Subgroups — #15762161

Order 3328.

518 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	#13115789	C₂	1	2	yes	maximal
2	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718139	C₂	1	2	yes	maximal
5	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	—	2	1	no	—
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₁₃×#494	—	2	2	no	—
10	C₁₃×#494	C₄	1	4	yes	complemented
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	—	2	1	no	—
13	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
16	#11718142	—	2	1	no	—
17	#11718139	—	2	4	no	—
18	#11718121	—	2	4	no	—
19	#11718050	—	2	2	no	—
20	C₂²×Q₈×C₁₃	—	2	2	no	—
21	C₂×C₁₃×Q₁₆	—	2	4	no	—
22	C₂×C₁₃×Q₁₆	—	2	4	no	—
23	C₂×C₁₃×Q₁₆	—	4	2	no	—
24	C₂×C₁₃×Q₁₆	—	4	2	no	—
25	C₂×C₁₃×Q₁₆	—	4	2	no	—
26	C₂×C₁₃×Q₁₆	—	4	2	no	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₁₃×C₄⋊Q₈	—	2	2	no	—
30	C₁₃×C₄⋊Q₈	—	2	2	no	—
31	C₁₃×C₄⋊Q₈	—	2	2	no	—
32	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₁₃	—	2	1	no	—
38	C₂²×#1055	—	2	1	no	—
39	C₂²×#1055	—	2	2	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	D₄	1	2	yes	—
42	#88071	—	2	1	no	—
43	C₈×Dic₁₃	—	4	2	no	—
44	C₄⋊#1055	—	2	1	no	—
45	C₄⋊#1055	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄⋊#1055	D₄	1	2	yes	—
48	#9733	—	13	1	no	maximal, Sylow
49	C₂×Q₈×C₁₃	—	2	4	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂×Q₈×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	2	2	no	—
60	C₁₃×Q₁₆	—	4	8	no	—
61	C₁₃×Q₁₆	—	4	8	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₁₃	D₈	1	2	yes	—
67	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃	#41	1	1	yes	characteristic
76	C₄⋊Dic₁₃	—	2	2	no	—
77	C₄⋊Dic₁₃	Q₁₆	1	4	yes	complemented
78	C₄×Dic₁₃	—	4	1	no	—
79	C₄×Dic₁₃	—	4	1	no	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	—	4	1	no	—
82	C₂×#1055	—	4	1	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#1055	—	4	4	no	—
85	C₂×#494	D₁₃	1	1	yes	characteristic
86	#1988	—	13	2	no	—
87	C₈×C₄⋊C₄	—	13	1	no	—
88	C₂×#335	—	13	1	no	—
89	C₂×#332	—	13	2	no	—
90	C₂³×C₁₃	#121	1	1	yes	characteristic
91	Q₈×C₁₃	—	4	4	no	—
92	Q₈×C₁₃	—	4	4	no	—
93	Q₈×C₁₃	—	8	2	no	—
94	Q₈×C₁₃	—	8	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	—	8	2	no	—
98	C₂×C₄×C₁₃	—	8	2	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	C₄×D₄	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	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	#135	1	2	yes	—
109	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×Dic₁₃	—	2	4	no	—
112	C₂×Dic₁₃	—	4	2	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	#102	1	4	yes	—
116	#1055	—	4	2	no	—
117	#1055	—	8	2	no	—
118	C₂²×Q₁₆	—	2	1	no	—
119	C₂²×Q₁₆	—	2	1	no	—
120	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
121	#494	—	2	2	no	—
122	#494	Dic₁₃	1	4	yes	complemented
123	C₂×C₄⋊C₈	—	13	1	no	—
124	C₂×C₄⋊C₈	—	13	2	no	—
125	C₂×C₄×C₈	—	26	1	no	—
126	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
127	C₄×C₄⋊C₄	—	13	1	no	—
128	#340	—	26	4	no	—
129	#336	—	26	1	no	—
130	#335	—	26	2	no	—
131	#332	—	26	4	no	—
132	C₂²×C₁₃	#390	1	1	yes	characteristic
133	C₂²×C₁₃	#459	1	1	yes	characteristic
134	C₂²×C₁₃	#460	1	1	yes	characteristic
135	C₂²×C₁₃	#467	1	2	yes	—
136	C₂²×C₁₃	#473	1	2	yes	—
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	8	4	no	—
140	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
141	C₄×C₁₃	#418	1	1	yes	characteristic
142	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
143	C₄×C₁₃	#459	1	1	yes	characteristic
144	C₄×C₁₃	#460	1	1	yes	characteristic
145	C₄×C₁₃	#494	1	1	yes	characteristic
146	C₄×C₁₃	#495	1	1	yes	characteristic
147	Dic₁₃	—	4	4	no	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂×Q₁₆	—	2	4	no	—
150	C₂×Q₁₆	—	2	4	no	—
151	C₂×Q₁₆	—	4	2	no	—
152	C₂×Q₁₆	—	4	2	no	—
153	C₂×Q₁₆	—	4	2	no	—
154	C₂×Q₁₆	—	4	2	no	—
155	C₂²×C₈	—	26	1	no	—
156	C₂²×C₈	—	26	2	no	—
157	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	C₂×Dic₁₃	1	4	yes	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	13	2	no	—
166	C₂×C₄²	—	26	1	no	—
167	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
168	C₄⋊C₈	—	13	2	no	—
169	C₄⋊C₈	—	26	2	no	—
170	C₄⋊C₈	—	26	1	no	—
171	C₄⋊C₈	—	26	2	no	—
172	C₄×C₈	—	52	2	no	—
173	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
174	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
175	#95	—	26	1	no	—
176	C₂×C₁₃	#1669	1	2	yes	—
177	C₂×C₁₃	#1670	1	2	yes	—
178	C₂×C₁₃	#1969	1	1	yes	characteristic
179	C₂×C₁₃	#1979	1	1	yes	characteristic
180	C₂×C₁₃	#1982	1	1	yes	characteristic
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	26	2	no	—
189	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
190	C₂²×C₄	#6129	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
192	Q₁₆	—	4	8	no	—
193	Q₁₆	—	4	8	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	26	2	no	—
197	C₂×C₈	—	52	1	no	—
198	C₂×C₈	—	52	1	no	—
199	C₂×C₈	—	52	4	no	—
200	C₂×C₈	—	52	2	no	—
201	C₂×C₈	#6104	1	2	yes	—
202	C₂×C₈	#6105	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
204	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	13	4	no	—
210	C₄⋊C₄	—	26	2	no	—
211	C₄²	—	52	1	no	—
212	C₄²	—	52	1	no	—
213	C₄²	D₄×D₁₃	1	1	yes	characteristic
214	C₄²	#6129	1	1	yes	characteristic
215	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	C₁₃	#9733	1	1	yes	characteristic, complemented, Sylow
218	C₂³	#88192	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	8	2	no	—
222	Q₈	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	26	4	no	—
228	C₂×C₄	—	52	2	no	—
229	C₂×C₄	—	52	2	no	—
230	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
231	C₂×C₄	#88173	1	2	yes	—
232	C₂×C₄	C₂×#6104	1	1	yes	characteristic
233	C₂×C₄	#88186	1	1	yes	characteristic
234	C₂×C₄	#88186	1	1	yes	characteristic
235	C₂×C₄	C₂×#6105	1	1	yes	characteristic
236	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
237	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
238	C₂×C₄	#88190	1	1	yes	characteristic
239	C₂×C₄	#88192	1	1	yes	characteristic
240	C₂×C₄	#88194	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	52	2	no	—
244	C₈	—	104	2	no	—
245	C₈	#88072	1	4	yes	—
246	C₂²	#11718733	1	1	yes	characteristic
247	C₂²	#11718736	1	1	yes	characteristic
248	C₂²	#11718849	1	2	yes	—
249	C₂²	#11718853	1	2	yes	—
250	C₂²	#11718882	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	4	no	—
254	C₄	—	52	4	no	—
255	C₄	#11718733	1	1	yes	characteristic
256	C₄	#11718736	1	1	yes	characteristic
257	C₄	#11718842	1	1	yes	characteristic
258	C₄	Q₁₆×Dic₁₃	1	2	yes	—
259	C₄	#11718852	1	1	yes	characteristic
260	C₄	C₂×#88072	1	1	yes	characteristic
261	C₄	#11718876	1	1	yes	characteristic
262	C₂	#13114198	1	2	yes	—
263	C₂	#13114203	1	2	yes	—
264	C₂	#13115698	1	1	yes	characteristic
265	C₂	#13115752	1	1	yes	characteristic
266	C₂	#13115764	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88192
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle, prime radical	C₂³×C₁₃	#121
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#9733
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
10. C₁₃×#494	C₄	semidirect
77. C₄⋊Dic₁₃	Q₁₆	semidirect
122. #494	Dic₁₃	semidirect
217. C₁₃	#9733	semidirect

Maximal subgroups

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

subgroup	index
#13115789	2
C₈×C₄⋊Dic₁₃	2
C₂×#11718050	2
C₂×#11718139	2
C₂×C₁₃×#494	2
#9733	13