Subgroups — #72309404

Order 4096.

At least 21091 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

The subgroup lattice has 198 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72309398	C₂	1	1	yes	maximal, characteristic, complemented
2	#72265763	C₂	1	1	yes	maximal, characteristic, complemented
3	#72265757	C₂	1	1	yes	maximal, characteristic, complemented
4	#72259092	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237629	C₂	1	1	yes	maximal, characteristic, complemented
6	#72236520	C₂	1	1	yes	maximal, characteristic, complemented
7	#72234600	C₂	1	1	yes	maximal, characteristic
8	C₂×#359428	C₂²	1	1	yes	characteristic
9	#72218518	C₂²	1	1	yes	characteristic
10	#72218375	C₂²	1	1	yes	characteristic
11	#72218291	C₂²	1	1	yes	characteristic
12	#72218180	C₄	1	2	yes	complemented
13	#72217875	C₂²	1	1	yes	characteristic
14	#72217789	C₂²	1	1	yes	characteristic
15	#72217511	C₂²	1	1	yes	characteristic
16	#72213862	C₄	1	2	yes	complemented
17	C₂×#62817	D₄	1	1	yes	characteristic
18	C₂²×#3484	C₂×C₄	1	1	yes	characteristic
19	C₂⁴×#120	C₂³	1	1	yes	characteristic
20	#7642244	D₄	1	2	yes	—
21	#7630939	C₂×C₄	1	2	yes	—
22	C₂×#35346	D₄	1	1	yes	characteristic
23	C₂×#35346	D₄	1	1	yes	characteristic
24	#7626202	D₄	1	1	yes	characteristic
25	C₂³×#379	D₄	1	1	yes	characteristic, complemented
26	C₂×#21746	D₄	1	1	yes	characteristic, complemented
27	C₂×#21746	D₄	1	1	yes	characteristic, complemented
28	C₂×#14679	D₄	1	1	yes	characteristic, complemented
29	#517411	C₂×C₄	1	1	yes	characteristic
30	#515285	D₄	1	1	yes	characteristic
31	#515285	D₄	1	1	yes	characteristic
32	#392109	C₂×C₄	1	2	yes	—
33	#392109	D₄	1	2	yes	—
34	#392109	D₄	1	2	yes	—
35	C₂×#10965	D₄	1	2	yes	complemented
36	#184773	D₄	1	2	yes	complemented
37	C₂⁸	C₂×D₄	1	1	yes	characteristic
38	C₂³×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic, complemented
39	C₂⁵×D₄	C₂²×C₄	1	1	yes	characteristic
40	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
41	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
42	C₂×#3484	C₂²⋊C₄	1	2	yes	—
43	C₂³×#120	C₂×D₄	1	1	yes	characteristic
44	C₂³×#120	C₂×D₄	1	1	yes	characteristic
45	C₂³×#120	C₂×D₄	1	1	yes	characteristic
46	C₂³×#120	C₂×D₄	1	1	yes	characteristic
47	C₂³×#120	C₂×D₄	1	1	yes	characteristic
48	C₂³×#120	C₂×D₄	1	1	yes	characteristic
49	C₂³×#120	#41	1	1	yes	characteristic
50	C₂³×#120	#41	1	1	yes	characteristic
51	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic, complemented
52	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
53	C₂⁴×C₂²⋊C₄	#41	1	1	yes	characteristic
54	C₂⁴×C₂²⋊C₄	#41	1	1	yes	characteristic
55	#34171	C₂²⋊C₄	1	2	yes	—
56	C₂×#2308	C₂²⋊C₄	1	1	yes	characteristic, complemented
57	C₂×#2308	C₂²⋊C₄	1	1	yes	characteristic, complemented
58	#14679	C₂²⋊C₄	1	4	yes	complemented
59	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂⁷	C₄×D₄	1	1	yes	characteristic
61	C₂⁷	#120	1	1	yes	characteristic
62	C₂⁷	#120	1	1	yes	characteristic
63	C₂⁷	#121	1	1	yes	characteristic
64	C₂⁷	#121	1	1	yes	characteristic
65	C₂⁷	#121	1	1	yes	characteristic
66	C₂⁷	#127	1	1	yes	characteristic
67	C₂²×H(5,𝔽₂)	#99	1	2	yes	complemented
68	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂⁴×D₄	C₂×C₂²⋊C₄	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	#120	1	1	yes	characteristic
76	C₂⁴×D₄	#121	1	1	yes	characteristic
77	C₂⁴×D₄	#121	1	1	yes	characteristic
78	C₂⁴×D₄	#124	1	1	yes	characteristic
79	C₂⁴×D₄	#127	1	1	yes	characteristic
80	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
81	C₂⁵×C₄	#121	1	1	yes	characteristic
82	C₂⁵×C₄	#124	1	1	yes	characteristic
83	C₂×#518	#99	1	2	yes	complemented
84	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
85	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
86	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
87	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
88	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
89	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
90	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
91	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
92	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
93	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
94	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
95	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
96	#2308	#99	1	4	yes	complemented
97	C₂⁶	#379	1	1	yes	characteristic
98	C₂⁶	#386	1	1	yes	characteristic
99	C₂⁶	#386	1	1	yes	characteristic
100	C₂⁶	#390	1	1	yes	characteristic
101	C₂⁶	#390	1	1	yes	characteristic
102	C₂⁶	#392	1	1	yes	characteristic
103	C₂⁶	#392	1	1	yes	characteristic
104	C₂⁶	#392	1	1	yes	characteristic
105	C₂⁶	#394	1	1	yes	characteristic
106	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
107	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
108	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
109	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
110	C₂⁶	B(4,𝔽₂)	1	2	yes	—
111	C₂⁶	B(4,𝔽₂)	1	2	yes	—
112	C₂³×D₄	C₂×#99	1	2	yes	—
113	C₂³×D₄	C₂×#99	1	2	yes	—
114	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
115	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
116	C₂³×D₄	#458	1	2	yes	—
117	C₂³×D₄	#458	1	2	yes	—
118	C₂⁴×C₄	#386	1	1	yes	characteristic
119	C₂⁴×C₄	#386	1	1	yes	characteristic
120	C₂⁴×C₄	#392	1	1	yes	characteristic
121	C₂⁴×C₄	#394	1	1	yes	characteristic
122	C₂⁴×C₄	#394	1	1	yes	characteristic
123	C₂⁴×C₄	#394	1	1	yes	characteristic
124	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
125	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
126	C₂⁵	#1438	1	1	yes	characteristic
127	C₂⁵	#1781	1	2	yes	—
128	C₂⁵	#1790	1	2	yes	—
129	C₂⁵	#1790	1	2	yes	—
130	C₂⁵	#1889	1	1	yes	characteristic
131	C₂⁵	#1889	1	2	yes	—
132	C₂⁵	#1889	1	1	yes	characteristic
133	C₂⁵	#1898	1	1	yes	characteristic
134	C₂⁵	#1898	1	1	yes	characteristic
135	C₂⁵	#1913	1	1	yes	characteristic
136	C₂⁵	#1913	1	1	yes	characteristic
137	C₂⁵	#2021	1	1	yes	characteristic
138	C₂⁵	#2021	1	1	yes	characteristic
139	C₂⁵	#2021	1	1	yes	characteristic
140	C₂⁵	#2021	1	1	yes	characteristic
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	2	yes	—
145	C₂⁵	#2021	1	1	yes	characteristic
146	C₂⁵	#2021	1	1	yes	characteristic
147	C₂⁵	#2021	1	1	yes	characteristic
148	C₂⁵	#2022	1	1	yes	characteristic
149	C₂⁵	#2022	1	1	yes	characteristic
150	C₂³×C₄	#1889	1	2	yes	—
151	C₂³×C₄	#1898	1	2	yes	—
152	C₂³×C₄	#2021	1	2	yes	—
153	C₂³×C₄	#2044	1	2	yes	—
154	C₂³×C₄	#2044	1	2	yes	—
155	C₂⁴	#8534	1	4	yes	—
156	C₂⁴	#9495	1	1	yes	characteristic
157	C₂⁴	#9495	1	1	yes	characteristic
158	C₂⁴	#9497	1	1	yes	characteristic
159	C₂⁴	#9497	1	1	yes	characteristic
160	C₂⁴	#10336	1	1	yes	characteristic
161	C₂⁴	#10336	1	1	yes	characteristic
162	C₂⁴	#10336	1	1	yes	characteristic
163	C₂⁴	#10336	1	1	yes	characteristic
164	C₂⁴	#10337	1	1	yes	characteristic
165	C₂⁴	#10337	1	1	yes	characteristic
166	C₂⁴	#10337	1	1	yes	characteristic
167	C₂⁴	#10337	1	1	yes	characteristic
168	C₂⁴	#10965	1	1	yes	characteristic
169	C₂⁴	#10972	1	1	yes	characteristic
170	C₂⁴	#10991	1	1	yes	characteristic
171	C₂⁴	#10991	1	1	yes	characteristic
172	C₂⁴	#10991	1	1	yes	characteristic
173	C₂⁴	#10995	1	1	yes	characteristic
174	C₂⁴	#10995	1	1	yes	characteristic
175	C₂³	#111458	1	1	yes	characteristic
176	C₂³	#111458	1	1	yes	characteristic
177	C₂³	#115133	1	1	yes	characteristic
178	C₂³	#115204	1	1	yes	characteristic
179	C₂³	#115204	1	1	yes	characteristic
180	C₂³	#115348	1	1	yes	characteristic
181	C₂³	#115690	1	1	yes	characteristic
182	C₂³	#139750	1	2	yes	—
183	C₂³	#144364	1	2	yes	—
184	C₂³	#144378	1	1	yes	characteristic
185	C₂³	#144379	1	1	yes	characteristic
186	C₂³	#144398	1	2	yes	—
187	C₂²	#72218387	1	1	yes	characteristic
188	C₂²	#72219158	1	1	yes	characteristic
189	C₂²	#72226366	1	1	yes	characteristic
190	C₂²	#72236517	1	1	yes	characteristic
191	C₂²	#72236636	1	1	yes	characteristic
192	C₂²	#72246450	1	1	yes	characteristic
193	C₂²	#72258532	1	1	yes	characteristic
194	C₂	#72226367	1	1	yes	characteristic
195	C₂	#72258230	1	1	yes	characteristic
196	C₂	#72266374	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72218387
commutator	C₂⁵×D₄	C₂²×C₄
Frattini, cosocle	C₂⁴×#120	C₂³
prime radical	#72234600	C₂
LCS	G ▷ C₂⁵×D₄ ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×#120 ◁ G
LECS	G ▷ C₂⁴×#120 ▷ C₂⁵ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72309398	C₂	semidirect
2. #72265763	C₂	semidirect
3. #72265757	C₂	semidirect
4. #72259092	C₂	semidirect
5. #72237629	C₂	semidirect
6. #72236520	C₂	semidirect
12. #72218180	C₄	semidirect
16. #72213862	C₄	semidirect
25. C₂³×#379	D₄	semidirect
26. C₂×#21746	D₄	semidirect
27. C₂×#21746	D₄	semidirect
28. C₂×#14679	D₄	semidirect
35. C₂×#10965	D₄	semidirect
36. #184773	D₄	semidirect
38. C₂³×H(5,𝔽₂)	C₂²⋊C₄	semidirect
51. C₂²×#518	C₂²⋊C₄	semidirect
56. C₂×#2308	C₂²⋊C₄	semidirect
57. C₂×#2308	C₂²⋊C₄	semidirect
58. #14679	C₂²⋊C₄	semidirect
67. C₂²×H(5,𝔽₂)	#99	semidirect
83. C₂×#518	#99	semidirect
96. #2308	#99	semidirect

Maximal subgroups

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

subgroup	index
#72309398	2
#72265763	2
#72265757	2
#72259092	2
#72237629	2
#72236520	2
#72234600	2