Finite Group Databasebeta

Subgroups — #16811336

Order 3392.

236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13133216	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133205	C₂	1	2	yes	maximal, complemented
3	#13133203	C₂	1	2	yes	maximal, complemented
4	#13133192	C₂	1	1	yes	maximal, characteristic, complemented
5	#13133191	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133190	C₂	1	2	yes	maximal, complemented
7	#13133179	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#13133175	C₂	1	1	yes	maximal, characteristic, complemented
10	#13133174	C₂	1	2	yes	maximal, complemented
11	#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
15	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
16	C₂×D₂₁₂	C₂²	1	2	yes	—
17	C₂×C₄×D₅₃	C₂²	1	2	yes	—
18	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₀₆	C₂²	1	2	yes	—
20	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₄²×C₅₃	C₂²	1	2	yes	—
24	C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
25	#11719921	C₂²	1	2	yes	—
26	#11719921	C₂²	1	2	yes	—
27	#11719921	C₂²	1	2	yes	—
28	#11719921	C₂²	1	1	yes	characteristic
29	#11719921	C₂²	1	2	yes	—
30	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
31	#11719919	C₂²	1	1	yes	characteristic
32	#11719919	C₂²	1	2	yes	—
33	C₄×Dic₅₃	C₂²	1	2	yes	—
34	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂³×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×D₅₃	C₂³	1	2	yes	—
37	C₂²×D₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²⋊D₅₃	—	2	2	no	—
44	C₂²⋊D₅₃	—	4	2	no	—
45	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂³	1	2	yes	—
47	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
49	D₂₁₂	—	2	2	no	—
50	D₂₁₂	—	4	2	no	—
51	C₄×D₅₃	—	2	2	no	—
52	C₄×D₅₃	—	2	4	no	—
53	Dic₁₀₆	—	4	2	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂×D₅₃	—	2	2	no	—
59	C₂×D₅₃	—	2	1	no	—
60	C₂×D₅₃	—	2	2	no	—
61	C₂×D₅₃	—	2	1	no	—
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	—	2	2	no	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	#41	1	2	yes	—
68	Dic₅₃	—	2	1	no	—
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	2	2	no	—
71	Dic₅₃	#41	1	2	yes	—
72	C₂×C₅₃	—	4	1	no	—
73	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
75	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
76	D₅₃	—	4	2	no	—
77	D₅₃	—	4	1	no	—
78	#555	—	53	1	no	maximal, Sylow
79	C₅₃	#555	1	1	yes	characteristic, complemented, Sylow
80	C₄⋊Q₈	—	53	1	no	—
81	#124	—	53	2	no	—
82	#124	—	53	2	no	—
83	#121	—	53	1	no	—
84	#121	—	53	1	no	—
85	#121	—	53	2	no	—
86	C₄×D₄	—	53	1	no	—
87	C₄×D₄	—	53	1	no	—
88	#117	—	53	1	no	—
89	#117	—	53	2	no	—
90	#117	D₅₃	1	1	yes	characteristic, complemented
91	C₂×Q₈	—	53	2	no	—
92	C₂×D₄	—	53	1	no	—
93	C₂×D₄	—	53	2	no	—
94	C₂×D₄	—	53	1	no	—
95	C₂×D₄	—	53	2	no	—
96	C₂²×C₄	—	53	2	no	—
97	C₂²×C₄	—	53	1	no	—
98	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
99	C₄⋊C₄	—	53	1	no	—
100	C₄⋊C₄	—	53	2	no	—
101	C₄⋊C₄	—	53	1	no	—
102	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
103	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
104	C₂²⋊C₄	—	53	1	no	—
105	C₂²⋊C₄	—	53	2	no	—
106	C₂²⋊C₄	—	53	2	no	—
107	C₂²⋊C₄	—	53	2	no	—
108	C₂²⋊C₄	—	53	2	no	—
109	C₂²⋊C₄	—	53	1	no	—
110	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
111	C₄²	—	53	2	no	—
112	C₄²	—	53	1	no	—
113	C₄²	C₂×D₅₃	1	2	yes	—
114	C₂³	—	53	2	no	—
115	C₂³	—	53	1	no	—
116	C₂³	C₂²×D₅₃	1	1	yes	characteristic
117	Q₈	—	212	2	no	—
118	D₄	—	106	2	no	—
119	D₄	—	106	2	no	—
120	D₄	—	212	2	no	—
121	D₄	—	212	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	53	1	no	—
124	C₂×C₄	—	53	1	no	—
125	C₂×C₄	—	53	1	no	—
126	C₂×C₄	—	53	2	no	—
127	C₂×C₄	—	106	4	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	C₂²×D₅₃	1	2	yes	—
130	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₅₃	1	2	yes	—
132	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	106	2	no	—
137	C₂²	—	106	1	no	—
138	C₂²	—	106	1	no	—
139	C₂²	—	106	2	no	—
140	C₂²	—	106	1	no	—
141	C₂²	—	106	2	no	—
142	C₂²	C₂³×D₅₃	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	2	no	—
146	C₄	—	53	2	no	—
147	C₄	—	106	1	no	—
148	C₄	—	106	2	no	—
149	C₄	—	106	1	no	—
150	C₄	#11719945	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	—	212	2	no	—
153	C₂	—	212	1	no	—
154	C₂	#41×D₅₃	1	1	yes	characteristic
155	C₂	C₂×#11719945	1	1	yes	characteristic
156	C₂	#13133263	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#117×C₅₃	C₂
hypercenter	#117	D₅₃
hypocenter	C₅₃	#555
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #13133216	C₂	semidirect
2. #13133205	C₂	semidirect
3. #13133203	C₂	semidirect
4. #13133192	C₂	semidirect
5. #13133191	C₂	semidirect
6. #13133190	C₂	semidirect
7. #13133179	C₂	semidirect
8. C₄×C₂²⋊D₅₃	C₂	semidirect
9. #13133175	C₂	semidirect
10. #13133174	C₂	semidirect
11. #117×C₅₃	C₂	semidirect
79. C₅₃	#555	semidirect
90. #117	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133216	2
#13133205	2
#13133203	2
#13133192	2
#13133191	2
#13133190	2
#13133179	2
C₄×C₂²⋊D₅₃	2
#13133175	2
#13133174	2
#117×C₅₃	2
#555	53