Subgroups — #15756677

Order 3328.

362 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114953	C₂	1	1	yes	maximal, characteristic
2	#13114582	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718071	C₂	1	2	yes	maximal
4	C₂×#11718072	C₂	1	2	yes	maximal
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#1055	C₂²	1	2	yes	—
9	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₁₃×#336	—	2	2	no	—
12	#11718116	—	2	2	no	—
13	#11718116	C₄	1	2	yes	—
14	#11718116	C₄	1	2	yes	—
15	#11718072	—	2	4	no	—
16	#11718071	—	2	4	no	—
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×#1055	—	2	2	no	—
24	C₂²×#1055	C₂×C₄	1	2	yes	—
25	C₂²×#1055	C₂×C₄	1	2	yes	—
26	C₁₃×C₄⋊C₈	—	2	2	no	—
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄×#1055	D₄	1	2	yes	—
31	C₄×#1055	D₄	1	2	yes	—
32	C₄×#1055	Q₈	1	2	yes	—
33	C₄×#1055	Q₈	1	2	yes	—
34	#8811	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	—	4	2	no	—
41	C₂×C₈×C₁₃	—	4	4	no	—
42	C₄²×C₁₃	—	2	2	no	—
43	C₄²×C₁₃	—	2	2	no	—
44	C₄²×C₁₃	—	2	4	no	—
45	C₄²×C₁₃	—	4	2	no	—
46	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	—	2	2	no	—
53	C₂×#1055	—	2	2	no	—
54	C₂×#1055	—	4	2	no	—
55	C₂×#1055	—	4	4	no	—
56	C₂×#1055	C₄⋊C₄	1	4	yes	—
57	C₂×#1055	C₄⋊C₄	1	4	yes	—
58	#1831	—	13	1	no	—
59	#1831	D₁₃	1	1	yes	characteristic
60	#1744	—	13	1	no	—
61	C₂×#335	—	13	2	no	—
62	C₂×#334	—	13	2	no	—
63	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#131	1	2	yes	—
78	C₂×C₄×C₁₃	#131	1	2	yes	—
79	C₂×C₄×C₁₃	#135	1	4	yes	—
80	C₈×C₁₃	—	8	2	no	—
81	#1055	—	2	4	no	—
82	#1055	—	2	4	no	—
83	#1055	—	8	2	no	—
84	C₂×C₄⋊C₈	—	13	2	no	—
85	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
86	C₂×C₄×C₈	—	13	1	no	—
87	C₂×C₄×C₈	—	13	1	no	—
88	C₄³	C₂×D₁₃	1	1	yes	characteristic
89	#336	—	2	2	no	—
90	#336	—	13	2	no	—
91	#336	—	13	2	no	—
92	#336	—	26	2	no	—
93	#335	—	26	4	no	—
94	#334	—	26	4	no	—
95	C₂²×C₁₃	#384	1	1	yes	characteristic
96	C₂²×C₁₃	#424	1	1	yes	characteristic
97	C₂²×C₁₃	#424	1	1	yes	characteristic
98	C₂²×C₁₃	#427	1	2	yes	—
99	C₂²×C₁₃	#430	1	2	yes	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	4	2	no	—
103	C₄×C₁₃	—	4	2	no	—
104	C₄×C₁₃	#424	1	2	yes	—
105	C₄×C₁₃	#424	1	2	yes	—
106	C₄×C₁₃	#495	1	2	yes	—
107	C₄×C₁₃	#499	1	2	yes	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	13	2	no	—
110	C₂²×C₈	—	13	2	no	—
111	C₂²×C₈	—	26	2	no	—
112	C₂×C₄²	Dic₂₆	1	2	yes	—
113	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
114	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
115	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
116	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	26	2	no	—
120	C₄⋊C₈	—	26	2	no	—
121	C₄×C₈	—	13	2	no	—
122	C₄×C₈	—	13	2	no	—
123	C₄×C₈	—	13	2	no	—
124	C₄×C₈	—	13	2	no	—
125	C₂×C₁₃	#1572	1	2	yes	—
126	C₂×C₁₃	#1572	1	2	yes	—
127	C₂×C₁₃	#1843	1	1	yes	characteristic
128	C₂×C₁₃	#1846	1	1	yes	characteristic
129	C₂×C₁₃	#1849	1	1	yes	characteristic
130	C₂²×C₄	#6101	1	2	yes	—
131	C₂²×C₄	#6101	1	2	yes	—
132	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	—	13	4	no	—
138	C₂×C₈	—	13	4	no	—
139	C₂×C₈	—	26	2	no	—
140	C₂×C₈	—	26	2	no	—
141	C₂×C₈	—	26	2	no	—
142	C₂×C₈	—	26	2	no	—
143	C₂×C₈	—	52	2	no	—
144	C₂×C₈	—	52	4	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	4	2	no	—
149	C₄²	D₄×D₁₃	1	1	yes	characteristic
150	C₄²	D₄×D₁₃	1	1	yes	characteristic
151	C₄²	Q₈×D₁₃	1	1	yes	characteristic
152	C₄²	Q₈×D₁₃	1	1	yes	characteristic
153	C₁₃	#8811	1	1	yes	characteristic, complemented, Sylow
154	C₂³	C₂×#6101	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	#88142	1	1	yes	characteristic
163	C₂×C₄	#88142	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
166	C₂×C₄	#88155	1	2	yes	—
167	C₂×C₄	#88161	1	2	yes	—
168	C₂×C₄	#88194	1	1	yes	characteristic
169	C₂×C₄	#88198	1	1	yes	characteristic
170	C₂×C₄	#88204	1	4	yes	—
171	C₈	—	8	2	no	—
172	C₈	—	26	4	no	—
173	C₈	—	26	4	no	—
174	C₈	—	104	2	no	—
175	C₂²	#11718637	1	1	yes	characteristic
176	C₂²	#11718659	1	2	yes	—
177	C₂²	#11718763	1	1	yes	characteristic
178	C₂²	#11718780	1	1	yes	characteristic
179	C₂²	#11718785	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	#11718500	1	2	yes	—
185	C₄	#11718500	1	2	yes	—
186	C₄	#11718720	1	2	yes	—
187	C₄	#11718725	1	2	yes	—
188	C₂	#13113614	1	2	yes	—
189	C₂	#13113617	1	2	yes	—
190	C₂	#13114989	1	1	yes	characteristic
191	C₂	#13114998	1	1	yes	characteristic
192	C₂	#13115010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#8811
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
153. C₁₃	#8811	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
#13114582	2
C₂×#11718071	2
C₂×#11718072	2
C₁₃×#1831	2
#8811	13