Subgroups — #15772053

Order 3328.

450 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116447	C₂	1	2	yes	maximal
2	#13112776	C₂	1	4	yes	maximal, complemented
3	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
9	C₂×C₁₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×#88052	—	2	2	no	—
12	C₂²⋊#6090	—	2	2	no	—
13	C₂²⋊#6090	C₂²	1	4	yes	—
14	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	—	2	1	no	—
18	C₂²×C₈×C₁₃	—	2	1	no	—
19	C₂²×C₈×C₁₃	—	4	2	no	—
20	C₂²×C₈×C₁₃	—	4	2	no	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	D₄	1	2	yes	—
24	C₂²×C₈×C₁₃	D₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	—	2	1	no	—
27	C₂×C₄²×C₁₃	—	2	1	no	—
28	C₂×C₄²×C₁₃	—	2	1	no	—
29	C₂×C₄²×C₁₃	—	2	1	no	—
30	C₂×C₄²×C₁₃	C₈	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₁₃×C₄⋊C₈	—	2	4	no	—
33	C₁₃×C₄⋊C₈	—	2	4	no	—
34	#88052	—	4	4	no	—
35	C₂×#6090	—	2	4	no	—
36	#11440	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	—	2	1	no	—
39	C₂²×C₄×C₁₃	—	2	1	no	—
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	—	4	1	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	—	4	1	no	—
59	C₂×C₈×C₁₃	—	4	2	no	—
60	C₂×C₈×C₁₃	—	4	2	no	—
61	C₂×C₈×C₁₃	—	4	2	no	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	#6090	—	4	4	no	—
69	C₂²×C₄⋊C₈	D₁₃	1	1	yes	characteristic
70	#2112	—	13	2	no	—
71	#1322	—	13	4	no	—
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
76	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
77	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#99	1	2	yes	—
96	C₂×C₄×C₁₃	#101	1	2	yes	—
97	C₂×C₄×C₁₃	M₃₂	1	4	yes	complemented
98	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
100	C₈×C₁₃	—	2	2	no	—
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	—	4	2	no	—
103	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
104	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
105	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
106	C₂×M₃₂	—	26	2	no	—
107	C₂×C₄⋊C₈	—	2	2	no	—
108	C₂×C₄⋊C₈	—	2	2	no	—
109	C₂×C₄⋊C₈	Dic₁₃	1	2	yes	—
110	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
111	C₂²⋊C₁₆	—	13	4	no	—
112	C₂²⋊C₁₆	—	26	2	no	—
113	C₂²×C₁₃	—	2	8	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	—	2	1	no	—
116	C₂²×C₁₃	—	2	1	no	—
117	C₂²×C₁₃	—	2	1	no	—
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	#324	1	2	yes	—
121	C₂²×C₁₃	#349	1	2	yes	—
122	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
124	C₄×C₁₃	—	2	4	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	2	1	no	—
127	C₄×C₁₃	—	4	2	no	—
128	C₄×C₁₃	#324	1	2	yes	—
129	C₄×C₁₃	C₂×#99	1	1	yes	characteristic
130	C₄×C₁₃	C₂×#101	1	1	yes	characteristic
131	C₂³×C₄	#1055	1	1	yes	characteristic
132	C₂³×C₄	#1055	1	1	yes	characteristic
133	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	—	4	2	no	—
137	C₂²×C₈	—	4	2	no	—
138	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
139	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
141	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
142	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	#1055	1	2	yes	—
148	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
149	M₃₂	—	52	4	no	—
150	C₂×C₁₆	—	26	4	no	—
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	2	4	no	—
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	—	2	2	no	—
155	C₂×C₁₃	—	2	2	no	—
156	C₂×C₁₃	C₂×#324	1	1	yes	characteristic
157	C₂×C₁₃	#2112	1	1	yes	characteristic
158	C₂×C₁₃	C₂×#349	1	1	yes	characteristic
159	C₂⁴	C₂×#1055	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
169	C₂²×C₄	C₂×#1055	1	2	yes	—
170	C₂²×C₄	C₂×#1055	1	2	yes	—
171	C₂²×C₄	#6099	1	1	yes	characteristic
172	C₂²×C₄	#6099	1	1	yes	characteristic
173	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
185	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₁₆	—	52	4	no	—
191	C₁₃	#11440	1	1	yes	characteristic, complemented, Sylow
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂²⋊#1055	1	1	yes	characteristic
196	C₂³	C₂²⋊#1055	1	1	yes	characteristic
197	C₂³	C₂²×#1055	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	#88052	1	4	yes	complemented
214	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
215	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
216	C₂×C₄	#88074	1	2	yes	—
217	C₂×C₄	#88076	1	2	yes	—
218	C₂×C₄	C₂×#6099	1	1	yes	characteristic
219	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₂²	—	2	8	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#11718120	1	2	yes	—
231	C₂²	#11718145	1	2	yes	—
232	C₂²	C₂×#88052	1	2	yes	—
233	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
234	C₄	—	2	4	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	4	2	no	—
238	C₄	#11718120	1	2	yes	—
239	C₄	C₂×#88074	1	1	yes	characteristic
240	C₄	C₂×#88076	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	2	no	—
243	C₂	—	2	2	no	—
244	C₂	C₂×#11718120	1	1	yes	characteristic
245	C₂	#13116447	1	1	yes	characteristic
246	C₂	C₂×#11718145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₃
commutator	C₂³×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂²×C₁₃×C₄⋊C₈	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#11440
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #13112776	C₂	semidirect
97. C₂×C₄×C₁₃	M₃₂	semidirect
191. C₁₃	#11440	semidirect
213. C₂×C₄	#88052	semidirect

Maximal subgroups

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

subgroup	index
#13116447	2
#13112776	2
C₂²×C₁₃×C₄⋊C₈	2
#11440	13