Subgroups — #44560891

Order 32128.

340 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632315	C₂	1	1	yes	maximal, characteristic
2	#30632097	C₂	1	2	yes	maximal, complemented
3	#30632093	C₂	1	2	yes	maximal, complemented
4	#30632086	C₂	1	2	yes	maximal, complemented
5	#30632083	C₂	1	2	yes	maximal, complemented
6	#30631730	C₂	1	1	yes	maximal, characteristic
7	#30631727	C₂	1	1	yes	maximal, characteristic
8	#30631723	C₂	1	1	yes	maximal, characteristic
9	C₂×#22548646	C₂	1	1	yes	maximal, characteristic
10	#30631643	C₂	1	1	yes	maximal, characteristic
11	#544×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#16860794	C₂²	1	1	yes	characteristic
13	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
14	#22548715	C₂²	1	2	yes	—
15	#22548714	—	2	1	no	—
16	#22548707	C₂²	1	2	yes	—
17	#22548706	—	2	1	no	—
18	#22548692	—	2	2	no	—
19	#22548674	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
21	#22548648	C₂²	1	1	yes	characteristic
22	#22548648	C₂²	1	1	yes	characteristic
23	#22548648	C₂²	1	2	yes	—
24	#22548648	C₂²	1	2	yes	—
25	#22548646	C₂²	1	2	yes	—
26	#22548646	C₂²	1	2	yes	—
27	#22548646	C₂²	1	1	yes	characteristic
28	#22548646	C₂²	1	1	yes	characteristic
29	#22548646	C₂²	1	2	yes	—
30	#22548646	C₂²	1	2	yes	—
31	#22548626	C₂²	1	2	yes	—
32	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₂₅₁	—	2	1	no	—
34	C₄⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
35	#125×C₂₅₁	—	2	1	no	—
36	#125×C₂₅₁	C₂²	1	2	yes	—
37	C₂²⋊Q₈×C₂₅₁	—	2	2	no	—
38	#117×C₂₅₁	C₂²	1	1	yes	characteristic
39	#117×C₂₅₁	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₅₀₂	—	2	2	no	—
42	C₂²×Dic₂₅₁	—	2	1	no	—
43	#16860794	Q₈	1	4	yes	complemented
44	C₂×C₂₅₁⋊C₈	Q₈	1	4	yes	complemented
45	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
47	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
48	C₄⋊Dic₂₅₁	—	2	1	no	—
49	C₄⋊Dic₂₅₁	—	2	2	no	—
50	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
51	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
53	Dic₂₅₁⋊C₄	—	2	2	no	—
54	Dic₂₅₁⋊C₄	—	2	2	no	—
55	C₂²⋊Dic₂₅₁	—	2	1	no	—
56	C₄×Dic₂₅₁	—	2	1	no	—
57	C₂×Q₈×C₂₅₁	—	2	2	no	—
58	C₂²×C₄×C₂₅₁	—	2	1	no	—
59	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₅₁	—	2	1	no	—
61	C₄⋊C₄×C₂₅₁	—	2	2	no	—
62	C₄⋊C₄×C₂₅₁	—	2	2	no	—
63	C₄⋊C₄×C₂₅₁	—	2	2	no	—
64	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₅₁	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
70	C₂²⋊C₄×C₂₅₁	D₄	1	2	yes	—
71	C₄²×C₂₅₁	—	2	1	no	—
72	C₄²×C₂₅₁	C₂³	1	2	yes	—
73	Dic₅₀₂	—	8	2	no	—
74	C₂×Dic₂₅₁	—	2	1	no	—
75	C₂×Dic₂₅₁	—	2	1	no	—
76	C₂×Dic₂₅₁	—	2	2	no	—
77	C₂×Dic₂₅₁	—	8	1	no	—
78	C₂₅₁⋊C₈	C₂×Q₈	1	2	yes	—
79	C₂₅₁⋊C₈	C₂×Q₈	1	2	yes	—
80	C₂₅₁⋊C₈	C₂×Q₈	1	4	yes	—
81	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₂₅₁	—	8	2	no	—
83	C₂×C₄×C₂₅₁	—	2	1	no	—
84	C₂×C₄×C₂₅₁	—	2	1	no	—
85	C₂×C₄×C₂₅₁	—	2	2	no	—
86	C₂×C₄×C₂₅₁	—	8	1	no	—
87	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
93	C₂²×C₂₅₁	—	2	1	no	—
94	C₂²×C₂₅₁	—	2	1	no	—
95	C₂²×C₂₅₁	C₄⋊Q₈	1	2	yes	—
96	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₂₅₁	—	2	2	no	—
98	C₄×C₂₅₁	—	2	2	no	—
99	C₄×C₂₅₁	—	4	1	no	—
100	C₄×C₂₅₁	—	4	2	no	—
101	C₄×C₂₅₁	—	4	1	no	—
102	C₄×C₂₅₁	C₄⋊Q₈	1	2	yes	—
103	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
104	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
105	Dic₂₅₁	—	4	1	no	—
106	Dic₂₅₁	—	4	2	no	—
107	Dic₂₅₁	—	4	1	no	—
108	C₂×C₂₅₁	—	2	2	no	—
109	C₂×C₂₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₂×C₂₅₁	#577	1	1	yes	characteristic
111	C₂×C₂₅₁	#577	1	1	yes	characteristic
112	C₂₅₁	#3163	1	1	yes	characteristic, complemented, Sylow
113	#3163	—	251	1	no	maximal, Sylow
114	#544	—	251	1	no	—
115	#544	D₂₅₁	1	1	yes	characteristic
116	#501	—	251	2	no	—
117	#501	—	251	2	no	—
118	#499	—	251	2	no	—
119	#498	—	251	2	no	—
120	#428	—	251	1	no	—
121	#428	—	251	1	no	—
122	#427	—	251	1	no	—
123	C₂×#106	—	251	1	no	—
124	#405	—	251	1	no	—
125	C₂×M₁₆	—	251	1	no	—
126	C₂²×C₈	—	251	1	no	—
127	C₄⋊Q₈	—	2	1	no	—
128	C₄⋊Q₈	—	502	1	no	—
129	C₄⋊Q₈	—	251	2	no	—
130	C₄⋊Q₈	C₂×D₂₅₁	1	2	yes	—
131	#125	—	2	1	no	—
132	#125	—	502	1	no	—
133	#125	—	251	2	no	—
134	#125	C₂×D₂₅₁	1	2	yes	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₂²⋊Q₈	—	502	2	no	—
137	#117	—	251	1	no	—
138	#117	C₂×D₂₅₁	1	1	yes	characteristic
139	#117	C₂×D₂₅₁	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	251	1	no	—
141	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
142	#107	—	251	1	no	—
143	#107	—	251	1	no	—
144	#107	—	251	2	no	—
145	#107	—	251	2	no	—
146	#106	—	251	2	no	—
147	#106	—	251	2	no	—
148	#106	—	251	1	no	—
149	#106	—	251	1	no	—
150	#106	—	251	2	no	—
151	#106	—	251	2	no	—
152	#97	—	251	2	no	—
153	C₄×C₈	—	251	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	502	2	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	502	1	no	—
158	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
159	M₁₆	—	251	4	no	—
160	C₂×C₈	—	251	4	no	—
161	C₂×C₈	—	251	1	no	—
162	C₂×C₈	—	251	2	no	—
163	C₂×C₈	—	251	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	502	1	no	—
169	C₄⋊C₄	—	502	2	no	—
170	C₄⋊C₄	—	502	2	no	—
171	C₄⋊C₄	—	502	2	no	—
172	C₄⋊C₄	—	251	2	no	—
173	C₄⋊C₄	—	251	1	no	—
174	C₄⋊C₄	—	251	1	no	—
175	C₄⋊C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
176	C₄⋊C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
177	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
178	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
179	C₄⋊C₄	C₂²×D₂₅₁	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	502	1	no	—
182	C₂²⋊C₄	C₂²⋊D₂₅₁	1	2	yes	—
183	C₄²	—	2	1	no	—
184	C₄²	—	502	1	no	—
185	C₄²	C₂²×D₂₅₁	1	2	yes	—
186	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
187	Q₈	—	8	2	no	—
188	Q₈	—	2008	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	2008	1	no	—
194	C₂×C₄	—	502	1	no	—
195	C₂×C₄	—	502	1	no	—
196	C₂×C₄	—	502	2	no	—
197	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
200	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
201	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
202	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
203	C₈	—	251	2	no	—
204	C₈	—	251	2	no	—
205	C₈	—	251	4	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#22548713	1	2	yes	—
209	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	1	no	—
215	C₄	—	1004	2	no	—
216	C₄	—	1004	1	no	—
217	C₄	—	1004	1	no	—
218	C₄	#22548713	1	2	yes	—
219	C₄	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
220	C₄	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#22548713	1	1	yes	characteristic
223	C₂	#30632495	1	1	yes	characteristic
224	C₂	#30632498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₅₁
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#544×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂³×C₂₅₁	C₂×D₄
hypercenter	#544	D₂₅₁
hypocenter	C₂₅₁	#3163
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
2. #30632097	C₂	semidirect
3. #30632093	C₂	semidirect
4. #30632086	C₂	semidirect
5. #30632083	C₂	semidirect
43. #16860794	Q₈	semidirect
44. C₂×C₂₅₁⋊C₈	Q₈	semidirect
112. C₂₅₁	#3163	semidirect

Maximal subgroups

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

subgroup	index
#30632315	2
#30632097	2
#30632093	2
#30632086	2
#30632083	2
#30631730	2
#30631727	2
#30631723	2
C₂×#22548646	2
#30631643	2
#544×C₂₅₁	2
#3163	251