Subgroups — #15772252

Order 3328.

360 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116639	C₂	1	1	yes	maximal, characteristic, complemented
2	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
3	#13112762	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#131	C₄	1	2	yes	—
6	C₁₃×#532	C₄	1	1	yes	characteristic
7	C₁₃×#436	—	2	1	no	—
8	C₁₃×#436	—	2	1	no	—
9	C₈×D₄×C₁₃	—	2	1	no	—
10	C₈×D₄×C₁₃	—	2	1	no	—
11	C₁₃×#423	C₄	1	1	yes	characteristic
12	C₁₃×#408	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²⋊#6090	C₂²	1	2	yes	—
15	C₂²⋊#6090	C₂²	1	2	yes	—
16	C₄⋊#6090	—	2	1	no	—
17	C₄⋊#6090	—	2	1	no	—
18	#11718053	—	2	1	no	—
19	C₄×#6090	—	2	1	no	—
20	C₂×C₁₃×#41	C₂×C₄	1	2	yes	—
21	C₁₃×#131	—	4	4	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
28	C₁₃×#127	C₈	1	1	yes	characteristic
29	C₁₃×#124	C₈	1	2	yes	—
30	C₁₃×#121	—	2	1	no	—
31	C₁₃×#121	—	2	1	no	—
32	C₄×D₄×C₁₃	—	2	1	no	—
33	C₄×D₄×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₁₃×C₄⋊C₈	—	2	1	no	—
36	C₁₃×C₄⋊C₈	—	2	1	no	—
37	C₁₃×C₂²⋊C₈	—	2	2	no	—
38	C₁₃×C₂²⋊C₈	—	2	2	no	—
39	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×#6090	—	2	2	no	—
44	C₂×#6090	—	2	2	no	—
45	#11492	—	13	1	no	maximal, Sylow
46	C₁₃×#41	—	4	2	no	—
47	C₁₃×#41	—	4	2	no	—
48	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	1	no	—
51	C₂×D₄×C₁₃	—	2	1	no	—
52	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₁₃×M₁₆	—	4	4	no	—
58	C₁₃×M₁₆	—	4	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	4	1	no	—
62	C₂×C₈×C₁₃	—	4	4	no	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	—	2	1	no	—
68	C₁₃×C₄⋊C₄	—	2	1	no	—
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
75	#6090	—	4	2	no	—
76	#6090	—	4	2	no	—
77	#2931	D₁₃	1	1	yes	characteristic
78	#2163	—	13	1	no	—
79	#2162	—	13	1	no	—
80	#1315	—	13	4	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	Q₈×C₁₃	—	4	2	no	—
84	D₄×C₁₃	—	4	2	no	—
85	D₄×C₁₃	—	4	2	no	—
86	D₄×C₁₃	—	4	4	no	—
87	D₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	1	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
101	C₈×C₁₃	—	2	1	no	—
102	C₈×C₁₃	—	2	1	no	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	—	4	2	no	—
105	C₈×C₁₃	#100	1	2	yes	—
106	C₂×#131	Dic₁₃	1	2	yes	—
107	#532	Dic₁₃	1	1	yes	characteristic
108	#436	—	2	1	no	—
109	#436	—	2	1	no	—
110	C₈×D₄	—	2	1	no	—
111	C₈×D₄	—	2	1	no	—
112	#423	Dic₁₃	1	1	yes	characteristic
113	#408	C₂×D₁₃	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
115	C₄⋊C₁₆	—	26	1	no	—
116	C₄⋊C₁₆	—	26	1	no	—
117	C₂²⋊C₁₆	—	13	2	no	—
118	C₂²⋊C₁₆	—	13	2	no	—
119	#346	—	26	1	no	—
120	C₄×C₁₆	—	26	1	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	4	2	no	—
125	C₂²×C₁₃	—	4	2	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	4	2	no	—
133	C₄×C₁₃	#350	1	2	yes	—
134	C₄×C₁₃	#410	1	1	yes	characteristic
135	C₄×C₁₃	C₂×#100	1	1	yes	characteristic
136	C₂×#41	C₂×Dic₁₃	1	2	yes	—
137	#131	—	4	4	no	—
138	C₂×M₁₆	—	2	2	no	—
139	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
143	C₄⋊Q₈	#1055	1	1	yes	characteristic
144	#127	#1055	1	1	yes	characteristic
145	#124	#1055	1	2	yes	—
146	#121	—	2	1	no	—
147	#121	—	2	1	no	—
148	C₄×D₄	—	2	1	no	—
149	C₄×D₄	—	2	1	no	—
150	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
151	C₂×C₁₆	—	26	2	no	—
152	C₂×C₁₆	—	26	2	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₂²⋊C₈	—	2	2	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂×C₁₃	—	4	1	no	—
162	C₂×C₁₃	—	8	2	no	—
163	C₂×C₁₃	#1460	1	1	yes	characteristic
164	C₂×C₁₃	C₂×#350	1	1	yes	characteristic
165	C₂×C₁₃	#2117	1	1	yes	characteristic
166	#41	—	4	2	no	—
167	#41	—	4	2	no	—
168	C₂×Q₈	C₂×#1055	1	2	yes	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	C₂×#1055	1	2	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	#6099	1	1	yes	characteristic
175	C₂²×C₄	#6099	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
177	M₁₆	—	4	4	no	—
178	M₁₆	—	4	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	C₂×#1055	1	1	yes	characteristic
192	C₄²	C₂×#1055	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
194	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
195	C₁₆	—	52	2	no	—
196	C₁₆	—	52	2	no	—
197	C₁₃	#11492	1	1	yes	characteristic, complemented, Sylow
198	C₂³	—	2	2	no	—
199	C₂³	C₂×#6099	1	1	yes	characteristic
200	Q₈	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
213	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
215	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
216	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	#88073	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	#11718153	1	2	yes	—
236	C₄	C₂×#88073	1	1	yes	characteristic
237	C₄	#11718901	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	8	2	no	—
240	C₂	#13113157	1	1	yes	characteristic
241	C₂	C₂×#11718153	1	1	yes	characteristic
242	C₂	#13116464	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2931	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₁₃×#532	C₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2931	D₁₃
hypocenter	C₁₃	#11492
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #13116639	C₂	semidirect
2. #13116635	C₂	semidirect
3. #13112762	C₂	semidirect
197. C₁₃	#11492	semidirect

Maximal subgroups

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

subgroup	index
#13116639	2
#13116635	2
#13112762	2
C₁₃×#2931	2
#11492	13