Subgroups — #72309402

Order 4096.

At least 21155 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

The subgroup lattice has 212 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	#72259108	C₂	1	1	yes	maximal, characteristic, complemented
3	#72259092	C₂	1	1	yes	maximal, characteristic, complemented
4	#72237038	C₂	1	1	yes	maximal, characteristic, complemented
5	#72225652	C₂	1	1	yes	maximal, characteristic, complemented
6	#72219295	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219242	C₂	1	1	yes	maximal, characteristic
8	#72219237	C₄	1	2	yes	complemented
9	C₂²×#15947	C₂²	1	1	yes	characteristic
10	C₂×#359428	C₂²	1	1	yes	characteristic
11	#72218518	C₂²	1	1	yes	characteristic
12	#72218375	C₂²	1	1	yes	characteristic
13	#72218291	C₂²	1	1	yes	characteristic
14	#72217789	C₂²	1	1	yes	characteristic
15	#72217511	C₂²	1	1	yes	characteristic
16	#72215858	C₄	1	2	yes	complemented
17	C₂×#62817	D₄	1	1	yes	characteristic
18	C₂⁴×#120	C₂×C₄	1	1	yes	characteristic
19	C₂⁴×#120	D₄	1	1	yes	characteristic
20	C₂⁴×#120	D₄	1	1	yes	characteristic
21	C₂⁴×#120	C₂³	1	1	yes	characteristic
22	#7642244	D₄	1	2	yes	—
23	#7630939	C₂×C₄	1	2	yes	—
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	C₂²×#1438	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1438	D₄	1	1	yes	characteristic
31	C₂²×#1438	D₄	1	1	yes	characteristic
32	#392109	C₂×C₄	1	2	yes	—
33	C₂×#10725	D₄	1	2	yes	complemented
34	C₂×#8547	D₄	1	2	yes	complemented
35	C₂×#8541	D₄	1	2	yes	—
36	C₂×#8541	D₄	1	2	yes	—
37	C₂⁸	C₂²⋊C₄	1	1	yes	characteristic, complemented
38	C₂⁸	C₂×D₄	1	1	yes	characteristic
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	C₂⁴×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
52	C₂⁴×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
53	C₂⁴×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
54	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
55	C₂⁴×C₂²⋊C₄	#41	1	1	yes	characteristic
56	C₂⁴×C₂²⋊C₄	#41	1	1	yes	characteristic
57	#34171	C₂²⋊C₄	1	2	yes	—
58	C₂×#1781	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	#120	1	1	yes	characteristic
77	C₂⁴×D₄	#121	1	1	yes	characteristic
78	C₂⁴×D₄	#121	1	1	yes	characteristic
79	C₂⁴×D₄	#123	1	1	yes	characteristic
80	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
81	C₂⁵×C₄	#121	1	1	yes	characteristic
82	C₂⁵×C₄	#123	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	#121	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	#123	1	1	yes	characteristic
92	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
93	C₂³×C₂²⋊C₄	#124	1	1	yes	characteristic
94	C₂³×C₂²⋊C₄	#124	1	1	yes	characteristic
95	C₂³×C₂²⋊C₄	#127	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	#386	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	2	yes	—
108	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
109	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
110	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
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	B(4,𝔽₂)	1	2	yes	—
117	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
118	C₂³×D₄	#458	1	2	yes	—
119	C₂³×D₄	#458	1	2	yes	—
120	C₂⁴×C₄	#386	1	1	yes	characteristic
121	C₂⁴×C₄	#390	1	1	yes	characteristic
122	C₂⁴×C₄	#392	1	1	yes	characteristic
123	C₂⁴×C₄	#394	1	1	yes	characteristic
124	C₂⁴×C₄	#394	1	1	yes	characteristic
125	C₂⁴×C₄	#394	1	1	yes	characteristic
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	2	yes	—
131	C₂⁵	#1889	1	1	yes	characteristic
132	C₂⁵	#1889	1	2	yes	—
133	C₂⁵	#1889	1	1	yes	characteristic
134	C₂⁵	#1898	1	1	yes	characteristic
135	C₂⁵	#1898	1	1	yes	characteristic
136	C₂⁵	#1913	1	1	yes	characteristic
137	C₂⁵	#1913	1	1	yes	characteristic
138	C₂⁵	#2021	1	2	yes	—
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	1	yes	characteristic
145	C₂⁵	#2021	1	1	yes	characteristic
146	C₂⁵	#2021	1	1	yes	characteristic
147	C₂⁵	#2021	1	1	yes	characteristic
148	C₂⁵	#2021	1	1	yes	characteristic
149	C₂⁵	#2022	1	1	yes	characteristic
150	C₂⁵	#2022	1	1	yes	characteristic
151	C₂⁵	#2196	1	2	yes	—
152	C₂⁵	#2197	1	2	yes	—
153	C₂⁵	#2199	1	2	yes	—
154	C₂⁵	#2200	1	2	yes	—
155	C₂³×C₄	#1898	1	2	yes	—
156	C₂³×C₄	#2021	1	2	yes	—
157	C₂³×C₄	#2044	1	2	yes	—
158	C₂³×C₄	#2044	1	2	yes	—
159	C₂⁴	#8534	1	4	yes	—
160	C₂⁴	#9495	1	1	yes	characteristic
161	C₂⁴	#9495	1	1	yes	characteristic
162	C₂⁴	#9497	1	1	yes	characteristic
163	C₂⁴	#9497	1	1	yes	characteristic
164	C₂⁴	#10336	1	1	yes	characteristic
165	C₂⁴	#10336	1	1	yes	characteristic
166	C₂⁴	#10336	1	1	yes	characteristic
167	C₂⁴	#10336	1	1	yes	characteristic
168	C₂⁴	#10337	1	1	yes	characteristic
169	C₂⁴	#10337	1	1	yes	characteristic
170	C₂⁴	#10337	1	1	yes	characteristic
171	C₂⁴	#10337	1	1	yes	characteristic
172	C₂⁴	#10965	1	1	yes	characteristic
173	C₂⁴	#10972	1	1	yes	characteristic
174	C₂⁴	#10991	1	1	yes	characteristic
175	C₂⁴	#10991	1	1	yes	characteristic
176	C₂⁴	#10991	1	1	yes	characteristic
177	C₂⁴	#10995	1	1	yes	characteristic
178	C₂⁴	#10995	1	1	yes	characteristic
179	C₂⁴	#13041	1	2	yes	—
180	C₂⁴	#13043	1	2	yes	—
181	C₂⁴	#13343	1	2	yes	—
182	C₂⁴	#13344	1	2	yes	—
183	C₂⁴	#13346	1	2	yes	—
184	C₂⁴	#13348	1	2	yes	—
185	C₂³	#111458	1	1	yes	characteristic
186	C₂³	#111458	1	1	yes	characteristic
187	C₂³	#115133	1	1	yes	characteristic
188	C₂³	#115204	1	1	yes	characteristic
189	C₂³	#115204	1	1	yes	characteristic
190	C₂³	#115348	1	1	yes	characteristic
191	C₂³	#115690	1	1	yes	characteristic
192	C₂³	#139750	1	2	yes	—
193	C₂³	#144366	1	1	yes	characteristic
194	C₂³	#144367	1	1	yes	characteristic
195	C₂³	#144378	1	2	yes	—
196	C₂³	#144412	1	2	yes	—
197	C₂³	#147749	1	2	yes	—
198	C₂³	#147752	1	2	yes	—
199	C₂²	#72218387	1	1	yes	characteristic
200	C₂²	#72218521	1	1	yes	characteristic
201	C₂²	#72219158	1	1	yes	characteristic
202	C₂²	#72219252	1	1	yes	characteristic
203	C₂²	#72226366	1	1	yes	characteristic
204	C₂²	#72237473	1	1	yes	characteristic
205	C₂²	#72258216	1	2	yes	—
206	C₂²	#72258219	1	2	yes	—
207	C₂²	#72258273	1	1	yes	characteristic
208	C₂	#72226367	1	1	yes	characteristic
209	C₂	#72258223	1	1	yes	characteristic
210	C₂	#72258401	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72218387
commutator	C₂⁵×D₄	C₂²×C₄
Frattini, cosocle	C₂⁴×#120	C₂³
prime radical	#72219242	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. #72259108	C₂	semidirect
3. #72259092	C₂	semidirect
4. #72237038	C₂	semidirect
5. #72225652	C₂	semidirect
6. #72219295	C₂	semidirect
8. #72219237	C₄	semidirect
16. #72215858	C₄	semidirect
25. C₂³×#379	D₄	semidirect
26. C₂×#21746	D₄	semidirect
27. C₂×#21746	D₄	semidirect
28. C₂×#14679	D₄	semidirect
33. C₂×#10725	D₄	semidirect
34. C₂×#8547	D₄	semidirect
37. C₂⁸	C₂²⋊C₄	semidirect
51. C₂⁴×C₂²⋊C₄	C₂²⋊C₄	semidirect
52. C₂⁴×C₂²⋊C₄	C₂²⋊C₄	semidirect
53. C₂⁴×C₂²⋊C₄	C₂²⋊C₄	semidirect
58. C₂×#1781	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
#72259108	2
#72259092	2
#72237038	2
#72225652	2
#72219295	2
#72219242	2