Subgroups — #3529×C₂₅₁ #44548116

Order 32128.

1058 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#563×C₂₅₁	C₂	1	3	yes	maximal, complemented
2	#562×C₂₅₁	C₂	1	6	yes	maximal, complemented
3	H(3,𝔽₄)×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#560×C₂₅₁	C₂	1	6	yes	maximal, complemented
5	#532×C₂₅₁	C₂	1	3	yes	maximal, complemented
6	#529×C₂₅₁	C₂	1	6	yes	maximal, complemented
7	#525×C₂₅₁	C₂	1	6	yes	maximal, complemented
8	C₂×#41×C₂₅₁	C₂²	1	3	yes	—
9	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₄×C₂₅₁	C₂²	1	2	yes	complemented
11	C₄⋊Q₈×C₂₅₁	C₂²	1	3	yes	complemented
12	#127×C₂₅₁	C₂²	1	3	yes	—
13	#126×C₂₅₁	C₂²	1	12	yes	complemented
14	#125×C₂₅₁	C₂²	1	6	yes	complemented
15	#124×C₂₅₁	C₂²	1	6	yes	—
16	#124×C₂₅₁	C₂²	1	3	yes	—
17	#124×C₂₅₁	C₂²	1	3	yes	complemented
18	#123×C₂₅₁	C₂²	1	6	yes	—
19	#123×C₂₅₁	C₂²	1	6	yes	complemented
20	#123×C₂₅₁	C₂²	1	12	yes	complemented
21	C₂²⋊Q₈×C₂₅₁	C₂²	1	12	yes	complemented
22	C₂²⋊Q₈×C₂₅₁	C₂²	1	6	yes	complemented
23	#121×C₂₅₁	C₂²	1	12	yes	—
24	#121×C₂₅₁	C₂²	1	6	yes	—
25	#121×C₂₅₁	C₂²	1	12	yes	complemented
26	#120×C₂₅₁	C₂²	1	6	yes	—
27	#120×C₂₅₁	C₂²	1	6	yes	complemented
28	C₄×Q₈×C₂₅₁	C₂²	1	2	yes	complemented
29	C₄×D₄×C₂₅₁	C₂²	1	6	yes	—
30	C₄×D₄×C₂₅₁	C₂²	1	12	yes	complemented
31	#117×C₂₅₁	C₂²	1	6	yes	complemented
32	C₂×C₄²×C₂₅₁	C₂²	1	3	yes	complemented
33	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
34	#41×C₂₅₁	—	4	6	no	—
35	#41×C₂₅₁	—	4	2	no	—
36	C₂×Q₈×C₂₅₁	C₂³	1	2	yes	—
37	C₂×Q₈×C₂₅₁	C₂³	1	3	yes	—
38	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
39	C₂×D₄×C₂₅₁	C₂³	1	6	yes	—
40	C₂×D₄×C₂₅₁	C₂³	1	12	yes	—
41	C₂×D₄×C₂₅₁	C₂³	1	3	yes	—
42	C₂×D₄×C₂₅₁	C₂³	1	6	yes	—
43	C₂×D₄×C₂₅₁	C₂³	1	3	yes	—
44	C₂²×C₄×C₂₅₁	—	4	4	no	—
45	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
46	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
47	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
48	C₂²×C₄×C₂₅₁	C₂³	1	6	yes	—
49	C₄⋊C₄×C₂₅₁	C₂³	1	12	yes	—
50	C₄⋊C₄×C₂₅₁	C₂³	1	6	yes	—
51	C₄⋊C₄×C₂₅₁	C₂³	1	12	yes	—
52	C₄⋊C₄×C₂₅₁	C₂³	1	6	yes	—
53	C₂²⋊C₄×C₂₅₁	C₂³	1	6	yes	—
54	C₂²⋊C₄×C₂₅₁	C₂³	1	6	yes	—
55	C₂²⋊C₄×C₂₅₁	C₂³	1	12	yes	—
56	C₂²⋊C₄×C₂₅₁	C₂³	1	12	yes	—
57	C₂²⋊C₄×C₂₅₁	C₂³	1	12	yes	—
58	C₄²×C₂₅₁	C₂³	1	3	yes	—
59	C₄²×C₂₅₁	C₂³	1	3	yes	—
60	C₄²×C₂₅₁	C₂³	1	6	yes	—
61	C₂³×C₂₅₁	—	4	2	no	—
62	C₂³×C₂₅₁	—	4	4	no	—
63	C₂³×C₂₅₁	C₂⁴	1	6	yes	—
64	C₂³×C₂₅₁	C₂⁴	1	3	yes	—
65	Q₈×C₂₅₁	—	4	2	no	—
66	Q₈×C₂₅₁	—	4	3	no	—
67	Q₈×C₂₅₁	—	4	1	no	—
68	D₄×C₂₅₁	—	4	12	no	—
69	D₄×C₂₅₁	—	4	6	no	—
70	D₄×C₂₅₁	—	4	6	no	—
71	D₄×C₂₅₁	—	4	3	no	—
72	D₄×C₂₅₁	—	4	3	no	—
73	C₂×C₄×C₂₅₁	—	2	12	no	—
74	C₂×C₄×C₂₅₁	—	2	6	no	—
75	C₂×C₄×C₂₅₁	—	4	6	no	—
76	C₂×C₄×C₂₅₁	—	4	6	no	—
77	C₂×C₄×C₂₅₁	C₂⁴	1	6	yes	—
78	C₂×C₄×C₂₅₁	C₂⁴	1	6	yes	—
79	C₂×C₄×C₂₅₁	C₂⁴	1	3	yes	—
80	C₂×C₄×C₂₅₁	C₂⁴	1	6	yes	—
81	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
82	C₂²×C₂₅₁	—	2	6	no	—
83	C₂²×C₂₅₁	—	2	3	no	—
84	C₂²×C₂₅₁	—	2	6	no	—
85	C₂²×C₂₅₁	—	2	12	no	—
86	C₂²×C₂₅₁	—	4	8	no	—
87	C₂²×C₂₅₁	C₂⁵	1	1	yes	characteristic
88	C₄×C₂₅₁	—	2	6	no	—
89	C₄×C₂₅₁	—	2	3	no	—
90	C₄×C₂₅₁	—	2	6	no	—
91	C₄×C₂₅₁	—	2	6	no	—
92	C₄×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
93	C₂×C₂₅₁	—	4	3	no	—
94	C₂×C₂₅₁	—	4	6	no	—
95	C₂×C₂₅₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
96	C₂×C₂₅₁	#585	1	2	yes	—
97	C₂₅₁	#3529	1	1	yes	characteristic, direct product factor, Sylow
98	#3529	C₂₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	#563	C₂×C₂₅₁	1	3	yes	complemented
100	#562	C₂×C₂₅₁	1	6	yes	complemented
101	H(3,𝔽₄)	C₂×C₂₅₁	1	1	yes	characteristic, complemented
102	#560	C₂×C₂₅₁	1	6	yes	complemented
103	#532	C₂×C₂₅₁	1	3	yes	complemented
104	#529	C₂×C₂₅₁	1	6	yes	complemented
105	#525	C₂×C₂₅₁	1	6	yes	complemented
106	C₂×#41	C₂²×C₂₅₁	1	3	yes	—
107	C₂×#41	C₂²×C₂₅₁	1	1	yes	characteristic, complemented
108	C₂³×C₄	C₂²×C₂₅₁	1	2	yes	complemented
109	C₄⋊Q₈	C₂²×C₂₅₁	1	3	yes	complemented
110	#127	C₂²×C₂₅₁	1	3	yes	—
111	#126	C₂²×C₂₅₁	1	12	yes	complemented
112	#125	C₂²×C₂₅₁	1	6	yes	complemented
113	#124	C₂²×C₂₅₁	1	6	yes	—
114	#124	C₂²×C₂₅₁	1	3	yes	—
115	#124	C₂²×C₂₅₁	1	3	yes	complemented
116	#123	C₂²×C₂₅₁	1	6	yes	—
117	#123	C₂²×C₂₅₁	1	6	yes	complemented
118	#123	C₂²×C₂₅₁	1	12	yes	complemented
119	C₂²⋊Q₈	C₂²×C₂₅₁	1	12	yes	complemented
120	C₂²⋊Q₈	C₂²×C₂₅₁	1	6	yes	complemented
121	#121	C₂²×C₂₅₁	1	12	yes	—
122	#121	C₂²×C₂₅₁	1	6	yes	—
123	#121	C₂²×C₂₅₁	1	12	yes	complemented
124	#120	C₂²×C₂₅₁	1	6	yes	—
125	#120	C₂²×C₂₅₁	1	6	yes	complemented
126	C₄×Q₈	C₂²×C₂₅₁	1	2	yes	complemented
127	C₄×D₄	C₂²×C₂₅₁	1	6	yes	—
128	C₄×D₄	C₂²×C₂₅₁	1	12	yes	complemented
129	#117	C₂²×C₂₅₁	1	6	yes	complemented
130	C₂×C₄²	C₂²×C₂₅₁	1	3	yes	complemented
131	C₂⁴	C₂³×C₂₅₁	1	2	yes	—
132	#41	—	4	6	no	—
133	#41	—	4	2	no	—
134	C₂×Q₈	C₂³×C₂₅₁	1	2	yes	—
135	C₂×Q₈	C₂³×C₂₅₁	1	3	yes	—
136	C₂×Q₈	C₂³×C₂₅₁	1	1	yes	characteristic
137	C₂×D₄	C₂³×C₂₅₁	1	6	yes	—
138	C₂×D₄	C₂³×C₂₅₁	1	12	yes	—
139	C₂×D₄	C₂³×C₂₅₁	1	3	yes	—
140	C₂×D₄	C₂³×C₂₅₁	1	6	yes	—
141	C₂×D₄	C₂³×C₂₅₁	1	3	yes	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₂³×C₂₅₁	1	3	yes	—
144	C₂²×C₄	C₂³×C₂₅₁	1	6	yes	—
145	C₂²×C₄	C₂³×C₂₅₁	1	6	yes	—
146	C₂²×C₄	C₂³×C₂₅₁	1	6	yes	—
147	C₄⋊C₄	C₂³×C₂₅₁	1	12	yes	—
148	C₄⋊C₄	C₂³×C₂₅₁	1	6	yes	—
149	C₄⋊C₄	C₂³×C₂₅₁	1	12	yes	—
150	C₄⋊C₄	C₂³×C₂₅₁	1	6	yes	—
151	C₂²⋊C₄	C₂³×C₂₅₁	1	6	yes	—
152	C₂²⋊C₄	C₂³×C₂₅₁	1	6	yes	—
153	C₂²⋊C₄	C₂³×C₂₅₁	1	12	yes	—
154	C₂²⋊C₄	C₂³×C₂₅₁	1	12	yes	—
155	C₂²⋊C₄	C₂³×C₂₅₁	1	12	yes	—
156	C₄²	C₂³×C₂₅₁	1	6	yes	—
157	C₄²	C₂³×C₂₅₁	1	3	yes	—
158	C₄²	C₂³×C₂₅₁	1	3	yes	—
159	C₂³	—	4	2	no	—
160	C₂³	—	4	4	no	—
161	C₂³	C₂⁴×C₂₅₁	1	6	yes	—
162	C₂³	C₂⁴×C₂₅₁	1	3	yes	—
163	Q₈	—	4	3	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	4	2	no	—
166	D₄	—	4	6	no	—
167	D₄	—	4	3	no	—
168	D₄	—	4	3	no	—
169	D₄	—	4	12	no	—
170	D₄	—	4	6	no	—
171	C₂×C₄	—	2	12	no	—
172	C₂×C₄	—	2	6	no	—
173	C₂×C₄	—	4	6	no	—
174	C₂×C₄	—	4	6	no	—
175	C₂×C₄	C₂⁴×C₂₅₁	1	6	yes	—
176	C₂×C₄	C₂⁴×C₂₅₁	1	6	yes	—
177	C₂×C₄	C₂⁴×C₂₅₁	1	3	yes	—
178	C₂×C₄	C₂⁴×C₂₅₁	1	6	yes	—
179	C₂×C₄	C₂⁴×C₂₅₁	1	1	yes	characteristic
180	C₂²	—	2	6	no	—
181	C₂²	—	2	3	no	—
182	C₂²	—	2	6	no	—
183	C₂²	—	2	12	no	—
184	C₂²	—	4	8	no	—
185	C₂²	C₂⁵×C₂₅₁	1	1	yes	characteristic
186	C₄	—	2	6	no	—
187	C₄	—	2	3	no	—
188	C₄	—	2	6	no	—
189	C₄	—	2	6	no	—
190	C₄	H(5,𝔽₂)×C₂₅₁	1	2	yes	—
191	C₂	—	4	6	no	—
192	C₂	—	4	3	no	—
193	C₂	C₂×H(5,𝔽₂)×C₂₅₁	1	1	yes	characteristic
194	C₂	#585×C₂₅₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₅₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₅₁
socle	C₂²×C₂₅₁	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #563×C₂₅₁	C₂	semidirect
2. #562×C₂₅₁	C₂	semidirect
3. H(3,𝔽₄)×C₂₅₁	C₂	semidirect
4. #560×C₂₅₁	C₂	semidirect
5. #532×C₂₅₁	C₂	semidirect
6. #529×C₂₅₁	C₂	semidirect
7. #525×C₂₅₁	C₂	semidirect
9. C₂×#41×C₂₅₁	C₂²	semidirect
10. C₂³×C₄×C₂₅₁	C₂²	semidirect
11. C₄⋊Q₈×C₂₅₁	C₂²	semidirect
13. #126×C₂₅₁	C₂²	semidirect
14. #125×C₂₅₁	C₂²	semidirect
17. #124×C₂₅₁	C₂²	semidirect
19. #123×C₂₅₁	C₂²	semidirect
20. #123×C₂₅₁	C₂²	semidirect
21. C₂²⋊Q₈×C₂₅₁	C₂²	semidirect
22. C₂²⋊Q₈×C₂₅₁	C₂²	semidirect
25. #121×C₂₅₁	C₂²	semidirect
27. #120×C₂₅₁	C₂²	semidirect
28. C₄×Q₈×C₂₅₁	C₂²	semidirect
30. C₄×D₄×C₂₅₁	C₂²	semidirect
31. #117×C₂₅₁	C₂²	semidirect
32. C₂×C₄²×C₂₅₁	C₂²	semidirect
97. C₂₅₁	#3529	direct (N × H)
98. #3529	C₂₅₁	direct (N × H)
99. #563	C₂×C₂₅₁	semidirect
100. #562	C₂×C₂₅₁	semidirect
101. H(3,𝔽₄)	C₂×C₂₅₁	semidirect
102. #560	C₂×C₂₅₁	semidirect
103. #532	C₂×C₂₅₁	semidirect
104. #529	C₂×C₂₅₁	semidirect
105. #525	C₂×C₂₅₁	semidirect
107. C₂×#41	C₂²×C₂₅₁	semidirect
108. C₂³×C₄	C₂²×C₂₅₁	semidirect
109. C₄⋊Q₈	C₂²×C₂₅₁	semidirect
111. #126	C₂²×C₂₅₁	semidirect
112. #125	C₂²×C₂₅₁	semidirect
115. #124	C₂²×C₂₅₁	semidirect
117. #123	C₂²×C₂₅₁	semidirect
118. #123	C₂²×C₂₅₁	semidirect
119. C₂²⋊Q₈	C₂²×C₂₅₁	semidirect
120. C₂²⋊Q₈	C₂²×C₂₅₁	semidirect
123. #121	C₂²×C₂₅₁	semidirect
125. #120	C₂²×C₂₅₁	semidirect
126. C₄×Q₈	C₂²×C₂₅₁	semidirect
128. C₄×D₄	C₂²×C₂₅₁	semidirect
129. #117	C₂²×C₂₅₁	semidirect
130. C₂×C₄²	C₂²×C₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#563×C₂₅₁	2
#562×C₂₅₁	2
H(3,𝔽₄)×C₂₅₁	2
#560×C₂₅₁	2
#532×C₂₅₁	2
#529×C₂₅₁	2
#525×C₂₅₁	2
#3529	251

Subgroups — #3529×C251 #44548116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3529×C₂₅₁ #44548116