Subgroups — #72262439

Order 4096.

At least 13775 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

The subgroup lattice has 209 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262373	C₂	1	2	yes	maximal, complemented
2	#72262254	C₂	1	1	yes	maximal, characteristic, complemented
3	#72256392	C₂	1	1	yes	maximal, characteristic, complemented
4	#72251598	C₂	1	1	yes	maximal, characteristic, complemented
5	#72251586	C₂²	1	1	yes	characteristic, complemented
6	#72251569	C₂²	1	1	yes	characteristic
7	#72249438	C₂	1	2	yes	maximal, complemented
8	#72249434	C₂²	1	1	yes	characteristic
9	#72237013	C₂	1	2	yes	maximal, complemented
10	C₂×#213906	C₂²	1	1	yes	characteristic
11	#72236923	C₂	1	2	yes	maximal, complemented
12	#72236921	C₂	1	2	yes	maximal, complemented
13	#72236901	C₂²	1	2	yes	complemented
14	#72236898	C₂²	1	2	yes	—
15	#72236875	C₂²	1	2	yes	complemented
16	#72236837	C₂²	1	2	yes	—
17	#72236816	C₂²	1	1	yes	characteristic, complemented
18	#72236814	C₂²	1	1	yes	characteristic, complemented
19	#72236754	C₂²	1	2	yes	complemented
20	#72235872	C₂	1	2	yes	maximal, complemented
21	#72235865	C₂²	1	2	yes	complemented
22	#72234825	C₂²	1	1	yes	characteristic, complemented
23	#72231822	C₂²	1	2	yes	complemented
24	#72231810	C₂²	1	2	yes	—
25	#72226038	C₂²	1	2	yes	—
26	#72225873	C₂²	1	2	yes	complemented
27	#72225733	C₂²	1	2	yes	—
28	#72219034	C₂²	1	2	yes	—
29	#72219034	C₂²	1	2	yes	—
30	C₂×#245225	C₂²	1	2	yes	—
31	C₂×#62652	D₄	1	1	yes	characteristic, complemented
32	#8934496	D₄	1	1	yes	characteristic, complemented
33	#8934400	D₄	1	2	yes	complemented
34	#8934377	D₄	1	1	yes	characteristic, complemented
35	#7642273	C₂³	1	2	yes	—
36	C₂×#35346	C₂³	1	2	yes	—
37	C₂×#35346	C₂³	1	1	yes	characteristic
38	C₂×#32898	D₄	1	2	yes	complemented
39	C₂×#32898	D₄	1	2	yes	complemented
40	C₂×#20585	D₄	1	1	yes	characteristic, complemented
41	#515285	C₂³	1	1	yes	characteristic
42	#515285	C₂³	1	1	yes	characteristic
43	#382422	C₂³	1	2	yes	—
44	#353314	D₄	1	1	yes	characteristic, complemented
45	#353314	D₄	1	1	yes	characteristic, complemented
46	C₂×#9495	D₄	1	2	yes	complemented
47	C₂×#9495	D₄	1	2	yes	complemented
48	C₂×#8547	C₂³	1	2	yes	—
49	C₂×#8547	C₂³	1	2	yes	—
50	C₂×#8541	C₂³	1	2	yes	—
51	#185255	D₄	1	4	yes	complemented
52	#133278	D₄	1	4	yes	complemented
53	#133278	D₄	1	4	yes	complemented
54	#106326	D₄	1	4	yes	complemented
55	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
56	C₂³×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
57	C₂×#3490	C₂×D₄	1	2	yes	—
58	C₂³×#120	C₂⁴	1	1	yes	characteristic
59	C₂²×#518	C₂×D₄	1	1	yes	characteristic
60	C₂²×#518	C₂×D₄	1	1	yes	characteristic
61	#35390	C₂×D₄	1	2	yes	—
62	#35390	#41	1	2	yes	—
63	#35367	C₂×D₄	1	2	yes	—
64	#35367	#41	1	2	yes	—
65	#35346	C₂×D₄	1	1	yes	characteristic
66	#35346	C₂×D₄	1	1	yes	characteristic
67	#35346	C₂×D₄	1	2	yes	—
68	#35346	C₂×D₄	1	2	yes	—
69	#35346	#41	1	1	yes	characteristic
70	#35346	#41	1	1	yes	characteristic
71	C₂×#2403	C₂×D₄	1	2	yes	—
72	C₂×#2403	C₂×D₄	1	2	yes	—
73	#34176	C₂×D₄	1	2	yes	—
74	C₂×#2308	C₂×D₄	1	2	yes	—
75	C₂×#2308	C₂×D₄	1	1	yes	characteristic
76	C₂×#2308	C₂×D₄	1	1	yes	characteristic
77	C₂×#2308	C₂×D₄	1	1	yes	characteristic
78	C₂×#1781	C₂×D₄	1	2	yes	—
79	C₂×#1781	C₂×D₄	1	2	yes	—
80	C₂×#1462	C₂×D₄	1	4	yes	—
81	#13868	C₂×D₄	1	4	yes	—
82	#10965	C₂×D₄	1	4	yes	—
83	#8547	C₂×D₄	1	4	yes	—
84	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
85	C₂²×H(5,𝔽₂)	#121	1	1	yes	characteristic
86	C₂²×H(5,𝔽₂)	#121	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	#127	1	1	yes	characteristic
88	C₂²×H(5,𝔽₂)	#127	1	1	yes	characteristic
89	C₂⁴×D₄	#120	1	1	yes	characteristic
90	C₂⁴×D₄	#120	1	1	yes	characteristic
91	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
92	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
93	C₂×#543	#121	1	2	yes	complemented
94	C₂×#534	#121	1	2	yes	complemented
95	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
96	C₂²×#120	C₂²×D₄	1	2	yes	—
97	C₂²×#120	C₂×#41	1	2	yes	—
98	C₂²×#120	C₂×#41	1	1	yes	characteristic
99	C₂²×#120	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×#120	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×#117	#120	1	2	yes	—
102	C₂³×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
103	C₂³×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
104	#2403	#120	1	2	yes	—
105	#2403	#120	1	2	yes	—
106	#2403	#120	1	2	yes	—
107	#2403	#120	1	2	yes	—
108	#2403	#121	1	2	yes	—
109	#2403	#121	1	2	yes	—
110	#2403	#121	1	2	yes	—
111	#2403	#121	1	2	yes	—
112	#2308	#121	1	1	yes	characteristic
113	#2308	#121	1	1	yes	characteristic
114	#2308	#127	1	2	yes	—
115	C₂×#381	#120	1	2	yes	—
116	C₂×#379	#120	1	2	yes	—
117	#2273	#121	1	2	yes	complemented
118	#2273	#121	1	2	yes	complemented
119	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
120	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
121	C₂⁶	C₂×#120	1	1	yes	characteristic
122	C₂⁶	#534	1	1	yes	characteristic
123	C₂⁶	#534	1	1	yes	characteristic
124	C₂⁶	#538	1	1	yes	characteristic
125	C₂⁶	D₄²	1	2	yes	—
126	C₂⁶	#546	1	2	yes	—
127	C₂³×D₄	#392	1	1	yes	characteristic
128	C₂³×D₄	#392	1	1	yes	characteristic
129	C₂³×D₄	#392	1	1	yes	characteristic
130	C₂³×D₄	#392	1	1	yes	characteristic
131	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
132	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
133	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
134	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
135	C₂³×D₄	C₂×#120	1	2	yes	—
136	C₂³×D₄	C₂×#121	1	1	yes	characteristic
137	C₂³×D₄	C₂×#121	1	2	yes	—
138	C₂³×D₄	#525	1	1	yes	characteristic
139	C₂³×D₄	C₂×#127	1	1	yes	characteristic
140	C₂³×D₄	#535	1	1	yes	characteristic
141	C₂³×D₄	#546	1	2	yes	—
142	C₂³×D₄	#546	1	2	yes	—
143	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
144	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
145	C₂⁴×C₄	C₂×#120	1	1	yes	characteristic
146	C₂⁴×C₄	D₄²	1	2	yes	—
147	C₂²×C₂²⋊C₄	C₂×#121	1	2	yes	—
148	C₂²×C₂²⋊C₄	#525	1	2	yes	—
149	C₂²×C₂²⋊C₄	#534	1	1	yes	characteristic
150	C₂²×C₂²⋊C₄	#538	1	1	yes	characteristic
151	C₂²×C₂²⋊C₄	D₄²	1	2	yes	—
152	C₂²×C₂²⋊C₄	D₄²	1	2	yes	—
153	C₂²×C₂²⋊C₄	#546	1	2	yes	—
154	C₂²×C₄²	B(4,𝔽₂)	1	2	yes	—
155	C₂²×C₄²	B(4,𝔽₂)	1	2	yes	—
156	#379	#392	1	2	yes	—
157	#379	#392	1	2	yes	—
158	C₂⁵	#2021	1	1	yes	characteristic
159	C₂⁵	#2021	1	1	yes	characteristic
160	C₂⁵	#2021	1	1	yes	characteristic
161	C₂⁵	#2021	1	1	yes	characteristic
162	C₂⁵	C₂×#392	1	1	yes	characteristic
163	C₂⁵	#2403	1	1	yes	characteristic
164	C₂⁵	#2404	1	1	yes	characteristic
165	C₂⁵	#2410	1	2	yes	—
166	C₂⁵	#2424	1	2	yes	—
167	C₂⁵	#2539	1	1	yes	characteristic
168	C₂⁵	#2617	1	1	yes	characteristic
169	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
170	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
171	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
172	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
173	C₂³×C₄	#2021	1	1	yes	characteristic
174	C₂³×C₄	#2021	1	1	yes	characteristic
175	C₂³×C₄	#2021	1	1	yes	characteristic
176	C₂³×C₄	#2021	1	1	yes	characteristic
177	C₂³×C₄	#2410	1	2	yes	—
178	C₂³×C₄	#2424	1	2	yes	—
179	C₂³×C₄	#2433	1	1	yes	characteristic
180	C₂³×C₄	#2847	1	1	yes	characteristic
181	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
182	C₂³×C₄	#3027	1	2	yes	—
183	C₂⁴	#14716	1	1	yes	characteristic
184	C₂⁴	C₂×#2021	1	1	yes	characteristic
185	C₂⁴	C₂×#2021	1	1	yes	characteristic
186	C₂⁴	#23895	1	1	yes	characteristic
187	C₂⁴	#23896	1	1	yes	characteristic
188	C₂⁴	#23900	1	1	yes	characteristic
189	C₂⁴	#23900	1	1	yes	characteristic
190	C₂⁴	#23900	1	1	yes	characteristic
191	C₂⁴	#23904	1	1	yes	characteristic
192	C₂⁴	#23984	1	1	yes	characteristic
193	C₂⁴	#23984	1	1	yes	characteristic
194	C₂²×C₄	#25771	1	2	yes	—
195	C₂²×C₄	#32926	1	2	yes	—
196	C₂³	#245225	1	1	yes	characteristic
197	C₂³	#245229	1	1	yes	characteristic
198	C₂³	#245247	1	1	yes	characteristic
199	C₂³	#245252	1	1	yes	characteristic
200	C₂³	#394167	1	1	yes	characteristic
201	C₂³	#394173	1	1	yes	characteristic
202	C₂³	#394174	1	1	yes	characteristic
203	C₂²	#72209844	1	1	yes	characteristic
204	C₂²	#72237355	1	1	yes	characteristic
205	C₂²	#72237500	1	1	yes	characteristic
206	C₂	#72251075	1	1	yes	characteristic
207	C₂	#72251601	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#72237500
commutator, Frattini, cosocle	C₂³×#120	C₂⁴
LCS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72262373	C₂	semidirect
2. #72262254	C₂	semidirect
3. #72256392	C₂	semidirect
4. #72251598	C₂	semidirect
5. #72251586	C₂²	semidirect
7. #72249438	C₂	semidirect
9. #72237013	C₂	semidirect
11. #72236923	C₂	semidirect
12. #72236921	C₂	semidirect
13. #72236901	C₂²	semidirect
15. #72236875	C₂²	semidirect
17. #72236816	C₂²	semidirect
18. #72236814	C₂²	semidirect
19. #72236754	C₂²	semidirect
20. #72235872	C₂	semidirect
21. #72235865	C₂²	semidirect
22. #72234825	C₂²	semidirect
23. #72231822	C₂²	semidirect
26. #72225873	C₂²	semidirect
31. C₂×#62652	D₄	semidirect
32. #8934496	D₄	semidirect
33. #8934400	D₄	semidirect
34. #8934377	D₄	semidirect
38. C₂×#32898	D₄	semidirect
39. C₂×#32898	D₄	semidirect
40. C₂×#20585	D₄	semidirect
44. #353314	D₄	semidirect
45. #353314	D₄	semidirect
46. C₂×#9495	D₄	semidirect
47. C₂×#9495	D₄	semidirect
51. #185255	D₄	semidirect
52. #133278	D₄	semidirect
53. #133278	D₄	semidirect
54. #106326	D₄	semidirect
93. C₂×#543	#121	semidirect
94. C₂×#534	#121	semidirect
117. #2273	#121	semidirect
118. #2273	#121	semidirect

Maximal subgroups

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

subgroup	index
#72262373	2
#72262254	2
#72256392	2
#72251598	2
#72249438	2
#72237013	2
#72236923	2
#72236921	2
#72235872	2