Subgroups — #13119871

Order 1664.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88124	C₂	1	2	yes	maximal
2	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#11718637	C₂	1	2	yes	maximal
4	#11718634	C₂	1	2	yes	maximal
5	#11718596	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	#11718589	C₂	1	2	yes	maximal
8	#11718318	C₂	1	4	yes	maximal
9	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×#6101	C₂²	1	4	yes	—
19	C₂×#6101	C₂²	1	4	yes	—
20	C₂×#6101	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
22	#88124	—	2	8	no	—
23	C₁₃×#95	C₂²	1	2	yes	—
24	#88071	C₂²	1	2	yes	—
25	#88071	C₂²	1	4	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₂³	1	4	yes	—
32	C₂²×Dic₁₃	C₂³	1	2	yes	—
33	C₂²×Dic₁₃	C₂³	1	2	yes	—
34	C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₄²×C₁₃	—	2	4	no	—
37	C₄²×C₁₃	Q₈	1	4	yes	—
38	C₄⋊Dic₁₃	—	2	4	no	—
39	C₄⋊Dic₁₃	—	2	4	no	—
40	C₄⋊Dic₁₃	—	4	2	no	—
41	C₄⋊Dic₁₃	D₄	1	4	yes	—
42	C₄⋊Dic₁₃	Q₈	1	4	yes	—
43	#6101	—	2	8	no	—
44	#6101	—	2	8	no	—
45	#6101	—	4	4	no	—
46	C₄×Dic₁₃	—	2	4	no	—
47	#2552	—	13	1	no	maximal, Sylow
48	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₁₃	—	2	1	no	—
50	C₂×C₄×C₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₁₃	#41	1	2	yes	—
63	C₂×C₄×C₁₃	#41	1	2	yes	—
64	C₂×Dic₁₃	—	2	2	no	—
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	—	2	2	no	—
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	—	2	4	no	—
69	C₂×Dic₁₃	—	2	4	no	—
70	C₂×Dic₁₃	—	2	4	no	—
71	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
72	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
73	C₂×#125	—	13	2	no	—
74	#398	—	13	4	no	—
75	#389	—	13	2	no	—
76	#389	—	13	2	no	—
77	#384	—	13	2	no	—
78	#384	—	13	1	no	—
79	C₄×C₄⋊C₄	—	13	1	no	—
80	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
81	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
82	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
85	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
86	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	4	2	no	—
90	C₄×C₁₃	—	4	2	no	—
91	C₄×C₁₃	#125	1	4	yes	complemented
92	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
93	Dic₁₃	—	2	4	no	—
94	Dic₁₃	—	2	4	no	—
95	Dic₁₃	—	4	4	no	—
96	#125	—	26	8	no	—
97	C₂×C₄⋊C₄	—	13	1	no	—
98	C₂×C₄⋊C₄	—	13	1	no	—
99	C₂×C₄⋊C₄	—	13	4	no	—
100	C₂×C₄⋊C₄	—	13	4	no	—
101	C₂×C₄⋊C₄	—	13	2	no	—
102	C₂×C₄⋊C₄	—	13	2	no	—
103	C₂×C₄⋊C₄	—	13	4	no	—
104	C₂×C₄⋊C₄	—	13	2	no	—
105	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
107	C₂×C₄²	—	13	2	no	—
108	C₂×C₄²	C₂×D₁₃	1	2	yes	—
109	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
110	#95	—	13	2	no	—
111	#95	—	13	4	no	—
112	#95	C₂×D₁₃	1	2	yes	—
113	C₂×C₁₃	C₂×#125	1	1	yes	characteristic
114	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₁₃	#544	1	1	yes	characteristic
116	C₂×C₁₃	#547	1	1	yes	characteristic
117	C₂×C₁₃	#550	1	1	yes	characteristic
118	C₂×C₁₃	#554	1	1	yes	characteristic
119	C₂×C₁₃	#557	1	1	yes	characteristic
120	C₂²×C₄	—	13	2	no	—
121	C₂²×C₄	—	13	2	no	—
122	C₂²×C₄	—	13	4	no	—
123	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
124	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	13	4	no	—
131	C₄⋊C₄	—	13	4	no	—
132	C₄⋊C₄	—	26	8	no	—
133	C₄⋊C₄	—	26	8	no	—
134	C₄⋊C₄	—	26	4	no	—
135	C₄⋊C₄	—	26	4	no	—
136	C₄⋊C₄	—	52	4	no	—
137	C₄⋊C₄	—	52	2	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	26	4	no	—
140	C₄²	Dic₂₆	1	4	yes	—
141	C₁₃	#2552	1	1	yes	characteristic, complemented, Sylow
142	C₂³	C₂³×D₁₃	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	13	4	no	—
152	C₂×C₄	—	13	4	no	—
153	C₂×C₄	—	26	2	no	—
154	C₂×C₄	—	26	2	no	—
155	C₂×C₄	—	26	2	no	—
156	C₂×C₄	—	26	2	no	—
157	C₂×C₄	—	26	4	no	—
158	C₂×C₄	—	26	4	no	—
159	C₂×C₄	—	26	4	no	—
160	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
161	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
162	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
163	C₂×C₄	#6127	1	2	yes	—
164	C₂×C₄	#6127	1	2	yes	—
165	C₂×C₄	D₄×D₁₃	1	2	yes	—
166	C₂×C₄	Q₈×D₁₃	1	2	yes	—
167	C₂²	C₂²×Dic₂₆	1	1	yes	characteristic
168	C₂²	C₂×#6127	1	1	yes	characteristic
169	C₂²	C₂×#6127	1	1	yes	characteristic
170	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
171	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
172	C₂²	#88256	1	1	yes	characteristic
173	C₂²	#88257	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	26	4	no	—
178	C₄	—	26	4	no	—
179	C₄	—	52	4	no	—
180	C₄	#88124	1	4	yes	complemented
181	C₄	#88142	1	4	yes	—
182	C₂	C₂×#88124	1	1	yes	characteristic
183	C₂	C₂×#88142	1	1	yes	characteristic
184	C₂	#11719224	1	1	yes	characteristic
185	C₂	#11719256	1	1	yes	characteristic
186	C₂	#11719258	1	1	yes	characteristic
187	C₂	#11719272	1	1	yes	characteristic
188	C₂	#11719284	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#2552
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
91. C₄×C₁₃	#125	semidirect
141. C₁₃	#2552	semidirect
180. C₄	#88124	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88124	2
C₄×C₁₃×C₄⋊C₄	2
#11718637	2
#11718634	2
#11718596	2
C₄×C₄⋊Dic₁₃	2
#11718589	2
#11718318	2
#2552	13