Subgroups — #72234162

Order 2048.

At least 6478 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

The subgroup lattice has 210 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234159	C₂	1	1	yes	maximal, characteristic, complemented
2	#72234148	C₂	1	1	yes	maximal, characteristic, complemented
3	#72234139	C₂	1	1	yes	maximal, characteristic, complemented
4	#72234129	C₂	1	1	yes	maximal, characteristic, complemented
5	#72233574	C₂	1	1	yes	maximal, characteristic, complemented
6	#72233342	C₂	1	2	yes	maximal, complemented
7	#72232784	C₂	1	2	yes	maximal, complemented
8	#72231707	C₂	1	1	yes	maximal, characteristic, complemented
9	#72216087	C₂	1	1	yes	maximal, characteristic, complemented
10	#72213256	C₂	1	2	yes	maximal, complemented
11	#72210600	C₂	1	2	yes	maximal, complemented
12	C₂⁴×#127	C₂²	1	1	yes	characteristic
13	C₂×#59126	C₂²	1	1	yes	characteristic
14	#7630939	C₂²	1	1	yes	characteristic, complemented
15	#6470225	C₂²	1	2	yes	complemented
16	C₂×#23906	C₂²	1	1	yes	characteristic
17	C₂×#23906	C₂²	1	1	yes	characteristic, complemented
18	#846785	C₂²	1	2	yes	—
19	#612424	C₂²	1	2	yes	—
20	#531766	C₂²	1	1	yes	characteristic
21	#496225	C₂²	1	2	yes	complemented
22	#496225	C₂²	1	2	yes	complemented
23	C₂×#12693	C₂²	1	1	yes	characteristic
24	C₂×#12693	C₂²	1	1	yes	characteristic
25	C₂×#12693	C₂²	1	2	yes	complemented
26	#392113	C₂²	1	1	yes	characteristic, complemented
27	#392109	C₂²	1	2	yes	complemented
28	#359458	C₂²	1	1	yes	characteristic
29	#359458	C₂²	1	1	yes	characteristic, complemented
30	#201664	C₂²	1	2	yes	complemented
31	#187928	C₂²	1	2	yes	complemented
32	#128073	C₂²	1	2	yes	complemented
33	#128073	C₂²	1	2	yes	complemented
34	#106766	C₂²	1	2	yes	complemented
35	C₂⁵×D₄	D₄	1	1	yes	characteristic, complemented
36	C₂⁵×D₄	C₂³	1	1	yes	characteristic
37	C₂×#3484	C₂³	1	1	yes	characteristic
38	C₂²×D₄²	D₄	1	1	yes	characteristic, complemented
39	C₂³×#127	D₄	1	1	yes	characteristic, complemented
40	C₂³×#127	D₄	1	1	yes	characteristic, complemented
41	C₂³×#127	C₂³	1	1	yes	characteristic
42	C₂³×#127	C₂³	1	1	yes	characteristic
43	C₂³×#120	D₄	1	2	yes	complemented
44	C₂⁴×C₄²	D₄	1	1	yes	characteristic, complemented
45	C₂²×#453	D₄	1	2	yes	complemented
46	C₂×#2669	D₄	1	2	yes	complemented
47	C₂×#2403	D₄	1	2	yes	complemented
48	C₂×#2403	C₂³	1	2	yes	—
49	#34171	C₂³	1	1	yes	characteristic
50	C₂×#2366	D₄	1	1	yes	characteristic, complemented
51	C₂²×#353	D₄	1	4	yes	complemented
52	C₂×#1887	D₄	1	2	yes	complemented
53	C₂×#1783	C₂³	1	1	yes	characteristic
54	C₂×#1783	C₂³	1	1	yes	characteristic
55	C₂×#1781	C₂³	1	2	yes	—
56	C₂×#1781	C₂³	1	2	yes	—
57	#15313	D₄	1	4	yes	complemented
58	#15258	D₄	1	4	yes	complemented
59	#14679	D₄	1	2	yes	complemented
60	#14419	D₄	1	2	yes	complemented
61	#14154	D₄	1	2	yes	complemented
62	#13867	D₄	1	2	yes	complemented
63	#13867	C₂³	1	2	yes	—
64	#10975	D₄	1	4	yes	complemented
65	C₂⁷	C₂×D₄	1	2	yes	—
66	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
67	C₂⁴×D₄	C₂×D₄	1	2	yes	—
68	C₂⁴×D₄	C₂×D₄	1	2	yes	—
69	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
70	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
71	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
72	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
73	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×#127	C₂×D₄	1	1	yes	characteristic
75	C₂²×#127	C₂×D₄	1	2	yes	—
76	C₂²×#127	C₂×D₄	1	2	yes	—
77	C₂²×#127	C₂×D₄	1	4	yes	—
78	C₂²×#127	C₂×D₄	1	1	yes	characteristic
79	C₂²×#127	C₂×D₄	1	1	yes	characteristic
80	C₂²×#127	C₂×D₄	1	1	yes	characteristic
81	C₂²×#127	C₂×D₄	1	2	yes	—
82	C₂²×#127	#41	1	2	yes	—
83	C₂²×#120	C₂×D₄	1	2	yes	—
84	C₂²×#120	C₂×D₄	1	2	yes	—
85	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
86	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
87	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
88	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₂²×#100	C₂×D₄	1	2	yes	complemented
90	C₂²×#100	C₂×D₄	1	2	yes	complemented
91	C₂²×#99	C₂×D₄	1	4	yes	complemented
92	#2309	C₂×D₄	1	2	yes	complemented
93	C₂×#386	C₂×D₄	1	2	yes	—
94	C₂×#386	C₂×D₄	1	2	yes	—
95	C₂×#379	C₂×D₄	1	2	yes	—
96	C₂×#379	C₂×D₄	1	2	yes	—
97	#1438	C₂×D₄	1	4	yes	—
98	#1438	C₂×D₄	1	4	yes	—
99	C₂⁶	#120	1	1	yes	characteristic
100	C₂⁶	#120	1	2	yes	—
101	C₂⁶	#120	1	2	yes	—
102	C₂⁶	C₂²×D₄	1	2	yes	—
103	C₂³×D₄	#120	1	1	yes	characteristic
104	C₂³×D₄	#120	1	2	yes	—
105	C₂³×D₄	#120	1	2	yes	—
106	C₂³×D₄	#120	1	1	yes	characteristic
107	C₂³×D₄	#121	1	2	yes	—
108	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
109	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
110	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
111	C₂³×D₄	C₂²×D₄	1	2	yes	—
112	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
113	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
114	C₂³×D₄	C₂×#41	1	1	yes	characteristic
115	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
116	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
117	C₂⁴×C₄	#120	1	1	yes	characteristic
118	C₂⁴×C₄	#120	1	2	yes	—
119	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
120	C₂×#127	#120	1	2	yes	—
121	C₂×#127	#120	1	2	yes	—
122	C₂×#127	#121	1	2	yes	—
123	C₂×#127	#121	1	2	yes	—
124	C₂×#127	#121	1	2	yes	—
125	C₂×#127	#121	1	2	yes	—
126	C₂×#127	#127	1	2	yes	—
127	C₂×#127	#127	1	2	yes	—
128	C₂²×C₂²⋊C₄	#120	1	2	yes	—
129	C₂²×C₂²⋊C₄	#120	1	2	yes	—
130	C₂²×C₂²⋊C₄	#120	1	4	yes	—
131	C₂²×C₂²⋊C₄	#120	1	4	yes	—
132	C₂²×C₄²	#120	1	1	yes	characteristic
133	C₂²×C₄²	#120	1	2	yes	—
134	C₂²×C₄²	#120	1	2	yes	—
135	C₂²×C₄²	#120	1	1	yes	characteristic
136	C₂²×C₄²	#121	1	2	yes	—
137	C₂⁵	B(4,𝔽₂)	1	2	yes	—
138	C₂⁵	C₂×#120	1	2	yes	—
139	C₂⁵	C₂×#120	1	2	yes	—
140	C₂⁵	C₂×#120	1	1	yes	characteristic
141	C₂⁵	D₄²	1	2	yes	—
142	C₂⁵	D₄²	1	2	yes	—
143	C₂⁵	D₄²	1	2	yes	—
144	C₂⁵	#546	1	2	yes	—
145	C₂²×D₄	#392	1	2	yes	—
146	C₂²×D₄	#392	1	2	yes	—
147	C₂²×D₄	C₂×#120	1	1	yes	characteristic
148	C₂²×D₄	C₂×#121	1	1	yes	characteristic
149	C₂²×D₄	C₂×#121	1	1	yes	characteristic
150	C₂²×D₄	C₂×#127	1	1	yes	characteristic
151	C₂²×D₄	#534	1	2	yes	—
152	C₂²×D₄	#535	1	2	yes	—
153	C₂²×D₄	D₄²	1	2	yes	—
154	C₂²×D₄	D₄²	1	2	yes	—
155	C₂²×D₄	D₄²	1	2	yes	—
156	C₂²×D₄	#546	1	2	yes	—
157	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
158	C₂³×C₄	C₂×#120	1	1	yes	characteristic
159	C₂³×C₄	C₂×#120	1	1	yes	characteristic
160	C₂³×C₄	C₂×#120	1	1	yes	characteristic
161	C₂³×C₄	C₂×#121	1	1	yes	characteristic
162	C₂³×C₄	#534	1	1	yes	characteristic
163	C₂³×C₄	#534	1	1	yes	characteristic
164	#127	B(4,𝔽₂)	1	8	yes	—
165	C₂×C₄²	#392	1	2	yes	—
166	C₂×C₄²	#392	1	2	yes	—
167	C₂⁴	#2196	1	4	yes	complemented
168	C₂⁴	#2403	1	1	yes	characteristic
169	C₂⁴	#2403	1	2	yes	—
170	C₂⁴	#2408	1	2	yes	—
171	C₂⁴	#2410	1	1	yes	characteristic
172	C₂⁴	#2410	1	2	yes	—
173	C₂⁴	#2433	1	2	yes	—
174	C₂⁴	#2846	1	1	yes	characteristic
175	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
176	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
177	C₂⁴	#3026	1	1	yes	characteristic
178	C₂⁴	#3026	1	1	yes	characteristic
179	C₂×D₄	#2021	1	4	yes	complemented
180	C₂×D₄	C₂×B(4,𝔽₂)	1	4	yes	—
181	C₂²×C₄	C₂×#392	1	1	yes	characteristic
182	C₂²×C₄	#2403	1	1	yes	characteristic
183	C₂²×C₄	#2410	1	1	yes	characteristic
184	C₂²×C₄	#2613	1	1	yes	characteristic
185	C₄²	#2021	1	4	yes	complemented
186	C₂³	#14715	1	1	yes	characteristic
187	C₂³	#23900	1	1	yes	characteristic
188	C₂³	#23900	1	4	yes	—
189	C₂³	#23903	1	1	yes	characteristic
190	C₂³	#24404	1	1	yes	characteristic
191	C₂³	#24404	1	1	yes	characteristic
192	C₂³	#32898	1	1	yes	characteristic
193	C₂³	#32898	1	1	yes	characteristic
194	C₂³	C₂×#2196	1	2	yes	—
195	C₂³	#33557	1	2	yes	—
196	C₂×C₄	C₂×#2021	1	2	yes	—
197	C₂×C₄	#23895	1	2	yes	—
198	C₂²	#394058	1	1	yes	characteristic
199	C₂²	#394064	1	1	yes	characteristic
200	C₂²	#394066	1	1	yes	characteristic
201	C₂²	#499843	1	2	yes	—
202	C₂²	#501084	1	2	yes	—
203	C₂²	#511771	1	1	yes	characteristic
204	C₂²	#511785	1	1	yes	characteristic
205	C₂²	#511811	1	2	yes	—
206	C₂	#72210600	1	1	yes	characteristic
207	C₂	#72234160	1	1	yes	characteristic
208	C₂	#72234161	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#394066
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. #72234159	C₂	semidirect
2. #72234148	C₂	semidirect
3. #72234139	C₂	semidirect
4. #72234129	C₂	semidirect
5. #72233574	C₂	semidirect
6. #72233342	C₂	semidirect
7. #72232784	C₂	semidirect
8. #72231707	C₂	semidirect
9. #72216087	C₂	semidirect
10. #72213256	C₂	semidirect
11. #72210600	C₂	semidirect
14. #7630939	C₂²	semidirect
15. #6470225	C₂²	semidirect
17. C₂×#23906	C₂²	semidirect
21. #496225	C₂²	semidirect
22. #496225	C₂²	semidirect
25. C₂×#12693	C₂²	semidirect
26. #392113	C₂²	semidirect
27. #392109	C₂²	semidirect
29. #359458	C₂²	semidirect
30. #201664	C₂²	semidirect
31. #187928	C₂²	semidirect
32. #128073	C₂²	semidirect
33. #128073	C₂²	semidirect
34. #106766	C₂²	semidirect
35. C₂⁵×D₄	D₄	semidirect
38. C₂²×D₄²	D₄	semidirect
39. C₂³×#127	D₄	semidirect
40. C₂³×#127	D₄	semidirect
43. C₂³×#120	D₄	semidirect
44. C₂⁴×C₄²	D₄	semidirect
45. C₂²×#453	D₄	semidirect
46. C₂×#2669	D₄	semidirect
47. C₂×#2403	D₄	semidirect
50. C₂×#2366	D₄	semidirect
51. C₂²×#353	D₄	semidirect
52. C₂×#1887	D₄	semidirect
57. #15313	D₄	semidirect
58. #15258	D₄	semidirect
59. #14679	D₄	semidirect
60. #14419	D₄	semidirect
61. #14154	D₄	semidirect
62. #13867	D₄	semidirect
64. #10975	D₄	semidirect
66. C₂²×H(5,𝔽₂)	C₂×D₄	semidirect
85. C₂×#518	C₂×D₄	semidirect
89. C₂²×#100	C₂×D₄	semidirect
90. C₂²×#100	C₂×D₄	semidirect
91. C₂²×#99	C₂×D₄	semidirect
92. #2309	C₂×D₄	semidirect
167. C₂⁴	#2196	semidirect
179. C₂×D₄	#2021	semidirect
185. C₄²	#2021	semidirect

Maximal subgroups

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

subgroup	index
#72234159	2
#72234148	2
#72234139	2
#72234129	2
#72233574	2
#72233342	2
#72232784	2
#72231707	2
#72216087	2
#72213256	2
#72210600	2