Subgroups — #15780287

Order 3328.

526 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	C₂³×#6090	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718156	C₂	1	2	yes	maximal, complemented
3	#13114865	C₂	1	1	yes	maximal, characteristic
4	#13112927	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#467	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
9	C₁₃×#467	—	2	2	no	—
10	C₁₃×#467	C₄	1	4	yes	complemented
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#103	—	2	1	no	—
13	#11718873	—	2	1	no	—
14	C₂×#88058	C₂²	1	2	yes	—
15	C₂²×#6090	—	2	4	no	—
16	C₂²×#6090	—	2	2	no	—
17	C₂²×#6090	C₂²	1	2	yes	—
18	#11718156	—	2	4	no	—
19	#11718143	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₁₃×Q₁₆	—	2	2	no	—
23	C₂×C₁₃×Q₁₆	—	2	1	no	—
24	C₂×C₁₃×Q₁₆	—	2	2	no	—
25	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×C₂²⋊Q₈	—	2	4	no	—
33	C₁₃×C₂²⋊Q₈	—	4	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
37	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₁₃×#107	—	2	1	no	—
40	C₁₃×#107	C₂×C₄	1	2	yes	—
41	C₁₃×#103	—	2	4	no	—
42	#88071	—	4	1	no	—
43	#88058	—	2	2	no	—
44	#88058	—	2	2	no	—
45	C₂×#6090	—	2	8	no	—
46	C₂×#6090	—	2	2	no	—
47	C₂×#6090	—	2	2	no	—
48	C₂×#6090	D₄	1	4	yes	—
49	#13060	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂×Q₈×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	4	1	no	—
58	C₂²×C₄×C₁₃	—	4	1	no	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	4	2	no	—
64	C₁₃×Q₁₆	—	2	4	no	—
65	C₁₃×Q₁₆	—	4	1	no	—
66	C₁₃×Q₁₆	—	4	1	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	2	yes	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	4	4	no	—
78	C₁₃×C₄⋊C₄	—	4	1	no	—
79	C₁₃×C₂²⋊C₄	—	4	4	no	—
80	C₂²⋊Dic₁₃	—	8	2	no	—
81	C₄⋊Dic₁₃	—	4	2	no	—
82	C₄⋊Dic₁₃	—	4	2	no	—
83	#6090	—	2	4	no	—
84	C₂³×C₁₆	—	13	1	no	—
85	C₂×#467	D₁₃	1	1	yes	characteristic
86	C₂×#358	—	13	2	no	—
87	#1808	—	13	1	no	—
88	#1380	—	13	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	#102	1	2	yes	—
92	C₂³×C₁₃	SD₃₂	1	2	yes	—
93	C₂³×C₁₃	Q₃₂	1	2	yes	—
94	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₁₃	—	4	2	no	—
96	Q₈×C₁₃	—	4	2	no	—
97	Q₈×C₁₃	—	8	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	#102	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#121	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#135	1	2	yes	—
111	C₂×Dic₁₃	—	8	2	no	—
112	C₂×Dic₁₃	—	8	2	no	—
113	C₂×Dic₁₃	—	8	2	no	—
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	C₄×D₄	1	2	yes	—
116	C₈×C₁₃	#120	1	1	yes	characteristic
117	C₈×C₁₃	#123	1	1	yes	characteristic
118	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
119	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
120	C₂×C₂²⋊Q₈	—	2	1	no	—
121	C₂²×C₁₆	—	13	2	no	—
122	C₂²×C₁₆	—	26	4	no	—
123	C₂²×C₁₆	—	26	2	no	—
124	#467	—	2	2	no	—
125	#467	Dic₁₃	1	4	yes	complemented
126	C₂×#107	—	13	2	no	—
127	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
128	C₂×#103	—	2	1	no	—
129	#380	—	26	1	no	—
130	#358	—	26	4	no	—
131	#340	—	26	1	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	2	4	no	—
135	C₂²×C₁₃	—	2	4	no	—
136	C₂²×C₁₃	#358	1	4	yes	complemented
137	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₁₃	#466	1	2	yes	—
139	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
141	C₂²×C₁₃	#508	1	2	yes	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	8	2	no	—
144	C₄×C₁₃	—	8	2	no	—
145	C₄×C₁₃	#386	1	1	yes	characteristic
146	C₄×C₁₃	#416	1	1	yes	characteristic
147	C₄×C₁₃	#465	1	2	yes	—
148	Dic₁₃	—	16	2	no	—
149	C₂²×Q₈	—	2	1	no	—
150	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	2	1	no	—
154	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
158	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
160	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
161	C₂²⋊Q₈	—	2	4	no	—
162	C₂²⋊Q₈	—	4	2	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	4	1	no	—
165	C₂×C₄⋊C₄	—	26	2	no	—
166	C₂×C₂²⋊C₄	—	4	1	no	—
167	C₂×C₂²⋊C₄	—	52	1	no	—
168	C₂×C₁₆	—	13	4	no	—
169	C₂×C₁₆	—	26	8	no	—
170	C₂×C₁₆	—	26	2	no	—
171	C₂×C₁₆	—	26	2	no	—
172	#107	—	2	1	no	—
173	#107	—	26	2	no	—
174	#107	—	26	2	no	—
175	#107	C₂×Dic₁₃	1	2	yes	—
176	#103	—	2	4	no	—
177	#95	—	52	1	no	—
178	C₂×C₁₃	—	2	4	no	—
179	C₂×C₁₃	#1893	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
181	C₂×C₁₃	#2138	1	1	yes	characteristic
182	C₂×C₁₃	#2216	1	2	yes	—
183	C₂×C₁₃	#2217	1	2	yes	—
184	C₂⁴	#6104	1	1	yes	characteristic
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	52	2	no	—
194	C₂²×C₄	#6104	1	1	yes	characteristic
195	C₂²×C₄	#6105	1	2	yes	—
196	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
197	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
198	Q₁₆	—	2	4	no	—
199	Q₁₆	—	4	1	no	—
200	Q₁₆	—	4	1	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
206	C₂×C₈	D₄×D₁₃	1	2	yes	—
207	C₂×C₈	#6129	1	2	yes	—
208	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	52	2	no	—
214	C₄⋊C₄	—	52	2	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	104	2	no	—
217	C₁₆	—	26	4	no	—
218	C₁₃	#13060	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#88068	1	2	yes	—
222	C₂³	#88069	1	2	yes	—
223	C₂³	#88072	1	2	yes	—
224	C₂³	C₂×#6104	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	8	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	104	2	no	—
237	C₂×C₄	—	104	2	no	—
238	C₂×C₄	—	104	2	no	—
239	C₂×C₄	#88072	1	2	yes	—
240	C₂×C₄	C₂×#6105	1	1	yes	characteristic
241	C₂×C₄	#88193	1	2	yes	—
242	C₂×C₄	#88204	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	D₄×Dic₁₃	1	2	yes	—
246	C₈	#88189	1	1	yes	characteristic
247	C₈	#88191	1	1	yes	characteristic
248	C₂²	—	2	4	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#11718156	1	4	yes	complemented
253	C₂²	#11718699	1	2	yes	—
254	C₂²	C₂×#88068	1	1	yes	characteristic
255	C₂²	C₂×#88069	1	1	yes	characteristic
256	C₂²	#11718855	1	2	yes	—
257	C₂²	C₂×#88072	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	208	2	no	—
262	C₄	#11718702	1	2	yes	—
263	C₄	#11718881	1	1	yes	characteristic
264	C₄	#11718894	1	1	yes	characteristic
265	C₂	—	2	4	no	—
266	C₂	#13115265	1	1	yes	characteristic
267	C₂	C₂×#11718156	1	1	yes	characteristic
268	C₂	#13116543	1	1	yes	characteristic
269	C₂	#13116925	1	2	yes	—
270	C₂	#13116934	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#467	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#467	D₁₃
hypocenter	C₁₃	#13060
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#11718156	C₂	semidirect
4. #13112927	C₂	semidirect
10. C₁₃×#467	C₄	semidirect
125. #467	Dic₁₃	semidirect
136. C₂²×C₁₃	#358	semidirect
218. C₁₃	#13060	semidirect
252. C₂²	#11718156	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#6090	2
C₂×#11718156	2
#13114865	2
#13112927	2
C₂×C₁₃×#467	2
#13060	13