Subgroups — #28328426

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291131	C₂	1	4	yes	maximal, complemented
2	#20291129	C₂	1	4	yes	maximal, complemented
3	#20291052	C₂	1	2	yes	maximal
4	#20291047	C₂	1	1	yes	maximal, characteristic
5	#20290926	C₂	1	2	yes	maximal
6	C₂×C₄×#11719910	C₂	1	1	yes	maximal, characteristic
7	#20283645	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133291	C₂²	1	2	yes	—
9	C₂²×#11719910	C₂²	1	1	yes	characteristic
10	C₂²×#11719910	C₂²	1	1	yes	characteristic
11	#16811544	C₂²	1	8	yes	—
12	C₄⋊#11719910	C₂²	1	4	yes	—
13	#16811512	C₂²	1	4	yes	—
14	#16811509	C₂²	1	4	yes	—
15	C₄×#11719910	C₂²	1	2	yes	—
16	C₄×#11719910	C₂²	1	2	yes	—
17	#16811180	C₄	1	1	yes	characteristic
18	#16810750	C₄	1	4	yes	—
19	#16810747	C₂²	1	4	yes	—
20	#16810651	C₄	1	2	yes	—
21	#16810581	C₂²	1	2	yes	—
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₅₃×#423	C₄	1	1	yes	characteristic
24	#13133291	—	2	4	no	—
25	C₂×#11719910	—	2	2	no	—
26	C₂×#11719910	Q₈	1	4	yes	complemented
27	C₂×#11719910	C₂³	1	4	yes	—
28	C₂×#11719910	C₂³	1	2	yes	—
29	C₂×#11719910	C₂³	1	1	yes	characteristic
30	C₂×#11719910	C₂³	1	1	yes	characteristic
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂³	1	1	yes	characteristic
33	#13133217	C₈	1	2	yes	—
34	#13133209	C₈	1	2	yes	—
35	#13133196	C₈	1	4	yes	—
36	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
37	#13133176	C₂×C₄	1	2	yes	—
38	#13133146	C₂×C₄	1	8	yes	—
39	#13133127	C₂×C₄	1	4	yes	—
40	C₄×#88322	C₂×C₄	1	2	yes	—
41	C₄×#88322	C₂³	1	2	yes	—
42	C₈×Dic₅₃	C₂³	1	4	yes	—
43	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
49	M₁₆×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
51	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
55	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
56	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
57	#11719919	C₂×C₈	1	8	yes	—
58	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
59	C₂×#88322	—	2	2	no	—
60	C₂×#88322	C₂²×C₄	1	2	yes	—
61	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
62	C₂×#88322	C₂×Q₈	1	2	yes	—
63	C₂×#88322	#41	1	2	yes	—
64	C₂×#88322	C₂⁴	1	1	yes	characteristic
65	#11719910	—	2	2	no	—
66	#11719910	—	2	4	no	—
67	#11719910	C₂×Q₈	1	2	yes	—
68	#11719910	C₂×Q₈	1	2	yes	—
69	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₅₃	#131	1	2	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
78	Dic₁₀₆	—	2	8	no	—
79	C₈×C₅₃	—	2	4	no	—
80	#88322	—	2	2	no	—
81	#88322	C₄×Q₈	1	2	yes	—
82	#88322	C₂²×Q₈	1	1	yes	characteristic
83	#88322	C₂×#41	1	1	yes	characteristic
84	#33474	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	2	no	—
86	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
87	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	C₈×Q₈	1	2	yes	—
90	C₄×C₅₃	#504	1	4	yes	—
91	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
93	Dic₅₃	—	2	8	no	—
94	#2960	—	53	1	no	—
95	#2183	—	53	4	no	—
96	Q₈×C₁₆	—	53	4	no	—
97	#2151	—	53	2	no	—
98	#2150	—	53	1	no	—
99	#2108	—	53	2	no	—
100	C₂×C₄×C₁₆	—	53	1	no	—
101	C₂×C₅₃	—	2	2	no	—
102	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
103	C₂×C₅₃	C₂×#504	1	2	yes	—
104	#533	—	53	1	no	—
105	C₂×M₃₂	—	53	2	no	—
106	C₂²×C₁₆	—	53	1	no	—
107	C₂²×C₁₆	—	53	1	no	—
108	#446	—	53	4	no	—
109	C₈×Q₈	—	53	4	no	—
110	#424	—	53	2	no	—
111	#423	#6157	1	1	yes	characteristic
112	#405	—	53	2	no	—
113	C₂×C₄×C₈	—	53	1	no	—
114	C₄⋊C₁₆	—	53	4	no	—
115	C₄⋊C₁₆	—	53	8	no	—
116	#346	—	53	4	no	—
117	C₄×C₁₆	—	53	2	no	—
118	C₄×C₁₆	—	53	2	no	—
119	C₄×C₁₆	—	53	4	no	—
120	C₅₃	#33474	1	1	yes	characteristic, complemented, Sylow
121	C₂×M₁₆	C₂×#6157	1	2	yes	—
122	C₂²×C₈	—	53	1	no	—
123	C₂²×C₈	—	53	1	no	—
124	C₄⋊Q₈	—	53	2	no	—
125	#125	—	53	2	no	—
126	C₂²⋊Q₈	—	53	4	no	—
127	C₄×Q₈	—	53	4	no	—
128	#117	—	53	2	no	—
129	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
130	M₃₂	—	106	4	no	—
131	C₂×C₁₆	—	53	1	no	—
132	C₂×C₁₆	—	53	1	no	—
133	C₂×C₁₆	—	53	4	no	—
134	C₂×C₁₆	—	53	2	no	—
135	C₂×C₁₆	—	53	4	no	—
136	C₂×C₁₆	—	106	2	no	—
137	C₄⋊C₈	—	53	8	no	—
138	C₄⋊C₈	C₂×#6157	1	4	yes	—
139	#97	—	53	4	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	53	2	no	—
142	C₄×C₈	—	53	2	no	—
143	C₂×Q₈	—	53	4	no	—
144	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
145	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	—	53	1	no	—
150	C₂×C₈	—	53	1	no	—
151	C₂×C₈	—	53	2	no	—
152	C₂×C₈	—	106	2	no	—
153	C₂×C₈	C₂²×#6157	1	4	yes	—
154	C₄⋊C₄	—	53	4	no	—
155	C₄⋊C₄	—	53	8	no	—
156	C₄⋊C₄	—	53	4	no	—
157	C₂²⋊C₄	—	53	4	no	—
158	C₄²	—	53	4	no	—
159	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
160	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
161	C₄²	C₂²×#6157	1	2	yes	—
162	C₁₆	—	53	2	no	—
163	C₁₆	—	53	2	no	—
164	C₁₆	—	106	2	no	—
165	C₁₆	—	106	4	no	—
166	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
167	Q₈	—	106	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	53	8	no	—
170	C₂×C₄	Q₈×#6157	1	2	yes	—
171	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
172	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
173	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
174	C₂×C₄	#13133304	1	2	yes	—
175	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
176	C₈	—	2	4	no	—
177	C₈	—	53	2	no	—
178	C₈	—	53	1	no	—
179	C₈	—	53	1	no	—
180	C₈	—	106	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	#16811672	1	2	yes	—
183	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	106	8	no	—
186	C₄	#16811672	1	2	yes	—
187	C₄	#16811682	1	4	yes	—
188	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
189	C₄	C₂×#13133304	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×#16811672	1	1	yes	characteristic
192	C₂	C₂×#16811682	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#423	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#423	#6157
hypocenter	C₅₃	#33474
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. #20291131	C₂	semidirect
2. #20291129	C₂	semidirect
26. C₂×#11719910	Q₈	semidirect
120. C₅₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#20291131	2
#20291129	2
#20291052	2
#20291047	2
#20290926	2
C₂×C₄×#11719910	2
#20283645	2
#33474	53