Subgroups — #72315298

Order 2048.

At least 6496 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

The subgroup lattice has 223 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314748	C₂	1	1	yes	maximal, characteristic, complemented
2	#72258019	C₂	1	1	yes	maximal, characteristic, complemented
3	#72255977	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255841	C₂	1	1	yes	maximal, characteristic, complemented
5	#72255778	C₂	1	1	yes	maximal, characteristic, complemented
6	#72251530	C₂	1	1	yes	maximal, characteristic, complemented
7	#72238342	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237402	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237045	C₂	1	1	yes	maximal, characteristic, complemented
10	#72236928	C₂	1	1	yes	maximal, characteristic, complemented
11	#72225742	C₂	1	1	yes	maximal, characteristic, complemented
12	#72218391	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218215	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#139750	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#144378	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	C₂×#23895	C₂²	1	1	yes	characteristic
18	C₂×#23895	C₂²	1	1	yes	characteristic
19	C₂×#23900	C₂²	1	1	yes	characteristic
20	C₂×#23900	C₂²	1	1	yes	characteristic
21	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
22	#6437445	C₂²	1	1	yes	characteristic
23	C₂×#21776	C₂²	1	1	yes	characteristic
24	C₂×#21776	C₂²	1	1	yes	characteristic
25	#616961	C₂²	1	1	yes	characteristic
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	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
35	#392030	C₂²	1	1	yes	characteristic, complemented
36	#382434	C₂²	1	1	yes	characteristic, complemented
37	#382422	C₂²	1	1	yes	characteristic, complemented
38	#188050	C₂²	1	1	yes	characteristic
39	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
40	#187944	C₂²	1	1	yes	characteristic
41	#187944	C₂²	1	1	yes	characteristic
42	#187944	C₂²	1	1	yes	characteristic, complemented
43	#128033	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	#128009	C₂²	1	1	yes	characteristic, complemented
50	#128009	C₂²	1	1	yes	characteristic, complemented
51	#62832	D₄	1	2	yes	complemented
52	#62824	D₄	1	2	yes	complemented
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂²×#535	C₂³	1	1	yes	characteristic
57	C₂³×#120	C₂³	1	1	yes	characteristic
58	C₂×#3026	D₄	1	2	yes	complemented
59	C₂×#2883	D₄	1	2	yes	complemented
60	C₂×#2846	D₄	1	2	yes	complemented
61	C₂×#1913	D₄	1	2	yes	complemented
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	C₂×#1781	C₂³	1	1	yes	characteristic
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	#14335	C₂³	1	1	yes	characteristic
74	#12707	D₄	1	4	yes	complemented
75	#12705	D₄	1	4	yes	complemented
76	#12694	D₄	1	2	yes	complemented
77	#12693	D₄	1	2	yes	complemented
78	#8534	D₄	1	4	yes	complemented
79	C₂⁷	C₂×D₄	1	1	yes	characteristic
80	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
81	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
82	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
83	C₂³×#41	#41	1	1	yes	characteristic
84	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
85	#3484	C₂×D₄	1	2	yes	—
86	#3484	C₂×D₄	1	2	yes	—
87	#3484	C₂×D₄	1	2	yes	—
88	#3484	C₂×D₄	1	2	yes	—
89	#3484	C₂×D₄	1	2	yes	—
90	#3484	#41	1	2	yes	—
91	C₂×#534	C₂×D₄	1	2	yes	—
92	C₂²×#127	C₂×D₄	1	1	yes	characteristic
93	C₂²×#127	C₂×D₄	1	1	yes	characteristic
94	C₂²×#124	C₂×D₄	1	1	yes	characteristic
95	C₂²×#124	C₂×D₄	1	1	yes	characteristic
96	C₂²×#120	C₂×D₄	1	2	yes	—
97	C₂×#518	C₂×D₄	1	1	yes	characteristic
98	C₂×#518	C₂×D₄	1	1	yes	characteristic
99	C₂×#518	#41	1	1	yes	characteristic
100	C₂²×#117	#41	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×#99	C₂×D₄	1	2	yes	—
103	C₂²×#99	C₂×D₄	1	2	yes	—
104	C₂²×#99	C₂×D₄	1	2	yes	—
105	C₂×#379	C₂×D₄	1	2	yes	—
106	#1783	C₂×D₄	1	2	yes	—
107	#1781	C₂×D₄	1	2	yes	—
108	#1781	C₂×D₄	1	2	yes	—
109	#1781	C₂×D₄	1	2	yes	—
110	#1781	C₂×D₄	1	2	yes	—
111	#1781	C₂×D₄	1	2	yes	—
112	#1781	C₂×D₄	1	2	yes	—
113	#1438	C₂×D₄	1	2	yes	—
114	C₂⁶	C₂²×D₄	1	1	yes	characteristic
115	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×H(5,𝔽₂)	#121	1	2	yes	—
117	C₂×H(5,𝔽₂)	#121	1	2	yes	complemented
118	C₂×H(5,𝔽₂)	#127	1	2	yes	—
119	C₂³×Q₈	#121	1	1	yes	characteristic, complemented
120	C₂³×D₄	#120	1	2	yes	—
121	C₂³×D₄	#121	1	1	yes	characteristic, complemented
122	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
123	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
124	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
125	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
126	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
127	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
128	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
129	C₂³×D₄	C₂×#41	1	1	yes	characteristic
130	C₂³×D₄	C₂×#41	1	1	yes	characteristic
131	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
135	#534	#120	1	4	yes	—
136	#534	#121	1	4	yes	—
137	C₂×#120	#120	1	4	yes	—
138	C₂×#120	#121	1	4	yes	—
139	#518	#121	1	2	yes	—
140	#518	#121	1	2	yes	complemented
141	#518	#127	1	2	yes	—
142	C₂²×C₂²⋊C₄	#120	1	2	yes	—
143	C₂²×C₄²	#121	1	1	yes	characteristic, complemented
144	C₂²×C₄²	#121	1	1	yes	characteristic, complemented
145	C₂×#99	#120	1	4	yes	—
146	C₂⁵	B(4,𝔽₂)	1	2	yes	—
147	C₂⁵	C₂×#120	1	1	yes	characteristic
148	C₂⁵	C₂×#120	1	1	yes	characteristic
149	C₂⁵	C₂×#121	1	1	yes	characteristic
150	C₂⁵	#534	1	1	yes	characteristic
151	C₂⁵	#535	1	1	yes	characteristic
152	C₂⁵	#538	1	1	yes	characteristic
153	C₂⁵	D₄²	1	1	yes	characteristic
154	C₂⁵	D₄²	1	1	yes	characteristic
155	C₂⁵	D₄²	1	1	yes	characteristic
156	C₂⁵	D₄²	1	1	yes	characteristic
157	C₂⁵	#546	1	1	yes	characteristic
158	C₂⁵	#546	1	1	yes	characteristic
159	C₂⁵	#546	1	1	yes	characteristic
160	C₂²×D₄	#392	1	4	yes	—
161	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
162	C₂²×D₄	C₂×#120	1	1	yes	characteristic
163	C₂²×D₄	C₂×#120	1	1	yes	characteristic
164	C₂²×D₄	C₂×#121	1	1	yes	characteristic
165	C₂²×D₄	C₂×#121	1	1	yes	characteristic
166	C₂²×D₄	C₂×#121	1	1	yes	characteristic
167	C₂²×D₄	#525	1	1	yes	characteristic
168	C₂²×D₄	C₂×#127	1	1	yes	characteristic
169	C₂²×D₄	#534	1	1	yes	characteristic
170	C₂²×D₄	#534	1	1	yes	characteristic
171	C₂²×D₄	#538	1	1	yes	characteristic
172	C₂²×D₄	D₄²	1	1	yes	characteristic
173	C₂²×D₄	D₄²	1	1	yes	characteristic
174	C₂²×D₄	D₄²	1	1	yes	characteristic
175	C₂²×D₄	D₄²	1	1	yes	characteristic
176	C₂²×D₄	#546	1	1	yes	characteristic
177	C₂²×D₄	#546	1	1	yes	characteristic
178	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
179	C₂³×C₄	C₂×#121	1	1	yes	characteristic
180	C₂³×C₄	#525	1	1	yes	characteristic
181	C₂³×C₄	#525	1	1	yes	characteristic
182	C₂³×C₄	#546	1	1	yes	characteristic
183	C₂³×C₄	#546	1	1	yes	characteristic
184	C₂³×C₄	#546	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	#392	1	4	yes	—
186	C₂⁴	#2021	1	4	yes	—
187	C₂⁴	C₂×#392	1	1	yes	characteristic
188	C₂⁴	#2403	1	1	yes	characteristic
189	C₂⁴	#2404	1	1	yes	characteristic
190	C₂⁴	#2410	1	1	yes	characteristic
191	C₂⁴	#2410	1	1	yes	characteristic
192	C₂⁴	#2410	1	1	yes	characteristic
193	C₂⁴	#2424	1	1	yes	characteristic
194	C₂⁴	#2424	1	1	yes	characteristic
195	C₂⁴	#2424	1	1	yes	characteristic
196	C₂⁴	#2433	1	1	yes	characteristic
197	C₂⁴	#2617	1	1	yes	characteristic
198	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
199	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
200	C₂²×C₄	#2021	1	4	yes	—
201	C₂²×C₄	#2410	1	1	yes	characteristic
202	C₂²×C₄	#2424	1	1	yes	characteristic
203	C₂²×C₄	#2539	1	1	yes	characteristic
204	C₂²×C₄	#2847	1	1	yes	characteristic
205	C₂³	#14716	1	1	yes	characteristic
206	C₂³	C₂×#2021	1	2	yes	—
207	C₂³	#23900	1	2	yes	—
208	C₂³	#23984	1	1	yes	characteristic
209	C₂³	#23984	1	1	yes	characteristic
210	C₂³	#23996	1	1	yes	characteristic
211	C₂³	#23996	1	1	yes	characteristic
212	C₂³	#23996	1	1	yes	characteristic
213	C₂³	#23996	1	1	yes	characteristic
214	C₂²	#251827	1	1	yes	characteristic
215	C₂²	#251854	1	1	yes	characteristic
216	C₂²	#251859	1	1	yes	characteristic
217	C₂²	#251859	1	1	yes	characteristic
218	C₂²	#394167	1	1	yes	characteristic
219	C₂²	#394273	1	1	yes	characteristic
220	C₂²	#394273	1	1	yes	characteristic
221	C₂	#72241287	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241287
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
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. #72314748	C₂	semidirect
2. #72258019	C₂	semidirect
3. #72255977	C₂	semidirect
4. #72255841	C₂	semidirect
5. #72255778	C₂	semidirect
6. #72251530	C₂	semidirect
7. #72238342	C₂	semidirect
8. #72237402	C₂	semidirect
9. #72237045	C₂	semidirect
10. #72236928	C₂	semidirect
11. #72225742	C₂	semidirect
12. #72218391	C₂	semidirect
13. #72218215	C₂	semidirect
14. C₂×#139750	C₂	semidirect
15. C₂×#144378	C₂	semidirect
21. C₂×#23900	C₂²	semidirect
28. C₂×#12707	C₂²	semidirect
34. C₂×#12687	C₂²	semidirect
35. #392030	C₂²	semidirect
36. #382434	C₂²	semidirect
37. #382422	C₂²	semidirect
39. C₂×#8541	C₂²	semidirect
42. #187944	C₂²	semidirect
43. #128033	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. #128009	C₂²	semidirect
50. #128009	C₂²	semidirect
51. #62832	D₄	semidirect
52. #62824	D₄	semidirect
58. C₂×#3026	D₄	semidirect
59. C₂×#2883	D₄	semidirect
60. C₂×#2846	D₄	semidirect
61. C₂×#1913	D₄	semidirect
62. C₂×#1898	D₄	semidirect
63. C₂×#1828	D₄	semidirect
74. #12707	D₄	semidirect
75. #12705	D₄	semidirect
76. #12694	D₄	semidirect
77. #12693	D₄	semidirect
78. #8534	D₄	semidirect
117. C₂×H(5,𝔽₂)	#121	semidirect
119. C₂³×Q₈	#121	semidirect
121. C₂³×D₄	#121	semidirect
140. #518	#121	semidirect
143. C₂²×C₄²	#121	semidirect
144. C₂²×C₄²	#121	semidirect

Maximal subgroups

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

subgroup	index
#72314748	2
#72258019	2
#72255977	2
#72255841	2
#72255778	2
#72251530	2
#72238342	2
#72237402	2
#72237045	2
#72236928	2
#72225742	2
#72218391	2
#72218215	2
C₂×#139750	2
C₂×#144378	2