Subgroups — C₂×#11718685 #13123764

Order 1664.

400 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	#11718685	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#1055	C₂	1	2	yes	maximal
4	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×#88043	C₂	1	2	yes	maximal
6	C₂×C₄×#1055	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₁₃	C₄	1	4	yes	—
9	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
10	C₂×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
14	C₂²×#1055	C₂²	1	1	yes	characteristic
15	C₂²×#1055	C₂²	1	2	yes	—
16	C₂²×#1055	C₂²	1	1	yes	characteristic
17	C₄⋊#1055	C₂²	1	8	yes	—
18	C₄⋊#1055	C₂²	1	4	yes	—
19	#88043	C₂²	1	8	yes	—
20	C₄×#1055	C₂²	1	4	yes	—
21	C₂×Q₈×C₁₃	—	2	4	no	—
22	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
29	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₄²×C₁₃	C₂×C₄	1	4	yes	—
31	C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₄²×C₁₃	C₂³	1	2	yes	—
33	C₄²×C₁₃	C₂³	1	4	yes	—
34	C₂×#1055	—	2	2	no	—
35	C₂×#1055	—	2	4	no	—
36	C₂×#1055	Q₈	1	4	yes	complemented
37	C₂×#1055	C₂³	1	2	yes	—
38	C₂×#1055	C₂³	1	4	yes	—
39	C₂×#1055	C₂³	1	2	yes	—
40	C₂×#446	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₁₃	—	2	8	no	—
43	C₂×C₄×C₁₃	—	2	4	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	4	no	—
46	C₂×C₄×C₁₃	M₁₆	1	4	yes	—
47	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₃	#41	1	2	yes	—
54	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
55	#1055	—	2	2	no	—
56	#1055	—	2	4	no	—
57	#1055	C₂×Q₈	1	4	yes	complemented
58	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic
59	#446	—	13	8	no	—
60	C₂×C₄⋊C₈	—	13	1	no	—
61	C₂×C₄⋊C₈	—	13	2	no	—
62	C₂×#97	—	13	2	no	—
63	C₂×C₄×C₈	—	13	1	no	—
64	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
65	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
66	C₂²×C₁₃	#131	1	2	yes	—
67	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
68	C₄×C₁₃	—	2	4	no	—
69	C₄×C₁₃	—	2	2	no	—
70	C₄×C₁₃	—	2	4	no	—
71	C₄×C₁₃	C₄×Q₈	1	2	yes	—
72	C₄×C₁₃	C₂×M₁₆	1	4	yes	—
73	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
74	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
75	C₂²×Q₈	Dic₁₃	1	1	yes	characteristic
76	C₂²×C₈	—	13	1	no	—
77	C₂²×C₈	—	13	1	no	—
78	C₂²×C₈	—	13	2	no	—
79	C₄×Q₈	Dic₁₃	1	4	yes	—
80	C₄×Q₈	C₂×D₁₃	1	4	yes	—
81	C₂×C₄⋊C₄	Dic₁₃	1	2	yes	—
82	C₂×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₁₃	1	2	yes	—
85	C₄⋊C₈	—	13	4	no	—
86	C₄⋊C₈	—	13	8	no	—
87	#97	—	13	8	no	—
88	C₄×C₈	—	13	4	no	—
89	C₂×C₁₃	#446	1	4	yes	complemented
90	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
92	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
93	C₂×Q₈	—	2	4	no	—
94	C₂×Q₈	C₂×Dic₁₃	1	4	yes	—
95	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
97	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
98	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
100	C₂×C₈	—	13	4	no	—
101	C₂×C₈	—	13	2	no	—
102	C₂×C₈	—	13	2	no	—
103	C₂×C₈	—	13	4	no	—
104	C₂×C₈	—	26	2	no	—
105	C₂×C₈	—	26	4	no	—
106	C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
107	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
108	C₄²	C₂×Dic₁₃	1	4	yes	—
109	C₄²	C₂×Dic₁₃	1	2	yes	—
110	C₄²	C₂²×D₁₃	1	2	yes	—
111	C₄²	C₂²×D₁₃	1	4	yes	—
112	C₁₃	C₂×#446	1	1	yes	characteristic, complemented, Sylow
113	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
114	Q₈	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#6099	1	4	yes	—
119	C₂×C₄	Q₈×D₁₃	1	2	yes	—
120	C₂×C₄	#6131	1	2	yes	—
121	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
122	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
123	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
124	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
125	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
126	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
127	C₈	—	13	4	no	—
128	C₈	—	26	2	no	—
129	C₈	—	26	4	no	—
130	C₂²	C₂×#6099	1	2	yes	—
131	C₂²	Q₈×Dic₁₃	1	2	yes	—
132	C₂²	#88202	1	2	yes	—
133	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	4	no	—
137	C₄	C₂×#6099	1	4	yes	—
138	C₄	Q₈×Dic₁₃	1	2	yes	—
139	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
140	C₄	C₂×#6131	1	1	yes	characteristic
141	C₂	#11718685	1	4	yes	direct product factor
142	C₂	C₂²×#6099	1	1	yes	characteristic
143	C₂	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
144	C₂	C₂×#88202	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂×C₄×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×#446
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #11718685	C₂	direct (N × H)
36. C₂×#1055	Q₈	semidirect
57. #1055	C₂×Q₈	semidirect
89. C₂×C₁₃	#446	semidirect
112. C₁₃	C₂×#446	semidirect
141. C₂	#11718685	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
#11718685	2
C₂×C₄⋊#1055	2
C₂×#88043	2
C₂×C₄×#1055	2
C₂×#446	13

Subgroups — C2×#11718685 #13123764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11718685 #13123764