Finite Group Databasebeta

Subgroups — #16811224

Order 3392.

240 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 95 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	#13133213	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133208	C₂	1	1	yes	maximal, characteristic, complemented
3	#13133200	C₂	1	2	yes	maximal, complemented
4	#13133199	C₂	1	2	yes	maximal, complemented
5	#13133192	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133191	C₂	1	2	yes	maximal, complemented
7	#13133190	C₂	1	2	yes	maximal, complemented
8	#13133188	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#13133174	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
17	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
18	C₂×D₂₁₂	C₂²	1	2	yes	—
19	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₅₃	C₂²	1	2	yes	—
21	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
25	#11719921	C₂²	1	1	yes	characteristic
26	#11719921	C₂²	1	1	yes	characteristic
27	#11719921	C₂²	1	2	yes	—
28	#11719921	C₂²	1	2	yes	—
29	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₅₃	C₂²	1	2	yes	—
31	#11719919	C₂²	1	1	yes	characteristic
32	#11719919	C₂²	1	1	yes	characteristic
33	#11719919	C₂²	1	2	yes	—
34	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂³×C₅₃	C₂³	1	2	yes	—
36	C₂²×D₅₃	C₂³	1	1	yes	characteristic
37	C₂²×D₅₃	C₂³	1	2	yes	—
38	D₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²⋊D₅₃	—	4	2	no	—
45	C₂²⋊D₅₃	—	4	2	no	—
46	C₂²⋊D₅₃	—	4	2	no	—
47	C₂×Dic₅₃	C₂³	1	2	yes	—
48	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
51	D₂₁₂	—	2	2	no	—
52	D₂₁₂	—	4	2	no	—
53	C₄×D₅₃	—	2	2	no	—
54	C₄×D₅₃	—	4	2	no	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	—	2	2	no	—
58	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₂×D₅₃	—	2	1	no	—
60	C₂×D₅₃	—	2	2	no	—
61	C₂×D₅₃	—	2	2	no	—
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	—	2	2	no	—
64	C₂×D₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	#41	1	2	yes	—
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	—	2	1	no	—
72	Dic₅₃	—	2	1	no	—
73	C₂×C₅₃	—	4	2	no	—
74	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
75	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
77	D₅₃	—	4	1	no	—
78	D₅₃	—	4	2	no	—
79	#540	—	53	1	no	maximal, Sylow
80	C₅₃	#540	1	1	yes	characteristic, complemented, Sylow
81	#127	—	53	1	no	—
82	#125	—	53	1	no	—
83	#123	—	53	2	no	—
84	#123	—	53	2	no	—
85	#121	—	53	1	no	—
86	#121	—	53	2	no	—
87	#121	—	53	1	no	—
88	#121	—	53	2	no	—
89	C₄×D₄	—	53	1	no	—
90	C₄×D₄	D₅₃	1	1	yes	characteristic, complemented
91	#117	—	53	1	no	—
92	C₂×D₄	—	53	2	no	—
93	C₂×D₄	—	53	2	no	—
94	C₂×D₄	—	53	2	no	—
95	C₂×D₄	—	53	1	no	—
96	C₂×D₄	—	53	2	no	—
97	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
98	C₂²×C₄	—	53	1	no	—
99	C₂²×C₄	—	53	2	no	—
100	C₂²×C₄	C₂×D₅₃	1	2	yes	—
101	C₄⋊C₄	—	53	1	no	—
102	C₄⋊C₄	—	53	1	no	—
103	C₄⋊C₄	—	53	1	no	—
104	C₄⋊C₄	—	53	2	no	—
105	C₄⋊C₄	—	53	2	no	—
106	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
107	C₂²⋊C₄	—	53	2	no	—
108	C₂²⋊C₄	—	53	1	no	—
109	C₂²⋊C₄	—	53	1	no	—
110	C₂²⋊C₄	—	53	2	no	—
111	C₂²⋊C₄	—	53	2	no	—
112	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
113	C₄²	—	53	1	no	—
114	C₄²	C₂×D₅₃	1	1	yes	characteristic
115	C₂³	—	53	1	no	—
116	C₂³	—	53	2	no	—
117	C₂³	C₂²×D₅₃	1	2	yes	—
118	D₄	—	2	2	no	—
119	D₄	—	106	2	no	—
120	D₄	—	212	2	no	—
121	D₄	—	212	2	no	—
122	D₄	—	212	2	no	—
123	D₄	—	212	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	53	1	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	—	106	2	no	—
130	C₂×C₄	—	212	2	no	—
131	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
132	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₅₃	1	2	yes	—
134	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	106	1	no	—
139	C₂²	—	106	2	no	—
140	C₂²	—	106	1	no	—
141	C₂²	—	106	2	no	—
142	C₂²	—	106	2	no	—
143	C₂²	—	106	1	no	—
144	C₂²	C₂³×D₅₃	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	106	1	no	—
149	C₄	—	106	1	no	—
150	C₄	—	106	2	no	—
151	C₄	—	106	1	no	—
152	C₄	#11719945	1	2	yes	—
153	C₂	—	4	2	no	—
154	C₂	—	212	1	no	—
155	C₂	—	212	2	no	—
156	C₂	C₂×#11719945	1	1	yes	characteristic
157	C₂	#13133262	1	1	yes	characteristic
158	C₂	#13133263	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₄×D₄×C₅₃	C₂
hypercenter	C₄×D₄	D₅₃
hypocenter	C₅₃	#540
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #13133213	C₂	semidirect
2. #13133208	C₂	semidirect
3. #13133200	C₂	semidirect
4. #13133199	C₂	semidirect
5. #13133192	C₂	semidirect
6. #13133191	C₂	semidirect
7. #13133190	C₂	semidirect
8. #13133188	C₂	semidirect
9. C₄×D₂₁₂	C₂	semidirect
10. #13133174	C₂	semidirect
11. C₄×D₄×C₅₃	C₂	semidirect
80. C₅₃	#540	semidirect
90. C₄×D₄	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133213	2
#13133208	2
#13133200	2
#13133199	2
#13133192	2
#13133191	2
#13133190	2
#13133188	2
C₄×D₂₁₂	2
#13133174	2
C₄×D₄×C₅₃	2
#540	53