Subgroups — #72337563

Order 2048.

Only the 306 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 149 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72337559	C₂	1	2	yes	maximal, complemented
2	#72311599	C₂	1	2	yes	maximal, complemented
3	#72311150	C₂	1	1	yes	maximal, characteristic, complemented
4	#72311058	C₂	1	1	yes	maximal, characteristic, complemented
5	#72310930	C₂	1	1	yes	maximal, characteristic, complemented
6	#72303055	C₂	1	2	yes	maximal, complemented
7	#72238013	C₂	1	2	yes	maximal, complemented
8	#72236925	C₂	1	1	yes	maximal, characteristic, complemented
9	#72227731	C₂	1	1	yes	maximal, characteristic, complemented
10	#72210035	C₂	1	1	yes	maximal, characteristic, complemented
11	#72210025	C₂	1	1	yes	maximal, characteristic
12	H(9,𝔽₂)	C₂²	1	1	yes	characteristic
13	#8932958	C₂²	1	1	yes	characteristic, complemented
14	#6628902	C₂²	1	1	yes	characteristic
15	#6628897	C₂²	1	1	yes	characteristic
16	#6628897	C₂²	1	2	yes	—
17	#6628896	C₂²	1	1	yes	characteristic
18	#6628896	C₂²	1	2	yes	—
19	#6628873	C₂²	1	2	yes	—
20	#496241	C₂²	1	2	yes	complemented
21	#496240	C₂²	1	2	yes	complemented
22	#496205	C₂²	1	2	yes	complemented
23	#392138	C₂²	1	2	yes	—
24	#392136	C₂²	1	1	yes	characteristic
25	#392135	C₂²	1	1	yes	characteristic
26	#392135	C₂²	1	2	yes	—
27	#392135	C₂²	1	1	yes	characteristic
28	#392135	C₂²	1	2	yes	—
29	#213862	C₂²	1	1	yes	characteristic, complemented
30	#213855	C₂²	1	1	yes	characteristic, complemented
31	#213854	C₂²	1	1	yes	characteristic, complemented
32	#128079	C₂²	1	2	yes	complemented
33	#128079	C₂²	1	2	yes	complemented
34	#128062	C₂²	1	2	yes	complemented
35	C₂×H(7,𝔽₂)	C₂³	1	1	yes	characteristic
36	C₂×H(7,𝔽₂)	C₂³	1	2	yes	—
37	C₂²×#458	D₄	1	4	yes	complemented
38	C₂²×B(4,𝔽₂)	D₄	1	4	yes	—
39	C₂×#3022	D₄	1	4	yes	complemented
40	#35616	D₄	1	2	yes	complemented
41	#35616	C₂³	1	1	yes	characteristic
42	#35446	D₄	1	2	yes	complemented
43	#35446	C₂³	1	1	yes	characteristic
44	#35346	D₄	1	2	yes	—
45	#35346	C₂³	1	1	yes	characteristic
46	#20615	D₄	1	4	yes	complemented
47	#20609	D₄	1	4	yes	—
48	#20609	D₄	1	4	yes	complemented
49	#20592	C₂³	1	2	yes	—
50	#20586	D₄	1	2	yes	complemented
51	#20586	C₂³	1	1	yes	characteristic
52	#20585	D₄	1	2	yes	—
53	#20585	D₄	1	2	yes	complemented
54	#20585	C₂³	1	1	yes	characteristic
55	#20585	C₂³	1	1	yes	characteristic
56	#20585	C₂³	1	2	yes	—
57	#20585	C₂³	1	2	yes	—
58	C₂×#585	C₂×D₄	1	4	yes	—
59	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
60	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
61	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
62	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
63	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
64	C₂²×H(5,𝔽₂)	C₂⁴	1	1	yes	characteristic
65	C₂×#536	C₂×D₄	1	2	yes	—
66	C₂²×#124	C₂×D₄	1	1	yes	characteristic
67	C₂²×#123	C₂×D₄	1	1	yes	characteristic
68	C₂²×#123	C₂×D₄	1	2	yes	—
69	C₂²×#120	C₂×D₄	1	1	yes	characteristic
70	C₂²×#120	C₂×D₄	1	2	yes	—
71	C₂×#413	C₂×D₄	1	4	yes	—
72	C₂²×#99	C₂×D₄	1	4	yes	—
73	C₂²×#99	C₂×D₄	1	4	yes	—
74	#2309	C₂×D₄	1	1	yes	characteristic
75	#2309	C₂×D₄	1	2	yes	—
76	#2308	C₂×D₄	1	1	yes	characteristic
77	#2308	C₂×D₄	1	2	yes	—
78	#2308	C₂×D₄	1	1	yes	characteristic
79	#2308	C₂×D₄	1	2	yes	—
80	#2308	C₂×D₄	1	2	yes	—
81	#2308	C₂×D₄	1	2	yes	—
82	#2297	C₂×D₄	1	2	yes	—
83	C₂⁶	#120	1	1	yes	characteristic
84	C₂×H(5,𝔽₂)	#120	1	8	yes	—
85	C₂²×#41	C₂²×D₄	1	2	yes	—
86	C₂²×#41	H(5,𝔽₂)	1	2	yes	—
87	C₂²×#41	H(5,𝔽₂)	1	2	yes	—
88	C₂³×Q₈	#120	1	1	yes	characteristic, complemented
89	C₂³×D₄	#120	1	4	yes	—
90	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
91	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
92	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
93	C₂³×D₄	C₂²×D₄	1	2	yes	—
94	C₂³×D₄	C₂²×D₄	1	2	yes	—
95	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
96	C₂⁴×C₄	#120	1	1	yes	characteristic, complemented
97	C₂×#117	#120	1	2	yes	complemented
98	C₂²×C₂²⋊C₄	#120	1	2	yes	—
99	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
100	C₂²×C₂²⋊C₄	#120	1	2	yes	complemented
101	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic, complemented
102	C₂²×C₄²	#120	1	1	yes	characteristic, complemented
103	C₂⁵	B(4,𝔽₂)	1	4	yes	—
104	C₂⁵	C₂×#120	1	1	yes	characteristic
105	C₂⁵	C₂×#120	1	1	yes	characteristic
106	C₂⁵	C₂×#120	1	1	yes	characteristic
107	C₂⁵	#534	1	1	yes	characteristic
108	C₂⁵	D₄²	1	2	yes	—
109	H(5,𝔽₂)	B(4,𝔽₂)	1	16	yes	—
110	C₂²×Q₈	#534	1	2	yes	—
111	C₂²×D₄	C₂×#120	1	2	yes	—
112	C₂²×D₄	#534	1	2	yes	—
113	C₂²×D₄	#534	1	2	yes	—
114	C₂²×D₄	D₄²	1	2	yes	—
115	C₂²×D₄	D₄²	1	2	yes	—
116	C₂²×D₄	D₄²	1	2	yes	—
117	C₂²×D₄	D₄²	1	2	yes	—
118	C₂²×D₄	D₄²	1	2	yes	—
119	C₂²×D₄	D₄²	1	2	yes	—
120	C₂³×C₄	B(4,𝔽₂)	1	4	yes	—
121	C₂³×C₄	C₂×#120	1	1	yes	characteristic
122	C₂³×C₄	#534	1	1	yes	characteristic
123	C₂³×C₄	#534	1	1	yes	characteristic
124	C₂³×C₄	D₄²	1	2	yes	—
125	C₂³×C₄	D₄²	1	2	yes	—
126	C₂³×C₄	H(3,𝔽₄)	1	2	yes	—
127	C₂⁴	#2403	1	1	yes	characteristic
128	C₂⁴	#2403	1	1	yes	characteristic
129	C₂⁴	#2403	1	1	yes	characteristic
130	C₂⁴	#2403	1	2	yes	—
131	C₂⁴	#2403	1	2	yes	—
132	C₂⁴	#2679	1	2	yes	—
133	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
134	C₂×D₄	C₂×B(4,𝔽₂)	1	8	yes	—
135	C₂²×C₄	#2403	1	2	yes	—
136	C₂²×C₄	#2679	1	2	yes	—
137	C₂²×C₄	#2679	1	2	yes	—
138	C₂³	#15947	1	1	yes	characteristic
139	C₂³	#23900	1	4	yes	—
140	C₂³	#23900	1	4	yes	—
141	C₂³	#23900	1	2	yes	—
142	C₂³	#23900	1	2	yes	—
143	C₂³	#23900	1	2	yes	—
144	C₂×C₄	#23900	1	4	yes	—
145	C₂²	#449475	1	2	yes	—
146	C₂²	#449475	1	1	yes	characteristic
147	C₂	#72209716	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72209716
commutator, Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂⁴
prime radical	#72210025	C₂
LCS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72337559	C₂	semidirect
2. #72311599	C₂	semidirect
3. #72311150	C₂	semidirect
4. #72311058	C₂	semidirect
5. #72310930	C₂	semidirect
6. #72303055	C₂	semidirect
7. #72238013	C₂	semidirect
8. #72236925	C₂	semidirect
9. #72227731	C₂	semidirect
10. #72210035	C₂	semidirect
13. #8932958	C₂²	semidirect
20. #496241	C₂²	semidirect
21. #496240	C₂²	semidirect
22. #496205	C₂²	semidirect
29. #213862	C₂²	semidirect
30. #213855	C₂²	semidirect
31. #213854	C₂²	semidirect
32. #128079	C₂²	semidirect
33. #128079	C₂²	semidirect
34. #128062	C₂²	semidirect
37. C₂²×#458	D₄	semidirect
39. C₂×#3022	D₄	semidirect
40. #35616	D₄	semidirect
42. #35446	D₄	semidirect
46. #20615	D₄	semidirect
48. #20609	D₄	semidirect
50. #20586	D₄	semidirect
53. #20585	D₄	semidirect
88. C₂³×Q₈	#120	semidirect
96. C₂⁴×C₄	#120	semidirect
97. C₂×#117	#120	semidirect
100. C₂²×C₂²⋊C₄	#120	semidirect
101. C₂²×C₂²⋊C₄	#120	semidirect
102. C₂²×C₄²	#120	semidirect

Maximal subgroups

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

subgroup	index
#72337559	2
#72311599	2
#72311150	2
#72311058	2
#72310930	2
#72303055	2
#72238013	2
#72236925	2
#72227731	2
#72210035	2
#72210025	2