Subgroups — #72266379

Order 2048.

At least 6011 conjugacy classes of subgroups; only the 388 normal subgroups are shown.

The subgroup lattice has 184 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266378	C₂	1	4	yes	maximal, complemented
2	#72263490	C₂	1	4	yes	maximal, complemented
3	#72259004	C₂	1	1	yes	maximal, characteristic, complemented
4	#72258393	C₂	1	1	yes	maximal, characteristic, complemented
5	#72236440	C₂	1	1	yes	maximal, characteristic, complemented
6	#72218306	C₂	1	1	yes	maximal, characteristic, complemented
7	#72213862	C₂	1	1	yes	maximal, characteristic, complemented
8	#72210555	C₂	1	1	yes	maximal, characteristic, complemented
9	#72210511	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#3484	C₂²	1	1	yes	characteristic
11	C₂²×#3484	C₂²	1	1	yes	characteristic
12	#7630939	C₂²	1	1	yes	characteristic
13	#6437621	C₂²	1	2	yes	complemented
14	#6437445	C₂²	1	2	yes	complemented
15	#763333	C₂²	1	4	yes	—
16	C₂×#13867	C₂²	1	1	yes	characteristic
17	#493370	C₂²	1	4	yes	complemented
18	#493357	C₂²	1	4	yes	complemented
19	#392113	C₂²	1	1	yes	characteristic
20	#392109	C₂²	1	1	yes	characteristic
21	#392109	C₂²	1	1	yes	characteristic
22	#359509	C₂²	1	2	yes	complemented
23	#359442	C₂²	1	2	yes	complemented
24	C₂×#8534	C₂²	1	2	yes	complemented
25	C₂×#7304	C₂²	1	4	yes	complemented
26	#108222	C₂²	1	2	yes	complemented
27	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic
28	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
29	C₂⁵×D₄	C₂³	1	1	yes	characteristic
30	C₂×#3484	D₄	1	2	yes	complemented
31	C₂×#3484	C₂³	1	1	yes	characteristic
32	C₂²×#535	C₂³	1	2	yes	—
33	C₂²×#534	C₂³	1	2	yes	—
34	C₂²×#525	D₄	1	2	yes	complemented
35	C₂³×#120	D₄	1	2	yes	—
36	C₂²×#518	D₄	1	1	yes	characteristic
37	C₂²×#518	D₄	1	1	yes	characteristic, complemented
38	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic, complemented
39	#34171	D₄	1	2	yes	complemented
40	#34171	C₂³	1	1	yes	characteristic
41	C₂²×#379	D₄	1	1	yes	characteristic, complemented
42	C₂×#1783	D₄	1	4	yes	complemented
43	C₂×#1783	C₂³	1	2	yes	—
44	C₂×#1781	D₄	1	4	yes	complemented
45	C₂×#1781	C₂³	1	2	yes	—
46	C₂×#1781	C₂³	1	2	yes	—
47	#15262	D₄	1	4	yes	complemented
48	#15261	D₄	1	4	yes	complemented
49	#15259	D₄	1	4	yes	complemented
50	#15258	D₄	1	4	yes	complemented
51	C₂×#1438	D₄	1	2	yes	complemented
52	C₂×#1438	C₂³	1	2	yes	—
53	C₂⁷	C₂×D₄	1	1	yes	characteristic
54	C₂⁷	C₂×D₄	1	1	yes	characteristic
55	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
56	C₂³×#41	C₂×D₄	1	2	yes	—
57	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
58	C₂⁴×D₄	C₂×D₄	1	2	yes	—
59	C₂⁴×D₄	C₂×D₄	1	2	yes	—
60	C₂⁴×D₄	C₂×D₄	1	2	yes	—
61	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
62	C₂⁴×D₄	C₂×D₄	1	2	yes	—
63	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
64	C₂⁴×D₄	#41	1	2	yes	—
65	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
66	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×#127	C₂×D₄	1	4	yes	—
68	C₂²×#124	C₂×D₄	1	4	yes	—
69	C₂²×#123	C₂×D₄	1	4	yes	—
70	C₂²×#120	C₂×D₄	1	4	yes	—
71	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
72	C₂×#518	C₂×D₄	1	2	yes	complemented
73	C₂²×#117	C₂×D₄	1	2	yes	—
74	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
75	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
76	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
77	C₂²×#100	C₂×D₄	1	4	yes	complemented
78	C₂²×#99	C₂×D₄	1	4	yes	complemented
79	#1438	C₂×D₄	1	4	yes	—
80	#1438	C₂×D₄	1	4	yes	—
81	C₂⁶	#120	1	1	yes	characteristic
82	C₂⁶	#120	1	1	yes	characteristic
83	C₂⁶	#120	1	2	yes	—
84	C₂⁶	#120	1	1	yes	characteristic
85	C₂⁶	#120	1	2	yes	—
86	C₂⁶	#121	1	2	yes	—
87	C₂⁶	#121	1	2	yes	—
88	C₂⁶	#127	1	2	yes	—
89	C₂⁶	C₂²×D₄	1	1	yes	characteristic
90	C₂⁶	C₂²×D₄	1	1	yes	characteristic
91	C₂³×Q₈	#120	1	2	yes	—
92	C₂³×D₄	#120	1	2	yes	—
93	C₂³×D₄	#120	1	2	yes	—
94	C₂³×D₄	#120	1	2	yes	—
95	C₂³×D₄	#121	1	2	yes	—
96	C₂³×D₄	#121	1	2	yes	—
97	C₂³×D₄	#121	1	2	yes	—
98	C₂³×D₄	#127	1	2	yes	—
99	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
100	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
101	C₂³×D₄	C₂²×D₄	1	4	yes	—
102	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
103	C₂³×D₄	C₂×#41	1	1	yes	characteristic
104	C₂³×D₄	H(5,𝔽₂)	1	4	yes	—
105	C₂⁴×C₄	#120	1	1	yes	characteristic
106	C₂⁴×C₄	#120	1	2	yes	—
107	C₂⁴×C₄	#120	1	2	yes	—
108	C₂⁴×C₄	#121	1	2	yes	—
109	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
110	C₂²×C₂²⋊C₄	#120	1	4	yes	—
111	C₂²×C₂²⋊C₄	#120	1	2	yes	—
112	C₂²×C₂²⋊C₄	#120	1	4	yes	—
113	C₂²×C₂²⋊C₄	#120	1	2	yes	—
114	C₂²×C₄²	#120	1	4	yes	—
115	C₂⁵	#392	1	2	yes	—
116	C₂⁵	#392	1	2	yes	—
117	C₂⁵	#392	1	2	yes	—
118	C₂⁵	B(4,𝔽₂)	1	2	yes	—
119	C₂⁵	B(4,𝔽₂)	1	2	yes	—
120	C₂⁵	C₂×#120	1	2	yes	—
121	C₂⁵	C₂×#120	1	1	yes	characteristic
122	C₂⁵	C₂×#120	1	2	yes	—
123	C₂⁵	C₂×#120	1	1	yes	characteristic
124	C₂⁵	C₂×#121	1	1	yes	characteristic
125	C₂⁵	C₂×#121	1	1	yes	characteristic
126	C₂⁵	C₂×#127	1	1	yes	characteristic
127	C₂⁵	#534	1	2	yes	—
128	C₂⁵	#535	1	2	yes	—
129	C₂²×D₄	B(4,𝔽₂)	1	8	yes	—
130	C₂²×D₄	B(4,𝔽₂)	1	8	yes	—
131	C₂²×D₄	D₄²	1	4	yes	—
132	C₂²×D₄	D₄²	1	4	yes	—
133	C₂²×D₄	D₄²	1	4	yes	—
134	C₂²×D₄	#546	1	4	yes	—
135	C₂³×C₄	#392	1	2	yes	—
136	C₂³×C₄	C₂×#120	1	2	yes	—
137	C₂³×C₄	C₂×#120	1	1	yes	characteristic
138	C₂³×C₄	C₂×#121	1	1	yes	characteristic
139	C₂³×C₄	#534	1	2	yes	—
140	C₂⁴	#2021	1	4	yes	—
141	C₂⁴	#2021	1	4	yes	—
142	C₂⁴	#2021	1	4	yes	—
143	C₂⁴	#2021	1	4	yes	—
144	C₂⁴	C₂×#392	1	1	yes	characteristic
145	C₂⁴	#2403	1	2	yes	—
146	C₂⁴	#2408	1	2	yes	—
147	C₂⁴	#2410	1	2	yes	—
148	C₂⁴	#2433	1	2	yes	—
149	C₂⁴	#2613	1	1	yes	characteristic
150	C₂⁴	#2846	1	1	yes	characteristic
151	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
152	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
153	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
154	C₂⁴	#3026	1	1	yes	characteristic
155	C₂⁴	#3026	1	1	yes	characteristic
156	C₂²×C₄	#2021	1	4	yes	—
157	C₂²×C₄	#2403	1	2	yes	—
158	C₂²×C₄	#2410	1	2	yes	—
159	C₂²×C₄	C₂×B(4,𝔽₂)	1	4	yes	—
160	C₂³	#10336	1	8	yes	—
161	C₂³	#14715	1	1	yes	characteristic
162	C₂³	C₂×#2021	1	1	yes	characteristic
163	C₂³	C₂×#2021	1	2	yes	—
164	C₂³	C₂×#2021	1	2	yes	—
165	C₂³	#23444	1	1	yes	characteristic
166	C₂³	#23900	1	4	yes	—
167	C₂³	#29634	1	1	yes	characteristic
168	C₂³	#29634	1	1	yes	characteristic
169	C₂³	#32898	1	1	yes	characteristic
170	C₂³	#32898	1	1	yes	characteristic
171	C₂×C₄	#23900	1	4	yes	—
172	C₂²	C₂×#10336	1	2	yes	—
173	C₂²	#233541	1	2	yes	—
174	C₂²	#393929	1	1	yes	characteristic
175	C₂²	#394058	1	1	yes	characteristic
176	C₂²	#394058	1	1	yes	characteristic
177	C₂²	#511790	1	1	yes	characteristic
178	C₂²	#511791	1	1	yes	characteristic
179	C₂²	#511793	1	2	yes	—
180	C₂	#72214549	1	1	yes	characteristic
181	C₂	#72235138	1	1	yes	characteristic
182	C₂	#72235244	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#393929
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266378	C₂	semidirect
2. #72263490	C₂	semidirect
3. #72259004	C₂	semidirect
4. #72258393	C₂	semidirect
5. #72236440	C₂	semidirect
6. #72218306	C₂	semidirect
7. #72213862	C₂	semidirect
8. #72210555	C₂	semidirect
9. #72210511	C₂	semidirect
13. #6437621	C₂²	semidirect
14. #6437445	C₂²	semidirect
17. #493370	C₂²	semidirect
18. #493357	C₂²	semidirect
22. #359509	C₂²	semidirect
23. #359442	C₂²	semidirect
24. C₂×#8534	C₂²	semidirect
25. C₂×#7304	C₂²	semidirect
26. #108222	C₂²	semidirect
28. C₂³×H(5,𝔽₂)	D₄	semidirect
30. C₂×#3484	D₄	semidirect
34. C₂²×#525	D₄	semidirect
37. C₂²×#518	D₄	semidirect
38. C₂⁴×C₂²⋊C₄	D₄	semidirect
39. #34171	D₄	semidirect
41. C₂²×#379	D₄	semidirect
42. C₂×#1783	D₄	semidirect
44. C₂×#1781	D₄	semidirect
47. #15262	D₄	semidirect
48. #15261	D₄	semidirect
49. #15259	D₄	semidirect
50. #15258	D₄	semidirect
51. C₂×#1438	D₄	semidirect
55. C₂²×H(5,𝔽₂)	C₂×D₄	semidirect
71. C₂×#518	C₂×D₄	semidirect
72. C₂×#518	C₂×D₄	semidirect
77. C₂²×#100	C₂×D₄	semidirect
78. C₂²×#99	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72266378	2
#72263490	2
#72259004	2
#72258393	2
#72236440	2
#72218306	2
#72213862	2
#72210555	2
#72210511	2