Subgroups — #15756038

Order 3328.

418 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114964	C₂	1	1	yes	maximal, characteristic
2	#13114841	C₂	1	2	yes	maximal
3	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
4	#13112720	C₂	1	2	yes	maximal
5	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₁₃×#446	—	2	2	no	—
8	C₁₃×#446	C₄	1	4	yes	complemented
9	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88075	—	2	2	no	—
14	#11718596	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	#11718143	—	2	1	no	—
17	#11718143	—	2	1	no	—
18	#11718121	—	2	2	no	—
19	#11718049	—	2	2	no	—
20	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	—	2	1	no	—
25	C₄×Q₈×C₁₃	—	2	1	no	—
26	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₄×Q₈×C₁₃	Q₈	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	—	2	1	no	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	C₂×C₄⋊Dic₁₃	—	2	2	no	—
34	C₂×C₄⋊Dic₁₃	—	2	2	no	—
35	C₂²×#1055	—	2	2	no	—
36	C₁₃×C₄⋊C₈	—	2	4	no	—
37	C₁₃×C₄⋊C₈	—	2	1	no	—
38	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₁₃×#97	—	2	4	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	#88075	—	4	4	no	—
43	C₄⋊#1055	—	2	2	no	—
44	C₄⋊#1055	—	2	2	no	—
45	#8657	—	13	1	no	maximal, Sylow
46	C₂×Q₈×C₁₃	—	2	1	no	—
47	C₂×Q₈×C₁₃	—	2	2	no	—
48	C₂×Q₈×C₁₃	—	2	2	no	—
49	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
55	C₂²×Dic₁₃	—	4	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	2	2	no	—
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	4	2	no	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₁₃	#41	1	1	yes	characteristic
72	C₄⋊Dic₁₃	—	4	2	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	C₄⋊Dic₁₃	—	8	2	no	—
77	C₂×#1055	—	4	2	no	—
78	C₂×#1055	—	4	2	no	—
79	C₂×#1055	—	4	2	no	—
80	C₂×#446	D₁₃	1	1	yes	characteristic
81	#1834	—	13	1	no	—
82	#1803	—	13	2	no	—
83	C₂×#334	—	13	1	no	—
84	#1296	—	13	2	no	—
85	C₂³×C₁₃	#121	1	1	yes	characteristic
86	Q₈×C₁₃	—	2	4	no	—
87	Q₈×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	4	1	no	—
92	C₂×C₄×C₁₃	—	4	1	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#135	1	2	yes	—
101	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₁₃	#137	1	2	yes	—
103	C₂×C₄×C₁₃	#137	1	2	yes	—
104	C₂×Dic₁₃	—	8	2	no	—
105	C₂×Dic₁₃	—	8	2	no	—
106	C₂×Dic₁₃	—	8	2	no	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	—	4	2	no	—
110	#1055	—	8	2	no	—
111	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
112	#446	—	2	2	no	—
113	#446	Dic₁₃	1	4	yes	complemented
114	C₂×C₄⋊C₈	—	2	1	no	—
115	C₂×C₄⋊C₈	—	13	2	no	—
116	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
117	C₂×#103	—	26	2	no	—
118	C₂×#97	—	2	1	no	—
119	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
120	#384	—	13	2	no	—
121	#340	—	26	1	no	—
122	#340	—	26	1	no	—
123	#340	—	26	2	no	—
124	#334	—	26	2	no	—
125	C₂²×C₁₃	#380	1	1	yes	characteristic
126	C₂²×C₁₃	#464	1	2	yes	—
127	C₂²×C₁₃	#469	1	1	yes	characteristic
128	C₂²×C₁₃	#470	1	1	yes	characteristic
129	C₂²×C₁₃	#471	1	2	yes	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	4	2	no	—
133	C₄×C₁₃	—	4	2	no	—
134	C₄×C₁₃	#416	1	1	yes	characteristic
135	C₄×C₁₃	#419	1	1	yes	characteristic
136	C₄×C₁₃	#428	1	2	yes	—
137	C₄×C₁₃	#468	1	2	yes	—
138	C₄×C₁₃	#479	1	2	yes	—
139	Dic₁₃	—	16	2	no	—
140	C₂²×Q₈	D₅₂	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	26	2	no	—
143	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
144	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	Dic₂₆	1	2	yes	—
148	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	26	2	no	—
151	C₂×C₄⋊C₄	—	26	2	no	—
152	C₂×C₄⋊C₄	—	26	2	no	—
153	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
156	C₄⋊C₈	—	2	4	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	26	2	no	—
159	C₄⋊C₈	—	26	2	no	—
160	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
161	#103	—	52	4	no	—
162	#97	—	2	4	no	—
163	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
164	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
165	C₂×C₁₃	#1702	1	2	yes	—
166	C₂×C₁₃	#1704	1	2	yes	—
167	C₂×C₁₃	#1805	1	1	yes	characteristic
168	C₂×C₁₃	#1809	1	1	yes	characteristic
169	C₂×C₁₃	#1815	1	1	yes	characteristic
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	52	2	no	—
177	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
178	C₂²×C₄	#6127	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	—	52	2	no	—
186	C₂×C₈	—	52	2	no	—
187	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
188	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	52	2	no	—
193	C₄⋊C₄	—	52	2	no	—
194	C₄⋊C₄	—	52	2	no	—
195	C₄⋊C₄	—	52	2	no	—
196	C₄⋊C₄	—	104	2	no	—
197	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
201	C₄²	#6127	1	1	yes	characteristic
202	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
203	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
204	C₁₃	#8657	1	1	yes	characteristic, complemented, Sylow
205	C₂³	#88184	1	1	yes	characteristic
206	Q₈	—	2	4	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	104	2	no	—
215	C₂×C₄	—	104	2	no	—
216	C₂×C₄	—	104	2	no	—
217	C₂×C₄	#88162	1	2	yes	—
218	C₂×C₄	#88163	1	2	yes	—
219	C₂×C₄	#88178	1	1	yes	characteristic
220	C₂×C₄	#88178	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
222	C₂×C₄	#88180	1	1	yes	characteristic
223	C₂×C₄	#88184	1	1	yes	characteristic
224	C₂×C₄	#88196	1	2	yes	—
225	C₂×C₄	#88204	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	4	2	no	—
230	C₈	—	104	2	no	—
231	C₂²	#11718689	1	2	yes	—
232	C₂²	#11718796	1	1	yes	characteristic
233	C₂²	#11718798	1	1	yes	characteristic
234	C₂²	#11718802	1	2	yes	—
235	C₂²	#11718873	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	208	2	no	—
241	C₄	#11718683	1	2	yes	—
242	C₄	#11718782	1	2	yes	—
243	C₄	#11718795	1	2	yes	—
244	C₄	#11718886	1	1	yes	characteristic
245	C₄	#11718894	1	1	yes	characteristic
246	C₂	#13114404	1	2	yes	—
247	C₂	#13114418	1	2	yes	—
248	C₂	#13114851	1	1	yes	characteristic
249	C₂	#13114869	1	1	yes	characteristic
250	C₂	#13114895	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle, prime radical	C₂³×C₁₃	#121
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	#8657
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
8. C₁₃×#446	C₄	semidirect
113. #446	Dic₁₃	semidirect
204. C₁₃	#8657	semidirect

Maximal subgroups

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

subgroup	index
#13114964	2
#13114841	2
C₂×#11718049	2
#13112720	2
C₂×C₁₃×#446	2
#8657	13