Subgroups — #15756583

Order 3328.

382 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114953	C₂	1	2	yes	maximal
2	#13112671	C₂	1	4	yes	maximal
3	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	C₂²	1	4	yes	—
10	#11718116	—	2	4	no	—
11	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	D₄	1	2	yes	—
16	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂²×#1055	—	2	4	no	—
19	C₄×C₈×C₁₃	—	2	4	no	—
20	C₄×C₈×C₁₃	—	2	2	no	—
21	C₄×C₈×C₁₃	—	2	4	no	—
22	C₄×C₈×C₁₃	C₈	1	4	yes	complemented
23	C₄⋊#1055	—	2	8	no	—
24	#8785	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	—	2	2	no	—
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	—	2	2	no	—
34	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
36	C₄²×C₁₃	—	2	4	no	—
37	C₄²×C₁₃	—	2	2	no	—
38	C₄²×C₁₃	—	2	4	no	—
39	C₄²×C₁₃	C₂×C₈	1	2	yes	—
40	C₄²×C₁₃	M₁₆	1	2	yes	—
41	C₄²×C₁₃	#41	1	4	yes	—
42	C₂×#1055	—	4	4	no	—
43	C₂×#1055	—	4	8	no	—
44	#1831	—	13	2	no	—
45	C₄²×C₈	D₁₃	1	1	yes	characteristic
46	#1277	—	13	4	no	—
47	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	—	2	2	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
54	C₂×C₄×C₁₃	#108	1	4	yes	—
55	C₂×C₄×C₁₃	#117	1	2	yes	—
56	C₂×C₄×C₁₃	#124	1	2	yes	—
57	C₂×C₄×C₁₃	#125	1	2	yes	—
58	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#135	1	4	yes	—
62	C₈×C₁₃	—	2	4	no	—
63	C₈×C₁₃	—	2	4	no	—
64	#1055	—	8	4	no	—
65	C₂×C₄⋊C₈	—	13	4	no	—
66	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
67	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
68	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
70	C₄³	C₂×D₁₃	1	1	yes	characteristic
71	#336	—	26	4	no	—
72	C₂²×C₁₃	#382	1	1	yes	characteristic
73	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₂²×C₁₃	#423	1	1	yes	characteristic
75	C₂²×C₁₃	#427	1	2	yes	—
76	C₂²×C₁₃	C₂×#108	1	2	yes	—
77	C₄×C₁₃	—	2	4	no	—
78	C₄×C₁₃	—	2	4	no	—
79	C₄×C₁₃	—	2	2	no	—
80	C₄×C₁₃	#431	1	2	yes	—
81	C₄×C₁₃	#432	1	2	yes	—
82	C₄×C₁₃	#488	1	4	yes	—
83	C₂²×C₈	—	26	4	no	—
84	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
85	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
86	C₂×C₄²	Dic₂₆	1	2	yes	—
87	C₂×C₄²	D₅₂	1	2	yes	—
88	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
89	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
90	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
91	C₄⋊C₈	—	26	8	no	—
92	C₄×C₈	—	2	4	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	—	2	4	no	—
95	C₄×C₈	#1055	1	4	yes	complemented
96	C₂×C₁₃	#1564	1	2	yes	—
97	C₂×C₁₃	#1568	1	2	yes	—
98	C₂×C₁₃	#1831	1	1	yes	characteristic
99	C₂×C₁₃	#1835	1	1	yes	characteristic
100	C₂×C₁₃	#1837	1	1	yes	characteristic
101	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
102	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
103	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
104	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
105	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	52	4	no	—
111	C₂×C₈	—	52	8	no	—
112	C₂×C₈	C₂×#1055	1	4	yes	—
113	C₂×C₈	#6099	1	4	yes	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	4	no	—
117	C₄²	C₂×#1055	1	2	yes	—
118	C₄²	#6099	1	2	yes	—
119	C₄²	#6127	1	4	yes	—
120	C₁₃	#8785	1	1	yes	characteristic, complemented, Sylow
121	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄⋊#1055	1	4	yes	—
128	C₂×C₄	#88059	1	4	yes	—
129	C₂×C₄	#88124	1	2	yes	—
130	C₂×C₄	#88129	1	2	yes	—
131	C₂×C₄	#88158	1	4	yes	—
132	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
133	C₂×C₄	C₂×#6099	1	2	yes	—
134	C₂×C₄	C₂×#6099	1	1	yes	characteristic
135	C₂×C₄	#88180	1	2	yes	—
136	C₈	—	2	4	no	—
137	C₈	—	2	4	no	—
138	C₈	—	104	4	no	—
139	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
140	C₂²	#11718585	1	1	yes	characteristic
141	C₂²	#11718597	1	1	yes	characteristic
142	C₂²	#11718767	1	2	yes	—
143	C₂²	C₂×#88059	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#11718358	1	4	yes	—
148	C₄	#11718586	1	2	yes	—
149	C₄	#11718587	1	2	yes	—
150	C₂	#13113584	1	2	yes	—
151	C₂	#13113599	1	2	yes	—
152	C₂	#13114953	1	1	yes	characteristic
153	C₂	#13114967	1	1	yes	characteristic
154	C₂	#13114974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#8785
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
22. C₄×C₈×C₁₃	C₈	semidirect
95. C₄×C₈	#1055	semidirect
120. C₁₃	#8785	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
#13112671	2
C₄²×C₈×C₁₃	2
#8785	13