Subgroups — #72316589

Order 4096.

At least 11456 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

The subgroup lattice has 172 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72316571	C₂	1	4	yes	maximal, complemented
2	#72316565	C₂	1	4	yes	maximal, complemented
3	#72314618	C₂	1	1	yes	maximal, characteristic, complemented
4	#72314614	C₂²	1	4	yes	complemented
5	#72314165	C₂	1	1	yes	maximal, characteristic, complemented
6	#72314150	C₂²	1	2	yes	complemented
7	#72313556	C₂	1	1	yes	maximal, characteristic, complemented
8	#72309804	C₂	1	1	yes	maximal, characteristic, complemented
9	#72309789	C₂²	1	2	yes	complemented
10	#72309784	C₂²	1	2	yes	complemented
11	#72265766	C₂²	1	1	yes	characteristic
12	#72258757	C₂	1	1	yes	maximal, characteristic, complemented
13	#72258606	C₂²	1	2	yes	complemented
14	#72237596	C₂	1	1	yes	maximal, characteristic, complemented
15	#72237593	C₂²	1	4	yes	complemented
16	#72237481	C₂	1	1	yes	maximal, characteristic, complemented
17	#72237479	C₂²	1	4	yes	complemented
18	#72237376	C₂²	1	4	yes	—
19	#72235718	C₂²	1	1	yes	characteristic
20	#72233247	C₂²	1	2	yes	complemented
21	#72233061	C₂²	1	2	yes	complemented
22	#72224136	C₂²	1	1	yes	characteristic
23	#72224136	C₂²	1	1	yes	characteristic
24	#72219311	C₂²	1	1	yes	characteristic
25	#72219310	C₂²	1	1	yes	characteristic
26	#72219257	C₂²	1	1	yes	characteristic
27	#10586404	D₄	1	1	yes	characteristic
28	#10586404	D₄	1	1	yes	characteristic, complemented
29	C₂²×#3484	D₄	1	1	yes	characteristic, complemented
30	C₂×#34980	C₂³	1	1	yes	characteristic
31	#6499871	C₂³	1	2	yes	—
32	#6499870	C₂³	1	2	yes	—
33	#6499869	D₄	1	2	yes	complemented
34	#6499867	D₄	1	2	yes	—
35	#6374128	D₄	1	1	yes	characteristic
36	#6374128	D₄	1	1	yes	characteristic, complemented
37	#4930495	D₄	1	1	yes	characteristic, complemented
38	#4930495	C₂³	1	1	yes	characteristic
39	#4930494	D₄	1	1	yes	characteristic, complemented
40	#4930494	C₂³	1	1	yes	characteristic
41	C₂²×#1438	D₄	1	1	yes	characteristic, complemented
42	#190147	D₄	1	2	yes	complemented
43	#190116	D₄	1	2	yes	complemented
44	C₂×#8547	D₄	1	1	yes	characteristic, complemented
45	C₂×#8541	D₄	1	1	yes	characteristic, complemented
46	#134036	C₂³	1	2	yes	—
47	#133825	D₄	1	2	yes	complemented
48	#133825	C₂³	1	2	yes	—
49	#115697	D₄	1	2	yes	complemented
50	#115690	D₄	1	2	yes	complemented
51	#115633	C₂³	1	2	yes	—
52	#115631	C₂³	1	2	yes	—
53	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
54	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
55	#62820	C₂×D₄	1	2	yes	—
56	#62817	C₂×D₄	1	2	yes	—
57	C₂²×#534	C₂×D₄	1	1	yes	characteristic
58	C₂²×#534	C₂×D₄	1	1	yes	characteristic
59	C₂³×#120	C₂×D₄	1	1	yes	characteristic
60	C₂²×#518	C₂×D₄	1	1	yes	characteristic, complemented
61	C₂×#2849	C₂×D₄	1	1	yes	characteristic
62	C₂×#2849	C₂×D₄	1	1	yes	characteristic
63	C₂×#2846	C₂×D₄	1	1	yes	characteristic
64	C₂×#2846	C₂×D₄	1	1	yes	characteristic
65	C₂×#2613	C₂×D₄	1	1	yes	characteristic
66	C₂×#2613	C₂×D₄	1	1	yes	characteristic
67	C₂×#2613	#41	1	1	yes	characteristic
68	C₂×#2613	C₂⁴	1	1	yes	characteristic
69	C₂×#2308	C₂×D₄	1	1	yes	characteristic
70	C₂×#2308	C₂×D₄	1	1	yes	characteristic
71	C₂²×#379	#41	1	1	yes	characteristic
72	#21747	C₂×D₄	1	2	yes	—
73	#21746	C₂×D₄	1	2	yes	—
74	C₂×#1438	C₂×D₄	1	1	yes	characteristic, complemented
75	C₂×#1438	C₂×D₄	1	1	yes	characteristic, complemented
76	#8547	C₂×D₄	1	2	yes	—
77	#8541	C₂×D₄	1	2	yes	—
78	#8114	C₂×D₄	1	4	yes	complemented
79	#8101	C₂×D₄	1	4	yes	complemented
80	C₂⁷	#120	1	1	yes	characteristic
81	C₂⁷	#121	1	1	yes	characteristic
82	C₂⁴×D₄	#120	1	1	yes	characteristic
83	C₂⁴×D₄	#120	1	1	yes	characteristic
84	C₂⁴×D₄	#120	1	1	yes	characteristic
85	C₂⁴×D₄	#121	1	1	yes	characteristic
86	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
87	C₂⁵×C₄	#120	1	1	yes	characteristic
88	C₂²×#123	#120	1	1	yes	characteristic
89	C₂²×#123	#120	1	1	yes	characteristic
90	C₂²×#123	#121	1	1	yes	characteristic
91	C₂²×#123	C₂²×D₄	1	1	yes	characteristic
92	C₂²×#120	#120	1	1	yes	characteristic
93	C₂²×#120	#120	1	1	yes	characteristic
94	C₂²×#120	#120	1	1	yes	characteristic
95	C₂²×#120	#121	1	1	yes	characteristic
96	C₂²×#120	#121	1	1	yes	characteristic
97	C₂²×#120	#127	1	1	yes	characteristic
98	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
100	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
103	C₂×#381	#120	1	1	yes	characteristic
104	C₂×#381	#121	1	1	yes	characteristic
105	C₂×#381	#127	1	1	yes	characteristic
106	C₂×#381	C₂²×D₄	1	1	yes	characteristic
107	C₂×#379	#120	1	1	yes	characteristic
108	C₂×#379	#121	1	1	yes	characteristic
109	C₂×#379	#121	1	1	yes	characteristic
110	C₂×#379	#121	1	1	yes	characteristic
111	C₂×#379	#121	1	1	yes	characteristic
112	C₂×#379	#127	1	1	yes	characteristic
113	C₂×#379	C₂²×D₄	1	1	yes	characteristic
114	C₂×#379	C₂×#41	1	1	yes	characteristic
115	C₂⁶	#392	1	1	yes	characteristic
116	C₂⁶	#392	1	1	yes	characteristic
117	C₂⁶	#392	1	1	yes	characteristic
118	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
119	C₂⁶	C₂×#120	1	1	yes	characteristic
120	C₂⁶	C₂×#121	1	1	yes	characteristic
121	C₂³×D₄	#392	1	1	yes	characteristic
122	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
123	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
124	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
125	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
126	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
127	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
128	C₂³×D₄	C₂×#120	1	1	yes	characteristic
129	C₂⁴×C₄	#392	1	1	yes	characteristic
130	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
131	C₂⁴×C₄	C₂×#120	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
133	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
134	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
135	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
136	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
137	C₂²×C₂²⋊C₄	C₂×#127	1	1	yes	characteristic
138	C₂⁵	#2021	1	1	yes	characteristic
139	C₂⁵	#2021	1	1	yes	characteristic
140	C₂⁵	#2021	1	2	yes	—
141	C₂⁵	#2021	1	1	yes	characteristic
142	C₂⁵	#2021	1	1	yes	characteristic
143	C₂⁵	#2021	1	1	yes	characteristic
144	C₂⁵	#2021	1	1	yes	characteristic
145	C₂⁵	#2021	1	1	yes	characteristic
146	C₂⁵	C₂×#392	1	1	yes	characteristic
147	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
148	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
149	C₂³×C₄	#2021	1	2	yes	—
150	C₂³×C₄	#2021	1	1	yes	characteristic
151	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
152	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
153	C₂⁴	#10336	1	1	yes	characteristic
154	C₂⁴	#10336	1	1	yes	characteristic
155	C₂⁴	#10336	1	1	yes	characteristic
156	C₂⁴	#10336	1	1	yes	characteristic
157	C₂⁴	C₂×#2021	1	1	yes	characteristic
158	C₂⁴	C₂×#2021	1	1	yes	characteristic
159	C₂⁴	C₂×#2021	1	1	yes	characteristic
160	C₂⁴	#33551	1	4	yes	—
161	C₂²×C₄	#33551	1	4	yes	—
162	C₂³	#144378	1	1	yes	characteristic
163	C₂³	#144378	1	1	yes	characteristic
164	C₂³	C₂×#10336	1	1	yes	characteristic
165	C₂³	#498886	1	2	yes	—
166	C₂³	#498886	1	2	yes	—
167	C₂²	C₂×#144378	1	1	yes	characteristic
168	C₂²	#72218838	1	1	yes	characteristic
169	C₂²	#72218838	1	1	yes	characteristic
170	C₂	#72266233	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266233
commutator, Frattini, cosocle	C₂×#2613	C₂⁴
LCS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72316571	C₂	semidirect
2. #72316565	C₂	semidirect
3. #72314618	C₂	semidirect
4. #72314614	C₂²	semidirect
5. #72314165	C₂	semidirect
6. #72314150	C₂²	semidirect
7. #72313556	C₂	semidirect
8. #72309804	C₂	semidirect
9. #72309789	C₂²	semidirect
10. #72309784	C₂²	semidirect
12. #72258757	C₂	semidirect
13. #72258606	C₂²	semidirect
14. #72237596	C₂	semidirect
15. #72237593	C₂²	semidirect
16. #72237481	C₂	semidirect
17. #72237479	C₂²	semidirect
20. #72233247	C₂²	semidirect
21. #72233061	C₂²	semidirect
28. #10586404	D₄	semidirect
29. C₂²×#3484	D₄	semidirect
33. #6499869	D₄	semidirect
36. #6374128	D₄	semidirect
37. #4930495	D₄	semidirect
39. #4930494	D₄	semidirect
41. C₂²×#1438	D₄	semidirect
42. #190147	D₄	semidirect
43. #190116	D₄	semidirect
44. C₂×#8547	D₄	semidirect
45. C₂×#8541	D₄	semidirect
47. #133825	D₄	semidirect
49. #115697	D₄	semidirect
50. #115690	D₄	semidirect
53. C₂³×H(5,𝔽₂)	C₂×D₄	semidirect
60. C₂²×#518	C₂×D₄	semidirect
74. C₂×#1438	C₂×D₄	semidirect
75. C₂×#1438	C₂×D₄	semidirect
78. #8114	C₂×D₄	semidirect
79. #8101	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72316571	2
#72316565	2
#72314618	2
#72314165	2
#72313556	2
#72309804	2
#72258757	2
#72237596	2
#72237481	2