Subgroups — #72323812

Order 4096.

At least 8991 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

The subgroup lattice has 161 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323226	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310472	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266039	C₂	1	1	yes	maximal, characteristic, complemented
4	#72259008	C₂²	1	1	yes	characteristic
5	#72258968	C₂	1	1	yes	maximal, characteristic
6	#72238601	C₂	1	1	yes	maximal, characteristic, complemented
7	#72238595	C₄	1	2	yes	complemented
8	#72237480	C₂	1	1	yes	maximal, characteristic, complemented
9	#72234668	C₂²	1	1	yes	characteristic
10	#72225997	C₂	1	1	yes	maximal, characteristic, complemented
11	#72225870	C₂²	1	1	yes	characteristic
12	#72224136	C₂²	1	1	yes	characteristic
13	#72219314	C₂²	1	1	yes	characteristic
14	#72219311	C₂²	1	1	yes	characteristic
15	#72219254	C₂²	1	1	yes	characteristic
16	#72209716	C₄	1	2	yes	complemented
17	#10586404	D₄	1	1	yes	characteristic
18	C₂⁴×#120	D₄	1	1	yes	characteristic
19	C₂×#34980	C₂³	1	1	yes	characteristic
20	#7626202	D₄	1	1	yes	characteristic, complemented
21	#6499870	D₄	1	2	yes	—
22	#6499867	C₂×C₄	1	2	yes	—
23	#6374128	D₄	1	1	yes	characteristic
24	C₂×#21746	D₄	1	1	yes	characteristic, complemented
25	#6342371	C₂×C₄	1	1	yes	characteristic
26	#6342371	D₄	1	1	yes	characteristic
27	#4308128	C₂×C₄	1	1	yes	characteristic
28	#4308128	D₄	1	1	yes	characteristic
29	#517411	D₄	1	1	yes	characteristic
30	#190116	D₄	1	2	yes	—
31	#188109	D₄	1	1	yes	characteristic, complemented
32	#188049	D₄	1	1	yes	characteristic, complemented
33	C₂×#8101	D₄	1	2	yes	complemented
34	#134036	D₄	1	2	yes	complemented
35	#115690	C₂×C₄	1	2	yes	—
36	#115631	D₄	1	2	yes	—
37	C₂⁸	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	C₂⁴×C₂²⋊C₄	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	C₂²×C₄	1	1	yes	characteristic
47	C₂×#2846	C₂×D₄	1	1	yes	characteristic
48	C₂×#2613	C₂×D₄	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	#41	1	1	yes	characteristic
54	C₂²×#379	C₂×D₄	1	1	yes	characteristic
55	#21746	C₂²⋊C₄	1	2	yes	—
56	#14679	C₂²⋊C₄	1	1	yes	characteristic, complemented
57	#14679	C₂²⋊C₄	1	1	yes	characteristic, complemented
58	#8101	C₂²⋊C₄	1	4	yes	complemented
59	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂⁷	#120	1	1	yes	characteristic
61	C₂⁴×D₄	#120	1	1	yes	characteristic
62	C₂⁴×D₄	#120	1	1	yes	characteristic
63	C₂⁴×D₄	#120	1	1	yes	characteristic
64	C₂⁴×D₄	#121	1	1	yes	characteristic
65	C₂⁴×D₄	#121	1	1	yes	characteristic
66	C₂⁵×C₄	#123	1	1	yes	characteristic
67	C₂²×#123	#121	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	C₄×D₄	1	1	yes	characteristic
74	C₂²×#120	#120	1	1	yes	characteristic
75	C₂²×#120	#120	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	#121	1	1	yes	characteristic
83	C₂×#381	#123	1	1	yes	characteristic
84	C₂×#381	#124	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	C₄×D₄	1	1	yes	characteristic
89	C₂×#379	#120	1	1	yes	characteristic
90	C₂×#379	#120	1	1	yes	characteristic
91	C₂×#379	#121	1	1	yes	characteristic
92	C₂×#379	#121	1	1	yes	characteristic
93	C₂×#379	#123	1	1	yes	characteristic
94	C₂⁶	#379	1	1	yes	characteristic
95	C₂⁶	#386	1	1	yes	characteristic
96	C₂⁶	#386	1	1	yes	characteristic
97	C₂⁶	#392	1	1	yes	characteristic
98	C₂⁶	#392	1	1	yes	characteristic
99	C₂⁶	B(4,𝔽₂)	1	2	yes	—
100	C₂⁶	B(4,𝔽₂)	1	2	yes	—
101	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
102	C₂³×D₄	#386	1	1	yes	characteristic
103	C₂³×D₄	#392	1	1	yes	characteristic
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	#394	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	2	yes	—
120	C₂⁵	#1889	1	1	yes	characteristic
121	C₂⁵	#2021	1	1	yes	characteristic
122	C₂⁵	#2021	1	1	yes	characteristic
123	C₂⁵	#2021	1	1	yes	characteristic
124	C₂⁵	#2021	1	2	yes	—
125	C₂⁵	#2021	1	1	yes	characteristic
126	C₂⁵	#2021	1	1	yes	characteristic
127	C₂⁵	#2021	1	1	yes	characteristic
128	C₂⁵	#2022	1	1	yes	characteristic
129	C₂⁵	#2196	1	2	yes	—
130	C₂⁵	#2197	1	2	yes	—
131	C₂⁵	#2199	1	2	yes	—
132	C₂⁵	#2200	1	2	yes	—
133	C₂³×C₄	#1889	1	1	yes	characteristic
134	C₂³×C₄	#2021	1	1	yes	characteristic
135	C₂³×C₄	#2044	1	2	yes	—
136	C₂⁴	#10336	1	1	yes	characteristic
137	C₂⁴	#10336	1	1	yes	characteristic
138	C₂⁴	#10337	1	1	yes	characteristic
139	C₂⁴	#10337	1	1	yes	characteristic
140	C₂⁴	#10991	1	1	yes	characteristic
141	C₂⁴	#10991	1	1	yes	characteristic
142	C₂⁴	#10995	1	1	yes	characteristic
143	C₂⁴	#12753	1	4	yes	—
144	C₂⁴	#13041	1	2	yes	—
145	C₂⁴	#13043	1	2	yes	—
146	C₂⁴	#13343	1	2	yes	—
147	C₂⁴	#13344	1	2	yes	—
148	C₂⁴	#13346	1	2	yes	—
149	C₂⁴	#13348	1	2	yes	—
150	C₂³	#111458	1	1	yes	characteristic
151	C₂³	#144366	1	1	yes	characteristic
152	C₂³	#144367	1	1	yes	characteristic
153	C₂³	#144412	1	2	yes	—
154	C₂³	#147749	1	2	yes	—
155	C₂³	#147815	1	2	yes	—
156	C₂²	#72218840	1	1	yes	characteristic
157	C₂²	#72219252	1	1	yes	characteristic
158	C₂²	#72237473	1	1	yes	characteristic
159	C₂	#72266316	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266316
commutator	C₂×#2846	C₂²×C₄
Frattini, cosocle	C₂×#34980	C₂³
prime radical	#72258968	C₂
LCS	G ▷ C₂×#2846 ▷ 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. #72323226	C₂	semidirect
2. #72310472	C₂	semidirect
3. #72266039	C₂	semidirect
6. #72238601	C₂	semidirect
7. #72238595	C₄	semidirect
8. #72237480	C₂	semidirect
10. #72225997	C₂	semidirect
16. #72209716	C₄	semidirect
20. #7626202	D₄	semidirect
24. C₂×#21746	D₄	semidirect
31. #188109	D₄	semidirect
32. #188049	D₄	semidirect
33. C₂×#8101	D₄	semidirect
34. #134036	D₄	semidirect
37. C₂⁸	C₂²⋊C₄	semidirect
43. C₂⁴×C₂²⋊C₄	C₂²⋊C₄	semidirect
56. #14679	C₂²⋊C₄	semidirect
57. #14679	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
#72323226	2
#72310472	2
#72266039	2
#72258968	2
#72238601	2
#72237480	2
#72225997	2