Subgroups — #28312352

Order 13568.

290 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811555	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289832	C₂	1	4	yes	maximal, complemented
4	C₂×#16811040	C₂	1	1	yes	maximal, characteristic
5	#16811555	—	2	1	no	—
6	#16811555	—	2	1	no	—
7	#16811555	—	2	1	no	—
8	#16811555	C₂²	1	2	yes	complemented
9	C₂×#13133267	—	2	1	no	—
10	C₂×#13133267	C₂²	1	2	yes	complemented
11	#16811040	—	2	1	no	—
12	#16811040	—	2	1	no	—
13	#16811040	C₄	1	2	yes	—
14	#16811040	C₂²	1	2	yes	—
15	C₂×#13133157	—	2	1	no	—
16	C₂×#13133157	C₄	1	1	yes	characteristic
17	C₂²×#11719908	—	2	1	no	—
18	C₂²×#11719908	C₂²	1	1	yes	characteristic
19	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
20	#13133267	—	2	2	no	—
21	#13133267	—	2	2	no	—
22	#13133157	—	2	1	no	—
23	#13133157	—	2	2	no	—
24	#13133157	—	2	2	no	—
25	#13133157	C₂×C₄	1	2	yes	—
26	C₂×#11719908	—	2	1	no	—
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	—	2	2	no	—
29	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719908	D₄	1	1	yes	characteristic
31	C₂×#11719908	D₄	1	1	yes	characteristic
32	C₂×#11719908	D₄	1	1	yes	characteristic
33	C₂×#11719908	D₄	1	1	yes	characteristic
34	C₂×#11719908	C₂³	1	1	yes	characteristic
35	C₂×#41×C₅₃	C₈	1	1	yes	characteristic
36	#131×C₅₃	—	2	1	no	—
37	#131×C₅₃	—	2	1	no	—
38	#131×C₅₃	C₈	1	2	yes	—
39	#131×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×M₁₆×C₅₃	—	2	1	no	—
41	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	#41×C₅₃	—	2	1	no	—
45	#41×C₅₃	—	2	1	no	—
46	#41×C₅₃	C₁₆	1	2	yes	—
47	#41×C₅₃	C₂×C₈	1	2	yes	—
48	C₂×Q₈×C₅₃	C₁₆	1	1	yes	characteristic
49	C₂×D₄×C₅₃	—	2	1	no	—
50	C₂×D₄×C₅₃	C₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	M₁₆×C₅₃	—	2	1	no	—
54	M₁₆×C₅₃	—	2	2	no	—
55	M₁₆×C₅₃	—	2	2	no	—
56	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
64	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	#11719908	—	2	2	no	—
67	#11719908	C₂²⋊C₄	1	1	yes	characteristic
68	#11719908	C₂²⋊C₄	1	1	yes	characteristic
69	#11719908	C₂×D₄	1	1	yes	characteristic
70	#11719908	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
73	Q₈×C₅₃	—	2	1	no	—
74	Q₈×C₅₃	C₂×C₁₆	1	2	yes	—
75	D₄×C₅₃	—	2	2	no	—
76	D₄×C₅₃	—	2	2	no	—
77	D₄×C₅₃	—	2	1	no	—
78	D₄×C₅₃	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	—	2	1	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
91	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	#13558	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	4	1	no	—
96	C₂²×C₅₃	—	4	2	no	—
97	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
98	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
100	C₄×C₅₃	—	2	2	no	—
101	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
103	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
105	C₂×#504	—	53	1	no	—
106	C₂×M₆₄	—	53	1	no	—
107	C₂×M₆₄	—	53	1	no	—
108	#1401	—	53	4	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	—	2	1	no	—
112	C₂×C₅₃	—	4	2	no	—
113	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
114	C₂×#131	#6157	1	1	yes	characteristic
115	#504	—	53	2	no	—
116	#504	—	53	2	no	—
117	#504	—	106	1	no	—
118	#504	—	106	1	no	—
119	C₂×M₃₂	—	53	1	no	—
120	C₂×M₃₂	—	106	1	no	—
121	C₂²×C₁₆	—	53	1	no	—
122	C₂²×C₁₆	—	106	1	no	—
123	M₆₄	—	53	2	no	—
124	M₆₄	—	106	1	no	—
125	M₆₄	—	106	1	no	—
126	M₆₄	—	106	1	no	—
127	C₂×C₃₂	—	53	2	no	—
128	C₂×C₃₂	—	106	1	no	—
129	C₅₃	#13558	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	#88324	1	1	yes	characteristic
131	#131	—	2	1	no	—
132	#131	—	2	1	no	—
133	#131	#88324	1	2	yes	—
134	#131	C₂×#6157	1	2	yes	—
135	C₂×M₁₆	—	2	1	no	—
136	C₂×M₁₆	#88324	1	1	yes	characteristic
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
139	M₃₂	—	53	2	no	—
140	M₃₂	—	106	2	no	—
141	M₃₂	—	106	2	no	—
142	M₃₂	—	106	1	no	—
143	C₂×C₁₆	—	53	1	no	—
144	C₂×C₁₆	—	53	1	no	—
145	C₂×C₁₆	—	53	1	no	—
146	C₂×C₁₆	—	53	1	no	—
147	C₂×C₁₆	—	53	1	no	—
148	C₂×C₁₆	—	53	1	no	—
149	C₂×C₁₆	—	106	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	C₂×C₁₆	—	106	1	no	—
152	C₃₂	—	106	2	no	—
153	C₃₂	—	106	2	no	—
154	#41	—	2	1	no	—
155	#41	—	2	1	no	—
156	#41	#11719910	1	2	yes	—
157	#41	C₂×#88324	1	2	yes	—
158	C₂×Q₈	#11719910	1	1	yes	characteristic
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	#11719910	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
163	M₁₆	—	2	1	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	—	2	2	no	—
166	M₁₆	C₂×#88324	1	2	yes	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂×#88324	1	1	yes	characteristic
171	C₂×C₈	#11719940	1	1	yes	characteristic
172	C₂×C₈	#11719940	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
174	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
175	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
176	C₁₆	—	53	1	no	—
177	C₁₆	—	53	1	no	—
178	C₁₆	—	53	1	no	—
179	C₁₆	—	53	1	no	—
180	C₁₆	—	106	2	no	—
181	C₂³	—	2	1	no	—
182	C₂³	#13133291	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	C₂×#11719910	1	2	yes	—
185	D₄	—	2	2	no	—
186	D₄	—	2	2	no	—
187	D₄	—	2	1	no	—
188	D₄	C₂×#11719910	1	2	yes	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
193	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
194	C₂×C₄	C₂×#11719910	1	1	yes	characteristic
195	C₂×C₄	C₂×#11719910	1	1	yes	characteristic
196	C₂×C₄	#13133291	1	1	yes	characteristic
197	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₂²⋊#88324	1	1	yes	characteristic
200	C₈	C₂²⋊#88324	1	1	yes	characteristic
201	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
202	C₈	C₂×#11719940	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	—	4	1	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²⋊#11719910	1	1	yes	characteristic
207	C₂²	C₂²⋊#11719910	1	1	yes	characteristic
208	C₂²	C₂×#13133291	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	C₂²⋊#11719910	1	1	yes	characteristic
211	C₄	C₂²⋊#11719910	1	1	yes	characteristic
212	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
213	C₄	C₂²×#11719910	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	4	2	no	—
218	C₂	C₂×C₂²⋊#11719910	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊#6157
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂×#131×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#41×C₅₃	C₈
cosocle	C₂×#11719908	C₂³
hypercenter	C₂×#131	#6157
hypocenter	C₅₃	#13558
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#16811555	C₂	semidirect
2. C₂×#16811555	C₂	semidirect
3. #20289832	C₂	semidirect
8. #16811555	C₂²	semidirect
10. C₂×#13133267	C₂²	semidirect
129. C₅₃	#13558	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811555	2
#20289832	2
C₂×#16811040	2
#13558	53