Subgroups — #72340433

Order 4096.

Only the 225 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 153 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72340432	C₂	1	1	yes	maximal, characteristic, complemented
2	#72337586	C₂	1	2	yes	maximal, complemented
3	#72337584	C₂	1	2	yes	maximal, complemented
4	#72337582	C₂	1	2	yes	maximal, complemented
5	#72337576	C₂	1	2	yes	maximal, complemented
6	#72337570	C₂	1	1	yes	maximal, characteristic, complemented
7	#72337568	C₂	1	1	yes	maximal, characteristic, complemented
8	#72311150	C₂²	1	2	yes	complemented
9	#72284823	C₂²	1	2	yes	—
10	#72241085	C₂	1	1	yes	maximal, characteristic, complemented
11	#72241006	C₂²	1	2	yes	—
12	#72241005	C₂²	1	1	yes	characteristic
13	#72236992	C₂	1	1	yes	maximal, characteristic, complemented
14	#72236927	C₂	1	1	yes	maximal, characteristic, complemented
15	#72236925	C₂²	1	1	yes	characteristic
16	#72236925	C₂²	1	2	yes	complemented
17	#72236925	C₂²	1	1	yes	characteristic, complemented
18	#72227588	C₂²	1	1	yes	characteristic, complemented
19	#72227300	C₂²	1	2	yes	complemented
20	#72227300	C₂²	1	2	yes	complemented
21	#72227297	C₂	1	1	yes	maximal, characteristic, complemented
22	#72226369	C₂²	1	1	yes	characteristic
23	#72226369	C₂²	1	1	yes	characteristic
24	#72226369	C₂²	1	2	yes	complemented
25	#72226369	C₂²	1	2	yes	complemented
26	#72226369	C₂²	1	2	yes	complemented
27	#72216396	C₂²	1	1	yes	characteristic
28	#72213462	C₂²	1	2	yes	complemented
29	#72213462	C₂²	1	1	yes	characteristic, complemented
30	#72210439	C₂²	1	1	yes	characteristic
31	#72210035	C₂²	1	2	yes	complemented
32	#72210025	C₂²	1	1	yes	characteristic
33	#72210025	C₂²	1	1	yes	characteristic, complemented
34	#72210025	C₂²	1	2	yes	complemented
35	H(9,𝔽₂)	C₂³	1	1	yes	characteristic
36	#8934366	D₄	1	1	yes	characteristic, complemented
37	#8932885	D₄	1	1	yes	characteristic, complemented
38	#8932880	D₄	1	1	yes	characteristic, complemented
39	#8926849	D₄	1	2	yes	complemented
40	#8926668	D₄	1	2	yes	complemented
41	#6628901	C₂³	1	1	yes	characteristic
42	#6628897	C₂³	1	2	yes	—
43	#6628896	C₂³	1	1	yes	characteristic
44	#6628896	C₂³	1	1	yes	characteristic
45	#6628896	C₂³	1	2	yes	—
46	#392136	C₂³	1	1	yes	characteristic
47	#392135	C₂³	1	2	yes	—
48	#392135	C₂³	1	2	yes	—
49	#392135	C₂³	1	1	yes	characteristic
50	#392135	C₂³	1	1	yes	characteristic
51	#213860	D₄	1	1	yes	characteristic, complemented
52	#213860	D₄	1	2	yes	complemented
53	#213855	D₄	1	2	yes	complemented
54	#213854	D₄	1	1	yes	characteristic, complemented
55	#213854	D₄	1	1	yes	characteristic, complemented
56	#213854	D₄	1	2	yes	complemented
57	#213830	D₄	1	2	yes	complemented
58	#213830	D₄	1	2	yes	complemented
59	#213828	D₄	1	1	yes	characteristic, complemented
60	#213828	D₄	1	2	yes	complemented
61	#213828	D₄	1	1	yes	characteristic, complemented
62	#213828	D₄	1	2	yes	complemented
63	#213828	D₄	1	2	yes	complemented
64	C₂×H(7,𝔽₂)	C₂⁴	1	1	yes	characteristic
65	#35527	C₂×D₄	1	2	yes	—
66	#35527	C₂×D₄	1	1	yes	characteristic
67	#35446	C₂×D₄	1	2	yes	—
68	#35446	C₂×D₄	1	2	yes	—
69	#35446	C₂×D₄	1	2	yes	—
70	#35346	C₂×D₄	1	1	yes	characteristic
71	#35346	C₂×D₄	1	1	yes	characteristic
72	#35346	C₂×D₄	1	2	yes	—
73	#35346	C₂×D₄	1	2	yes	—
74	#35346	C₂×D₄	1	2	yes	—
75	#35346	C₂×D₄	1	2	yes	—
76	#35346	C₂×D₄	1	1	yes	characteristic
77	#35346	C₂×D₄	1	1	yes	characteristic
78	#20586	C₂×D₄	1	1	yes	characteristic
79	#20586	C₂×D₄	1	2	yes	—
80	#20585	C₂×D₄	1	1	yes	characteristic
81	#20585	C₂×D₄	1	2	yes	—
82	#20585	C₂×D₄	1	2	yes	—
83	#20585	C₂×D₄	1	2	yes	—
84	#20585	C₂×D₄	1	1	yes	characteristic
85	#20585	C₂×D₄	1	1	yes	characteristic
86	#20585	C₂×D₄	1	1	yes	characteristic
87	#20585	C₂×D₄	1	2	yes	—
88	#20585	C₂×D₄	1	2	yes	—
89	#20585	C₂×D₄	1	2	yes	—
90	#20585	C₂×D₄	1	2	yes	—
91	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
92	C₂²×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
93	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
94	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
95	C₂²×H(5,𝔽₂)	C₂²×D₄	1	2	yes	—
96	C₂²×#127	#120	1	1	yes	characteristic
97	C₂²×#123	#120	1	2	yes	—
98	C₂²×#120	#120	1	1	yes	characteristic
99	C₂²×#120	#120	1	2	yes	—
100	C₂²×#120	#120	1	1	yes	characteristic
101	#2309	#120	1	2	yes	—
102	#2309	#120	1	1	yes	characteristic
103	#2308	#120	1	1	yes	characteristic
104	#2308	#120	1	2	yes	—
105	#2308	#120	1	1	yes	characteristic
106	#2308	#120	1	2	yes	—
107	#2308	#120	1	2	yes	—
108	#2308	#120	1	2	yes	—
109	#2308	#120	1	1	yes	characteristic
110	#2308	#120	1	1	yes	characteristic
111	#2308	#120	1	2	yes	—
112	#2308	#120	1	2	yes	—
113	#2308	#120	1	2	yes	—
114	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
115	C₂⁶	B(4,𝔽₂)	1	2	yes	—
116	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic, complemented
117	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic, complemented
118	C₂³×D₄	C₂×#120	1	1	yes	characteristic
119	C₂³×D₄	C₂×#120	1	2	yes	—
120	C₂³×D₄	C₂×#120	1	1	yes	characteristic
121	C₂³×D₄	C₂×#120	1	1	yes	characteristic
122	C₂³×D₄	C₂×#120	1	2	yes	—
123	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	complemented
124	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
126	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	1	yes	characteristic, complemented
127	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	complemented
128	C₂²×C₄²	B(4,𝔽₂)	1	1	yes	characteristic, complemented
129	C₂⁵	#2846	1	1	yes	characteristic
130	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
131	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
132	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
133	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
134	C₂⁵	#3026	1	1	yes	characteristic
135	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
136	C₂³×C₄	#3026	1	2	yes	—
137	C₂³×C₄	#3026	1	1	yes	characteristic
138	C₂³×C₄	#3026	1	1	yes	characteristic
139	C₂³×C₄	#3026	1	2	yes	—
140	C₂⁴	#32898	1	1	yes	characteristic
141	C₂⁴	#32898	1	2	yes	—
142	C₂⁴	#32898	1	1	yes	characteristic
143	C₂⁴	#32898	1	2	yes	—
144	C₂⁴	#32898	1	1	yes	characteristic
145	C₂³	#347042	1	2	yes	—
146	C₂³	#347042	1	1	yes	characteristic
147	C₂³	#347042	1	1	yes	characteristic
148	C₂³	#347042	1	1	yes	characteristic
149	C₂³	#347042	1	2	yes	—
150	C₂²	#72216432	1	1	yes	characteristic
151	C₂	#72216433	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72216433
commutator, Frattini, cosocle	C₂×H(7,𝔽₂)	C₂⁴
LCS	G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁵ ◁ C₂×H(7,𝔽₂) ◁ G
LECS	G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72340432	C₂	semidirect
2. #72337586	C₂	semidirect
3. #72337584	C₂	semidirect
4. #72337582	C₂	semidirect
5. #72337576	C₂	semidirect
6. #72337570	C₂	semidirect
7. #72337568	C₂	semidirect
8. #72311150	C₂²	semidirect
10. #72241085	C₂	semidirect
13. #72236992	C₂	semidirect
14. #72236927	C₂	semidirect
16. #72236925	C₂²	semidirect
17. #72236925	C₂²	semidirect
18. #72227588	C₂²	semidirect
19. #72227300	C₂²	semidirect
20. #72227300	C₂²	semidirect
21. #72227297	C₂	semidirect
24. #72226369	C₂²	semidirect
25. #72226369	C₂²	semidirect
26. #72226369	C₂²	semidirect
28. #72213462	C₂²	semidirect
29. #72213462	C₂²	semidirect
31. #72210035	C₂²	semidirect
33. #72210025	C₂²	semidirect
34. #72210025	C₂²	semidirect
36. #8934366	D₄	semidirect
37. #8932885	D₄	semidirect
38. #8932880	D₄	semidirect
39. #8926849	D₄	semidirect
40. #8926668	D₄	semidirect
51. #213860	D₄	semidirect
52. #213860	D₄	semidirect
53. #213855	D₄	semidirect
54. #213854	D₄	semidirect
55. #213854	D₄	semidirect
56. #213854	D₄	semidirect
57. #213830	D₄	semidirect
58. #213830	D₄	semidirect
59. #213828	D₄	semidirect
60. #213828	D₄	semidirect
61. #213828	D₄	semidirect
62. #213828	D₄	semidirect
63. #213828	D₄	semidirect
116. C₂⁶	B(4,𝔽₂)	semidirect
117. C₂³×D₄	B(4,𝔽₂)	semidirect
123. C₂⁴×C₄	B(4,𝔽₂)	semidirect
126. C₂²×C₂²⋊C₄	B(4,𝔽₂)	semidirect
127. C₂²×C₂²⋊C₄	B(4,𝔽₂)	semidirect
128. C₂²×C₄²	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72340432	2
#72337586	2
#72337584	2
#72337582	2
#72337576	2
#72337570	2
#72337568	2
#72241085	2
#72236992	2
#72236927	2
#72227297	2