Subgroups — #72316556

Order 4096.

At least 9400 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

The subgroup lattice has 150 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314610	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314165	C₂	1	1	yes	maximal, characteristic, complemented
3	#72313608	C₂	1	1	yes	maximal, characteristic, complemented
4	#72309483	C₂	1	1	yes	maximal, characteristic, complemented
5	#72265766	C₂²	1	1	yes	characteristic
6	#72258757	C₂	1	1	yes	maximal, characteristic
7	#72258606	C₄	1	2	yes	complemented
8	#72237908	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237596	C₂	1	1	yes	maximal, characteristic, complemented
10	#72235718	C₂²	1	1	yes	characteristic
11	#72233061	C₄	1	2	yes	complemented
12	#72225713	C₂²	1	1	yes	characteristic
13	#72224136	C₂²	1	1	yes	characteristic
14	#72219311	C₂²	1	1	yes	characteristic
15	#72219311	C₂²	1	1	yes	characteristic
16	#72219257	C₂²	1	1	yes	characteristic
17	#10586404	D₄	1	1	yes	characteristic
18	C₂²×#3484	D₄	1	1	yes	characteristic
19	C₂×#34980	C₂³	1	1	yes	characteristic
20	#6499870	C₂×C₄	1	2	yes	—
21	#6499867	D₄	1	2	yes	—
22	#6374251	D₄	1	1	yes	characteristic, complemented
23	#6374128	D₄	1	1	yes	characteristic
24	#6374128	D₄	1	1	yes	characteristic, complemented
25	#4930495	C₂×C₄	1	1	yes	characteristic
26	#4930495	D₄	1	1	yes	characteristic
27	#4930494	C₂×C₄	1	1	yes	characteristic
28	#4930494	D₄	1	1	yes	characteristic
29	C₂²×#1438	D₄	1	1	yes	characteristic
30	#190116	D₄	1	2	yes	—
31	#190116	D₄	1	2	yes	complemented
32	C₂×#8547	D₄	1	1	yes	characteristic, complemented
33	C₂×#8541	D₄	1	1	yes	characteristic, complemented
34	#134036	D₄	1	2	yes	complemented
35	#115690	D₄	1	2	yes	—
36	#115631	C₂×C₄	1	2	yes	—
37	C₂³×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic, complemented
38	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
39	#62817	C₂²⋊C₄	1	2	yes	—
40	C₂²×#534	C₂×D₄	1	1	yes	characteristic
41	C₂²×#534	C₂×D₄	1	1	yes	characteristic
42	C₂³×#120	C₂×D₄	1	1	yes	characteristic
43	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic, complemented
44	C₂×#2849	#41	1	1	yes	characteristic
45	C₂×#2849	#41	1	1	yes	characteristic
46	C₂×#2846	C₂×D₄	1	1	yes	characteristic
47	C₂×#2846	C₂×D₄	1	1	yes	characteristic
48	C₂×#2613	C₂²×C₄	1	1	yes	characteristic
49	C₂×#2613	C₂×D₄	1	1	yes	characteristic
50	C₂×#2613	C₂×D₄	1	1	yes	characteristic
51	C₂×#2613	#41	1	1	yes	characteristic
52	C₂×#2308	C₂×D₄	1	1	yes	characteristic
53	C₂×#2308	C₂×D₄	1	1	yes	characteristic
54	C₂²×#379	#41	1	1	yes	characteristic
55	#21746	C₂²⋊C₄	1	2	yes	—
56	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic, complemented
57	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic, complemented
58	#8101	C₂²⋊C₄	1	4	yes	complemented
59	C₂⁷	#120	1	1	yes	characteristic
60	C₂⁷	#121	1	1	yes	characteristic
61	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂⁴×D₄	#120	1	1	yes	characteristic
63	C₂⁴×D₄	#120	1	1	yes	characteristic
64	C₂⁴×D₄	#120	1	1	yes	characteristic
65	C₂⁴×D₄	#121	1	1	yes	characteristic
66	C₂⁵×C₄	#123	1	1	yes	characteristic
67	C₂²×#123	C₄×D₄	1	1	yes	characteristic
68	C₂²×#123	#121	1	1	yes	characteristic
69	C₂²×#123	#121	1	1	yes	characteristic
70	C₂²×#123	#123	1	1	yes	characteristic
71	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×#120	C₄×D₄	1	1	yes	characteristic
73	C₂²×#120	#120	1	1	yes	characteristic
74	C₂²×#120	#120	1	1	yes	characteristic
75	C₂²×#120	#121	1	1	yes	characteristic
76	C₂²×#120	#121	1	1	yes	characteristic
77	C₂²×#120	#121	1	1	yes	characteristic
78	C₂²×#120	#127	1	1	yes	characteristic
79	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
80	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
81	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
82	C₂×#381	C₄×D₄	1	1	yes	characteristic
83	C₂×#381	#121	1	1	yes	characteristic
84	C₂×#381	#121	1	1	yes	characteristic
85	C₂×#381	#124	1	1	yes	characteristic
86	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#379	C₄×D₄	1	1	yes	characteristic
88	C₂×#379	#120	1	1	yes	characteristic
89	C₂×#379	#120	1	1	yes	characteristic
90	C₂×#379	#121	1	1	yes	characteristic
91	C₂×#379	#123	1	1	yes	characteristic
92	C₂×#379	#123	1	1	yes	characteristic
93	C₂×#379	#124	1	1	yes	characteristic
94	C₂⁶	#379	1	1	yes	characteristic
95	C₂⁶	#386	1	1	yes	characteristic
96	C₂⁶	#392	1	1	yes	characteristic
97	C₂⁶	#392	1	1	yes	characteristic
98	C₂⁶	#394	1	1	yes	characteristic
99	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
100	C₂³×D₄	#386	1	1	yes	characteristic
101	C₂³×D₄	#392	1	1	yes	characteristic
102	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
103	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
104	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
105	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
106	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
107	C₂³×D₄	#458	1	2	yes	—
108	C₂⁴×C₄	#386	1	1	yes	characteristic
109	C₂⁴×C₄	#394	1	1	yes	characteristic
110	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
112	C₂²×C₂²⋊C₄	#390	1	1	yes	characteristic
113	C₂²×C₂²⋊C₄	#390	1	1	yes	characteristic
114	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
115	C₂²×C₂²⋊C₄	#394	1	1	yes	characteristic
116	C₂²×C₂²⋊C₄	#394	1	1	yes	characteristic
117	C₂⁵	#1438	1	1	yes	characteristic
118	C₂⁵	#1790	1	2	yes	—
119	C₂⁵	#1889	1	1	yes	characteristic
120	C₂⁵	#2021	1	1	yes	characteristic
121	C₂⁵	#2021	1	1	yes	characteristic
122	C₂⁵	#2021	1	2	yes	—
123	C₂⁵	#2021	1	1	yes	characteristic
124	C₂⁵	#2021	1	1	yes	characteristic
125	C₂⁵	#2021	1	1	yes	characteristic
126	C₂⁵	#2021	1	1	yes	characteristic
127	C₂⁵	#2022	1	1	yes	characteristic
128	C₂³×C₄	#1889	1	2	yes	—
129	C₂³×C₄	#1889	1	1	yes	characteristic
130	C₂³×C₄	#2021	1	1	yes	characteristic
131	C₂³×C₄	#2044	1	2	yes	—
132	C₂⁴	#10336	1	1	yes	characteristic
133	C₂⁴	#10336	1	1	yes	characteristic
134	C₂⁴	#10337	1	1	yes	characteristic
135	C₂⁴	#10337	1	1	yes	characteristic
136	C₂⁴	#10991	1	1	yes	characteristic
137	C₂⁴	#10991	1	1	yes	characteristic
138	C₂⁴	#10995	1	1	yes	characteristic
139	C₂⁴	#12753	1	4	yes	—
140	C₂³	#111458	1	1	yes	characteristic
141	C₂³	#144378	1	1	yes	characteristic
142	C₂³	#144378	1	1	yes	characteristic
143	C₂³	#147752	1	2	yes	—
144	C₂³	#147815	1	2	yes	—
145	C₂²	#72218521	1	1	yes	characteristic
146	C₂²	#72218840	1	1	yes	characteristic
147	C₂²	#72258273	1	1	yes	characteristic
148	C₂	#72266234	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266234
commutator	C₂×#2613	C₂²×C₄
Frattini, cosocle	C₂×#34980	C₂³
prime radical	#72258757	C₂
LCS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#34980 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72314610	C₂	semidirect
2. #72314165	C₂	semidirect
3. #72313608	C₂	semidirect
4. #72309483	C₂	semidirect
7. #72258606	C₄	semidirect
8. #72237908	C₂	semidirect
9. #72237596	C₂	semidirect
11. #72233061	C₄	semidirect
22. #6374251	D₄	semidirect
24. #6374128	D₄	semidirect
31. #190116	D₄	semidirect
32. C₂×#8547	D₄	semidirect
33. C₂×#8541	D₄	semidirect
34. #134036	D₄	semidirect
37. C₂³×H(5,𝔽₂)	C₂²⋊C₄	semidirect
43. C₂²×#518	C₂²⋊C₄	semidirect
56. C₂×#1438	C₂²⋊C₄	semidirect
57. C₂×#1438	C₂²⋊C₄	semidirect
58. #8101	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72314610	2
#72314165	2
#72313608	2
#72309483	2
#72258757	2
#72237908	2
#72237596	2