—
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	—	2	1	no	—
120	C₈×C₁₃	—	4	2	no	—
121	C₈×C₁₃	—	4	2	no	—
122	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
123	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
124	#1055	—	8	2	no	—
125	C₂²×Q₁₆	—	13	1	no	—
126	C₂×C₄×Q₈	—	13	2	no	—
127	C₄×Q₁₆	—	26	1	no	—
128	C₄×Q₁₆	—	26	1	no	—
129	C₄×Q₁₆	—	26	1	no	—
130	C₄×Q₁₆	—	26	1	no	—
131	#431	—	13	2	no	—
132	#431	C₂×D₁₃	1	1	yes	characteristic
133	#431	C₂×D₁₃	1	1	yes	characteristic
134	C₂×#107	—	13	1	no	—
135	C₂×#103	—	13	2	no	—
136	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
137	#335	—	26	1	no	—
138	#334	—	26	1	no	—
139	Q₈⋊C₈	—	26	4	no	—
140	#324	—	26	4	no	—
141	#322	—	2	1	no	—
142	C₈²	—	2	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	#390	1	1	yes	characteristic
146	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
147	C₂²×C₁₃	#436	1	1	yes	characteristic
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	C₄×C₁₃	#408	1	1	yes	characteristic
153	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
154	C₄×C₁₃	#436	1	1	yes	characteristic
155	C₄×C₁₃	#473	1	2	yes	—
156	Dic₁₃	—	4	4	no	—
157	Dic₁₃	—	8	2	no	—
158	C₂²×Q₈	—	13	2	no	—
159	C₂×Q₁₆	—	13	4	no	—
160	C₂×Q₁₆	—	26	2	no	—
161	C₂×Q₁₆	—	26	1	no	—
162	C₂×Q₁₆	—	26	1	no	—
163	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
165	C₄×Q₈	—	26	2	no	—
166	C₄×Q₈	—	26	2	no	—
167	C₄×Q₈	—	26	2	no	—
168	C₄×Q₈	—	26	2	no	—
169	C₂×C₄⋊C₄	—	13	2	no	—
170	C₂×C₄⋊C₄	—	26	2	no	—
171	C₂×C₄²	—	26	2	no	—
172	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
173	#107	—	26	1	no	—
174	#107	—	26	1	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	26	2	no	—
177	C₄⋊C₈	D₅₂	1	2	yes	—
178	#103	—	26	2	no	—
179	#103	—	26	2	no	—
180	C₂²⋊C₈	—	2	1	no	—
181	C₂²⋊C₈	—	26	2	no	—
182	C₂²⋊C₈	D₅₂	1	2	yes	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	26	2	no	—
186	C₄×C₈	D₅₂	1	1	yes	characteristic
187	C₄×C₈	D₅₂	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
190	C₂×C₁₃	—	2	1	no	—
191	C₂×C₁₃	—	2	1	no	—
192	C₂×C₁₃	#1954	1	1	yes	characteristic
193	C₂×C₁₃	#1976	1	1	yes	characteristic
194	C₂×C₁₃	#1979	1	1	yes	characteristic
195	C₂×Q₈	—	13	4	no	—
196	C₂×Q₈	—	26	2	no	—
197	C₂×Q₈	—	26	2	no	—
198	C₂×Q₈	—	26	2	no	—
199	C₂×Q₈	—	26	2	no	—
200	C₂×Q₈	—	26	2	no	—
201	C₂²×C₄	—	26	2	no	—
202	C₂²×C₄	—	26	2	no	—
203	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
205	C₂²×C₄	#6127	1	1	yes	characteristic
206	Q₁₆	—	26	4	no	—
207	Q₁₆	—	26	4	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	52	2	no	—
213	C₂×C₈	C₈×D₁₃	1	2	yes	—
214	C₂×C₈	#6094	1	2	yes	—
215	C₂×C₈	#6103	1	1	yes	characteristic
216	C₂×C₈	#6103	1	1	yes	characteristic
217	C₂×C₈	#6103	1	1	yes	characteristic
218	C₂×C₈	#6103	1	1	yes	characteristic
219	C₄⋊C₄	—	26	2	no	—
220	C₄⋊C₄	—	26	2	no	—
221	C₄⋊C₄	—	52	2	no	—
222	C₄⋊C₄	—	52	2	no	—
223	C₄²	—	52	2	no	—
224	C₄²	—	52	2	no	—
225	C₄²	C₂×D₅₂	1	1	yes	characteristic
226	C₄²	C₂×D₅₂	1	1	yes	characteristic
227	C₄²	#6127	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₁₃	#9669	1	1	yes	characteristic, complemented, Sylow
230	C₂³	C₄×D₅₂	1	1	yes	characteristic
231	Q₈	—	26	4	no	—
232	Q₈	—	26	4	no	—
233	Q₈	—	52	2	no	—
234	Q₈	—	52	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	26	4	no	—
238	C₂×C₄	—	52	2	no	—
239	C₂×C₄	—	52	2	no	—
240	C₂×C₄	—	52	2	no	—
241	C₂×C₄	—	104	2	no	—
242	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
243	C₂×C₄	#88128	1	1	yes	characteristic
244	C₂×C₄	#88129	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#6094	1	1	yes	characteristic
247	C₂×C₄	#88155	1	1	yes	characteristic
248	C₂×C₄	#88155	1	1	yes	characteristic
249	C₂×C₄	C₂×#6103	1	1	yes	characteristic
250	C₂×C₄	#88184	1	1	yes	characteristic
251	C₂×C₄	#88184	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	104	2	no	—
257	C₈	#88060	1	2	yes	—
258	C₈	#88060	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	C₈×D₅₂	1	1	yes	characteristic
262	C₂²	#11718349	1	1	yes	characteristic
263	C₂²	#11718600	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	52	4	no	—
268	C₄	—	104	2	no	—
269	C₄	C₈×D₅₂	1	1	yes	characteristic
270	C₄	#11718349	1	1	yes	characteristic
271	C₄	C₂×#88060	1	1	yes	characteristic
272	C₄	#11718771	1	1	yes	characteristic
273	C₄	#11718804	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#13115630	1	1	yes	characteristic
277	C₂	#13115734	1	1	yes	characteristic
278	C₂	#13115751	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1450	C₂
socle	C₂²×C₁₃	#390
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1450	D₁₃
hypocenter	C₁₃	#9669
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
34. C₂×Dic₅₂	C₈	semidirect
229. C₁₃	#9669	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
#13113584	2
#13113230	2
#13113159	2
C₁₃×#1450	2
#9669	13