Subgroups — #72266286

Order 4096.

At least 11811 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

The subgroup lattice has 187 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266135	C₂	1	1	yes	maximal, characteristic, complemented
2	#72258939	C₂	1	1	yes	maximal, characteristic, complemented
3	#72258662	C₂	1	1	yes	maximal, characteristic
4	#72258649	C₄	1	1	yes	characteristic, complemented
5	#72258447	C₂	1	1	yes	maximal, characteristic, complemented
6	#72240619	C₄	1	1	yes	characteristic, complemented
7	#72238447	C₂	1	1	yes	maximal, characteristic, complemented
8	#72236906	C₂	1	1	yes	maximal, characteristic, complemented
9	#72236899	C₂	1	1	yes	maximal, characteristic, complemented
10	#72235937	C₂²	1	1	yes	characteristic
11	#72235937	C₂²	1	1	yes	characteristic
12	#72233780	C₄	1	1	yes	characteristic, complemented
13	#72225783	C₄	1	1	yes	characteristic, complemented
14	#72225665	C₂²	1	1	yes	characteristic
15	#72225621	C₂²	1	1	yes	characteristic
16	#72219377	C₂²	1	1	yes	characteristic
17	#72219307	C₂²	1	1	yes	characteristic
18	C₂×#7631286	C₂²	1	1	yes	characteristic
19	C₂⁴×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic
20	C₂³×D₄²	C₂³	1	1	yes	characteristic
21	C₂×#35250	D₄	1	2	yes	—
22	C₂×#34980	D₄	1	1	yes	characteristic
23	C₂×#34736	D₄	1	2	yes	—
24	C₂²×#2403	C₂×C₄	1	1	yes	characteristic
25	C₂²×#2403	D₄	1	1	yes	characteristic
26	C₂×#34171	C₂×C₄	1	1	yes	characteristic
27	#556752	D₄	1	1	yes	characteristic
28	#517411	C₂×C₄	1	1	yes	characteristic
29	#517411	C₂×C₄	1	1	yes	characteristic
30	#517411	C₂×C₄	1	1	yes	characteristic
31	#517411	D₄	1	1	yes	characteristic
32	#392109	D₄	1	2	yes	—
33	#382422	D₄	1	2	yes	—
34	#382422	D₄	1	2	yes	complemented
35	#184796	D₄	1	2	yes	complemented
36	#184796	D₄	1	2	yes	complemented
37	#154094	D₄	1	2	yes	complemented
38	C₂⁴×#41	C₂²⋊C₄	1	2	yes	—
39	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
40	C₂⁵×D₄	C₂²×C₄	1	1	yes	characteristic
41	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
42	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
43	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
44	C₂³×#127	#41	1	1	yes	characteristic
45	C₂³×#121	C₂×D₄	1	1	yes	characteristic
46	C₂³×#121	C₂×D₄	1	1	yes	characteristic
47	C₂³×#121	C₂×D₄	1	1	yes	characteristic
48	C₂³×#121	#41	1	1	yes	characteristic
49	C₂³×#120	C₂×D₄	1	1	yes	characteristic
50	C₂³×#120	C₂×D₄	1	1	yes	characteristic
51	C₂³×#120	C₂×D₄	1	1	yes	characteristic
52	C₂³×#120	#41	1	1	yes	characteristic
53	C₂³×C₄×D₄	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₄×D₄	#41	1	1	yes	characteristic
55	C₂×#2280	C₂²⋊C₄	1	2	yes	—
56	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
59	#14679	C₂²⋊C₄	1	2	yes	—
60	#14679	C₂²⋊C₄	1	2	yes	—
61	C₂⁷	#99	1	1	yes	characteristic
62	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂⁷	#120	1	1	yes	characteristic
65	C₂⁷	#121	1	1	yes	characteristic
66	C₂⁴×Q₈	#99	1	2	yes	complemented
67	C₂⁴×D₄	#99	1	2	yes	complemented
68	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
69	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
70	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
71	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
72	C₂⁴×D₄	#120	1	1	yes	characteristic
73	C₂⁴×D₄	#120	1	1	yes	characteristic
74	C₂⁴×D₄	#120	1	1	yes	characteristic
75	C₂⁴×D₄	#121	1	1	yes	characteristic
76	C₂⁴×D₄	#121	1	1	yes	characteristic
77	C₂⁴×D₄	#121	1	1	yes	characteristic
78	C₂⁴×D₄	#121	1	1	yes	characteristic
79	C₂⁴×D₄	#121	1	1	yes	characteristic
80	C₂⁴×D₄	#123	1	1	yes	characteristic
81	C₂⁴×D₄	#124	1	1	yes	characteristic
82	C₂⁴×D₄	#127	1	1	yes	characteristic
83	C₂⁴×D₄	#127	1	1	yes	characteristic
84	C₂⁵×C₄	#99	1	1	yes	characteristic
85	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂⁵×C₄	#120	1	1	yes	characteristic
87	C₂⁵×C₄	#121	1	1	yes	characteristic
88	C₂⁵×C₄	#123	1	1	yes	characteristic
89	C₂³×C₄⋊C₄	#121	1	1	yes	characteristic
90	C₂³×C₄⋊C₄	#121	1	1	yes	characteristic
91	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic
92	C₂³×C₂²⋊C₄	#99	1	2	yes	—
93	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic
94	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
95	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
96	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
97	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
98	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
99	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
100	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	#124	1	1	yes	characteristic
102	C₂³×C₄²	#123	1	1	yes	characteristic
103	C₂×#380	#99	1	4	yes	complemented
104	C₂×#379	#99	1	2	yes	—
105	C₂×#379	#99	1	4	yes	complemented
106	C₂⁶	#379	1	1	yes	characteristic
107	C₂⁶	#386	1	1	yes	characteristic
108	C₂⁶	#386	1	1	yes	characteristic
109	C₂⁶	#390	1	1	yes	characteristic
110	C₂⁶	#390	1	1	yes	characteristic
111	C₂⁶	#392	1	1	yes	characteristic
112	C₂⁶	#392	1	1	yes	characteristic
113	C₂⁶	#394	1	1	yes	characteristic
114	C₂⁶	C₂×#99	1	1	yes	characteristic
115	C₂⁶	C₂×#99	1	1	yes	characteristic
116	C₂⁶	C₂×#99	1	1	yes	characteristic
117	C₂⁶	C₂×#99	1	1	yes	characteristic
118	C₂⁶	C₂×#99	1	2	yes	—
119	C₂⁶	#410	1	2	yes	—
120	C₂⁶	B(4,𝔽₂)	1	4	yes	—
121	C₂⁴×C₄	#386	1	1	yes	characteristic
122	C₂⁴×C₄	#386	1	1	yes	characteristic
123	C₂⁴×C₄	#392	1	1	yes	characteristic
124	C₂⁴×C₄	#392	1	1	yes	characteristic
125	C₂⁴×C₄	#394	1	1	yes	characteristic
126	C₂⁴×C₄	#394	1	1	yes	characteristic
127	C₂⁴×C₄	#394	1	1	yes	characteristic
128	C₂⁴×C₄	C₂×#99	1	2	yes	—
129	C₂⁴×C₄	C₂×#99	1	2	yes	—
130	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
131	C₂⁵	#1438	1	1	yes	characteristic
132	C₂⁵	#1781	1	1	yes	characteristic
133	C₂⁵	#1781	1	1	yes	characteristic
134	C₂⁵	#1781	1	1	yes	characteristic
135	C₂⁵	#1781	1	1	yes	characteristic
136	C₂⁵	#1781	1	1	yes	characteristic
137	C₂⁵	#1782	1	1	yes	characteristic
138	C₂⁵	#1889	1	2	yes	—
139	C₂⁵	#1889	1	2	yes	—
140	C₂⁵	#1898	1	1	yes	characteristic
141	C₂⁵	#1898	1	1	yes	characteristic
142	C₂⁵	#1898	1	1	yes	characteristic
143	C₂⁵	#1898	1	1	yes	characteristic
144	C₂⁵	#1913	1	2	yes	—
145	C₂⁵	#1913	1	2	yes	—
146	C₂⁵	#1964	1	2	yes	—
147	C₂⁵	#1964	1	2	yes	—
148	C₂⁵	#2021	1	2	yes	—
149	C₂⁵	#2021	1	2	yes	—
150	C₂³×C₄	#1898	1	2	yes	—
151	C₂³×C₄	#1898	1	2	yes	—
152	C₂³×C₄	#2021	1	2	yes	—
153	C₂³×C₄	#2021	1	2	yes	—
154	C₂⁴	#8541	1	1	yes	characteristic
155	C₂⁴	#8542	1	1	yes	characteristic
156	C₂⁴	#8542	1	1	yes	characteristic
157	C₂⁴	#8546	1	1	yes	characteristic
158	C₂⁴	#9495	1	2	yes	—
159	C₂⁴	#9495	1	2	yes	—
160	C₂⁴	#9497	1	2	yes	—
161	C₂⁴	#9497	1	2	yes	—
162	C₂⁴	#10965	1	1	yes	characteristic
163	C₂⁴	#10965	1	1	yes	characteristic
164	C₂⁴	#10972	1	1	yes	characteristic
165	C₂⁴	#10972	1	1	yes	characteristic
166	C₂⁴	#10980	1	1	yes	characteristic
167	C₂⁴	#10991	1	1	yes	characteristic
168	C₂⁴	#10991	1	1	yes	characteristic
169	C₂³	#111516	1	1	yes	characteristic
170	C₂³	#115133	1	1	yes	characteristic
171	C₂³	#115133	1	1	yes	characteristic
172	C₂³	#115163	1	1	yes	characteristic
173	C₂³	#115163	1	1	yes	characteristic
174	C₂³	#115631	1	1	yes	characteristic
175	C₂³	#115690	1	1	yes	characteristic
176	C₂³	#133839	1	2	yes	—
177	C₂³	#139750	1	4	yes	—
178	C₂²	#72219158	1	2	yes	—
179	C₂²	#72219443	1	1	yes	characteristic
180	C₂²	#72251552	1	1	yes	characteristic
181	C₂²	#72264770	1	1	yes	characteristic
182	C₂²	#72265929	1	2	yes	—
183	C₂	#72219455	1	1	yes	characteristic
184	C₂	#72265787	1	1	yes	characteristic
185	C₂	#72265930	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72219443
commutator	C₂⁵×D₄	C₂²×C₄
Frattini, cosocle	C₂³×D₄²	C₂³
prime radical	#72258662	C₂
LCS	G ▷ C₂⁵×D₄ ▷ 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. #72266135	C₂	semidirect
2. #72258939	C₂	semidirect
4. #72258649	C₄	semidirect
5. #72258447	C₂	semidirect
6. #72240619	C₄	semidirect
7. #72238447	C₂	semidirect
8. #72236906	C₂	semidirect
9. #72236899	C₂	semidirect
12. #72233780	C₄	semidirect
13. #72225783	C₄	semidirect
34. #382422	D₄	semidirect
35. #184796	D₄	semidirect
36. #184796	D₄	semidirect
37. #154094	D₄	semidirect
66. C₂⁴×Q₈	#99	semidirect
67. C₂⁴×D₄	#99	semidirect
103. C₂×#380	#99	semidirect
105. C₂×#379	#99	semidirect

Maximal subgroups

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

subgroup	index
#72266135	2
#72258939	2
#72258662	2
#72258447	2
#72238447	2
#72236906	2
#72236899	2