Subgroups — #15889509

Order 3328.

664 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	#13115773	C₂	1	4	yes	maximal
2	#13115395	C₂	1	8	yes	maximal
3	#13114979	C₂	1	2	yes	maximal
4	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₁₃×#391	C₄	1	2	yes	—
8	C₁₃×#389	C₄	1	4	yes	—
9	C₁₃×#384	C₄	1	1	yes	characteristic
10	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₄³×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	C₂²	1	4	yes	—
13	C₂×C₄×#1055	C₂²	1	4	yes	—
14	#11718116	C₂²	1	8	yes	—
15	#11718116	C₂²	1	8	yes	—
16	#11718116	C₂²	1	4	yes	—
17	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₁₃×C₄⋊Q₈	C₈	1	8	yes	complemented
19	C₄×Q₈×C₁₃	—	4	2	no	—
20	C₄×Q₈×C₁₃	—	4	2	no	—
21	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×#1055	C₂³	1	8	yes	—
32	C₁₃×#95	C₂×C₄	1	8	yes	—
33	C₄⋊#1055	—	4	4	no	—
34	C₄×#1055	—	4	4	no	—
35	#23363	—	13	1	no	maximal, Sylow
36	C₂×Q₈×C₁₃	—	4	2	no	—
37	C₂×Q₈×C₁₃	—	4	2	no	—
38	C₂×Q₈×C₁₃	C₂×C₈	1	8	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
46	C₁₃×C₄⋊C₄	—	4	4	no	—
47	C₁₃×C₄⋊C₄	—	4	2	no	—
48	C₁₃×C₄⋊C₄	—	4	4	no	—
49	C₁₃×C₄⋊C₄	C₂×C₈	1	16	yes	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	—	2	4	no	—
52	C₄²×C₁₃	—	2	4	no	—
53	C₄²×C₁₃	—	2	4	no	—
54	C₄²×C₁₃	—	4	2	no	—
55	C₄²×C₁₃	—	4	1	no	—
56	C₄²×C₁₃	C₂×C₈	1	4	yes	—
57	C₂×#1055	—	2	8	no	—
58	C₂×#1055	—	2	16	no	—
59	C₄×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
60	#1984	—	13	4	no	—
61	#1916	—	13	8	no	—
62	#1839	—	13	2	no	—
63	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
64	Q₈×C₁₃	—	4	8	no	—
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	4	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	8	no	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
81	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
82	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
83	#1055	—	4	8	no	—
84	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
85	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
86	C₂×C₄⋊C₈	—	13	4	no	—
87	C₂×C₄×C₈	—	13	4	no	—
88	#391	Dic₁₃	1	2	yes	—
89	#389	Dic₁₃	1	4	yes	—
90	#384	Dic₁₃	1	1	yes	characteristic
91	C₄×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
92	C₄³	C₂×D₁₃	1	1	yes	characteristic
93	#336	—	13	8	no	—
94	#336	—	13	4	no	—
95	#336	—	13	8	no	—
96	C₂²×C₁₃	#518	1	1	yes	characteristic
97	C₂²×C₁₃	#519	1	2	yes	—
98	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
99	C₂²×C₁₃	#568	1	1	yes	characteristic
100	C₂²×C₁₃	#568	1	2	yes	—
101	C₄×C₁₃	—	2	8	no	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	2	4	no	—
104	C₄×C₁₃	—	4	1	no	—
105	C₄×C₁₃	—	4	4	no	—
106	C₄×C₁₃	—	4	2	no	—
107	C₄×C₁₃	#563	1	4	yes	—
108	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
109	C₂²×C₈	—	13	8	no	—
110	C₄⋊Q₈	#1055	1	8	yes	complemented
111	C₄×Q₈	—	4	2	no	—
112	C₄×Q₈	—	4	2	no	—
113	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
114	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
115	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
116	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
117	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
118	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
119	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
121	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
122	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
123	C₄⋊C₈	—	52	4	no	—
124	C₄×C₈	—	52	4	no	—
125	#95	C₂×Dic₁₃	1	8	yes	—
126	C₂×C₁₃	#2381	1	1	yes	characteristic
127	C₂×C₁₃	#2972	1	1	yes	characteristic
128	C₂×C₁₃	#2980	1	1	yes	characteristic
129	C₂×C₁₃	#2982	1	2	yes	—
130	C₂×C₁₃	#2987	1	2	yes	—
131	C₂×Q₈	—	4	2	no	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	C₂×#1055	1	8	yes	—
134	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
135	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
136	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
138	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
140	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
141	C₂×C₈	—	26	8	no	—
142	C₂×C₈	—	26	16	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	C₂×#1055	1	16	yes	—
147	C₄²	—	2	4	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	4	2	no	—
152	C₄²	—	4	1	no	—
153	C₄²	C₂×#1055	1	4	yes	—
154	C₁₃	#23363	1	1	yes	characteristic, complemented, Sylow
155	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
156	Q₈	—	4	8	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	C₂²×#1055	1	4	yes	—
171	C₂×C₄	C₂²×#1055	1	2	yes	—
172	C₂×C₄	C₂²×#1055	1	8	yes	—
173	C₂×C₄	#88251	1	2	yes	—
174	C₂×C₄	#88254	1	4	yes	—
175	C₈	—	52	8	no	—
176	C₂²	C₂³×#1055	1	1	yes	characteristic
177	C₂²	#11719503	1	1	yes	characteristic
178	C₂²	#11719517	1	2	yes	—
179	C₂²	#11719524	1	1	yes	characteristic
180	C₂²	#11719524	1	2	yes	—
181	C₄	—	2	8	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	2	no	—
187	C₄	#11719440	1	4	yes	—
188	C₂	#13118017	1	1	yes	characteristic
189	C₂	#13123860	1	1	yes	characteristic
190	C₂	#13123907	1	1	yes	characteristic
191	C₂	#13123917	1	2	yes	—
192	C₂	#13123942	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Fitting	C₄×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#23363
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
18. C₁₃×C₄⋊Q₈	C₈	semidirect
110. C₄⋊Q₈	#1055	semidirect
154. C₁₃	#23363	semidirect

Maximal subgroups

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

subgroup	index
#13115773	2
#13115395	2
#13114979	2
C₄×C₁₃×C₄⋊Q₈	2
#23363	13