Subgroups — C₂²×#11718344 #16732513

Order 3328.

1966 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88124	C₂	1	2	yes	maximal
2	C₂×#11718344	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#88058	C₂	1	2	yes	maximal
4	C₂²×#88057	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
10	C₂²×#6101	—	2	4	no	—
11	C₂×#88124	—	2	12	no	—
12	C₂×#88124	C₂²	1	24	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	12	yes	—
14	C₂×#88058	C₂²	1	24	yes	—
15	C₂×#88057	C₂²	1	24	yes	—
16	#11718344	C₂²	1	64	yes	direct product factor
17	C₂³×C₄×C₁₃	D₄	1	2	yes	—
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂³×Dic₁₃	—	2	4	no	—
20	C₂²×C₈×C₁₃	Q₈	1	8	yes	—
21	C₂²×C₈×C₁₃	Q₈	1	8	yes	—
22	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
23	C₂²×C₈×C₁₃	C₂³	1	6	yes	—
24	C₂×C₄²×C₁₃	D₄	1	6	yes	—
25	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
26	C₂×C₄⋊Dic₁₃	—	2	12	no	—
27	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
28	C₂×#6101	—	2	48	no	—
29	#88124	—	2	48	no	—
30	#88124	C₂³	1	32	yes	—
31	C₄×C₈×C₁₃	C₂³	1	16	yes	—
32	#88058	C₂³	1	32	yes	—
33	#88057	C₂³	1	32	yes	—
34	C₂²×#499	—	13	1	no	maximal, Sylow
35	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	—	2	8	no	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
40	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
42	C₂²×Dic₁₃	—	2	24	no	—
43	C₂²×Dic₁₃	—	4	16	no	—
44	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	—
45	C₂×C₈×C₁₃	C₂×Q₈	1	24	yes	—
46	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
47	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
48	C₄²×C₁₃	C₂×D₄	1	12	yes	—
49	C₄²×C₁₃	C₂⁴	1	4	yes	—
50	C₄⋊Dic₁₃	—	2	24	no	—
51	C₄⋊Dic₁₃	C₂⁴	1	16	yes	—
52	#6101	—	2	64	no	—
53	C₂²×#125	—	13	2	no	—
54	C₂×#499	—	13	24	no	—
55	C₂²×#107	—	13	2	no	—
56	C₂²×#106	—	13	2	no	—
57	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
58	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
59	C₂³×C₁₃	#135	1	8	yes	—
60	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
61	C₂×C₄×C₁₃	—	2	24	no	—
62	C₂×C₄×C₁₃	C₄⋊Q₈	1	12	yes	—
63	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
64	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
65	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
66	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
67	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
68	C₂×Dic₁₃	—	2	16	no	—
69	C₂×Dic₁₃	—	4	48	no	—
70	C₈×C₁₃	C₂²×Q₈	1	8	yes	—
71	C₈×C₁₃	C₂²×Q₈	1	8	yes	—
72	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
73	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
74	C₂×#125	—	13	24	no	—
75	C₂×#125	—	26	12	no	—
76	C₂²×C₄⋊C₄	—	13	4	no	—
77	C₂²×C₄⋊C₄	—	26	4	no	—
78	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
79	#499	—	13	64	no	—
80	C₂×#107	—	13	24	no	—
81	C₂×#106	—	13	24	no	—
82	C₂×C₄×C₈	C₂×D₁₃	1	12	yes	—
83	C₂²×C₁₃	#499	1	16	yes	complemented
84	C₂²×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
85	C₂²×C₁₃	C₂×#135	1	12	yes	—
86	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
87	C₄×C₁₃	—	2	8	no	—
88	C₄×C₁₃	C₂×C₄⋊Q₈	1	6	yes	—
89	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
90	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
91	Dic₁₃	—	4	16	no	—
92	C₂³×C₄	—	26	4	no	—
93	C₂³×C₄	D₅₂	1	2	yes	—
94	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
95	C₂²×C₈	Dic₂₆	1	8	yes	—
96	C₂²×C₈	Dic₂₆	1	8	yes	—
97	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
98	C₂²×C₈	C₂²×D₁₃	1	6	yes	—
99	#125	—	13	32	no	—
100	#125	—	26	48	no	—
101	C₂×C₄⋊C₄	—	13	24	no	—
102	C₂×C₄⋊C₄	—	26	12	no	—
103	C₂×C₄⋊C₄	—	26	48	no	—
104	C₂×C₄²	D₅₂	1	6	yes	—
105	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
106	#107	—	13	32	no	—
107	#106	—	13	32	no	—
108	C₄×C₈	C₂²×D₁₃	1	16	yes	—
109	C₂×C₁₃	C₂×#499	1	12	yes	complemented
110	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₁₃	C₂²×#135	1	2	yes	—
112	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	—	26	24	no	—
115	C₂²×C₄	—	52	16	no	—
116	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
117	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
118	C₂²×C₄	C₂×D₅₂	1	12	yes	—
119	C₂²×C₄	C₂×D₅₂	1	3	yes	—
120	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
121	C₂×C₈	C₂×Dic₂₆	1	24	yes	—
122	C₂×C₈	C₂×Dic₂₆	1	24	yes	—
123	C₂×C₈	C₂³×D₁₃	1	4	yes	—
124	C₂×C₈	C₂³×D₁₃	1	4	yes	—
125	C₄⋊C₄	—	13	16	no	—
126	C₄⋊C₄	—	26	24	no	—
127	C₄⋊C₄	—	26	64	no	—
128	C₄²	C₂×D₅₂	1	12	yes	—
129	C₄²	C₂³×D₁₃	1	4	yes	—
130	C₁₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
131	C₂³	#88123	1	4	yes	—
132	C₂³	#88158	1	8	yes	—
133	C₂³	C₂²×D₅₂	1	3	yes	—
134	C₂×C₄	—	2	24	no	—
135	C₂×C₄	—	26	16	no	—
136	C₂×C₄	—	52	48	no	—
137	C₂×C₄	#88123	1	12	yes	—
138	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
139	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
140	C₂×C₄	C₂²×D₅₂	1	8	yes	—
141	C₂×C₄	C₂²×D₅₂	1	3	yes	—
142	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
143	C₈	C₂²×Dic₂₆	1	8	yes	—
144	C₈	C₂²×Dic₂₆	1	8	yes	—
145	C₂²	#11718344	1	16	yes	direct product factor
146	C₂²	C₂×#88123	1	6	yes	—
147	C₂²	C₂×#88158	1	12	yes	—
148	C₂²	C₂³×D₅₂	1	1	yes	characteristic
149	C₄	—	2	8	no	—
150	C₄	—	52	16	no	—
151	C₄	C₂×#88123	1	6	yes	—
152	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
153	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
154	C₂	C₂×#11718344	1	12	yes	direct product factor
155	C₂	C₂²×#88123	1	1	yes	characteristic
156	C₂	C₂²×#88158	1	2	yes	—

	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×#499
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×#11718344	C₂	direct (N × H)
16. #11718344	C₂²	direct (N × H)
83. C₂²×C₁₃	#499	semidirect
109. C₂×C₁₃	C₂×#499	semidirect
130. C₁₃	C₂²×#499	semidirect
145. C₂²	#11718344	direct (N × H)
154. C₂	C₂×#11718344	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×#88124	2
C₂×#11718344	2
C₂²×#88058	2
C₂²×#88057	2
C₂²×C₄×C₈×C₁₃	2
C₂²×#499	13

Subgroups — C22×#11718344 #16732513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11718344 #16732513