Subgroups — #15768840

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88059	C₂	1	2	yes	maximal
2	#13114575	C₂	1	2	yes	maximal
3	C₂×#11718152	C₂	1	2	yes	maximal
4	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₂²×#6099	C₂²	1	4	yes	—
9	C₁₃×#405	—	2	4	no	—
10	C₂×C₁₃×#97	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
12	C₂×#88059	—	2	4	no	—
13	C₂×#88059	—	2	4	no	—
14	C₂×#88059	C₄	1	8	yes	complemented
15	#11718152	—	2	4	no	—
16	#11718122	—	2	4	no	—
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	—	2	2	no	—
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
21	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₂²×C₈×C₁₃	D₄	1	4	yes	—
28	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
29	C₁₃×#117	—	2	2	no	—
30	C₁₃×#117	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
32	C₂×C₁₃×C₂²⋊C₄	Q₈	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×#6099	—	2	4	no	—
35	C₂×#6099	—	2	4	no	—
36	C₂×#6099	—	2	4	no	—
37	C₂×#6099	—	2	4	no	—
38	C₂×#6099	—	2	4	no	—
39	C₂×#6099	C₂×C₄	1	8	yes	—
40	C₂²×#1055	—	2	4	no	—
41	C₁₃×#97	—	2	4	no	—
42	C₄×C₈×C₁₃	—	2	4	no	—
43	#88059	—	2	16	no	—
44	#10867	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	#41	1	2	yes	—
56	C₂²×C₄×C₁₃	#41	1	2	yes	—
57	C₁₃×M₁₆	—	2	4	no	—
58	C₁₃×M₁₆	—	2	4	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	C₄²	1	2	yes	—
62	C₂×C₈×C₁₃	C₄²	1	2	yes	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	8	yes	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
68	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
70	C₁₃×C₄⋊C₄	—	2	4	no	—
71	C₁₃×C₂²⋊C₄	—	2	4	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	2	no	—
74	#6099	—	2	8	no	—
75	#6099	—	2	8	no	—
76	#6099	—	4	4	no	—
77	#6099	—	4	4	no	—
78	C₂×#1055	—	2	8	no	—
79	C₂×#1055	—	4	4	no	—
80	C₂×#1055	—	4	4	no	—
81	C₂²×#108	—	13	2	no	—
82	C₂×#405	D₁₃	1	1	yes	characteristic
83	#1742	—	13	2	no	—
84	C₂×#338	—	13	2	no	—
85	C₂³×C₁₃	—	4	2	no	—
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
88	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
89	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
90	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃	#121	1	2	yes	—
105	C₂×C₄×C₁₃	#121	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₃	#124	1	2	yes	—
108	C₂×C₄×C₁₃	#125	1	2	yes	—
109	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
111	C₈×C₁₃	—	2	4	no	—
112	C₈×C₁₃	#95	1	4	yes	—
113	C₈×C₁₃	#95	1	4	yes	—
114	#1055	—	4	8	no	—
115	C₂²×M₁₆	—	13	4	no	—
116	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
117	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
118	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
119	C₂×#108	—	13	8	no	—
120	C₂×#108	—	26	4	no	—
121	C₂×#108	—	26	4	no	—
122	#405	—	2	4	no	—
123	C₂×#97	Dic₁₃	1	2	yes	—
124	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
125	#343	—	26	4	no	—
126	#338	—	26	4	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	4	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	—	4	1	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₁₃	#380	1	1	yes	characteristic
134	C₂²×C₁₃	#384	1	1	yes	characteristic
135	C₂²×C₁₃	#389	1	2	yes	—
136	C₂²×C₁₃	#391	1	2	yes	—
137	C₄×C₁₃	—	4	4	no	—
138	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
139	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₄×C₁₃	#380	1	1	yes	characteristic
141	C₄×C₁₃	#382	1	1	yes	characteristic
142	C₄×C₁₃	#389	1	2	yes	—
143	C₄×C₁₃	#390	1	2	yes	—
144	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	13	8	no	—
148	C₂×M₁₆	—	26	4	no	—
149	C₂×M₁₆	—	26	4	no	—
150	C₂×M₁₆	—	26	4	no	—
151	C₂×M₁₆	—	26	4	no	—
152	C₂×M₁₆	—	26	4	no	—
153	C₂×M₁₆	Dic₂₆	1	2	yes	—
154	C₂×M₁₆	D₅₂	1	2	yes	—
155	C₂²×C₈	—	26	4	no	—
156	C₂²×C₈	Dic₂₆	1	2	yes	—
157	C₂²×C₈	C₄×D₁₃	1	4	yes	—
158	C₂²×C₈	D₅₂	1	2	yes	—
159	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
160	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
161	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
166	C₂×C₂²⋊C₄	Dic₂₆	1	2	yes	—
167	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
168	#108	—	26	16	no	—
169	#97	—	2	4	no	—
170	C₄×C₈	—	2	4	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	2	1	no	—
173	C₂×C₁₃	—	2	2	no	—
174	C₂×C₁₃	—	2	2	no	—
175	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
176	C₂×C₁₃	#1813	1	2	yes	—
177	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
179	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
182	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
185	C₂²×C₄	#6127	1	2	yes	—
186	C₂²×C₄	#6127	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	26	8	no	—
192	M₁₆	—	26	8	no	—
193	M₁₆	—	52	4	no	—
194	M₁₆	—	52	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	26	8	no	—
198	C₂×C₈	—	52	4	no	—
199	C₂×C₈	—	52	4	no	—
200	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
201	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
202	C₂×C₈	#6101	1	8	yes	—
203	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
204	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
205	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
206	C₂×C₈	#6103	1	8	yes	—
207	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
208	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
209	C₄⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₁₃	#10867	1	1	yes	characteristic, complemented, Sylow
214	C₂³	—	4	2	no	—
215	C₂³	C₄×Dic₂₆	1	2	yes	—
216	C₂³	#88123	1	1	yes	characteristic
217	C₂³	#88123	1	1	yes	characteristic
218	C₂³	#88178	1	2	yes	—
219	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
223	C₂×C₄	#88123	1	1	yes	characteristic
224	C₂×C₄	#88124	1	2	yes	—
225	C₂×C₄	C₄×D₅₂	1	2	yes	—
226	C₂×C₄	C₄×D₅₂	1	2	yes	—
227	C₂×C₄	#88128	1	1	yes	characteristic
228	C₂×C₄	#88129	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
230	C₂×C₄	C₂×#6101	1	2	yes	—
231	C₂×C₄	#88178	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
234	C₂×C₄	#88180	1	1	yes	characteristic
235	C₂×C₄	#88180	1	1	yes	characteristic
236	C₂×C₄	C₂×#6103	1	2	yes	—
237	C₂×C₄	#88184	1	2	yes	—
238	C₂×C₄	#88184	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	52	8	no	—
242	C₈	#88071	1	4	yes	—
243	C₈	#88071	1	4	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	#11718589	1	2	yes	—
250	C₂²	#11718590	1	2	yes	—
251	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
252	C₂²	#11718596	1	1	yes	characteristic
253	C₂²	#11718873	1	1	yes	characteristic
254	C₄	—	4	4	no	—
255	C₄	#11718589	1	2	yes	—
256	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
257	C₄	#11718597	1	1	yes	characteristic
258	C₄	#11718600	1	2	yes	—
259	C₄	C₂×#88071	1	1	yes	characteristic
260	C₄	#11718873	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	2	1	no	—
264	C₂	—	2	1	no	—
265	C₂	#13113066	1	1	yes	characteristic
266	C₂	#13114886	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88180
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#10867
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
14. C₂×#88059	C₄	semidirect
213. C₁₃	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88059	2
#13114575	2
C₂×#11718152	2
C₂×C₁₃×#405	2
#10867	13