Subgroups — C₁₁₃×#2960 #28387756

Order 14464.

364 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	#446×C₁₁₃	C₂	1	4	yes	maximal, complemented
3	C₈×Q₈×C₁₁₃	C₂	1	4	yes	maximal, complemented
4	#424×C₁₁₃	C₂	1	2	yes	maximal
5	#423×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#405×C₁₁₃	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×M₁₆×C₁₁₃	C₂²	1	2	yes	—
9	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₁₁₃	C₄	1	2	yes	—
12	#125×C₁₁₃	C₄	1	2	yes	—
13	C₂²⋊Q₈×C₁₁₃	C₄	1	4	yes	—
14	C₄×Q₈×C₁₁₃	C₂²	1	4	yes	—
15	#117×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₁₃	C₂²	1	4	yes	—
18	C₄⋊C₈×C₁₁₃	C₂²	1	8	yes	—
19	#97×C₁₁₃	C₂²	1	4	yes	—
20	C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
21	C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
22	C₄×C₈×C₁₁₃	C₂²	1	4	yes	—
23	C₂×Q₈×C₁₁₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₁₃	—	2	4	no	—
27	C₂×C₈×C₁₁₃	—	2	2	no	—
28	C₂×C₈×C₁₁₃	Q₈	1	4	yes	complemented
29	C₂×C₈×C₁₁₃	C₂³	1	2	yes	—
30	C₂×C₈×C₁₁₃	C₂³	1	4	yes	—
31	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
34	C₄⋊C₄×C₁₁₃	C₂×C₄	1	8	yes	—
35	C₄⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
36	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	4	yes	—
37	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₁₁₃	C₂³	1	2	yes	—
40	C₄²×C₁₁₃	C₂³	1	4	yes	—
41	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₁₁₃	—	2	8	no	—
43	C₂×C₄×C₁₁₃	—	2	4	no	—
44	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₁₃	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₁₃	#41	1	2	yes	—
50	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₁₃	—	2	2	no	—
52	C₈×C₁₁₃	—	2	4	no	—
53	C₈×C₁₁₃	C₂×Q₈	1	2	yes	—
54	C₈×C₁₁₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₁₁₃	—	2	2	no	—
56	C₂²×C₁₁₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₁₁₃	—	2	2	no	—
59	C₄×C₁₁₃	—	2	8	no	—
60	C₄×C₁₁₃	C₄×Q₈	1	2	yes	—
61	C₄×C₁₁₃	#131	1	4	yes	—
62	C₄×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
64	C₂×C₁₁₃	—	2	2	no	—
65	C₂×C₁₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₁₁₃	C₂×#131	1	2	yes	—
67	#2960	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₁₁₃	#2960	1	1	yes	characteristic, direct product factor, Sylow
69	#533	C₂×C₁₁₃	1	1	yes	characteristic
70	#446	C₂×C₁₁₃	1	4	yes	complemented
71	C₈×Q₈	C₂×C₁₁₃	1	4	yes	complemented
72	#424	C₂×C₁₁₃	1	2	yes	—
73	#423	C₂×C₁₁₃	1	1	yes	characteristic
74	#405	C₂×C₁₁₃	1	2	yes	—
75	C₂×C₄×C₈	C₂×C₁₁₃	1	1	yes	characteristic
76	C₂×M₁₆	C₂²×C₁₁₃	1	2	yes	—
77	C₂²×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
78	C₂²×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
79	C₄⋊Q₈	C₄×C₁₁₃	1	2	yes	—
80	#125	C₄×C₁₁₃	1	2	yes	—
81	C₂²⋊Q₈	C₄×C₁₁₃	1	4	yes	—
82	C₄×Q₈	C₂²×C₁₁₃	1	4	yes	—
83	#117	C₂²×C₁₁₃	1	2	yes	—
84	C₂×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
85	C₄⋊C₈	C₂²×C₁₁₃	1	4	yes	—
86	C₄⋊C₈	C₂²×C₁₁₃	1	8	yes	—
87	#97	C₂²×C₁₁₃	1	4	yes	—
88	C₄×C₈	C₂²×C₁₁₃	1	2	yes	—
89	C₄×C₈	C₂²×C₁₁₃	1	2	yes	—
90	C₄×C₈	C₂²×C₁₁₃	1	4	yes	—
91	C₂×Q₈	C₂×C₄×C₁₁₃	1	4	yes	—
92	C₂²×C₄	C₂×C₄×C₁₁₃	1	2	yes	—
93	C₂²×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
94	M₁₆	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	Q₈×C₁₁₃	1	4	yes	complemented
97	C₂×C₈	C₂³×C₁₁₃	1	2	yes	—
98	C₂×C₈	C₂³×C₁₁₃	1	4	yes	—
99	C₂×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
100	C₂×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
101	C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
102	C₄⋊C₄	C₂×C₄×C₁₁₃	1	8	yes	—
103	C₄⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
104	C₂²⋊C₄	C₂×C₄×C₁₁₃	1	4	yes	—
105	C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
106	C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic
107	C₄²	C₂³×C₁₁₃	1	2	yes	—
108	C₄²	C₂³×C₁₁₃	1	4	yes	—
109	C₂³	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
113	C₂×C₄	C₂²×C₄×C₁₁₃	1	8	yes	—
114	C₂×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
115	C₂×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
116	C₂×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
117	C₂×C₄	#41×C₁₁₃	1	2	yes	—
118	C₂×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	2	4	no	—
121	C₈	C₂×Q₈×C₁₁₃	1	2	yes	—
122	C₈	C₂×Q₈×C₁₁₃	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Q₈×C₁₁₃	1	2	yes	—
125	C₂²	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	C₄×Q₈×C₁₁₃	1	2	yes	—
129	C₄	#131×C₁₁₃	1	4	yes	—
130	C₄	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
131	C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Q₈×C₁₁₃	1	1	yes	characteristic
134	C₂	C₂×#131×C₁₁₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₁₃	C₂³
commutator	C₂²	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₁₃
socle	C₂²×C₁₁₃	C₂³×C₄
prime radical	C₂³×C₁₁₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #446×C₁₁₃	C₂	semidirect
3. C₈×Q₈×C₁₁₃	C₂	semidirect
28. C₂×C₈×C₁₁₃	Q₈	semidirect
67. #2960	C₁₁₃	direct (N × H)
68. C₁₁₃	#2960	direct (N × H)
70. #446	C₂×C₁₁₃	semidirect
71. C₈×Q₈	C₂×C₁₁₃	semidirect
96. C₂×C₈	Q₈×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#533×C₁₁₃	2
#446×C₁₁₃	2
C₈×Q₈×C₁₁₃	2
#424×C₁₁₃	2
#423×C₁₁₃	2
#405×C₁₁₃	2
C₂×C₄×C₈×C₁₁₃	2
#2960	113

Subgroups — C113×#2960 #28387756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#2960 #28387756