Subgroups — #72314746

Order 2048.

At least 6078 conjugacy classes of subgroups; only the 258 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	#72314739	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314734	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310319	C₂	1	1	yes	maximal, characteristic, complemented
4	#72309478	C₂	1	1	yes	maximal, characteristic, complemented
5	#72266834	C₂	1	1	yes	maximal, characteristic, complemented
6	#72260584	C₂	1	1	yes	maximal, characteristic, complemented
7	#72258094	C₂	1	1	yes	maximal, characteristic, complemented
8	#72255874	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251527	C₂	1	1	yes	maximal, characteristic, complemented
10	#72251140	C₂	1	1	yes	maximal, characteristic, complemented
11	#72238350	C₂	1	1	yes	maximal, characteristic, complemented
12	#72238342	C₂	1	1	yes	maximal, characteristic
13	#72237632	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219160	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#139743	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	#6440460	C₂²	1	1	yes	characteristic
18	#6440460	C₂²	1	1	yes	characteristic
19	#6440460	C₂²	1	1	yes	characteristic, complemented
20	C₂×#23895	C₂²	1	1	yes	characteristic
21	C₂×#23895	C₂²	1	1	yes	characteristic
22	#6437445	C₂²	1	1	yes	characteristic
23	C₂×#21776	C₂²	1	1	yes	characteristic, complemented
24	#763514	C₂²	1	1	yes	characteristic
25	#493879	C₂²	1	1	yes	characteristic, complemented
26	C₂×#12707	C₂²	1	1	yes	characteristic
27	C₂×#12707	C₂²	1	1	yes	characteristic
28	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
29	C₂×#12705	C₂²	1	1	yes	characteristic
30	C₂×#12705	C₂²	1	1	yes	characteristic
31	#493370	C₂²	1	1	yes	characteristic
32	#493370	C₂²	1	1	yes	characteristic
33	#493370	C₂²	1	1	yes	characteristic
34	#392030	C₂²	1	1	yes	characteristic, complemented
35	#382707	C₂²	1	1	yes	characteristic, complemented
36	#382434	C₂²	1	1	yes	characteristic
37	#382434	C₂²	1	1	yes	characteristic
38	#190116	C₂²	1	1	yes	characteristic, complemented
39	#188050	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
43	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂²×#535	C₂³	1	1	yes	characteristic
55	C₂×#3026	D₄	1	2	yes	—
56	#34992	D₄	1	2	yes	complemented
57	C₂×#2308	C₂³	1	1	yes	characteristic
58	#23907	D₄	1	2	yes	—
59	#23906	D₄	1	2	yes	—
60	#23895	D₄	1	4	yes	—
61	C₂×#1913	D₄	1	2	yes	—
62	C₂×#1898	D₄	1	2	yes	complemented
63	C₂×#1828	D₄	1	2	yes	complemented
64	C₂×#1783	C₂³	1	1	yes	characteristic
65	#20585	D₄	1	2	yes	complemented
66	C₂×#1781	C₂³	1	1	yes	characteristic
67	C₂×#1781	C₂³	1	1	yes	characteristic
68	C₂×#1781	C₂³	1	1	yes	characteristic
69	C₂×#1781	C₂³	1	1	yes	characteristic
70	C₂×#1781	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	C₂×#1438	C₂³	1	1	yes	characteristic
75	#8546	D₄	1	2	yes	complemented
76	#8544	D₄	1	2	yes	complemented
77	#8534	D₄	1	2	yes	complemented
78	#8534	D₄	1	2	yes	complemented
79	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
80	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
81	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
82	C₂³×#41	#41	1	1	yes	characteristic
83	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
84	#3484	C₂×D₄	1	2	yes	—
85	#3484	C₂×D₄	1	2	yes	—
86	#3484	C₂×D₄	1	2	yes	—
87	#3484	C₂×D₄	1	2	yes	—
88	C₂×#534	C₂×D₄	1	2	yes	—
89	C₂²×#127	C₂×D₄	1	1	yes	characteristic
90	C₂²×#127	C₂×D₄	1	1	yes	characteristic
91	C₂²×#124	C₂×D₄	1	1	yes	characteristic
92	C₂²×#124	C₂×D₄	1	1	yes	characteristic
93	C₂²×#120	C₂×D₄	1	2	yes	—
94	C₂×#518	C₂×D₄	1	1	yes	characteristic
95	C₂×#518	C₂×D₄	1	1	yes	characteristic
96	C₂×#518	#41	1	1	yes	characteristic
97	C₂²×#117	#41	1	1	yes	characteristic
98	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×#99	C₂×D₄	1	2	yes	—
101	C₂²×#99	C₂×D₄	1	2	yes	—
102	C₂²×#99	C₂×D₄	1	2	yes	—
103	C₂×#379	C₂×D₄	1	2	yes	—
104	#1783	C₂×D₄	1	2	yes	—
105	#1781	C₂×D₄	1	2	yes	—
106	#1781	C₂×D₄	1	2	yes	—
107	#1781	C₂×D₄	1	2	yes	—
108	#1781	C₂×D₄	1	2	yes	—
109	#1438	C₂×D₄	1	1	yes	characteristic
110	#1438	C₂×D₄	1	1	yes	characteristic
111	C₂⁶	C₂²×D₄	1	1	yes	characteristic
112	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×H(5,𝔽₂)	#120	1	2	yes	—
114	C₂×H(5,𝔽₂)	#121	1	2	yes	—
115	C₂³×Q₈	#121	1	1	yes	characteristic
116	C₂³×D₄	#120	1	2	yes	—
117	C₂³×D₄	#121	1	1	yes	characteristic
118	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
119	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
120	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
121	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
122	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
123	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
124	C₂³×D₄	C₂×#41	1	1	yes	characteristic
125	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
131	#518	#120	1	2	yes	—
132	#518	#121	1	2	yes	—
133	C₂²×C₂²⋊C₄	#120	1	2	yes	—
134	C₂²×C₄²	#121	1	1	yes	characteristic
135	C₂²×C₄²	#121	1	1	yes	characteristic
136	#379	#120	1	4	yes	—
137	C₂⁵	B(4,𝔽₂)	1	2	yes	—
138	C₂⁵	C₂×#120	1	1	yes	characteristic
139	C₂⁵	C₂×#120	1	1	yes	characteristic
140	C₂⁵	#534	1	1	yes	characteristic
141	C₂⁵	#534	1	1	yes	characteristic
142	C₂⁵	#535	1	1	yes	characteristic
143	C₂⁵	#535	1	1	yes	characteristic
144	C₂⁵	D₄²	1	1	yes	characteristic
145	C₂⁵	D₄²	1	1	yes	characteristic
146	C₂⁵	D₄²	1	1	yes	characteristic
147	C₂⁵	#546	1	1	yes	characteristic
148	C₂⁵	#546	1	1	yes	characteristic
149	C₂⁵	#546	1	1	yes	characteristic
150	C₂⁵	#560	1	1	yes	characteristic
151	C₂²×D₄	C₂×#120	1	1	yes	characteristic
152	C₂²×D₄	C₂×#121	1	1	yes	characteristic
153	C₂²×D₄	#534	1	1	yes	characteristic
154	C₂²×D₄	#534	1	1	yes	characteristic
155	C₂²×D₄	#535	1	1	yes	characteristic
156	C₂²×D₄	#538	1	1	yes	characteristic
157	C₂²×D₄	D₄²	1	1	yes	characteristic
158	C₂²×D₄	D₄²	1	1	yes	characteristic
159	C₂²×D₄	D₄²	1	1	yes	characteristic
160	C₂²×D₄	D₄²	1	1	yes	characteristic
161	C₂²×D₄	D₄²	1	1	yes	characteristic
162	C₂²×D₄	D₄²	1	1	yes	characteristic
163	C₂²×D₄	D₄²	1	1	yes	characteristic
164	C₂²×D₄	#546	1	1	yes	characteristic
165	C₂²×D₄	#546	1	1	yes	characteristic
166	C₂²×D₄	#560	1	1	yes	characteristic
167	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
168	C₂³×C₄	C₂×#121	1	1	yes	characteristic
169	C₂³×C₄	#525	1	1	yes	characteristic
170	C₂³×C₄	#538	1	1	yes	characteristic
171	C₂³×C₄	#546	1	1	yes	characteristic
172	C₂³×C₄	#546	1	1	yes	characteristic
173	C₂³×C₄	#546	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
175	C₂⁴	#2403	1	1	yes	characteristic
176	C₂⁴	#2403	1	1	yes	characteristic
177	C₂⁴	#2404	1	1	yes	characteristic
178	C₂⁴	#2408	1	1	yes	characteristic
179	C₂⁴	#2408	1	1	yes	characteristic
180	C₂⁴	#2410	1	1	yes	characteristic
181	C₂⁴	#2668	1	1	yes	characteristic
182	C₂⁴	#2668	1	1	yes	characteristic
183	C₂⁴	#2669	1	1	yes	characteristic
184	C₂⁴	#2669	1	1	yes	characteristic
185	C₂⁴	#2678	1	1	yes	characteristic
186	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
187	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
188	C₂²×C₄	#2410	1	1	yes	characteristic
189	C₂²×C₄	#2424	1	1	yes	characteristic
190	C₂²×C₄	#2424	1	1	yes	characteristic
191	C₂²×C₄	#2678	1	1	yes	characteristic
192	C₂³	#15257	1	1	yes	characteristic
193	C₂³	#23900	1	2	yes	—
194	C₂³	#23900	1	2	yes	—
195	C₂³	#23984	1	1	yes	characteristic
196	C₂³	#23984	1	1	yes	characteristic
197	C₂³	#23996	1	1	yes	characteristic
198	C₂³	#23996	1	1	yes	characteristic
199	C₂³	#23997	1	1	yes	characteristic
200	C₂³	#23997	1	1	yes	characteristic
201	C₂²	#251837	1	1	yes	characteristic
202	C₂²	#251838	1	1	yes	characteristic
203	C₂²	#251871	1	1	yes	characteristic
204	C₂²	#251872	1	1	yes	characteristic
205	C₂²	#417493	1	1	yes	characteristic
206	C₂²	#417538	1	1	yes	characteristic
207	C₂²	#417543	1	1	yes	characteristic
208	C₂	#72241197	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241197
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72238342	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×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. #72314739	C₂	semidirect
2. #72314734	C₂	semidirect
3. #72310319	C₂	semidirect
4. #72309478	C₂	semidirect
5. #72266834	C₂	semidirect
6. #72260584	C₂	semidirect
7. #72258094	C₂	semidirect
8. #72255874	C₂	semidirect
9. #72251527	C₂	semidirect
10. #72251140	C₂	semidirect
11. #72238350	C₂	semidirect
13. #72237632	C₂	semidirect
14. #72219160	C₂	semidirect
15. C₂×#139743	C₂	semidirect
19. #6440460	C₂²	semidirect
23. C₂×#21776	C₂²	semidirect
25. #493879	C₂²	semidirect
28. C₂×#12707	C₂²	semidirect
34. #392030	C₂²	semidirect
35. #382707	C₂²	semidirect
38. #190116	C₂²	semidirect
42. C₂×#8534	C₂²	semidirect
43. C₂×#7304	C₂²	semidirect
44. C₂×#7316	C₂²	semidirect
45. C₂×#7314	C₂²	semidirect
46. C₂×#7310	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7310	C₂²	semidirect
49. C₂×#7307	C₂²	semidirect
50. C₂×#7307	C₂²	semidirect
56. #34992	D₄	semidirect
62. C₂×#1898	D₄	semidirect
63. C₂×#1828	D₄	semidirect
65. #20585	D₄	semidirect
75. #8546	D₄	semidirect
76. #8544	D₄	semidirect
77. #8534	D₄	semidirect
78. #8534	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72314739	2
#72314734	2
#72310319	2
#72309478	2
#72266834	2
#72260584	2
#72258094	2
#72255874	2
#72251527	2
#72251140	2
#72238350	2
#72238342	2
#72237632	2
#72219160	2
C₂×#139743	2