Subgroups — #44559281

Order 32128.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548737	C₂	1	1	yes	maximal, characteristic
2	#30632453	C₂	1	2	yes	maximal, complemented
3	#30631808	C₂	1	4	yes	maximal, complemented
4	#30631803	C₂	1	4	yes	maximal, complemented
5	#30631703	C₂	1	2	yes	maximal, complemented
6	#30631642	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#108×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16860810	C₂²	1	2	yes	—
9	#22548737	C₄	1	2	yes	complemented
10	#22548737	C₄	1	2	yes	complemented
11	#22548737	C₂²	1	2	yes	—
12	#22548737	C₂²	1	2	yes	—
13	C₂×Dic₁₀₀₄	C₄	1	1	yes	characteristic, complemented
14	C₂×#16860799	C₄	1	2	yes	complemented
15	C₂×D₂₀₀₈	C₄	1	1	yes	characteristic, complemented
16	#22548723	C₂²	1	8	yes	complemented
17	#22548721	—	2	4	no	—
18	C₂×#16860794	C₂²	1	1	yes	characteristic
19	M₁₆×D₂₅₁	—	2	4	no	—
20	#22548674	C₂²	1	1	yes	characteristic
21	#22548641	C₂²	1	8	yes	complemented
22	#22548625	C₂²	1	2	yes	complemented
23	C₈×Dic₂₅₁	C₂²	1	2	yes	complemented
24	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
25	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
26	#108×C₂₅₁	C₂²	1	2	yes	complemented
27	#108×C₂₅₁	C₂²	1	2	yes	complemented
28	#16860810	—	2	2	no	—
29	#16860810	—	2	2	no	—
30	#16860810	C₂×C₄	1	4	yes	—
31	#16860810	C₂³	1	4	yes	—
32	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
33	C₂×D₁₀₀₄	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊D₂₅₁	—	2	2	no	—
35	C₂×C₄×D₂₅₁	—	2	2	no	—
36	Dic₁₀₀₄	C₂×C₄	1	2	yes	—
37	Dic₁₀₀₄	C₂×C₄	1	2	yes	—
38	#16860799	C₂×C₄	1	4	yes	—
39	#16860799	C₂×C₄	1	4	yes	—
40	D₂₀₀₈	C₂×C₄	1	2	yes	—
41	D₂₀₀₈	C₂×C₄	1	2	yes	—
42	#16860794	D₄	1	2	yes	—
43	#16860794	D₄	1	2	yes	—
44	#16860793	—	2	8	no	—
45	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
46	C₈×D₂₅₁	—	2	8	no	—
47	C₄⋊Dic₂₅₁	—	2	1	no	—
48	C₂²⋊Dic₂₅₁	—	2	1	no	—
49	C₄×Dic₂₅₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
51	M₁₆×C₂₅₁	—	2	2	no	—
52	M₁₆×C₂₅₁	C₂³	1	4	yes	—
53	C₂×C₈×C₂₅₁	—	2	2	no	—
54	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
60	C₂²×D₂₅₁	—	2	2	no	—
61	Dic₅₀₂	—	2	2	no	—
62	Dic₅₀₂	C₂²×C₄	1	4	yes	—
63	D₁₀₀₄	—	2	2	no	—
64	D₁₀₀₄	C₂²×C₄	1	4	yes	—
65	C₂²⋊D₂₅₁	—	2	4	no	—
66	C₂²⋊D₂₅₁	—	2	4	no	—
67	C₂×Dic₂₅₁	—	2	2	no	—
68	C₂×Dic₂₅₁	—	2	2	no	—
69	C₄×D₂₅₁	—	2	4	no	—
70	C₄×D₂₅₁	—	2	4	no	—
71	C₂₅₁⋊C₈	C₂×D₄	1	4	yes	—
72	C₂³×C₂₅₁	#41	1	1	yes	characteristic
73	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
78	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
79	C₈×C₂₅₁	—	2	4	no	—
80	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
83	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₂₅₁	—	4	1	no	—
85	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₂₅₁	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
88	C₂×D₂₅₁	—	2	4	no	—
89	C₂×D₂₅₁	—	8	2	no	—
90	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
91	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
92	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
94	Dic₂₅₁	—	2	4	no	—
95	Dic₂₅₁	—	4	2	no	—
96	C₂×C₂₅₁	—	2	1	no	—
97	C₂×C₂₅₁	—	2	1	no	—
98	C₂×C₂₅₁	—	2	1	no	—
99	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
100	D₂₅₁	—	4	4	no	—
101	C₂₅₁	#2957	1	1	yes	characteristic, complemented, Sylow
102	#2957	—	251	1	no	maximal, Sylow
103	C₂×#135	—	251	1	no	—
104	#568	—	251	2	no	—
105	#444	—	251	4	no	—
106	#443	—	251	4	no	—
107	C₂×#108	D₂₅₁	1	1	yes	characteristic, complemented
108	#421	—	251	2	no	—
109	#405	—	251	1	no	—
110	C₂×#41	—	251	2	no	—
111	#135	—	251	2	no	—
112	#135	—	251	2	no	—
113	#135	—	251	2	no	—
114	#135	—	251	2	no	—
115	C₂×Q₁₆	—	251	1	no	—
116	C₂×SD₁₆	—	251	2	no	—
117	C₂×D₈	—	251	1	no	—
118	#131	—	502	4	no	—
119	#131	—	251	8	no	—
120	C₂×M₁₆	—	502	4	no	—
121	C₂×M₁₆	—	251	1	no	—
122	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
123	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
124	#117	—	251	1	no	—
125	#108	C₂×D₂₅₁	1	2	yes	complemented
126	#108	C₂×D₂₅₁	1	2	yes	complemented
127	#104	—	251	8	no	—
128	#97	—	251	2	no	—
129	C₄×C₈	—	251	2	no	—
130	#41	—	502	2	no	—
131	#41	—	502	2	no	—
132	#41	—	251	4	no	—
133	#41	—	251	4	no	—
134	C₂×Q₈	—	251	2	no	—
135	C₂×D₄	—	502	2	no	—
136	C₂×D₄	—	251	2	no	—
137	C₂²×C₄	—	502	2	no	—
138	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
139	Q₁₆	—	251	2	no	—
140	Q₁₆	—	251	2	no	—
141	SD₁₆	—	251	4	no	—
142	SD₁₆	—	251	4	no	—
143	D₈	—	251	2	no	—
144	D₈	—	251	2	no	—
145	M₁₆	—	2	2	no	—
146	M₁₆	—	502	8	no	—
147	M₁₆	—	251	2	no	—
148	M₁₆	—	251	2	no	—
149	M₁₆	C₂²×D₂₅₁	1	4	yes	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	502	8	no	—
152	C₂×C₈	—	251	2	no	—
153	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
154	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
155	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
156	C₂×C₈	C₄×D₂₅₁	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
159	C₄⋊C₄	—	502	1	no	—
160	C₂²⋊C₄	—	502	1	no	—
161	C₄²	—	251	2	no	—
162	C₂³	—	502	2	no	—
163	C₂³	#16860811	1	1	yes	characteristic
164	Q₈	—	502	2	no	—
165	Q₈	—	251	4	no	—
166	D₄	—	502	4	no	—
167	D₄	—	502	4	no	—
168	D₄	—	502	2	no	—
169	D₄	—	251	4	no	—
170	C₂×C₄	—	502	2	no	—
171	C₂×C₄	—	502	2	no	—
172	C₂×C₄	—	502	4	no	—
173	C₂×C₄	—	502	4	no	—
174	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
176	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
177	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
178	C₂×C₄	#16860811	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₈	—	251	4	no	—
182	C₈	C₂×C₄×D₂₅₁	1	1	yes	characteristic
183	C₈	C₂×C₄×D₂₅₁	1	1	yes	characteristic
184	C₈	C₂×C₄×D₂₅₁	1	1	yes	characteristic
185	C₈	C₂×C₄×D₂₅₁	1	1	yes	characteristic
186	C₂²	—	4	1	no	—
187	C₂²	—	2008	2	no	—
188	C₂²	—	502	4	no	—
189	C₂²	#22548677	1	1	yes	characteristic
190	C₂²	#22548677	1	1	yes	characteristic
191	C₂²	C₂×#16860811	1	1	yes	characteristic
192	C₄	—	1004	2	no	—
193	C₄	—	502	4	no	—
194	C₄	#22548677	1	1	yes	characteristic
195	C₄	#22548677	1	1	yes	characteristic
196	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
197	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	1004	4	no	—
202	C₂	C₂×#22548677	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×#108×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂×C₄×D₄
prime radical	C₂×#22548737	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#108	D₂₅₁
hypocenter	C₂₅₁	#2957
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
2. #30632453	C₂	semidirect
3. #30631808	C₂	semidirect
4. #30631803	C₂	semidirect
5. #30631703	C₂	semidirect
6. #30631642	C₂	semidirect
7. C₂×#108×C₂₅₁	C₂	semidirect
9. #22548737	C₄	semidirect
10. #22548737	C₄	semidirect
13. C₂×Dic₁₀₀₄	C₄	semidirect
14. C₂×#16860799	C₄	semidirect
15. C₂×D₂₀₀₈	C₄	semidirect
16. #22548723	C₂²	semidirect
21. #22548641	C₂²	semidirect
22. #22548625	C₂²	semidirect
23. C₈×Dic₂₅₁	C₂²	semidirect
26. #108×C₂₅₁	C₂²	semidirect
27. #108×C₂₅₁	C₂²	semidirect
101. C₂₅₁	#2957	semidirect
107. C₂×#108	D₂₅₁	semidirect
125. #108	C₂×D₂₅₁	semidirect
126. #108	C₂×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548737	2
#30632453	2
#30631808	2
#30631803	2
#30631703	2
#30631642	2
C₂×#108×C₂₅₁	2
#2957	251