Subgroups — #72238604

Order 4096.

At least 11111 conjugacy classes of subgroups; only the 329 normal subgroups are shown.

The subgroup lattice has 200 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72238601	C₂	1	1	yes	maximal, characteristic, complemented
2	#72238598	C₂	1	1	yes	maximal, characteristic, complemented
3	#72237776	C₂	1	4	yes	maximal, complemented
4	#72237748	C₂	1	4	yes	maximal, complemented
5	#72237747	C₂²	1	4	yes	complemented
6	#72237746	C₂²	1	4	yes	—
7	#72237596	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237594	C₂²	1	4	yes	complemented
9	#72237391	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237387	C₂²	1	4	yes	complemented
11	#72235740	C₂	1	1	yes	maximal, characteristic, complemented
12	#72235732	C₂²	1	2	yes	complemented
13	#72235718	C₂²	1	1	yes	characteristic
14	#72234789	C₂	1	1	yes	maximal, characteristic, complemented
15	#72234668	C₂²	1	1	yes	characteristic
16	#72234639	C₂	1	1	yes	maximal, characteristic, complemented
17	#72234636	C₂²	1	1	yes	characteristic
18	#72234598	C₂²	1	2	yes	complemented
19	#72234597	C₂²	1	2	yes	complemented
20	#72232613	C₂²	1	2	yes	complemented
21	#72231851	C₂²	1	2	yes	complemented
22	#72231809	C₂²	1	2	yes	complemented
23	#72225870	C₂²	1	1	yes	characteristic
24	#72224136	C₂²	1	1	yes	characteristic
25	#72224136	C₂²	1	1	yes	characteristic
26	#72219311	C₂²	1	1	yes	characteristic
27	#10586404	D₄	1	1	yes	characteristic
28	#10586404	D₄	1	1	yes	characteristic, complemented
29	C₂×#46996	C₂³	1	1	yes	characteristic
30	#7630939	D₄	1	2	yes	—
31	C₂×#34980	C₂³	1	1	yes	characteristic
32	#7626202	D₄	1	1	yes	characteristic, complemented
33	#6451874	C₂³	1	2	yes	—
34	C₂×#23984	D₄	1	2	yes	complemented
35	#6374210	C₂³	1	2	yes	—
36	#6374128	D₄	1	1	yes	characteristic
37	#6374128	D₄	1	1	yes	characteristic, complemented
38	C₂×#21746	D₄	1	1	yes	characteristic, complemented
39	#5024691	D₄	1	1	yes	characteristic, complemented
40	#5020751	D₄	1	1	yes	characteristic, complemented
41	#4430405	D₄	1	1	yes	characteristic, complemented
42	#648360	D₄	1	1	yes	characteristic, complemented
43	#576160	C₂³	1	1	yes	characteristic
44	#480250	C₂³	1	2	yes	—
45	#480250	C₂³	1	2	yes	—
46	#449090	D₄	1	4	yes	complemented
47	#436464	D₄	1	4	yes	complemented
48	#432839	D₄	1	4	yes	complemented
49	#418911	D₄	1	4	yes	complemented
50	#359705	D₄	1	2	yes	complemented
51	#359705	C₂³	1	2	yes	—
52	#190116	C₂³	1	2	yes	—
53	#134036	D₄	1	2	yes	complemented
54	#134035	D₄	1	2	yes	complemented
55	#115690	D₄	1	2	yes	complemented
56	#115631	D₄	1	2	yes	complemented
57	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
58	C₂×#3490	C₂×D₄	1	2	yes	—
59	C₂×#3484	C₂×D₄	1	2	yes	—
60	C₂²×#534	C₂×D₄	1	1	yes	characteristic
61	C₂²×#534	C₂×D₄	1	1	yes	characteristic
62	C₂³×#120	C₂×D₄	1	1	yes	characteristic
63	C₂×#2840	C₂×D₄	1	1	yes	characteristic, complemented
64	C₂×#2849	C₂×D₄	1	1	yes	characteristic
65	C₂×#2849	#41	1	1	yes	characteristic
66	C₂×#2846	C₂×D₄	1	1	yes	characteristic
67	C₂×#2846	C₂×D₄	1	1	yes	characteristic
68	C₂×#2403	C₂×D₄	1	1	yes	characteristic, complemented
69	C₂×#2613	C₂×D₄	1	1	yes	characteristic
70	C₂×#2613	C₂×D₄	1	1	yes	characteristic
71	C₂×#2613	C₂×D₄	1	1	yes	characteristic
72	C₂×#2613	#41	1	1	yes	characteristic
73	#34176	C₂×D₄	1	2	yes	—
74	#34171	C₂×D₄	1	2	yes	—
75	C₂×#2308	C₂×D₄	1	1	yes	characteristic
76	C₂×#2308	C₂⁴	1	1	yes	characteristic
77	C₂²×#379	C₂×D₄	1	1	yes	characteristic
78	#23468	C₂×D₄	1	4	yes	—
79	#23444	C₂×D₄	1	4	yes	—
80	#20689	C₂×D₄	1	4	yes	—
81	C₂×#1819	C₂×D₄	1	2	yes	—
82	C₂×#1819	C₂×D₄	1	2	yes	—
83	#20585	C₂×D₄	1	4	yes	—
84	#13827	C₂×D₄	1	1	yes	characteristic, complemented
85	C₂²⋊C₄²	C₂×D₄	1	1	yes	characteristic, complemented
86	#11500	C₂×D₄	1	4	yes	complemented
87	#10938	C₂×D₄	1	4	yes	—
88	#10938	C₂×D₄	1	4	yes	—
89	#8101	C₂×D₄	1	4	yes	complemented
90	C₂⁷	#120	1	1	yes	characteristic
91	C₂⁷	#120	1	1	yes	characteristic
92	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
93	C₂⁴×D₄	#120	1	1	yes	characteristic
94	C₂⁴×D₄	#120	1	1	yes	characteristic
95	C₂⁴×D₄	#120	1	1	yes	characteristic
96	C₂⁴×D₄	#121	1	1	yes	characteristic
97	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
98	C₂⁵×C₄	#121	1	1	yes	characteristic
99	C₂×#543	#120	1	2	yes	—
100	C₂×#534	#120	1	2	yes	—
101	C₂²×#123	#120	1	1	yes	characteristic
102	C₂²×#123	#121	1	1	yes	characteristic
103	C₂²×#123	#121	1	1	yes	characteristic
104	C₂²×#123	#121	1	1	yes	characteristic
105	C₂²×#120	#120	1	1	yes	characteristic
106	C₂²×#120	#120	1	1	yes	characteristic
107	C₂²×#120	#120	1	1	yes	characteristic
108	C₂²×#120	#121	1	1	yes	characteristic
109	C₂²×#120	#121	1	1	yes	characteristic
110	C₂²×#120	#121	1	1	yes	characteristic
111	C₂²×#120	#127	1	1	yes	characteristic
112	C₂²×#120	#127	1	1	yes	characteristic
113	C₂×#518	#120	1	2	yes	—
114	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
115	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
117	#2308	#120	1	4	yes	—
118	#2308	C₂²×D₄	1	4	yes	—
119	#2308	H(5,𝔽₂)	1	4	yes	—
120	C₂×#381	#121	1	1	yes	characteristic
121	C₂×#381	#127	1	1	yes	characteristic
122	C₂×#381	C₂²×D₄	1	1	yes	characteristic
123	C₂×#381	C₂×#41	1	1	yes	characteristic
124	C₂×#379	#120	1	1	yes	characteristic
125	C₂×#379	#120	1	1	yes	characteristic
126	C₂×#379	#121	1	1	yes	characteristic
127	C₂×#379	#121	1	1	yes	characteristic
128	C₂×#379	#121	1	1	yes	characteristic
129	C₂×#379	#127	1	1	yes	characteristic
130	C₂×#379	C₂²×D₄	1	1	yes	characteristic
131	C₂×#379	C₂²×D₄	1	1	yes	characteristic
132	#2273	#120	1	2	yes	—
133	#2273	#120	1	2	yes	—
134	C₂×#342	#120	1	4	yes	—
135	C₂×#342	#120	1	4	yes	—
136	C₂⁶	#392	1	1	yes	characteristic
137	C₂⁶	#392	1	1	yes	characteristic
138	C₂⁶	#392	1	1	yes	characteristic
139	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
140	C₂⁶	C₂×#120	1	1	yes	characteristic
141	C₂⁶	C₂×#120	1	1	yes	characteristic
142	C₂³×D₄	#392	1	1	yes	characteristic
143	C₂³×D₄	#392	1	1	yes	characteristic
144	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
145	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
146	C₂³×D₄	C₂×#120	1	2	yes	—
147	C₂³×D₄	#534	1	2	yes	—
148	C₂⁴×C₄	#392	1	1	yes	characteristic
149	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
150	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
151	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
152	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
154	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
155	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
156	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
157	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	C₂×#127	1	1	yes	characteristic
159	C₂²×C₂²⋊C₄	#534	1	2	yes	—
160	C₂²×C₂²⋊C₄	#535	1	2	yes	—
161	#381	D₄²	1	4	yes	—
162	#381	#546	1	4	yes	—
163	#379	D₄²	1	4	yes	—
164	#379	D₄²	1	4	yes	—
165	C₂⁵	#2021	1	1	yes	characteristic
166	C₂⁵	#2021	1	1	yes	characteristic
167	C₂⁵	C₂×#392	1	1	yes	characteristic
168	C₂⁵	#2403	1	2	yes	—
169	C₂⁵	#2613	1	1	yes	characteristic
170	C₂⁵	#2846	1	1	yes	characteristic
171	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
172	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
173	C₂⁵	#3026	1	1	yes	characteristic
174	C₂⁵	#3026	1	1	yes	characteristic
175	C₂³×C₄	#2021	1	1	yes	characteristic
176	C₂³×C₄	#2021	1	1	yes	characteristic
177	C₂³×C₄	#2410	1	2	yes	—
178	C₂×C₂²⋊C₄	#2403	1	2	yes	—
179	C₂×C₂²⋊C₄	#2408	1	2	yes	—
180	C₂×C₂²⋊C₄	#2410	1	2	yes	—
181	C₂×C₂²⋊C₄	#2433	1	2	yes	—
182	C₂⁴	#14715	1	1	yes	characteristic
183	C₂⁴	C₂×#2021	1	1	yes	characteristic
184	C₂⁴	#23444	1	1	yes	characteristic
185	C₂⁴	#23897	1	4	yes	—
186	C₂⁴	#29634	1	1	yes	characteristic
187	C₂⁴	#29634	1	1	yes	characteristic
188	C₂⁴	#32898	1	1	yes	characteristic
189	C₂⁴	#32898	1	1	yes	characteristic
190	C₂³	#393929	1	1	yes	characteristic
191	C₂³	#393975	1	2	yes	—
192	C₂³	#511790	1	2	yes	—
193	C₂³	#511790	1	1	yes	characteristic
194	C₂³	#511791	1	1	yes	characteristic
195	C₂²	#72218121	1	1	yes	characteristic
196	C₂²	#72233800	1	1	yes	characteristic
197	C₂²	#72235138	1	1	yes	characteristic
198	C₂	#72235872	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72235872
commutator, Frattini, cosocle	C₂×#2308	C₂⁴
LCS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72238601	C₂	semidirect
2. #72238598	C₂	semidirect
3. #72237776	C₂	semidirect
4. #72237748	C₂	semidirect
5. #72237747	C₂²	semidirect
7. #72237596	C₂	semidirect
8. #72237594	C₂²	semidirect
9. #72237391	C₂	semidirect
10. #72237387	C₂²	semidirect
11. #72235740	C₂	semidirect
12. #72235732	C₂²	semidirect
14. #72234789	C₂	semidirect
16. #72234639	C₂	semidirect
18. #72234598	C₂²	semidirect
19. #72234597	C₂²	semidirect
20. #72232613	C₂²	semidirect
21. #72231851	C₂²	semidirect
22. #72231809	C₂²	semidirect
28. #10586404	D₄	semidirect
32. #7626202	D₄	semidirect
34. C₂×#23984	D₄	semidirect
37. #6374128	D₄	semidirect
38. C₂×#21746	D₄	semidirect
39. #5024691	D₄	semidirect
40. #5020751	D₄	semidirect
41. #4430405	D₄	semidirect
42. #648360	D₄	semidirect
46. #449090	D₄	semidirect
47. #436464	D₄	semidirect
48. #432839	D₄	semidirect
49. #418911	D₄	semidirect
50. #359705	D₄	semidirect
53. #134036	D₄	semidirect
54. #134035	D₄	semidirect
55. #115690	D₄	semidirect
56. #115631	D₄	semidirect
63. C₂×#2840	C₂×D₄	semidirect
68. C₂×#2403	C₂×D₄	semidirect
84. #13827	C₂×D₄	semidirect
85. C₂²⋊C₄²	C₂×D₄	semidirect
86. #11500	C₂×D₄	semidirect
89. #8101	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72238601	2
#72238598	2
#72237776	2
#72237748	2
#72237596	2
#72237391	2
#72235740	2
#72234789	2
#72234639	2