Subgroups — #44558973

Order 32128.

440 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632453	C₂	1	2	yes	maximal
2	C₂×#22548721	C₂	1	1	yes	maximal, characteristic
3	#30631735	C₂	1	2	yes	maximal, complemented
4	#30631734	C₂	1	4	yes	maximal, complemented
5	C₂×#22548651	C₂	1	1	yes	maximal, characteristic, complemented
6	#108×D₂₅₁	C₂	1	4	yes	maximal, complemented
7	C₂×#108×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16860810	C₂²	1	1	yes	characteristic
9	#22548723	—	2	4	no	—
10	#22548721	—	2	4	no	—
11	#22548721	C₄	1	4	yes	—
12	#22548721	C₄	1	4	yes	—
13	C₂×#16860794	C₂²	1	2	yes	—
14	C₂×#16860794	C₂²	1	1	yes	characteristic
15	C₂×#16860793	C₂²	1	2	yes	—
16	M₁₆×D₂₅₁	C₂²	1	8	yes	—
17	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
18	#22548651	C₂²	1	2	yes	—
19	#22548651	C₂²	1	2	yes	—
20	#22548650	C₂²	1	8	yes	—
21	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
22	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
23	#108×C₂₅₁	C₂²	1	2	yes	—
24	#108×C₂₅₁	C₂²	1	2	yes	—
25	#16860810	C₂×C₄	1	2	yes	—
26	#16860810	C₂×C₄	1	2	yes	—
27	#16860810	D₄	1	2	yes	—
28	#16860810	D₄	1	2	yes	—
29	C₂×Dic₅₀₂	Q₈	1	1	yes	characteristic
30	C₂×D₁₀₀₄	Q₈	1	1	yes	characteristic
31	C₂×C₂²⋊D₂₅₁	Q₈	1	2	yes	—
32	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
33	#16860794	—	2	2	no	—
34	#16860794	C₂×C₄	1	2	yes	—
35	#16860794	C₂×C₄	1	2	yes	—
36	#16860794	C₂³	1	4	yes	—
37	#16860793	—	2	8	no	—
38	#16860793	C₂×C₄	1	8	yes	—
39	C₂×C₂₅₁⋊C₈	—	2	2	no	—
40	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
41	C₈×D₂₅₁	—	2	8	no	—
42	C₈×D₂₅₁	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
44	M₁₆×C₂₅₁	—	2	2	no	—
45	M₁₆×C₂₅₁	C₂³	1	4	yes	—
46	C₂×C₈×C₂₅₁	—	2	2	no	—
47	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
48	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
49	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
51	C₂²×D₂₅₁	C₂×Q₈	1	2	yes	—
52	Dic₅₀₂	C₄⋊C₄	1	2	yes	—
53	Dic₅₀₂	C₄⋊C₄	1	2	yes	—
54	D₁₀₀₄	C₄⋊C₄	1	2	yes	—
55	D₁₀₀₄	C₄⋊C₄	1	2	yes	—
56	C₂²⋊D₂₅₁	C₄⋊C₄	1	4	yes	—
57	C₂²⋊D₂₅₁	C₄⋊C₄	1	4	yes	—
58	C₂×Dic₂₅₁	C₂×Q₈	1	2	yes	—
59	C₄×D₂₅₁	C₂²×C₄	1	4	yes	—
60	C₄×D₂₅₁	C₂×D₄	1	4	yes	—
61	C₂₅₁⋊C₈	—	2	4	no	—
62	C₂₅₁⋊C₈	C₂²×C₄	1	4	yes	—
63	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
70	C₈×C₂₅₁	—	2	4	no	—
71	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
72	C₈×C₂₅₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₂₅₁	—	4	1	no	—
74	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
77	C₂×D₂₅₁	—	4	2	no	—
78	C₂×D₂₅₁	C₂×C₄⋊C₄	1	4	yes	—
79	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
82	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
83	Dic₂₅₁	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₂₅₁	—	2	1	no	—
85	C₂×C₂₅₁	—	2	1	no	—
86	C₂×C₂₅₁	—	2	1	no	—
87	C₂×C₂₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	D₂₅₁	—	2	4	no	—
89	C₂₅₁	#2916	1	1	yes	characteristic, complemented, Sylow
90	#2916	—	251	1	no	maximal, Sylow
91	#568	—	251	2	no	—
92	C₂×#131	—	251	1	no	—
93	#430	—	251	2	no	—
94	#430	—	251	4	no	—
95	C₂×#108	—	251	1	no	—
96	C₂×#108	—	251	4	no	—
97	C₂×#108	D₂₅₁	1	1	yes	characteristic, complemented
98	C₂×#41	—	251	1	no	—
99	#131	—	502	4	no	—
100	#131	—	502	4	no	—
101	#131	—	251	4	no	—
102	#131	—	251	4	no	—
103	C₂×M₁₆	—	251	2	no	—
104	C₂×M₁₆	—	251	8	no	—
105	C₂×M₁₆	—	251	1	no	—
106	C₂×M₁₆	—	251	2	no	—
107	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
108	C₂²×C₈	—	251	2	no	—
109	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
110	#108	—	251	8	no	—
111	#108	—	251	2	no	—
112	#108	—	251	2	no	—
113	#108	C₂×D₂₅₁	1	2	yes	—
114	#108	C₂×D₂₅₁	1	2	yes	—
115	#41	—	251	2	no	—
116	#41	—	251	2	no	—
117	#41	—	251	2	no	—
118	#41	—	251	2	no	—
119	C₂×Q₈	—	251	1	no	—
120	C₂×D₄	—	251	2	no	—
121	C₂×D₄	—	251	1	no	—
122	C₂²×C₄	—	251	2	no	—
123	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	502	8	no	—
126	M₁₆	—	502	2	no	—
127	M₁₆	—	251	4	no	—
128	M₁₆	—	251	2	no	—
129	M₁₆	—	251	2	no	—
130	M₁₆	—	251	8	no	—
131	M₁₆	C₂²×D₂₅₁	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	502	8	no	—
134	C₂×C₈	—	502	2	no	—
135	C₂×C₈	—	251	2	no	—
136	C₂×C₈	—	251	8	no	—
137	C₂×C₈	C₄×D₂₅₁	1	2	yes	—
138	C₂×C₈	C₄×D₂₅₁	1	2	yes	—
139	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
141	C₂³	—	251	2	no	—
142	C₂³	Q₈×D₂₅₁	1	1	yes	characteristic
143	Q₈	—	251	2	no	—
144	Q₈	—	251	2	no	—
145	D₄	—	251	4	no	—
146	D₄	—	251	4	no	—
147	D₄	—	251	2	no	—
148	D₄	—	251	2	no	—
149	C₂×C₄	—	251	2	no	—
150	C₂×C₄	—	251	4	no	—
151	C₂×C₄	—	251	4	no	—
152	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
154	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
155	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₂₅₁	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	502	4	no	—
160	C₈	—	251	4	no	—
161	C₈	C₂×C₄×D₂₅₁	1	2	yes	—
162	C₈	C₂×C₄×D₂₅₁	1	2	yes	—
163	C₂²	—	4	1	no	—
164	C₂²	—	1004	2	no	—
165	C₂²	—	251	4	no	—
166	C₂²	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
167	C₂²	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
169	C₄	—	251	4	no	—
170	C₄	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
171	C₄	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
172	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
173	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	502	4	no	—
178	C₂	C₂×C₄⋊C₄×D₂₅₁	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	C₂²×C₄⋊C₄
prime radical	C₂×#16860810	C₂²
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#108	D₂₅₁
hypocenter	C₂₅₁	#2916
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
3. #30631735	C₂	semidirect
4. #30631734	C₂	semidirect
5. C₂×#22548651	C₂	semidirect
6. #108×D₂₅₁	C₂	semidirect
7. C₂×#108×C₂₅₁	C₂	semidirect
89. C₂₅₁	#2916	semidirect
97. C₂×#108	D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632453	2
C₂×#22548721	2
#30631735	2
#30631734	2
C₂×#22548651	2
#108×D₂₅₁	2
C₂×#108×C₂₅₁	2
#2916	251