Subgroups — #72317206

Order 4096.

At least 8424 conjugacy classes of subgroups; only the 303 normal subgroups are shown.

The subgroup lattice has 192 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72317196	C₂	1	4	yes	maximal, complemented
2	#72317192	C₂	1	4	yes	maximal, complemented
3	#72314618	C₂	1	1	yes	maximal, characteristic, complemented
4	#72314614	C₂²	1	4	yes	complemented
5	#72314447	C₂	1	1	yes	maximal, characteristic, complemented
6	#72310472	C₂	1	1	yes	maximal, characteristic, complemented
7	#72310469	C₂²	1	4	yes	complemented
8	#72309929	C₂	1	1	yes	maximal, characteristic, complemented
9	#72266165	C₂	1	1	yes	maximal, characteristic, complemented
10	#72266164	C₂²	1	2	yes	complemented
11	#72265766	C₂²	1	1	yes	characteristic
12	#72263077	C₂²	1	2	yes	complemented
13	#72262471	C₂²	1	4	yes	—
14	#72259008	C₂²	1	1	yes	characteristic
15	#72237585	C₂	1	1	yes	maximal, characteristic, complemented
16	#72237579	C₂²	1	2	yes	complemented
17	#72237576	C₂²	1	4	yes	—
18	#72235147	C₂²	1	2	yes	complemented
19	#72234639	C₂	1	1	yes	maximal, characteristic
20	#72234638	C₂²	1	2	yes	—
21	#72234637	C₂²	1	2	yes	—
22	#72234636	C₂²	1	1	yes	characteristic
23	#72234598	C₄	1	2	yes	complemented
24	#72234597	C₄	1	2	yes	complemented
25	#72224136	C₂²	1	1	yes	characteristic
26	#72219314	C₂²	1	1	yes	characteristic
27	#72219311	C₂²	1	1	yes	characteristic
28	#72219310	C₂²	1	1	yes	characteristic
29	#72211186	C₄	1	2	yes	complemented
30	#72210578	C₄	1	2	yes	complemented
31	#10586404	D₄	1	1	yes	characteristic
32	C₂×#46996	C₂×C₄	1	1	yes	characteristic, complemented
33	#7630939	D₄	1	2	yes	—
34	C₂×#34980	C₂³	1	1	yes	characteristic
35	#6499870	D₄	1	2	yes	—
36	#6499870	C₂³	1	2	yes	—
37	#6499867	C₂×C₄	1	2	yes	—
38	#6499867	D₄	1	2	yes	—
39	#6374210	C₂×C₄	1	2	yes	—
40	#6374128	D₄	1	1	yes	characteristic
41	#5024691	C₂³	1	1	yes	characteristic
42	#5020751	C₂×C₄	1	1	yes	characteristic, complemented
43	#4430405	C₂³	1	1	yes	characteristic
44	#648360	C₂×C₄	1	1	yes	characteristic, complemented
45	#576160	C₂×C₄	1	1	yes	characteristic, complemented
46	#238077	C₂×C₄	1	4	yes	complemented
47	#233541	C₂×C₄	1	4	yes	complemented
48	#190116	C₂×C₄	1	2	yes	complemented
49	#188109	D₄	1	1	yes	characteristic, complemented
50	C₂×#8547	D₄	1	1	yes	characteristic, complemented
51	#188049	D₄	1	1	yes	characteristic, complemented
52	C₂×#8541	D₄	1	1	yes	characteristic, complemented
53	#133826	C₂³	1	2	yes	—
54	#133825	D₄	1	2	yes	complemented
55	#133825	C₂³	1	2	yes	—
56	#133824	D₄	1	2	yes	complemented
57	#115690	C₂×C₄	1	2	yes	complemented
58	#115633	C₂³	1	2	yes	—
59	#115631	C₂³	1	2	yes	—
60	#115631	C₂³	1	2	yes	—
61	#110180	C₂×C₄	1	4	yes	complemented
62	#110086	C₂×C₄	1	4	yes	complemented
63	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
64	#62817	C₂²⋊C₄	1	2	yes	—
65	#62817	C₂×D₄	1	2	yes	—
66	C₂×#3484	C₂²⋊C₄	1	2	yes	—
67	C₂²×#534	C₂×D₄	1	1	yes	characteristic
68	C₂²×#534	C₂×D₄	1	1	yes	characteristic
69	C₂³×#120	C₂×D₄	1	1	yes	characteristic
70	C₂×#2840	C₂²×C₄	1	1	yes	characteristic, complemented
71	C₂×#2849	C₂×D₄	1	1	yes	characteristic
72	C₂×#2849	#41	1	1	yes	characteristic
73	C₂×#2846	C₂²⋊C₄	1	4	yes	—
74	C₂×#2846	C₂²×C₄	1	1	yes	characteristic
75	C₂×#2846	C₂×D₄	1	1	yes	characteristic
76	#34980	C₂×D₄	1	4	yes	—
77	C₂×#2403	C₂²×C₄	1	1	yes	characteristic, complemented
78	C₂×#2613	C₂×D₄	1	1	yes	characteristic
79	C₂×#2613	C₂×D₄	1	1	yes	characteristic
80	C₂×#2613	#41	1	1	yes	characteristic
81	C₂×#2613	C₂⁴	1	1	yes	characteristic
82	#34171	C₂²⋊C₄	1	2	yes	—
83	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
84	C₂×#2308	#41	1	1	yes	characteristic
85	C₂²×#379	#41	1	1	yes	characteristic
86	#21746	C₂²⋊C₄	1	4	yes	—
87	#21746	C₂²⋊C₄	1	2	yes	—
88	#21746	C₂²×C₄	1	4	yes	—
89	#21746	C₂×D₄	1	2	yes	—
90	#13827	C₂²×C₄	1	1	yes	characteristic, complemented
91	C₂²⋊C₄²	C₂²×C₄	1	1	yes	characteristic, complemented
92	#8548	C₂×D₄	1	2	yes	—
93	#8547	C₂×D₄	1	2	yes	—
94	#8543	C₂×D₄	1	2	yes	—
95	#8541	C₂×D₄	1	2	yes	—
96	#8101	C₂²×C₄	1	4	yes	complemented
97	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂⁷	#121	1	1	yes	characteristic
99	C₂²×H(5,𝔽₂)	#99	1	2	yes	—
100	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂⁴×D₄	#120	1	1	yes	characteristic
102	C₂⁴×D₄	#121	1	1	yes	characteristic
103	C₂⁴×D₄	#121	1	1	yes	characteristic
104	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
105	C₂⁵×C₄	#121	1	1	yes	characteristic
106	C₂×#534	#99	1	4	yes	—
107	C₂²×#123	#120	1	1	yes	characteristic
108	C₂²×#123	#121	1	1	yes	characteristic
109	C₂²×#123	#123	1	1	yes	characteristic
110	C₂²×#123	C₂²×D₄	1	1	yes	characteristic
111	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂²×#120	C₄×D₄	1	1	yes	characteristic
114	C₂²×#120	C₄×D₄	1	1	yes	characteristic
115	C₂²×#120	#120	1	1	yes	characteristic
116	C₂²×#120	#120	1	1	yes	characteristic
117	C₂²×#120	#121	1	1	yes	characteristic
118	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
119	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
120	C₂×#518	#99	1	2	yes	—
121	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
123	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
124	C₂×#381	C₄×D₄	1	1	yes	characteristic
125	C₂×#381	C₄×D₄	1	1	yes	characteristic
126	C₂×#381	#121	1	1	yes	characteristic
127	C₂×#381	C₂×#41	1	1	yes	characteristic
128	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
129	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×#379	C₄×D₄	1	1	yes	characteristic
131	C₂×#379	C₄×D₄	1	1	yes	characteristic
132	C₂×#379	#121	1	1	yes	characteristic
133	C₂×#379	#123	1	1	yes	characteristic
134	C₂×#379	#123	1	1	yes	characteristic
135	C₂×#379	C₂³×C₄	1	1	yes	characteristic
136	C₂×#379	C₂×#41	1	1	yes	characteristic
137	C₂⁶	#386	1	1	yes	characteristic
138	C₂⁶	#386	1	1	yes	characteristic
139	C₂⁶	#386	1	1	yes	characteristic
140	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
141	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₂⁶	C₂×#121	1	1	yes	characteristic
143	C₂³×D₄	#386	1	1	yes	characteristic
144	C₂³×D₄	C₂×#99	1	4	yes	—
145	C₂³×D₄	C₂×#99	1	2	yes	—
146	C₂³×D₄	C₂×#99	1	2	yes	—
147	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
148	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
149	C₂³×D₄	C₂×#120	1	1	yes	characteristic
150	C₂⁴×C₄	#386	1	1	yes	characteristic
151	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
152	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
154	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
156	C₂²×C₂²⋊C₄	C₂×#99	1	4	yes	—
157	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
160	C₂⁵	#1889	1	1	yes	characteristic
161	C₂⁵	#1889	1	1	yes	characteristic
162	C₂⁵	#1889	1	1	yes	characteristic
163	C₂⁵	#1898	1	1	yes	characteristic
164	C₂⁵	#1898	1	1	yes	characteristic
165	C₂⁵	#1913	1	1	yes	characteristic
166	C₂⁵	#1913	1	1	yes	characteristic
167	C₂⁵	#2127	1	4	yes	complemented
168	C₂⁵	C₂×#386	1	1	yes	characteristic
169	C₂⁵	C₂²×#99	1	2	yes	—
170	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
171	C₂³×C₄	#1889	1	1	yes	characteristic
172	C₂³×C₄	#1898	1	2	yes	—
173	C₂³×C₄	#2127	1	4	yes	complemented
174	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
175	C₂⁴	#9495	1	1	yes	characteristic
176	C₂⁴	#9495	1	1	yes	characteristic
177	C₂⁴	#9495	1	1	yes	characteristic
178	C₂⁴	#9495	1	1	yes	characteristic
179	C₂⁴	C₂×#1889	1	1	yes	characteristic
180	C₂⁴	C₂×#1898	1	1	yes	characteristic
181	C₂⁴	C₂×#1913	1	1	yes	characteristic
182	C₂⁴	C₂×#2127	1	4	yes	—
183	C₂³	#139743	1	1	yes	characteristic
184	C₂³	#139743	1	1	yes	characteristic
185	C₂³	C₂×#9495	1	1	yes	characteristic
186	C₂³	#494713	1	2	yes	—
187	C₂²	C₂×#139743	1	1	yes	characteristic
188	C₂²	#72240132	1	1	yes	characteristic
189	C₂²	#72255841	1	1	yes	characteristic
190	C₂	#72266244	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266244
commutator	C₂×#379	C₂³×C₄
Frattini, cosocle	C₂×#2613	C₂⁴
prime radical	#72234639	C₂
LCS	G ▷ C₂×#379 ▷ 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. #72317196	C₂	semidirect
2. #72317192	C₂	semidirect
3. #72314618	C₂	semidirect
4. #72314614	C₂²	semidirect
5. #72314447	C₂	semidirect
6. #72310472	C₂	semidirect
7. #72310469	C₂²	semidirect
8. #72309929	C₂	semidirect
9. #72266165	C₂	semidirect
10. #72266164	C₂²	semidirect
12. #72263077	C₂²	semidirect
15. #72237585	C₂	semidirect
16. #72237579	C₂²	semidirect
18. #72235147	C₂²	semidirect
23. #72234598	C₄	semidirect
24. #72234597	C₄	semidirect
29. #72211186	C₄	semidirect
30. #72210578	C₄	semidirect
32. C₂×#46996	C₂×C₄	semidirect
42. #5020751	C₂×C₄	semidirect
44. #648360	C₂×C₄	semidirect
45. #576160	C₂×C₄	semidirect
46. #238077	C₂×C₄	semidirect
47. #233541	C₂×C₄	semidirect
48. #190116	C₂×C₄	semidirect
49. #188109	D₄	semidirect
50. C₂×#8547	D₄	semidirect
51. #188049	D₄	semidirect
52. C₂×#8541	D₄	semidirect
54. #133825	D₄	semidirect
56. #133824	D₄	semidirect
57. #115690	C₂×C₄	semidirect
61. #110180	C₂×C₄	semidirect
62. #110086	C₂×C₄	semidirect
70. C₂×#2840	C₂²×C₄	semidirect
77. C₂×#2403	C₂²×C₄	semidirect
90. #13827	C₂²×C₄	semidirect
91. C₂²⋊C₄²	C₂²×C₄	semidirect
96. #8101	C₂²×C₄	semidirect
167. C₂⁵	#2127	semidirect
173. C₂³×C₄	#2127	semidirect

Maximal subgroups

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

subgroup	index
#72317196	2
#72317192	2
#72314618	2
#72314447	2
#72310472	2
#72309929	2
#72266165	2
#72237585	2
#72234639	2