Subgroups — C₂²×#11718343 #16732517

Order 3328.

2398 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88123	C₂	1	2	yes	maximal
2	C₂×#11718343	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88058	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₆	—	2	2	no	—
6	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₁₃	—	2	2	no	—
9	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
10	C₂×#88123	—	2	12	no	—
11	C₂×#88123	C₂²	1	24	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	—
13	C₂×#88058	C₂²	1	48	yes	—
14	#11718343	C₂²	1	64	yes	direct product factor
15	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₃	—	2	4	no	—
19	C₂²×Dic₂₆	—	2	24	no	—
20	C₂²×Dic₂₆	—	4	16	no	—
21	C₂²×C₈×C₁₃	Q₈	1	16	yes	complemented
22	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
23	C₂×C₄²×C₁₃	D₄	1	6	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
25	C₂×C₄⋊Dic₁₃	—	2	12	no	—
26	C₂×C₄⋊Dic₁₃	—	2	24	no	—
27	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
28	#88123	—	2	48	no	—
29	#88123	C₂³	1	32	yes	—
30	C₄×C₈×C₁₃	C₂³	1	16	yes	—
31	#88058	C₂³	1	64	yes	—
32	C₂²×#500	—	13	1	no	maximal, Sylow
33	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	D₈	1	8	yes	—
35	C₂²×C₄×C₁₃	Q₁₆	1	8	yes	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
39	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
40	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
41	C₂²×Dic₁₃	—	2	24	no	—
42	C₂²×Dic₁₃	—	4	16	no	—
43	C₂×Dic₂₆	—	2	32	no	—
44	C₂×Dic₂₆	—	4	96	no	—
45	C₂×C₈×C₁₃	C₂×Q₈	1	48	yes	complemented
46	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
47	C₄²×C₁₃	C₂×D₄	1	12	yes	—
48	C₄²×C₁₃	C₂⁴	1	4	yes	—
49	C₄⋊Dic₁₃	—	2	24	no	—
50	C₄⋊Dic₁₃	—	2	32	no	—
51	C₄⋊Dic₁₃	C₂⁴	1	16	yes	—
52	C₂²×C₄⋊Q₈	—	13	2	no	—
53	C₂×#500	—	13	24	no	—
54	C₂²×#107	—	13	4	no	—
55	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
56	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
57	C₂³×C₁₃	C₂×D₈	1	4	yes	—
58	C₂³×C₁₃	C₂×Q₁₆	1	4	yes	—
59	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
60	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
61	C₂×C₄×C₁₃	C₂×D₈	1	24	yes	—
62	C₂×C₄×C₁₃	C₂×Q₁₆	1	24	yes	—
63	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
64	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
65	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
66	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
67	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
68	C₂×Dic₁₃	—	2	16	no	—
69	C₂×Dic₁₃	—	4	48	no	—
70	Dic₂₆	—	4	64	no	—
71	C₈×C₁₃	C₂²×Q₈	1	16	yes	complemented
72	C₂³×Q₈	—	26	2	no	—
73	C₂³×C₈	C₂×D₁₃	1	2	yes	—
74	C₂×C₄⋊Q₈	—	13	24	no	—
75	C₂×C₄⋊Q₈	—	26	12	no	—
76	C₂²×C₄⋊C₄	—	13	4	no	—
77	C₂²×C₄⋊C₄	—	26	2	no	—
78	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
79	#500	—	13	64	no	—
80	C₂×#107	—	13	48	no	—
81	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	—
82	C₂²×C₁₃	#500	1	16	yes	complemented
83	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
84	C₂²×C₁₃	C₂²×D₈	1	6	yes	—
85	C₂²×C₁₃	C₂²×Q₁₆	1	6	yes	—
86	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
87	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
88	C₄×C₁₃	C₂²×D₈	1	8	yes	—
89	C₄×C₁₃	C₂²×Q₁₆	1	8	yes	—
90	C₄×C₁₃	C₂³×Q₈	1	2	yes	—
91	Dic₁₃	—	4	16	no	—
92	C₂²×Q₈	—	26	24	no	—
93	C₂²×Q₈	—	52	16	no	—
94	C₂³×C₄	—	26	4	no	—
95	C₂³×C₄	D₅₂	1	1	yes	characteristic
96	C₂³×C₄	D₅₂	1	1	yes	characteristic
97	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
98	C₂²×C₈	Dic₂₆	1	16	yes	complemented
99	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
100	C₄⋊Q₈	—	13	32	no	—
101	C₄⋊Q₈	—	26	48	no	—
102	C₂×C₄⋊C₄	—	13	24	no	—
103	C₂×C₄⋊C₄	—	26	12	no	—
104	C₂×C₄⋊C₄	—	26	24	no	—
105	C₂×C₄²	D₅₂	1	6	yes	—
106	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
107	#107	—	13	64	no	—
108	C₄×C₈	C₂²×D₁₃	1	16	yes	—
109	C₂×C₁₃	C₂×#500	1	12	yes	complemented
110	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
112	C₂×C₁₃	C₂³×Q₁₆	1	1	yes	characteristic
113	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
114	C₂×Q₈	—	26	32	no	—
115	C₂×Q₈	—	52	96	no	—
116	C₂²×C₄	—	26	24	no	—
117	C₂²×C₄	—	52	16	no	—
118	C₂²×C₄	D₁₀₄	1	8	yes	—
119	C₂²×C₄	Dic₅₂	1	8	yes	—
120	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
121	C₂²×C₄	C₂×D₅₂	1	3	yes	—
122	C₂²×C₄	C₂×D₅₂	1	6	yes	—
123	C₂²×C₄	C₂×D₅₂	1	6	yes	—
124	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
125	C₂×C₈	C₂×Dic₂₆	1	48	yes	complemented
126	C₂×C₈	C₂³×D₁₃	1	8	yes	—
127	C₄⋊C₄	—	13	16	no	—
128	C₄⋊C₄	—	26	24	no	—
129	C₄⋊C₄	—	26	32	no	—
130	C₄²	C₂×D₅₂	1	12	yes	—
131	C₄²	C₂³×D₁₃	1	4	yes	—
132	C₁₃	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
133	C₂³	#88123	1	4	yes	—
134	C₂³	C₂×D₁₀₄	1	4	yes	—
135	C₂³	C₂×Dic₅₂	1	4	yes	—
136	C₂³	C₂²×D₅₂	1	3	yes	—
137	Q₈	—	52	64	no	—
138	C₂×C₄	—	26	16	no	—
139	C₂×C₄	—	52	48	no	—
140	C₂×C₄	#88123	1	12	yes	—
141	C₂×C₄	C₂×D₁₀₄	1	24	yes	—
142	C₂×C₄	C₂×Dic₅₂	1	24	yes	—
143	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
144	C₂×C₄	C₂²×D₅₂	1	3	yes	—
145	C₂×C₄	C₂²×D₅₂	1	4	yes	—
146	C₂×C₄	C₂²×D₅₂	1	4	yes	—
147	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
148	C₈	C₂²×Dic₂₆	1	16	yes	complemented
149	C₂²	#11718343	1	16	yes	direct product factor
150	C₂²	C₂×#88123	1	6	yes	—
151	C₂²	C₂²×D₁₀₄	1	6	yes	—
152	C₂²	C₂²×Dic₅₂	1	6	yes	—
153	C₂²	C₂³×D₅₂	1	1	yes	characteristic
154	C₄	—	52	16	no	—
155	C₄	C₂×#88123	1	6	yes	—
156	C₄	C₂²×D₁₀₄	1	8	yes	—
157	C₄	C₂²×Dic₅₂	1	8	yes	—
158	C₄	C₂³×Dic₂₆	1	2	yes	—
159	C₂	C₂×#11718343	1	12	yes	direct product factor
160	C₂	C₂²×#88123	1	1	yes	characteristic
161	C₂	C₂³×D₁₀₄	1	1	yes	characteristic
162	C₂	C₂³×Dic₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂²×#500
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718343	C₂	direct (N × H)
14. #11718343	C₂²	direct (N × H)
21. C₂²×C₈×C₁₃	Q₈	semidirect
45. C₂×C₈×C₁₃	C₂×Q₈	semidirect
71. C₈×C₁₃	C₂²×Q₈	semidirect
82. C₂²×C₁₃	#500	semidirect
98. C₂²×C₈	Dic₂₆	semidirect
109. C₂×C₁₃	C₂×#500	semidirect
125. C₂×C₈	C₂×Dic₂₆	semidirect
132. C₁₃	C₂²×#500	semidirect
148. C₈	C₂²×Dic₂₆	semidirect
149. C₂²	#11718343	direct (N × H)
159. C₂	C₂×#11718343	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×#88123	2
C₂×#11718343	2
C₂²×#88058	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#500	13

Subgroups — C22×#11718343 #16732517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718343 #16732517