Subgroups — #72311187

Order 4096.

At least 6281 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

The subgroup lattice has 140 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72311168	C₂	1	1	yes	maximal, characteristic, complemented
2	#72311155	C₂	1	1	yes	maximal, characteristic, complemented
3	#72311067	C₂	1	1	yes	maximal, characteristic, complemented
4	#72311058	C₂²	1	1	yes	characteristic
5	#72311052	C₂	1	2	yes	maximal, complemented
6	#72311044	C₂	1	2	yes	maximal, complemented
7	#72311038	C₂²	1	2	yes	—
8	#72311034	C₂	1	2	yes	maximal, complemented
9	#72311026	C₂	1	2	yes	maximal, complemented
10	#72311024	C₂²	1	2	yes	complemented
11	#72311023	C₂²	1	2	yes	—
12	#72310930	C₂²	1	1	yes	characteristic, complemented
13	#72310923	C₂	1	1	yes	maximal, characteristic, complemented
14	#72310922	C₂²	1	2	yes	complemented
15	#72310916	C₂²	1	1	yes	characteristic, complemented
16	#72310913	C₂	1	1	yes	maximal, characteristic, complemented
17	#72285275	C₂²	1	1	yes	characteristic
18	#72284657	C₂²	1	2	yes	complemented
19	#72284221	C₂²	1	2	yes	complemented
20	#72283936	C₂²	1	1	yes	characteristic, complemented
21	#72283640	C₂²	1	1	yes	characteristic, complemented
22	#72283582	C₂²	1	2	yes	complemented
23	#72283582	C₂²	1	2	yes	complemented
24	#72241362	C₂	1	1	yes	maximal, characteristic, complemented
25	#72241361	C₂	1	1	yes	maximal, characteristic, complemented
26	#72241347	C₂²	1	1	yes	characteristic
27	#72236998	C₂²	1	1	yes	characteristic
28	#72236998	C₂²	1	1	yes	characteristic
29	#72236934	C₂²	1	2	yes	—
30	#72236934	C₂²	1	2	yes	—
31	#72213804	C₂²	1	2	yes	—
32	#72213786	C₂²	1	2	yes	—
33	#72213523	C₂²	1	1	yes	characteristic
34	#72213513	C₂²	1	1	yes	characteristic
35	#8933820	D₄	1	2	yes	complemented
36	#8932958	D₄	1	1	yes	characteristic
37	#6628902	C₂³	1	1	yes	characteristic
38	#6628897	D₄	1	1	yes	characteristic
39	#6628896	D₄	1	1	yes	characteristic
40	#6437630	C₂³	1	1	yes	characteristic
41	#356056	C₂³	1	1	yes	characteristic
42	#346288	D₄	1	2	yes	complemented
43	#346284	D₄	1	1	yes	characteristic
44	#346284	D₄	1	1	yes	characteristic
45	#346267	D₄	1	2	yes	complemented
46	#346265	D₄	1	2	yes	complemented
47	#346259	D₄	1	2	yes	—
48	#346239	D₄	1	2	yes	—
49	#251833	D₄	1	2	yes	complemented
50	#251833	C₂³	1	2	yes	—
51	#251828	D₄	1	2	yes	complemented
52	#251828	C₂³	1	2	yes	—
53	#245243	D₄	1	2	yes	complemented
54	#245236	D₄	1	1	yes	characteristic
55	#245225	D₄	1	1	yes	characteristic
56	#213863	D₄	1	2	yes	complemented
57	#213862	D₄	1	1	yes	characteristic
58	#213855	C₂³	1	1	yes	characteristic
59	#213854	C₂³	1	1	yes	characteristic
60	#149742	C₂³	1	2	yes	—
61	#149742	C₂³	1	2	yes	—
62	#128062	C₂³	1	1	yes	characteristic
63	#128062	C₂³	1	1	yes	characteristic
64	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
65	#46998	C₂×D₄	1	2	yes	—
66	#46384	C₂×D₄	1	2	yes	—
67	#35616	C₂×D₄	1	1	yes	characteristic
68	#35616	C₂×D₄	1	2	yes	—
69	#35446	C₂×D₄	1	1	yes	characteristic
70	#35346	C₂×D₄	1	1	yes	characteristic
71	#20586	C₂×D₄	1	2	yes	—
72	#20586	C₂⁴	1	1	yes	characteristic
73	#20585	C₂×D₄	1	1	yes	characteristic
74	#20585	C₂×D₄	1	1	yes	characteristic
75	#10392	C₂×D₄	1	2	yes	—
76	#10392	C₂×D₄	1	2	yes	—
77	#10390	C₂×D₄	1	2	yes	—
78	#10390	C₂×D₄	1	2	yes	—
79	#10338	C₂×D₄	1	2	yes	—
80	#10338	C₂×D₄	1	2	yes	—
81	#10336	C₂×D₄	1	2	yes	—
82	#10336	C₂×D₄	1	2	yes	—
83	#8748	C₂×D₄	1	2	yes	—
84	#8748	C₂×D₄	1	2	yes	—
85	#8747	C₂×D₄	1	2	yes	—
86	#8747	C₂×D₄	1	1	yes	characteristic
87	#8747	C₂×D₄	1	1	yes	characteristic
88	#8543	C₂×D₄	1	2	yes	—
89	#8541	C₂×D₄	1	1	yes	characteristic
90	#8541	C₂×D₄	1	1	yes	characteristic
91	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
92	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
93	C₂²×#124	#120	1	1	yes	characteristic
94	C₂²×#123	#120	1	1	yes	characteristic
95	C₂²×#120	#120	1	1	yes	characteristic
96	#2842	#120	1	2	yes	—
97	#2840	#120	1	2	yes	—
98	#2483	#120	1	2	yes	—
99	#2403	#120	1	2	yes	—
100	#2309	#120	1	1	yes	characteristic
101	#2309	C₂²×D₄	1	2	yes	—
102	#2308	#120	1	1	yes	characteristic
103	#2308	#120	1	1	yes	characteristic
104	#1781	#120	1	1	yes	characteristic
105	#1781	#120	1	2	yes	—
106	#1781	#120	1	1	yes	characteristic
107	#1781	C₂²×D₄	1	2	yes	—
108	#1781	C₂²×D₄	1	1	yes	characteristic
109	#1781	C₂²×D₄	1	1	yes	characteristic
110	#1304	#120	1	2	yes	—
111	#1304	#120	1	2	yes	—
112	#1304	#120	1	2	yes	—
113	#1304	#120	1	2	yes	—
114	C₂⁶	#453	1	1	yes	characteristic
115	C₂³×Q₈	B(4,𝔽₂)	1	1	yes	characteristic
116	C₂³×D₄	C₂×#120	1	1	yes	characteristic
117	C₂³×D₄	C₂×#120	1	1	yes	characteristic
118	C₂³×D₄	C₂×#120	1	1	yes	characteristic
119	C₂⁴×C₄	#453	1	1	yes	characteristic
120	C₂²×C₂²⋊C₄	#453	1	1	yes	characteristic
121	C₂²×C₂²⋊C₄	#453	1	1	yes	characteristic
122	C₂²×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
123	#379	C₂×#120	1	2	yes	—
124	#379	C₂×#120	1	2	yes	—
125	C₂⁵	#2846	1	1	yes	characteristic
126	C₂⁵	C₂×#453	1	1	yes	characteristic
127	C₂⁵	C₂×#453	1	1	yes	characteristic
128	C₂⁵	#3026	1	1	yes	characteristic
129	C₂³×C₄	#3019	1	1	yes	characteristic
130	C₂³×C₄	#3019	1	1	yes	characteristic
131	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
132	C₂⁴	#32888	1	1	yes	characteristic
133	C₂⁴	#32888	1	1	yes	characteristic
134	C₂⁴	#32898	1	1	yes	characteristic
135	C₂³	#346446	1	1	yes	characteristic
136	C₂³	#511790	1	2	yes	—
137	C₂²	#72216464	1	1	yes	characteristic
138	C₂	#72216466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72216466
commutator, Frattini, cosocle	#20586	C₂⁴
LCS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6628902 ◁ G
LECS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72311168	C₂	semidirect
2. #72311155	C₂	semidirect
3. #72311067	C₂	semidirect
5. #72311052	C₂	semidirect
6. #72311044	C₂	semidirect
8. #72311034	C₂	semidirect
9. #72311026	C₂	semidirect
10. #72311024	C₂²	semidirect
12. #72310930	C₂²	semidirect
13. #72310923	C₂	semidirect
14. #72310922	C₂²	semidirect
15. #72310916	C₂²	semidirect
16. #72310913	C₂	semidirect
18. #72284657	C₂²	semidirect
19. #72284221	C₂²	semidirect
20. #72283936	C₂²	semidirect
21. #72283640	C₂²	semidirect
22. #72283582	C₂²	semidirect
23. #72283582	C₂²	semidirect
24. #72241362	C₂	semidirect
25. #72241361	C₂	semidirect
35. #8933820	D₄	semidirect
42. #346288	D₄	semidirect
45. #346267	D₄	semidirect
46. #346265	D₄	semidirect
49. #251833	D₄	semidirect
51. #251828	D₄	semidirect
53. #245243	D₄	semidirect
56. #213863	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72311168	2
#72311155	2
#72311067	2
#72311052	2
#72311044	2
#72311034	2
#72311026	2
#72310923	2
#72310913	2
#72241362	2
#72241361	2