Subgroups — #72311738

Order 8192.

At least 13729 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

The subgroup lattice has 141 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310584	C₂	1	1	yes	maximal, characteristic
2	#72310564	C₄	1	2	yes	complemented
3	#72310558	C₄	1	2	yes	complemented
4	#72310498	C₂	1	1	yes	maximal, characteristic, complemented
5	#72310246	C₂	1	2	yes	maximal
6	#72309737	C₄	1	2	yes	complemented
7	#72309645	C₄	1	2	yes	complemented
8	#72309490	C₄	1	2	yes	complemented
9	#72309421	C₄	1	2	yes	complemented
10	#72309405	C₂	1	2	yes	maximal, complemented
11	#72309397	C₂²	1	1	yes	characteristic
12	C₂×#97555	C₂×C₄	1	2	yes	—
13	#72302114	C₂	1	1	yes	maximal, characteristic, complemented
14	#72259029	C₂²	1	2	yes	—
15	#72259008	C₂×C₄	1	2	yes	—
16	#72235847	C₂×C₄	1	2	yes	—
17	#72234688	C₂²	1	2	yes	—
18	#72225781	C₂²	1	1	yes	characteristic
19	#72225778	C₂×C₄	1	2	yes	—
20	#72225777	D₄	1	1	yes	characteristic
21	#72225776	D₄	1	1	yes	characteristic
22	#72225738	C₂²	1	1	yes	characteristic
23	#72225736	C₂×C₄	1	1	yes	characteristic
24	#72225735	C₂×C₄	1	1	yes	characteristic
25	#72225713	C₂×C₄	1	2	yes	—
26	#72224136	C₂×C₄	1	2	yes	—
27	#72219311	C₂×C₄	1	2	yes	—
28	#72219311	C₂×C₄	1	2	yes	—
29	#72219301	D₄	1	2	yes	—
30	#72219301	D₄	1	2	yes	—
31	C₂×#7670203	C₂³	1	1	yes	characteristic
32	#72218518	D₄	1	2	yes	—
33	#72218518	D₄	1	2	yes	—
34	C₂×#133822	D₄	1	2	yes	—
35	#10586404	C₄²	1	1	yes	characteristic, complemented
36	C₂²×#3484	C₂×D₄	1	2	yes	—
37	C₂⁴×#120	C₂×D₄	1	2	yes	—
38	C₂²×#2846	C₂×D₄	1	1	yes	characteristic
39	C₂×#35390	C₂²×C₄	1	1	yes	characteristic
40	C₂×#35348	#41	1	2	yes	—
41	C₂×#35346	C₂×D₄	1	1	yes	characteristic
42	C₂×#34981	#41	1	2	yes	—
43	C₂×#34980	C₂²×C₄	1	2	yes	—
44	C₂×#34198	#41	1	2	yes	—
45	#6374251	C₄²	1	1	yes	characteristic, complemented
46	#6374128	C₄²	1	2	yes	complemented
47	#392109	C₂²⋊C₄	1	2	yes	—
48	#392109	C₂²⋊C₄	1	2	yes	—
49	#392109	C₂²⋊C₄	1	2	yes	—
50	#392109	C₂²⋊C₄	1	2	yes	—
51	C₂×#8547	C₂²⋊C₄	1	2	yes	—
52	C₂×#8541	C₂²⋊C₄	1	2	yes	—
53	C₂⁸	#127	1	1	yes	characteristic
54	C₂³×H(5,𝔽₂)	#120	1	1	yes	characteristic
55	C₂⁵×D₄	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×#534	C₂×C₄²	1	1	yes	characteristic
57	C₂²×#534	C₄×D₄	1	2	yes	—
58	C₂²×#534	C₄×D₄	1	2	yes	—
59	C₂³×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂³×#120	C₄×D₄	1	2	yes	—
61	C₂³×#120	C₄×D₄	1	2	yes	—
62	C₂³×#120	#121	1	2	yes	—
63	C₂³×#120	#121	1	2	yes	—
64	C₂³×#120	#121	1	2	yes	—
65	C₂²×#518	#123	1	2	yes	—
66	C₂⁴×C₂²⋊C₄	#124	1	2	yes	—
67	C₂×#2315	#126	1	2	yes	—
68	C₂×#2308	#117	1	2	yes	—
69	C₂×#2308	C₄×D₄	1	2	yes	—
70	C₂×#2308	#123	1	1	yes	characteristic
71	C₂²×#379	C₄×D₄	1	2	yes	—
72	C₂²×#379	#124	1	1	yes	characteristic
73	C₂×#2273	#117	1	1	yes	characteristic
74	C₂×#1781	#99	1	4	yes	complemented
75	#14679	#99	1	4	yes	complemented
76	#8541	#99	1	4	yes	complemented
77	C₂⁷	#390	1	2	yes	—
78	C₂⁷	#390	1	1	yes	characteristic
79	C₂⁷	C₂×#99	1	2	yes	—
80	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
81	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	2	yes	—
82	C₂²×H(5,𝔽₂)	B(4,𝔽₂)	1	1	yes	characteristic
83	C₂⁴×D₄	C₄×C₂²⋊C₄	1	2	yes	—
84	C₂⁴×D₄	#386	1	2	yes	—
85	C₂⁴×D₄	#386	1	1	yes	characteristic
86	C₂⁴×D₄	C₂×#99	1	2	yes	—
87	C₂⁴×D₄	#410	1	2	yes	—
88	C₂⁵×C₄	#410	1	2	yes	—
89	C₂²×#120	C₂×#99	1	2	yes	—
90	C₂²×#120	#410	1	2	yes	—
91	C₂³×C₂²⋊C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₂²⋊C₄	#388	1	2	yes	—
93	C₂³×C₂²⋊C₄	#388	1	2	yes	—
94	C₂³×C₂²⋊C₄	#388	1	2	yes	—
95	C₂⁶	#1437	1	1	yes	characteristic
96	C₂⁶	C₄×#99	1	2	yes	—
97	C₂⁶	#1898	1	2	yes	—
98	C₂⁶	#1913	1	1	yes	characteristic
99	C₂⁶	#1964	1	2	yes	—
100	C₂⁶	#1964	1	2	yes	—
101	C₂⁶	#1964	1	2	yes	—
102	C₂³×D₄	#1889	1	2	yes	—
103	C₂³×D₄	#1889	1	1	yes	characteristic
104	C₂³×D₄	#1889	1	1	yes	characteristic
105	C₂³×D₄	#1889	1	2	yes	—
106	C₂³×D₄	#1898	1	1	yes	characteristic
107	C₂³×D₄	#1899	1	1	yes	characteristic
108	C₂³×D₄	#1913	1	2	yes	—
109	C₂³×D₄	#1913	1	2	yes	—
110	C₂⁴×C₄	C₄×#99	1	2	yes	—
111	C₂⁴×C₄	#1899	1	2	yes	—
112	C₂⁴×C₄	#1913	1	1	yes	characteristic
113	C₂²×C₂²⋊C₄	C₄×#99	1	2	yes	—
114	C₂⁵	#9495	1	2	yes	—
115	C₂⁵	#9495	1	1	yes	characteristic
116	C₂⁵	#9497	1	2	yes	—
117	C₂⁵	#9498	1	1	yes	characteristic
118	C₂⁵	#10935	1	1	yes	characteristic
119	C₂⁵	#10938	1	2	yes	—
120	C₂⁵	#10938	1	1	yes	characteristic
121	C₂⁵	#10960	1	2	yes	—
122	C₂⁵	#10960	1	1	yes	characteristic
123	C₂³×C₄	#9496	1	2	yes	—
124	C₂³×C₄	#9496	1	1	yes	characteristic
125	C₂³×C₄	#9497	1	1	yes	characteristic
126	C₂³×C₄	#9498	1	2	yes	—
127	C₂⁴	#110086	1	1	yes	characteristic
128	C₂⁴	#110086	1	2	yes	—
129	C₂⁴	#110180	1	1	yes	characteristic
130	C₂⁴	#110401	1	1	yes	characteristic
131	C₂⁴	#110401	1	2	yes	—
132	C₂⁴	#139743	1	4	yes	—
133	C₂³	#72246286	1	2	yes	—
134	C₂³	#72255986	1	2	yes	—
135	C₂³	#72264951	1	1	yes	characteristic
136	C₂³	#72265908	1	2	yes	—
137	C₂²	#72265612	1	2	yes	—
138	C₂²	#72265909	1	1	yes	characteristic
139	C₂	#72266186	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266186
commutator	C₂²×#534	C₂×C₄²
Frattini, cosocle	C₂×#7670203	C₂³
prime radical	#72225738	C₂²
LCS	G ▷ C₂²×#534 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#7670203 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72310564	C₄	semidirect
3. #72310558	C₄	semidirect
4. #72310498	C₂	semidirect
6. #72309737	C₄	semidirect
7. #72309645	C₄	semidirect
8. #72309490	C₄	semidirect
9. #72309421	C₄	semidirect
10. #72309405	C₂	semidirect
13. #72302114	C₂	semidirect
35. #10586404	C₄²	semidirect
45. #6374251	C₄²	semidirect
46. #6374128	C₄²	semidirect
74. C₂×#1781	#99	semidirect
75. #14679	#99	semidirect
76. #8541	#99	semidirect

Maximal subgroups

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

subgroup	index
#72310584	2
#72310498	2
#72310246	2
#72309405	2
#72302114	2