Subgroups — #72337693

Order 4096.

At least 14672 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

The subgroup lattice has 142 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72337692	C₂	1	1	yes	maximal, characteristic, complemented
2	#72337691	C₂	1	2	yes	maximal, complemented
3	#72337686	C₂	1	1	yes	maximal, characteristic, complemented
4	#72337685	C₂	1	1	yes	maximal, characteristic, complemented
5	#72337684	C₂²	1	1	yes	characteristic, complemented
6	#72337683	C₂²	1	1	yes	characteristic, complemented
7	#72337682	C₂	1	2	yes	maximal, complemented
8	#72337681	C₂²	1	2	yes	complemented
9	#72337680	C₂²	1	2	yes	complemented
10	#72337679	C₂²	1	1	yes	characteristic, complemented
11	#72241412	C₂	1	2	yes	maximal, complemented
12	#72241411	C₂	1	2	yes	maximal, complemented
13	#72241312	C₂	1	1	yes	maximal, characteristic, complemented
14	#72241307	C₂²	1	2	yes	complemented
15	#72241306	C₂	1	1	yes	maximal, characteristic
16	#72228070	C₂²	1	2	yes	complemented
17	#72227589	C₂²	1	2	yes	complemented
18	#72227588	C₂²	1	1	yes	characteristic
19	#72226293	C₂²	1	2	yes	—
20	#72219530	C₂	1	1	yes	maximal, characteristic, complemented
21	#72219524	C₂	1	1	yes	maximal, characteristic, complemented
22	#72215968	C₂²	1	2	yes	—
23	#72213758	C₂²	1	1	yes	characteristic, complemented
24	#72213746	C₂²	1	2	yes	complemented
25	#72213651	C₂²	1	2	yes	complemented
26	#72213638	C₂²	1	2	yes	complemented
27	#72213500	C₂²	1	1	yes	characteristic, complemented
28	#72213423	C₂²	1	1	yes	characteristic
29	#72213406	C₂²	1	2	yes	complemented
30	#72213400	C₂²	1	2	yes	complemented
31	#72213269	C₂²	1	1	yes	characteristic, complemented
32	#72213260	C₂²	1	1	yes	characteristic, complemented
33	#72210529	C₂²	1	1	yes	characteristic, complemented
34	#72210524	C₂²	1	1	yes	characteristic
35	H(9,𝔽₂)	D₄	1	1	yes	characteristic
36	#10576178	D₄	1	2	yes	—
37	#8933848	C₂³	1	1	yes	characteristic, complemented
38	#7672751	D₄	1	1	yes	characteristic
39	#6628901	C₂³	1	1	yes	characteristic
40	#6628872	C₂³	1	2	yes	—
41	#6437630	D₄	1	1	yes	characteristic
42	#6437610	D₄	1	2	yes	—
43	#496222	C₂³	1	2	yes	complemented
44	#449492	D₄	1	1	yes	characteristic
45	#449492	C₂³	1	1	yes	characteristic
46	#449491	D₄	1	2	yes	—
47	#418909	D₄	1	2	yes	—
48	#418906	D₄	1	2	yes	—
49	#418884	D₄	1	2	yes	—
50	#417807	D₄	1	1	yes	characteristic
51	#417807	C₂³	1	1	yes	characteristic
52	#417805	D₄	1	2	yes	—
53	#417798	C₂³	1	2	yes	—
54	#392138	D₄	1	2	yes	—
55	#392136	D₄	1	1	yes	characteristic
56	#374020	D₄	1	2	yes	—
57	#374009	D₄	1	1	yes	characteristic
58	#356069	D₄	1	2	yes	—
59	#356056	D₄	1	1	yes	characteristic
60	#213861	C₂³	1	1	yes	characteristic, complemented
61	#134056	C₂³	1	2	yes	complemented
62	#133835	C₂³	1	1	yes	characteristic, complemented
63	#97460	C₂³	1	1	yes	characteristic, complemented
64	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
65	C₂×H(7,𝔽₂)	C₂×D₄	1	2	yes	—
66	#60644	C₂×D₄	1	2	yes	—
67	C₂×#3021	C₂×D₄	1	2	yes	—
68	C₂×#3021	C₂×D₄	1	2	yes	—
69	C₂×#3019	C₂×D₄	1	2	yes	—
70	C₂×#2867	C₂×D₄	1	1	yes	characteristic
71	#35527	C₂×D₄	1	1	yes	characteristic
72	#35527	C₂×D₄	1	1	yes	characteristic
73	#35527	C₂×D₄	1	1	yes	characteristic
74	#35527	C₂⁴	1	1	yes	characteristic
75	#23445	C₂×D₄	1	2	yes	—
76	#23430	C₂×D₄	1	2	yes	—
77	#23406	C₂×D₄	1	2	yes	—
78	#20615	C₂×D₄	1	2	yes	—
79	#20615	C₂×D₄	1	2	yes	—
80	#20592	C₂×D₄	1	2	yes	—
81	#20586	C₂×D₄	1	1	yes	characteristic
82	#20586	C₂×D₄	1	1	yes	characteristic
83	#20586	C₂×D₄	1	1	yes	characteristic
84	#11006	C₂×D₄	1	2	yes	—
85	#10989	C₂×D₄	1	2	yes	—
86	#10981	C₂×D₄	1	1	yes	characteristic
87	#10981	C₂×D₄	1	1	yes	characteristic
88	#10959	C₂×D₄	1	2	yes	—
89	#10939	C₂×D₄	1	2	yes	—
90	#10923	C₂×D₄	1	2	yes	—
91	C₂×#585	#120	1	2	yes	—
92	C₂×#585	#120	1	2	yes	—
93	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
94	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
95	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
96	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
97	#3449	#120	1	2	yes	—
98	C₂×#535	C₂²×D₄	1	2	yes	—
99	C₂²×#127	#120	1	1	yes	characteristic
100	C₂²×#127	C₂²×D₄	1	1	yes	characteristic
101	C₂×#413	#120	1	2	yes	—
102	C₂×#413	#120	1	2	yes	—
103	C₂²×#100	#120	1	2	yes	—
104	#2873	#120	1	2	yes	—
105	#2309	#120	1	1	yes	characteristic
106	#2309	#120	1	1	yes	characteristic
107	#2309	#120	1	1	yes	characteristic
108	#2309	C₂²×D₄	1	1	yes	characteristic
109	#2297	C₂²×D₄	1	2	yes	—
110	C₂×C₄³	#120	1	1	yes	characteristic
111	#1742	#120	1	2	yes	—
112	#1740	#120	1	2	yes	—
113	#1728	#120	1	2	yes	—
114	C₂×2₋¹⁺⁴	B(4,𝔽₂)	1	4	yes	—
115	C₂²×#41	C₂×#120	1	2	yes	—
116	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
117	C₂³×D₄	C₂×#120	1	1	yes	characteristic
118	C₂³×D₄	C₂×#120	1	1	yes	characteristic
119	C₂×#117	C₂×#120	1	2	yes	—
120	C₂²×C₄²	C₂×#120	1	1	yes	characteristic
121	C₂⁵	#2196	1	2	yes	—
122	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
123	C₂⁵	#3026	1	1	yes	characteristic
124	2₋¹⁺⁴	#2196	1	8	yes	—
125	C₂²×Q₈	C₂×B(4,𝔽₂)	1	2	yes	—
126	C₂²×D₄	#3026	1	2	yes	—
127	C₂³×C₄	#2196	1	2	yes	—
128	C₂³×C₄	#2846	1	1	yes	characteristic
129	C₂⁴	#32898	1	1	yes	characteristic
130	C₂⁴	C₂×#2196	1	2	yes	—
131	C₂×Q₈	C₂×#2196	1	4	yes	—
132	C₂²×C₄	#32898	1	2	yes	—
133	C₂³	#347042	1	1	yes	characteristic
134	C₂³	#511771	1	1	yes	characteristic
135	C₂³	#511790	1	1	yes	characteristic
136	C₂×C₄	#511771	1	2	yes	—
137	C₂×C₄	#511790	1	2	yes	—
138	C₂²	#72215683	1	2	yes	—
139	C₂²	#72215683	1	1	yes	characteristic
140	C₂	#72215688	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72215688
commutator, Frattini, cosocle	#35527	C₂⁴
prime radical	#72241306	C₂
LCS	G ▷ #35527 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35527 ◁ G
LECS	G ▷ #35527 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72337692	C₂	semidirect
2. #72337691	C₂	semidirect
3. #72337686	C₂	semidirect
4. #72337685	C₂	semidirect
5. #72337684	C₂²	semidirect
6. #72337683	C₂²	semidirect
7. #72337682	C₂	semidirect
8. #72337681	C₂²	semidirect
9. #72337680	C₂²	semidirect
10. #72337679	C₂²	semidirect
11. #72241412	C₂	semidirect
12. #72241411	C₂	semidirect
13. #72241312	C₂	semidirect
14. #72241307	C₂²	semidirect
16. #72228070	C₂²	semidirect
17. #72227589	C₂²	semidirect
20. #72219530	C₂	semidirect
21. #72219524	C₂	semidirect
23. #72213758	C₂²	semidirect
24. #72213746	C₂²	semidirect
25. #72213651	C₂²	semidirect
26. #72213638	C₂²	semidirect
27. #72213500	C₂²	semidirect
29. #72213406	C₂²	semidirect
30. #72213400	C₂²	semidirect
31. #72213269	C₂²	semidirect
32. #72213260	C₂²	semidirect
33. #72210529	C₂²	semidirect
37. #8933848	C₂³	semidirect
43. #496222	C₂³	semidirect
60. #213861	C₂³	semidirect
61. #134056	C₂³	semidirect
62. #133835	C₂³	semidirect
63. #97460	C₂³	semidirect

Maximal subgroups

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

subgroup	index
#72337692	2
#72337691	2
#72337686	2
#72337685	2
#72337682	2
#72241412	2
#72241411	2
#72241312	2
#72241306	2
#72219530	2
#72219524	2