Subgroups — #72337607

Order 8192.

Only the 240 normal subgroups are shown; non-normal subgroups are omitted.

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	#72323220	C₂	1	2	yes	maximal, complemented
2	#72314603	C₂	1	1	yes	maximal, characteristic, complemented
3	#72302114	C₂	1	1	yes	maximal, characteristic, complemented
4	#72265845	C₂	1	2	yes	maximal
5	#72265825	C₄	1	2	yes	complemented
6	#72265785	C₂²	1	2	yes	—
7	#72265763	C₄	1	2	yes	complemented
8	#72265678	C₄	1	2	yes	complemented
9	#72234688	C₂²	1	2	yes	—
10	#72226360	C₂	1	1	yes	maximal, characteristic
11	#72226350	C₄	1	2	yes	complemented
12	#72225883	C₂²	1	1	yes	characteristic
13	#72225781	C₂²	1	1	yes	characteristic
14	#72225778	D₄	1	2	yes	—
15	#72225777	D₄	1	1	yes	characteristic
16	#72225776	D₄	1	1	yes	characteristic
17	#72219311	D₄	1	2	yes	—
18	#72219311	D₄	1	2	yes	—
19	#72219301	D₄	1	2	yes	—
20	#72219301	D₄	1	2	yes	—
21	C₂×#72217546	C₂²	1	1	yes	characteristic
22	C₂×#374022	C₂×C₄	1	1	yes	characteristic
23	C₂×#359428	C₂×C₄	1	2	yes	—
24	C₂×#7670203	C₂³	1	1	yes	characteristic
25	#72218836	C₄	1	2	yes	complemented
26	C₂×#72210547	C₄	1	2	yes	complemented
27	#72218518	C₂×C₄	1	2	yes	—
28	#72218518	C₂×C₄	1	2	yes	—
29	#72218518	C₂×C₄	1	2	yes	—
30	#72218291	C₂×C₄	1	2	yes	—
31	C₂×#134033	C₂×C₄	1	2	yes	—
32	C₂×#133822	C₂×C₄	1	2	yes	—
33	C₂³×#2846	C₂×C₄	1	2	yes	—
34	C₂×#6342371	C₂×C₄	1	1	yes	characteristic
35	C₂⁹	C₄²	1	1	yes	characteristic, complemented
36	C₂²×#3484	C₂×D₄	1	2	yes	—
37	C₂⁴×#120	C₂²×C₄	1	2	yes	—
38	C₂²×#2846	C₂²×C₄	1	1	yes	characteristic
39	C₂×#35390	C₂×D₄	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	C₂×D₄	1	2	yes	—
44	C₂×#34198	#41	1	2	yes	—
45	C₂³×#379	C₄²	1	2	yes	complemented
46	C₂×#14679	C₄²	1	1	yes	characteristic, 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	C₂×C₄²	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	#121	1	2	yes	—
57	C₂²×#534	#121	1	2	yes	—
58	C₂²×#534	#127	1	1	yes	characteristic
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	C₄×D₄	1	2	yes	—
63	C₂³×#120	C₄×D₄	1	2	yes	—
64	C₂³×#120	C₄×D₄	1	2	yes	—
65	C₂²×#518	#123	1	2	yes	—
66	C₂⁴×C₂²⋊C₄	#117	1	2	yes	—
67	C₂×#2315	#126	1	2	yes	—
68	C₂×#2308	#121	1	2	yes	—
69	C₂×#2308	#123	1	1	yes	characteristic
70	C₂×#2308	#124	1	2	yes	—
71	C₂²×#379	#117	1	1	yes	characteristic
72	C₂²×#379	C₄×D₄	1	2	yes	—
73	C₂×#2273	#124	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	C₄×C₂²⋊C₄	1	2	yes	—
78	C₂⁷	C₄×C₂²⋊C₄	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	#386	1	2	yes	—
84	C₂⁴×D₄	#386	1	1	yes	characteristic
85	C₂⁴×D₄	#390	1	2	yes	—
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	#388	1	2	yes	—
92	C₂³×C₂²⋊C₄	#388	1	2	yes	—
93	C₂³×C₂²⋊C₄	#388	1	2	yes	—
94	C₂³×C₂²⋊C₄	#390	1	1	yes	characteristic
95	C₂⁶	#1437	1	1	yes	characteristic
96	C₂⁶	C₄×#99	1	2	yes	—
97	C₂⁶	C₄×#99	1	2	yes	—
98	C₂⁶	C₄×#99	1	2	yes	—
99	C₂⁶	#1898	1	2	yes	—
100	C₂⁶	#1913	1	1	yes	characteristic
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	#1899	1	2	yes	—
111	C₂⁴×C₄	#1913	1	1	yes	characteristic
112	C₂⁴×C₄	#1964	1	2	yes	—
113	C₂²×C₂²⋊C₄	#1964	1	2	yes	—
114	C₂⁵	#9495	1	1	yes	characteristic
115	C₂⁵	#9495	1	2	yes	—
116	C₂⁵	#9497	1	2	yes	—
117	C₂⁵	#9498	1	1	yes	characteristic
118	C₂⁵	#10935	1	1	yes	characteristic
119	C₂⁵	#10938	1	1	yes	characteristic
120	C₂⁵	#10938	1	2	yes	—
121	C₂⁵	#10960	1	1	yes	characteristic
122	C₂⁵	#10960	1	2	yes	—
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	2	yes	—
131	C₂⁴	#110401	1	1	yes	characteristic
132	C₂⁴	#139750	1	4	yes	—
133	C₂³	#72246299	1	2	yes	—
134	C₂³	#72256001	1	2	yes	—
135	C₂³	#72264951	1	1	yes	characteristic
136	C₂³	#72265908	1	2	yes	—
137	C₂²	#72265793	1	2	yes	—
138	C₂²	#72265909	1	1	yes	characteristic
139	C₂	#72266681	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266681
commutator	C₂⁸	C₂×C₄²
Frattini, cosocle	C₂×#7670203	C₂³
prime radical	C₂×#72217546	C₂²
LCS	G ▷ C₂⁸ ▷ 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
1. #72323220	C₂	semidirect
2. #72314603	C₂	semidirect
3. #72302114	C₂	semidirect
5. #72265825	C₄	semidirect
7. #72265763	C₄	semidirect
8. #72265678	C₄	semidirect
11. #72226350	C₄	semidirect
25. #72218836	C₄	semidirect
26. C₂×#72210547	C₄	semidirect
35. C₂⁹	C₄²	semidirect
45. C₂³×#379	C₄²	semidirect
46. C₂×#14679	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
#72323220	2
#72314603	2
#72302114	2
#72265845	2
#72226360	2