Subgroups — #72266740

Order 8192.

Only the 354 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 199 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266723	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266689	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266681	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266474	C₂	1	2	yes	maximal, complemented
5	#72266450	C₂²	1	1	yes	characteristic
6	#72266322	C₂	1	2	yes	maximal, complemented
7	#72266310	C₂²	1	1	yes	characteristic
8	#72266303	C₂	1	2	yes	maximal, complemented
9	#72266293	C₂²	1	2	yes	complemented
10	#72266289	C₂²	1	2	yes	complemented
11	#72266248	C₂	1	1	yes	maximal, characteristic, complemented
12	#72266242	C₂²	1	2	yes	complemented
13	#72266238	C₂	1	1	yes	maximal, characteristic, complemented
14	#72266200	C₂²	1	1	yes	characteristic, complemented
15	#72266190	C₂²	1	2	yes	complemented
16	#72266186	C₂	1	1	yes	maximal, characteristic, complemented
17	#72266170	C₂²	1	1	yes	characteristic
18	#72266169	D₄	1	2	yes	complemented
19	#72266168	D₄	1	2	yes	complemented
20	#72266156	C₂²	1	1	yes	characteristic, complemented
21	#72266136	C₂²	1	2	yes	complemented
22	#72266128	C₂	1	1	yes	maximal, characteristic, complemented
23	#72266085	C₂²	1	1	yes	characteristic, complemented
24	#72262702	C₂²	1	1	yes	characteristic
25	#72262701	D₄	1	1	yes	characteristic, complemented
26	#72262637	D₄	1	1	yes	characteristic, complemented
27	#72258971	C₂	1	2	yes	maximal, complemented
28	#72258946	C₂²	1	2	yes	—
29	#72258939	C₂²	1	2	yes	—
30	#72258929	C₂²	1	1	yes	characteristic
31	#72258752	D₄	1	4	yes	complemented
32	#72258662	C₂²	1	2	yes	—
33	#72258649	D₄	1	2	yes	complemented
34	#72255646	C₂²	1	2	yes	—
35	#72240619	D₄	1	2	yes	complemented
36	C₂×#72233348	C₂²	1	1	yes	characteristic, complemented
37	#72236110	C₂²	1	1	yes	characteristic
38	#72235054	D₄	1	4	yes	complemented
39	#72234749	C₂²	1	2	yes	complemented
40	#72234675	C₂³	1	1	yes	characteristic
41	#72234668	D₄	1	1	yes	characteristic, complemented
42	#72234625	D₄	1	1	yes	characteristic, complemented
43	#72233780	D₄	1	2	yes	complemented
44	#72226359	C₂²	1	1	yes	characteristic
45	#72226349	D₄	1	2	yes	complemented
46	#72225881	C₂³	1	1	yes	characteristic
47	#72225783	D₄	1	2	yes	complemented
48	C₂×#188035	C₂³	1	1	yes	characteristic
49	#72225773	C₂²	1	2	yes	—
50	#72225737	C₂³	1	1	yes	characteristic
51	#72225711	C₂²	1	2	yes	—
52	#72225665	C₂³	1	1	yes	characteristic
53	#72225621	C₂³	1	2	yes	—
54	#72219423	D₄	1	4	yes	complemented
55	#72219377	C₂³	1	1	yes	characteristic
56	#72219377	C₂³	1	2	yes	—
57	#72219307	C₂³	1	2	yes	—
58	C₂×#7631286	C₂³	1	2	yes	—
59	C₂×#647656	C₂³	1	1	yes	characteristic
60	#72218518	D₄	1	4	yes	complemented
61	C₂×#133822	D₄	1	2	yes	complemented
62	C₂⁴×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
63	C₂×#62817	C₂×D₄	1	1	yes	characteristic
64	C₂³×D₄²	C₂⁴	1	1	yes	characteristic
65	C₂⁴×#120	C₂×D₄	1	2	yes	—
66	C₂²×#2846	C₂×D₄	1	1	yes	characteristic
67	C₂×#35527	C₂×D₄	1	1	yes	characteristic
68	C₂×#35346	C₂×D₄	1	2	yes	—
69	C₂×#34980	C₂×D₄	1	1	yes	characteristic
70	C₂²×#2403	C₂×D₄	1	2	yes	—
71	#7626202	C₂×D₄	1	1	yes	characteristic
72	C₂×#34171	C₂×D₄	1	2	yes	—
73	C₂×#21746	C₂×D₄	1	1	yes	characteristic
74	C₂×#21746	C₂×D₄	1	1	yes	characteristic
75	#556752	C₂×D₄	1	1	yes	characteristic
76	#555905	C₂×D₄	1	1	yes	characteristic
77	#555894	C₂×D₄	1	2	yes	—
78	#517411	C₂×D₄	1	2	yes	—
79	#517411	C₂×D₄	1	2	yes	—
80	#517411	C₂×D₄	1	2	yes	—
81	C₂×#14134	C₂×D₄	1	1	yes	characteristic
82	C₂×#13867	C₂×D₄	1	2	yes	—
83	#392109	C₂×D₄	1	4	yes	—
84	#392109	C₂×D₄	1	4	yes	—
85	#382422	C₂×D₄	1	4	yes	—
86	C₂×#10965	C₂×D₄	1	4	yes	—
87	C₂×#8547	C₂×D₄	1	2	yes	—
88	C₂×#8546	C₂×D₄	1	2	yes	—
89	C₂×#8546	C₂×D₄	1	2	yes	—
90	C₂×#8541	C₂×D₄	1	2	yes	—
91	C₂⁸	#127	1	1	yes	characteristic, complemented
92	C₂³×H(5,𝔽₂)	#127	1	1	yes	characteristic, complemented
93	C₂⁵×D₄	#120	1	1	yes	characteristic
94	C₂⁵×D₄	C₂²×D₄	1	2	yes	—
95	C₂⁵×D₄	C₂²×D₄	1	2	yes	—
96	C₂³×#127	C₂²×D₄	1	1	yes	characteristic
97	C₂³×#121	H(5,𝔽₂)	1	2	yes	—
98	C₂³×#121	H(5,𝔽₂)	1	2	yes	—
99	C₂³×#120	C₂²×D₄	1	1	yes	characteristic
100	C₂³×#120	C₂²×D₄	1	1	yes	characteristic
101	C₂³×#120	C₂²×D₄	1	2	yes	—
102	C₂³×C₄×D₄	H(5,𝔽₂)	1	2	yes	—
103	C₂⁴×C₂²⋊C₄	#127	1	2	yes	complemented
104	C₂×#2309	#127	1	1	yes	characteristic, complemented
105	C₂×#2308	#127	1	2	yes	complemented
106	C₂²×#379	#120	1	2	yes	—
107	C₂²×#379	#120	1	1	yes	characteristic
108	C₂²×#379	#127	1	1	yes	characteristic, complemented
109	C₂×#1781	#120	1	2	yes	—
110	C₂×#1781	#120	1	2	yes	—
111	#14679	#120	1	4	yes	—
112	#14679	#120	1	4	yes	—
113	C₂⁷	B(4,𝔽₂)	1	1	yes	characteristic
114	C₂⁷	C₂×#120	1	1	yes	characteristic
115	C₂⁷	C₂×#120	1	2	yes	—
116	C₂⁷	C₂×#127	1	1	yes	characteristic
117	C₂⁴×D₄	C₂×#120	1	1	yes	characteristic
118	C₂⁴×D₄	C₂×#127	1	1	yes	characteristic
119	C₂⁴×D₄	D₄²	1	2	yes	—
120	C₂⁴×D₄	D₄²	1	2	yes	—
121	C₂⁴×D₄	D₄²	1	2	yes	—
122	C₂⁴×D₄	D₄²	1	2	yes	—
123	C₂⁴×D₄	D₄²	1	2	yes	—
124	C₂⁴×D₄	D₄²	1	2	yes	—
125	C₂⁴×D₄	D₄²	1	2	yes	—
126	C₂⁵×C₄	B(4,𝔽₂)	1	1	yes	characteristic
127	C₂⁵×C₄	#534	1	2	yes	—
128	C₂⁵×C₄	#535	1	2	yes	—
129	C₂³×C₄⋊C₄	#560	1	2	yes	—
130	C₂³×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
131	C₂³×C₂²⋊C₄	#534	1	1	yes	characteristic
132	C₂³×C₂²⋊C₄	#535	1	1	yes	characteristic
133	C₂³×C₂²⋊C₄	#535	1	2	yes	—
134	C₂³×C₂²⋊C₄	D₄²	1	2	yes	—
135	C₂³×C₂²⋊C₄	D₄²	1	2	yes	—
136	C₂³×C₄²	#535	1	1	yes	characteristic
137	C₂×#379	B(4,𝔽₂)	1	2	yes	—
138	C₂×#379	B(4,𝔽₂)	1	8	yes	complemented
139	#1781	B(4,𝔽₂)	1	4	yes	complemented
140	C₂⁶	#2403	1	2	yes	—
141	C₂⁶	#2403	1	1	yes	characteristic
142	C₂⁶	#2433	1	2	yes	—
143	C₂⁶	#2433	1	1	yes	characteristic
144	C₂⁶	#2433	1	2	yes	—
145	C₂⁶	C₂×B(4,𝔽₂)	1	1	yes	characteristic
146	C₂⁶	C₂×B(4,𝔽₂)	1	2	yes	—
147	C₂⁶	C₂×B(4,𝔽₂)	1	1	yes	characteristic
148	C₂⁶	C₂×B(4,𝔽₂)	1	4	yes	—
149	C₂⁶	#3027	1	4	yes	—
150	C₂³×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
151	C₂³×D₄	#3027	1	2	yes	—
152	C₂⁴×C₄	#2433	1	1	yes	characteristic
153	C₂⁴×C₄	#2669	1	2	yes	—
154	C₂⁴×C₄	#2669	1	2	yes	—
155	C₂⁴×C₄	#2669	1	2	yes	—
156	C₂⁵	#15258	1	1	yes	characteristic
157	C₂⁵	#23895	1	2	yes	—
158	C₂⁵	#23895	1	1	yes	characteristic
159	C₂⁵	#23895	1	2	yes	—
160	C₂⁵	#23895	1	1	yes	characteristic
161	C₂⁵	#23900	1	2	yes	—
162	C₂⁵	#23900	1	1	yes	characteristic
163	C₂⁵	#23900	1	2	yes	—
164	C₂⁵	#23900	1	1	yes	characteristic
165	C₂⁵	#23900	1	1	yes	characteristic
166	C₂⁵	#23900	1	2	yes	—
167	C₂⁵	#23904	1	1	yes	characteristic
168	C₂⁵	#23904	1	2	yes	—
169	C₂⁵	#25771	1	2	yes	—
170	C₂⁵	#25771	1	4	yes	—
171	C₂⁵	#25771	1	4	yes	—
172	C₂³×C₄	#25771	1	2	yes	—
173	C₂⁴	#245225	1	2	yes	—
174	C₂⁴	#245225	1	1	yes	characteristic
175	C₂⁴	#245233	1	2	yes	—
176	C₂⁴	#245233	1	1	yes	characteristic
177	C₂⁴	#245249	1	2	yes	—
178	C₂⁴	#245249	1	1	yes	characteristic
179	C₂⁴	#245252	1	2	yes	—
180	C₂⁴	#245252	1	1	yes	characteristic
181	C₂⁴	#417744	1	1	yes	characteristic
182	C₂⁴	#417821	1	1	yes	characteristic
183	C₂⁴	#417821	1	2	yes	—
184	C₂⁴	#417824	1	2	yes	—
185	C₂⁴	#417824	1	1	yes	characteristic
186	C₂³	B(5,𝔽₂)	1	4	yes	—
187	C₂³	#72210578	1	2	yes	—
188	C₂³	#72210578	1	1	yes	characteristic
189	C₂³	#72234637	1	1	yes	characteristic
190	C₂³	#72237501	1	2	yes	—
191	C₂³	#72237501	1	1	yes	characteristic
192	C₂²	#72256024	1	2	yes	—
193	C₂²	#72264779	1	2	yes	—
194	C₂²	#72265084	1	1	yes	characteristic
195	C₂²	#72265989	1	2	yes	—
196	C₂	#72265820	1	2	yes	—
197	C₂	#72265990	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72265084
commutator, Frattini, cosocle	C₂³×D₄²	C₂⁴
LCS	G ▷ C₂³×D₄² ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×D₄² ◁ G
LECS	G ▷ C₂³×D₄² ▷ C₂⁵ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266723	C₂	semidirect
2. #72266689	C₂	semidirect
3. #72266681	C₂	semidirect
4. #72266474	C₂	semidirect
6. #72266322	C₂	semidirect
8. #72266303	C₂	semidirect
9. #72266293	C₂²	semidirect
10. #72266289	C₂²	semidirect
11. #72266248	C₂	semidirect
12. #72266242	C₂²	semidirect
13. #72266238	C₂	semidirect
14. #72266200	C₂²	semidirect
15. #72266190	C₂²	semidirect
16. #72266186	C₂	semidirect
18. #72266169	D₄	semidirect
19. #72266168	D₄	semidirect
20. #72266156	C₂²	semidirect
21. #72266136	C₂²	semidirect
22. #72266128	C₂	semidirect
23. #72266085	C₂²	semidirect
25. #72262701	D₄	semidirect
26. #72262637	D₄	semidirect
27. #72258971	C₂	semidirect
31. #72258752	D₄	semidirect
33. #72258649	D₄	semidirect
35. #72240619	D₄	semidirect
36. C₂×#72233348	C₂²	semidirect
38. #72235054	D₄	semidirect
39. #72234749	C₂²	semidirect
41. #72234668	D₄	semidirect
42. #72234625	D₄	semidirect
43. #72233780	D₄	semidirect
45. #72226349	D₄	semidirect
47. #72225783	D₄	semidirect
54. #72219423	D₄	semidirect
60. #72218518	D₄	semidirect
61. C₂×#133822	D₄	semidirect
91. C₂⁸	#127	semidirect
92. C₂³×H(5,𝔽₂)	#127	semidirect
103. C₂⁴×C₂²⋊C₄	#127	semidirect
104. C₂×#2309	#127	semidirect
105. C₂×#2308	#127	semidirect
108. C₂²×#379	#127	semidirect
138. C₂×#379	B(4,𝔽₂)	semidirect
139. #1781	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
#72266723	2
#72266689	2
#72266681	2
#72266474	2
#72266322	2
#72266303	2
#72266248	2
#72266238	2
#72266186	2
#72266128	2
#72258971	2