Subgroups — C₂×#11719175 #13128127

Order 1664.

1168 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	C₂²×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#88180	C₂	1	2	yes	maximal
3	C₂×#88178	C₂	1	4	yes	maximal
4	#11719175	C₂	1	16	yes	maximal, direct product factor
5	C₂×#88124	C₂	1	2	yes	maximal, complemented
6	C₂×#88123	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Dic₂₆	C₂	1	4	yes	maximal, complemented
8	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
9	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₂₆	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂²	1	8	yes	—
15	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
16	#88180	C₂²	1	16	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
19	#88178	C₂²	1	32	yes	—
20	C₂×#6101	C₂²	1	8	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
22	#88124	C₂²	1	16	yes	complemented
23	#88123	C₂²	1	16	yes	complemented
24	C₄×Dic₂₆	C₂²	1	32	yes	complemented
25	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	—	2	8	no	—
27	C₂²×C₄×C₁₃	Q₈	1	8	yes	complemented
28	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	8	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₃	C₂³	1	8	yes	—
35	C₂×Dic₂₆	—	2	16	no	—
36	C₂×Dic₂₆	C₂³	1	16	yes	—
37	C₄²×C₁₃	C₂³	1	8	yes	—
38	C₄²×C₁₃	C₂³	1	4	yes	—
39	C₄²×C₁₃	C₂³	1	4	yes	—
40	C₂²⋊Dic₁₃	C₂³	1	16	yes	—
41	C₄⋊Dic₁₃	C₂³	1	16	yes	—
42	C₄⋊Dic₁₃	C₂³	1	16	yes	—
43	#6101	C₂³	1	32	yes	—
44	C₄×Dic₁₃	C₂³	1	16	yes	—
45	C₂×#533	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	—	2	4	no	—
47	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
48	C₂³×C₁₃	C₂⁴	1	2	yes	—
49	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	—	2	16	no	—
52	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
54	C₂×C₄×C₁₃	C₂×Q₈	1	16	yes	complemented
55	C₂×C₄×C₁₃	#41	1	8	yes	—
56	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
57	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
58	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
59	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
60	C₂×Dic₁₃	—	2	16	no	—
61	C₂×Dic₁₃	C₂⁴	1	16	yes	—
62	Dic₂₆	—	2	32	no	—
63	#533	—	13	16	no	—
64	C₂×C₄⋊Q₈	—	13	2	no	—
65	C₂×#125	—	13	2	no	—
66	C₂×C₂²⋊Q₈	—	13	4	no	—
67	C₂×C₄×Q₈	—	13	4	no	—
68	C₂×#117	—	13	2	no	—
69	C₂²×C₄²	D₁₃	1	1	yes	characteristic
70	C₂²×C₁₃	—	2	8	no	—
71	C₂²×C₁₃	—	2	4	no	—
72	C₂²×C₁₃	C₂²×Q₈	1	4	yes	—
73	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
74	C₂²×C₁₃	C₂×#41	1	4	yes	—
75	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
76	C₄×C₁₃	—	2	4	no	—
77	C₄×C₁₃	C₂²×Q₈	1	4	yes	—
78	C₄×C₁₃	C₂²×Q₈	1	4	yes	—
79	C₄×C₁₃	C₂×#41	1	8	yes	—
80	Dic₁₃	—	2	16	no	—
81	C₂²×Q₈	—	13	4	no	—
82	C₂³×C₄	C₂×D₁₃	1	2	yes	—
83	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
84	C₄⋊Q₈	—	13	16	no	—
85	#125	—	13	16	no	—
86	C₂²⋊Q₈	—	13	32	no	—
87	C₄×Q₈	—	13	32	no	—
88	#117	—	13	16	no	—
89	C₂×C₄⋊C₄	—	13	4	no	—
90	C₂×C₄⋊C₄	—	13	8	no	—
91	C₂×C₄⋊C₄	—	13	4	no	—
92	C₂×C₂²⋊C₄	—	13	4	no	—
93	C₂×C₄²	—	13	4	no	—
94	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₁₃	1	8	yes	—
97	C₂×C₄²	C₂×D₁₃	1	2	yes	—
98	C₂×C₁₃	—	2	4	no	—
99	C₂×C₁₃	#533	1	4	yes	complemented
100	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
101	C₂×C₁₃	C₂²×#41	1	2	yes	—
102	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
103	C₂×Q₈	—	13	16	no	—
104	C₂×Q₈	—	26	16	no	—
105	C₂²×C₄	—	2	8	no	—
106	C₂²×C₄	—	13	8	no	—
107	C₂²×C₄	Dic₂₆	1	8	yes	complemented
108	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₃	1	8	yes	—
110	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
111	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
112	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
113	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
114	C₄⋊C₄	—	13	16	no	—
115	C₄⋊C₄	—	13	32	no	—
116	C₄⋊C₄	—	13	16	no	—
117	C₂²⋊C₄	—	13	16	no	—
118	C₄²	—	13	16	no	—
119	C₄²	C₂²×D₁₃	1	8	yes	—
120	C₄²	C₂²×D₁₃	1	4	yes	—
121	C₄²	C₂²×D₁₃	1	4	yes	—
122	C₁₃	C₂×#533	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	2	4	no	—
124	C₂³	C₂×Dic₂₆	1	4	yes	—
125	C₂³	C₂³×D₁₃	1	2	yes	—
126	C₂³	C₂³×D₁₃	1	1	yes	characteristic
127	Q₈	—	26	32	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	16	no	—
130	C₂×C₄	—	13	16	no	—
131	C₂×C₄	—	26	16	no	—
132	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
133	C₂×C₄	C₂×Dic₂₆	1	4	yes	—
134	C₂×C₄	C₂×Dic₂₆	1	16	yes	complemented
135	C₂×C₄	#6127	1	8	yes	—
136	C₂×C₄	C₂³×D₁₃	1	4	yes	—
137	C₂×C₄	C₂³×D₁₃	1	2	yes	—
138	C₂×C₄	C₂³×D₁₃	1	2	yes	—
139	C₂×C₄	C₂³×D₁₃	1	4	yes	—
140	C₂²	—	2	8	no	—
141	C₂²	—	2	4	no	—
142	C₂²	C₂²×Dic₂₆	1	4	yes	—
143	C₂²	C₂²×Dic₂₆	1	2	yes	—
144	C₂²	C₂×#6127	1	4	yes	—
145	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	26	16	no	—
148	C₄	C₂²×Dic₂₆	1	4	yes	—
149	C₄	C₂²×Dic₂₆	1	4	yes	—
150	C₄	C₂×#6127	1	8	yes	—
151	C₂	—	2	4	no	—
152	C₂	#11719175	1	4	yes	direct product factor
153	C₂	C₂³×Dic₂₆	1	1	yes	characteristic
154	C₂	C₂²×#6127	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂²×C₄²×C₁₃	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	C₂²×C₄²	D₁₃
hypocenter	C₁₃	C₂×#533
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. #11719175	C₂	direct (N × H)
5. C₂×#88124	C₂	semidirect
6. C₂×#88123	C₂	semidirect
7. C₂×C₄×Dic₂₆	C₂	semidirect
22. #88124	C₂²	semidirect
23. #88123	C₂²	semidirect
24. C₄×Dic₂₆	C₂²	semidirect
27. C₂²×C₄×C₁₃	Q₈	semidirect
54. C₂×C₄×C₁₃	C₂×Q₈	semidirect
99. C₂×C₁₃	#533	semidirect
107. C₂²×C₄	Dic₂₆	semidirect
122. C₁₃	C₂×#533	semidirect
134. C₂×C₄	C₂×Dic₂₆	semidirect
152. C₂	#11719175	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄²×C₁₃	2
C₂×#88180	2
C₂×#88178	2
#11719175	2
C₂×#88124	2
C₂×#88123	2
C₂×C₄×Dic₂₆	2
C₂×#533	13

Subgroups — C2×#11719175 #13128127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719175 #13128127