Finite Group Databasebeta

← C₂×#16811480 #20290163

Subgroups — C₂×#16811480 #20290163

Order 6784.

232 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133272	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133272	C₂	1	1	yes	maximal, characteristic
3	#16811480	C₂	1	4	yes	maximal, direct product factor
4	C₂×#13133208	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88324	—	2	1	no	—
6	C₂²×#88324	—	2	1	no	—
7	#13133272	—	2	1	no	—
8	#13133272	—	2	1	no	—
9	#13133272	C₂²	1	2	yes	—
10	#13133272	C₂²	1	2	yes	—
11	#13133208	—	2	1	no	—
12	#13133208	—	2	1	no	—
13	#13133208	C₄	1	2	yes	—
14	#13133208	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₅₃	—	2	1	no	—
16	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
17	C₂×#11719919	—	2	1	no	—
18	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	—	2	1	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	1	no	—
23	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
24	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
25	C₂×#88324	—	2	4	no	—
26	C₂×#88324	—	2	4	no	—
27	C₂×#88324	—	2	2	no	—
28	C₂×#88324	—	2	2	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	2	no	—
33	C₄⋊Dic₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
35	#11719919	—	2	2	no	—
36	#11719919	—	2	2	no	—
37	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₅₃	D₄	1	1	yes	characteristic
39	C₄×Dic₅₃	D₄	1	1	yes	characteristic
40	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
41	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	2	2	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	—	2	2	no	—
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	—	4	2	no	—
51	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
55	#88324	—	2	4	no	—
56	#88324	—	2	4	no	—
57	C₂²×C₅₃	#101	1	2	yes	—
58	C₂²×C₅₃	#104	1	2	yes	—
59	C₂²×C₅₃	#104	1	2	yes	—
60	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	—	4	2	no	—
64	Dic₅₃	—	2	1	no	—
65	Dic₅₃	—	2	1	no	—
66	Dic₅₃	—	2	1	no	—
67	Dic₅₃	—	2	1	no	—
68	Dic₅₃	—	4	2	no	—
69	C₂×#330	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	#330	1	4	yes	complemented
71	C₂×C₅₃	C₂×#101	1	1	yes	characteristic
72	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
73	C₂×C₅₃	C₂×#104	1	1	yes	characteristic
74	C₂×#125	—	53	1	no	—
75	C₂×#97	—	53	1	no	—
76	C₂×#97	—	53	1	no	—
77	#330	—	53	4	no	—
78	C₅₃	C₂×#330	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	106	1	no	—
80	C₂²×C₈	—	106	1	no	—
81	#125	—	53	2	no	—
82	#125	—	53	2	no	—
83	#125	—	106	1	no	—
84	#125	—	106	1	no	—
85	C₂×C₄⋊C₄	—	53	1	no	—
86	C₂×C₄⋊C₄	—	106	1	no	—
87	C₂×C₄⋊C₄	—	106	1	no	—
88	C₂×C₄⋊C₄	#6157	1	1	yes	characteristic
89	C₂×C₄²	—	53	1	no	—
90	#97	—	53	2	no	—
91	#97	—	53	2	no	—
92	#97	—	106	1	no	—
93	#97	—	106	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	—	53	1	no	—
96	C₂²×C₄	—	53	1	no	—
97	C₂²×C₄	—	106	1	no	—
98	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
99	C₂×C₈	—	106	2	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂×C₈	—	106	4	no	—
102	C₂×C₈	—	106	4	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	53	2	no	—
105	C₄⋊C₄	—	106	2	no	—
106	C₄⋊C₄	—	106	2	no	—
107	C₄⋊C₄	—	106	2	no	—
108	C₄⋊C₄	—	106	2	no	—
109	C₄⋊C₄	—	106	1	no	—
110	C₄⋊C₄	C₂×#6157	1	2	yes	—
111	C₄²	—	53	1	no	—
112	C₄²	—	53	1	no	—
113	C₄²	—	53	1	no	—
114	C₄²	—	53	1	no	—
115	C₂³	C₂²⋊#6157	1	1	yes	characteristic
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	1	no	—
123	C₂×C₄	—	106	2	no	—
124	C₂×C₄	—	106	2	no	—
125	C₂×C₄	—	106	2	no	—
126	C₂×C₄	—	212	2	no	—
127	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
128	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
129	C₈	—	106	4	no	—
130	C₈	—	106	4	no	—
131	C₂²	#13133278	1	2	yes	—
132	C₂²	#13133281	1	2	yes	—
133	C₂²	#13133282	1	2	yes	—
134	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	2	no	—
138	C₄	—	106	1	no	—
139	C₄	—	106	1	no	—
140	C₄	—	106	1	no	—
141	C₄	—	106	1	no	—
142	C₄	—	212	2	no	—
143	C₂	#16811480	1	4	yes	direct product factor
144	C₂	C₂×#13133278	1	1	yes	characteristic
145	C₂	C₂×#13133281	1	1	yes	characteristic
146	C₂	C₂×#13133282	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂×C₄⋊C₄×C₅₃	C₄
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	C₂×C₄⋊C₄	#6157
hypocenter	C₅₃	C₂×#330
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #16811480	C₂	direct (N × H)
70. C₂×C₅₃	#330	semidirect
78. C₅₃	C₂×#330	semidirect
143. C₂	#16811480	direct (N × H)