Subgroups — C₂²×#11718260 #16806990

Order 3328.

862 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	C₂³×#6092	C₂	1	2	yes	maximal
2	C₂×#11718260	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂²×#6092	—	2	8	no	—
8	C₂²×#6092	C₂²	1	12	yes	—
9	C₂×C₈×Dic₁₃	C₄	1	6	yes	—
10	C₂×C₈×Dic₁₃	C₂²	1	6	yes	—
11	#11718260	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂³×Dic₁₃	C₈	1	1	yes	characteristic
14	C₂³×Dic₁₃	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	—	2	4	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
17	C₂×C₄×Dic₁₃	C₈	1	6	yes	—
18	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
19	C₂²×#1055	C₂×C₄	1	3	yes	—
20	C₂²×#1055	D₄	1	4	yes	—
21	C₂²×#1055	Q₈	1	4	yes	—
22	C₂²×#1055	C₂³	1	3	yes	—
23	C₂×#6092	—	2	24	no	—
24	C₂×#6092	C₂³	1	8	yes	—
25	C₈×Dic₁₃	C₂×C₄	1	12	yes	—
26	C₈×Dic₁₃	C₂³	1	4	yes	—
27	C₂²×C₄⋊C₁₆	—	13	1	no	maximal, Sylow
28	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
30	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
31	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
32	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
33	C₂²×Dic₁₃	—	2	4	no	—
34	C₂²×Dic₁₃	C₁₆	1	8	yes	complemented
35	C₂²×Dic₁₃	C₂×C₈	1	6	yes	—
36	C₂²×Dic₁₃	C₂×C₈	1	6	yes	—
37	C₂×C₈×C₁₃	—	2	12	no	—
38	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
39	C₄×Dic₁₃	C₂×C₈	1	12	yes	—
40	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
41	C₂×#1055	C₄⋊C₄	1	12	yes	—
42	C₂×#1055	C₂²×C₄	1	3	yes	—
43	C₂×#1055	C₂×D₄	1	6	yes	—
44	C₂×#1055	C₂×Q₈	1	6	yes	—
45	C₂×#1055	C₂⁴	1	1	yes	characteristic
46	#6092	—	2	8	no	—
47	C₂³×C₁₆	—	13	2	no	—
48	C₂²×C₄×C₈	—	13	1	no	—
49	C₂×C₄⋊C₁₆	—	13	12	no	—
50	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
51	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
52	C₂³×C₁₃	M₃₂	1	4	yes	—
53	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
54	C₂×C₄×C₁₃	C₄⋊C₈	1	12	yes	—
55	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
56	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
57	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
58	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
59	C₂×Dic₁₃	—	2	12	no	—
60	C₂×Dic₁₃	C₂×C₁₆	1	24	yes	complemented
61	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
62	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
63	C₈×C₁₃	—	2	4	no	—
64	#1055	C₂×C₄⋊C₄	1	6	yes	—
65	#1055	C₂²×D₄	1	1	yes	characteristic
66	#1055	C₂²×Q₈	1	1	yes	characteristic
67	C₂³×C₈	—	13	1	no	—
68	C₂³×C₈	#261	1	1	yes	characteristic
69	C₂²×C₄²	—	13	1	no	—
70	C₂²×C₁₆	—	13	12	no	—
71	C₂²×C₁₆	—	26	8	no	—
72	C₂×C₄×C₈	—	13	6	no	—
73	C₂×C₄×C₈	—	13	6	no	—
74	C₄⋊C₁₆	—	13	16	no	—
75	C₂²×C₁₃	C₄⋊C₁₆	1	16	yes	complemented
76	C₂²×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
77	C₂²×C₁₃	C₂²×C₁₆	1	6	yes	—
78	C₂²×C₁₃	C₂×M₃₂	1	6	yes	—
79	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
81	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
83	Dic₁₃	—	2	4	no	—
84	Dic₁₃	C₂²×C₁₆	1	8	yes	complemented
85	C₂³×C₄	—	13	1	no	—
86	C₂³×C₄	—	13	1	no	—
87	C₂³×C₄	C₂×#261	1	1	yes	characteristic
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	—	13	4	no	—
90	C₂²×C₈	—	13	4	no	—
91	C₂²×C₈	—	13	3	no	—
92	C₂²×C₈	—	13	3	no	—
93	C₂²×C₈	C₂×#261	1	6	yes	—
94	C₂×C₄²	—	13	6	no	—
95	C₂×C₄²	—	13	6	no	—
96	C₂×C₁₆	—	13	8	no	—
97	C₂×C₁₆	—	26	24	no	—
98	C₄×C₈	—	13	4	no	—
99	C₄×C₈	—	13	12	no	—
100	C₂×C₁₃	C₂×C₄⋊C₁₆	1	12	yes	complemented
101	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
103	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
104	C₂⁴	D₁₃⋊C₈	1	1	yes	characteristic
105	C₂²×C₄	—	13	8	no	—
106	C₂²×C₄	—	13	6	no	—
107	C₂²×C₄	—	13	6	no	—
108	C₂²×C₄	—	26	4	no	—
109	C₂²×C₄	D₁₃⋊C₈	1	3	yes	—
110	C₂²×C₄	C₄⋊#261	1	4	yes	—
111	C₂²×C₄	#6122	1	4	yes	—
112	C₂²×C₄	C₂²×#261	1	3	yes	—
113	C₂×C₈	—	2	12	no	—
114	C₂×C₈	—	13	6	no	—
115	C₂×C₈	—	13	12	no	—
116	C₂×C₈	—	13	6	no	—
117	C₂×C₈	—	13	1	no	—
118	C₂×C₈	—	13	3	no	—
119	C₂×C₈	C₂²×#261	1	4	yes	—
120	C₄²	—	13	12	no	—
121	C₄²	—	13	4	no	—
122	C₁₆	—	26	8	no	—
123	C₁₃	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
124	C₂³	D₁₃⋊C₁₆	1	4	yes	—
125	C₂³	#88098	1	4	yes	—
126	C₂³	#88112	1	4	yes	—
127	C₂³	C₂×D₁₃⋊C₈	1	3	yes	—
128	C₂×C₄	—	13	24	no	—
129	C₂×C₄	—	13	4	no	—
130	C₂×C₄	—	13	4	no	—
131	C₂×C₄	—	26	12	no	—
132	C₂×C₄	#88112	1	12	yes	—
133	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
134	C₂×C₄	C₂×C₄⋊#261	1	6	yes	—
135	C₂×C₄	C₂×#6122	1	6	yes	—
136	C₂×C₄	C₂³×#261	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	—	13	1	no	—
139	C₈	—	13	6	no	—
140	C₈	—	13	1	no	—
141	C₂²	#11718260	1	16	yes	direct product factor
142	C₂²	C₂×D₁₃⋊C₁₆	1	6	yes	—
143	C₂²	C₂×#88098	1	6	yes	—
144	C₂²	C₂×#88112	1	6	yes	—
145	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
146	C₄	—	13	8	no	—
147	C₄	—	26	4	no	—
148	C₄	C₂×#88112	1	6	yes	—
149	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
150	C₄	C₂²×#6122	1	1	yes	characteristic
151	C₂	C₂×#11718260	1	12	yes	direct product factor
152	C₂	C₂²×#88112	1	1	yes	characteristic
153	C₂	C₂²×D₁₃⋊C₁₆	1	1	yes	characteristic
154	C₂	C₂²×#88098	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#261
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂³×C₈×C₁₃	C₄
socle, prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718260	C₂	direct (N × H)
11. #11718260	C₂²	direct (N × H)
34. C₂²×Dic₁₃	C₁₆	semidirect
60. C₂×Dic₁₃	C₂×C₁₆	semidirect
75. C₂²×C₁₃	C₄⋊C₁₆	semidirect
84. Dic₁₃	C₂²×C₁₆	semidirect
100. C₂×C₁₃	C₂×C₄⋊C₁₆	semidirect
123. C₁₃	C₂²×C₄⋊C₁₆	semidirect
141. C₂²	#11718260	direct (N × H)
151. C₂	C₂×#11718260	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6092	2
C₂×#11718260	2
C₂²×C₈×Dic₁₃	2
C₂²×C₄⋊C₁₆	13

Subgroups — C22×#11718260 #16806990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718260 #16806990