Subgroups — #15881752

Order 3328.

750 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	#13117793	C₂	1	2	yes	maximal
2	#13115606	C₂	1	4	yes	maximal
3	#13115595	C₂	1	2	yes	maximal
4	#13115000	C₂	1	2	yes	maximal
5	#13114993	C₂	1	1	yes	maximal, characteristic
6	C₄×#88057	C₂	1	1	yes	maximal, characteristic
7	C₄×#88057	C₂	1	2	yes	maximal
8	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₁₃	C₄	1	8	yes	complemented
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	#11718891	—	2	2	no	—
16	C₂×#88058	C₂²	1	4	yes	—
17	C₂×#88057	C₂²	1	2	yes	—
18	C₂×#88057	C₂²	1	1	yes	characteristic
19	C₂×#88057	C₂²	1	2	yes	—
20	C₂×#88057	C₂²	1	1	yes	characteristic
21	#11718642	—	2	4	no	—
22	C₄⋊C₄×Dic₁₃	—	2	2	no	—
23	#11718597	C₂²	1	4	yes	—
24	#11718596	C₂²	1	2	yes	—
25	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
27	#11718143	C₂²	1	4	yes	—
28	#11718143	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
35	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
36	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	4	no	—
46	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
50	#88071	—	2	4	no	—
51	#88071	—	2	2	no	—
52	#88071	—	2	4	no	—
53	#88071	—	2	4	no	—
54	#88058	—	4	4	no	—
55	#88057	—	2	4	no	—
56	#88057	—	2	2	no	—
57	#88057	—	4	2	no	—
58	#88057	Q₈	1	4	yes	complemented
59	#22725	—	13	1	no	maximal, Sylow
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	2	2	no	—
68	C₂²×Dic₁₃	—	2	2	no	—
69	C₂²×Dic₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
80	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
84	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	2	4	no	—
89	C₄⋊Dic₁₃	—	2	2	no	—
90	C₄⋊Dic₁₃	—	4	4	no	—
91	C₄⋊Dic₁₃	—	8	2	no	—
92	C₄⋊Dic₁₃	C₂×Q₈	1	4	yes	—
93	C₄×Dic₁₃	—	4	4	no	—
94	C₄×Dic₁₃	—	8	4	no	—
95	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
96	#2355	—	13	2	no	—
97	#1949	—	13	4	no	—
98	#1947	—	13	2	no	—
99	#1846	—	13	2	no	—
100	#1844	—	13	1	no	—
101	C₄×#106	—	13	2	no	—
102	C₄×#106	—	13	1	no	—
103	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃	#135	1	4	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×Dic₁₃	—	2	4	no	—
120	C₂×Dic₁₃	—	4	2	no	—
121	C₂×Dic₁₃	—	4	2	no	—
122	C₂×Dic₁₃	—	4	4	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	4	4	no	—
127	C₂×Dic₁₃	—	4	2	no	—
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	4	no	—
130	C₈×C₁₃	C₄×Q₈	1	4	yes	—
131	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
132	C₈×Q₈	Dic₁₃	1	8	yes	complemented
133	C₂×#107	—	13	4	no	—
134	C₂×#106	—	13	1	no	—
135	C₂×#106	—	13	2	no	—
136	C₂×#106	—	13	1	no	—
137	C₂×#106	—	13	2	no	—
138	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
140	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
142	#391	—	26	2	no	—
143	#387	—	26	4	no	—
144	#384	—	13	2	no	—
145	#382	—	13	4	no	—
146	C₄×C₄⋊C₄	—	13	2	no	—
147	C₄×C₄⋊C₄	—	13	4	no	—
148	C₄×C₄⋊C₄	—	26	2	no	—
149	#340	—	13	4	no	—
150	#340	—	13	2	no	—
151	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
153	C₂²×C₁₃	#550	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×#135	1	2	yes	—
155	C₂²×C₁₃	#577	1	2	yes	—
156	C₄×C₁₃	—	2	4	no	—
157	C₄×C₁₃	—	2	4	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	#427	1	4	yes	—
160	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₄×C₁₃	#520	1	1	yes	characteristic
162	C₄×C₁₃	#557	1	1	yes	characteristic
163	C₄×C₁₃	Q₈²	1	1	yes	characteristic
164	Dic₁₃	—	4	4	no	—
165	Dic₁₃	—	8	2	no	—
166	Dic₁₃	—	8	4	no	—
167	C₂²×Q₈	D₅₂	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
170	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₄×Q₈	Dic₂₆	1	4	yes	—
172	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
173	C₂×C₄⋊C₄	—	13	2	no	—
174	C₂×C₄⋊C₄	—	13	2	no	—
175	C₂×C₄⋊C₄	—	13	4	no	—
176	C₂×C₄⋊C₄	—	26	2	no	—
177	C₂×C₄⋊C₄	—	26	2	no	—
178	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
179	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
180	C₂×C₄²	—	26	2	no	—
181	C₂×C₄²	—	26	4	no	—
182	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
183	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
184	#107	—	52	4	no	—
185	#106	—	13	4	no	—
186	#106	—	26	4	no	—
187	#106	—	26	2	no	—
188	#106	—	52	2	no	—
189	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
190	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
191	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
192	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
193	#95	—	26	2	no	—
194	#95	—	26	4	no	—
195	#95	—	26	4	no	—
196	#95	—	26	4	no	—
197	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
199	C₂×C₁₃	#2912	1	1	yes	characteristic
200	C₂×C₁₃	#3381	1	2	yes	—
201	C₂×C₁₃	#3394	1	2	yes	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
204	C₂²×C₄	—	26	2	no	—
205	C₂²×C₄	—	26	2	no	—
206	C₂²×C₄	—	26	4	no	—
207	C₂²×C₄	C₂×D₅₂	1	2	yes	—
208	C₂²×C₄	C₂×D₅₂	1	2	yes	—
209	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
211	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	Q₈×D₁₃	1	2	yes	—
217	C₂×C₈	#6131	1	2	yes	—
218	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
219	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
220	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
221	C₄⋊C₄	—	13	4	no	—
222	C₄⋊C₄	—	26	2	no	—
223	C₄⋊C₄	—	26	2	no	—
224	C₄⋊C₄	—	26	4	no	—
225	C₄⋊C₄	—	26	4	no	—
226	C₄⋊C₄	—	26	2	no	—
227	C₄⋊C₄	—	52	4	no	—
228	C₄⋊C₄	—	104	2	no	—
229	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
230	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
231	C₄²	—	52	4	no	—
232	C₄²	—	104	4	no	—
233	C₄²	C₂×Dic₂₆	1	4	yes	—
234	C₄²	C₂×Dic₂₆	1	2	yes	—
235	C₄²	C₂²×Dic₁₃	1	4	yes	—
236	C₄²	C₂²×Dic₁₃	1	2	yes	—
237	C₁₃	#22725	1	1	yes	characteristic, complemented, Sylow
238	C₂³	C₂²×D₅₂	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	26	4	no	—
244	C₂×C₄	—	52	2	no	—
245	C₂×C₄	—	52	2	no	—
246	C₂×C₄	—	52	2	no	—
247	C₂×C₄	—	52	4	no	—
248	C₂×C₄	—	52	4	no	—
249	C₂×C₄	—	52	2	no	—
250	C₂×C₄	—	52	4	no	—
251	C₂×C₄	—	52	2	no	—
252	C₂×C₄	#88158	1	4	yes	—
253	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
254	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
256	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
257	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂×#6131	1	1	yes	characteristic
260	C₂×C₄	#88256	1	1	yes	characteristic
261	C₂×C₄	#88257	1	1	yes	characteristic
262	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	Q₈×Dic₁₃	1	4	yes	—
266	C₂²	Q₈×D₅₂	1	1	yes	characteristic
267	C₂²	#11719282	1	1	yes	characteristic
268	C₂²	C₂×#88158	1	2	yes	—
269	C₂²	#11719456	1	2	yes	—
270	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	52	4	no	—
275	C₄	—	104	2	no	—
276	C₄	—	104	4	no	—
277	C₄	#11718767	1	4	yes	—
278	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
279	C₄	#11719275	1	1	yes	characteristic
280	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
281	C₄	#11719532	1	1	yes	characteristic
282	C₂	Q₈×C₄⋊Dic₁₃	1	1	yes	characteristic
283	C₂	C₂×#11718767	1	1	yes	characteristic
284	C₂	#13123431	1	1	yes	characteristic
285	C₂	#13127713	1	2	yes	—
286	C₂	#13127834	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	#22725
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. C₈×Q₈×C₁₃	C₄	semidirect
58. #88057	Q₈	semidirect
132. C₈×Q₈	Dic₁₃	semidirect
237. C₁₃	#22725	semidirect

Maximal subgroups

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

subgroup	index
#13117793	2
#13115606	2
#13115595	2
#13115000	2
#13114993	2
C₄×#88057	2
C₂×C₈×Q₈×C₁₃	2
#22725	13