Subgroups — #72314779

Order 2048.

At least 6113 conjugacy classes of subgroups; only the 364 normal subgroups are shown.

The subgroup lattice has 231 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	#72312216	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310366	C₂	1	1	yes	maximal, characteristic, complemented
4	#72258334	C₂	1	1	yes	maximal, characteristic, complemented
5	#72246299	C₂	1	1	yes	maximal, characteristic, complemented
6	#72238974	C₂	1	1	yes	maximal, characteristic, complemented
7	#72225743	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#139748	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#493474	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#139716	C₂	1	1	yes	maximal, characteristic, complemented
11	#72219162	C₂	1	1	yes	maximal, characteristic, complemented
12	#72218475	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218317	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#139731	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#347042	C₂	1	1	yes	maximal, characteristic
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	C₂×#23900	C₂²	1	1	yes	characteristic
18	C₂×#23900	C₂²	1	1	yes	characteristic
19	C₂×#21778	C₂²	1	1	yes	characteristic
20	C₂×#21776	C₂²	1	1	yes	characteristic
21	C₂×#21746	C₂²	1	1	yes	characteristic, complemented
22	C₂×#15947	C₂²	1	1	yes	characteristic
23	C₂×#12707	C₂²	1	1	yes	characteristic
24	C₂×#12705	C₂²	1	1	yes	characteristic
25	C₂×#12705	C₂²	1	1	yes	characteristic
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	C₂×#12687	C₂²	1	1	yes	characteristic
28	C₂×#12687	C₂²	1	1	yes	characteristic
29	C₂×#12687	C₂²	1	1	yes	characteristic
30	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
31	#392030	C₂²	1	1	yes	characteristic, complemented
32	#359558	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic
34	#359501	C₂²	1	1	yes	characteristic
35	#359501	C₂²	1	1	yes	characteristic, complemented
36	#355096	C₂²	1	1	yes	characteristic, complemented
37	#347042	C₄	1	8	yes	complemented
38	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
39	C₂×#8541	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	#146475	C₂²	1	1	yes	characteristic, complemented
43	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
51	#107130	C₂²	1	1	yes	characteristic, complemented
52	#62817	C₂×C₄	1	4	yes	complemented
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂³×#120	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂×#2883	D₄	1	2	yes	complemented
58	C₂×#2846	D₄	1	2	yes	—
59	#34171	C₂³	1	1	yes	characteristic
60	#23900	C₂×C₄	1	4	yes	—
61	#23900	C₂×C₄	1	4	yes	complemented
62	#23900	D₄	1	4	yes	—
63	C₂×#1913	D₄	1	2	yes	complemented
64	C₂×#1898	D₄	1	2	yes	—
65	#20960	D₄	1	2	yes	complemented
66	C₂×#1828	D₄	1	2	yes	complemented
67	C₂×#1783	C₂³	1	1	yes	characteristic
68	C₂×#1783	C₂³	1	1	yes	characteristic
69	C₂×#1783	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×#1781	C₂³	1	1	yes	characteristic
75	C₂×#1781	C₂³	1	1	yes	characteristic
76	#15947	C₂×C₄	1	4	yes	complemented
77	C₂×#1438	C₂³	1	1	yes	characteristic
78	#12705	D₄	1	4	yes	complemented
79	#12687	D₄	1	4	yes	complemented
80	#8541	C₂×C₄	1	4	yes	complemented
81	#8534	C₂×C₄	1	4	yes	complemented
82	#8534	C₂×C₄	1	4	yes	complemented
83	C₂⁷	C₂×D₄	1	1	yes	characteristic
84	C₂⁷	#41	1	1	yes	characteristic
85	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
86	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
87	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
88	#3484	C₂²×C₄	1	2	yes	—
89	#3484	C₂²×C₄	1	2	yes	—
90	#3484	C₂×D₄	1	2	yes	—
91	#3484	C₂×D₄	1	2	yes	—
92	#3484	#41	1	2	yes	—
93	#3484	#41	1	2	yes	—
94	C₂×#534	C₂×D₄	1	2	yes	—
95	C₂²×#120	C₂²×C₄	1	2	yes	—
96	C₂²×#120	C₂²×C₄	1	2	yes	—
97	C₂²×#120	C₂×D₄	1	2	yes	—
98	C₂²×#120	C₂×D₄	1	2	yes	—
99	C₂²×#120	C₂×D₄	1	2	yes	—
100	C₂×#518	C₂×D₄	1	1	yes	characteristic
101	C₂×#518	C₂×D₄	1	1	yes	characteristic
102	C₂×#518	C₂×D₄	1	1	yes	characteristic
103	C₂×#518	#41	1	1	yes	characteristic
104	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	#41	1	1	yes	characteristic
106	C₂²×#99	C₂×D₄	1	2	yes	—
107	C₂²×#99	C₂×D₄	1	2	yes	—
108	C₂²×#99	C₂×D₄	1	2	yes	—
109	#2846	C₂²⋊C₄	1	8	yes	complemented
110	C₂×#379	C₂×D₄	1	2	yes	—
111	#1898	C₂²⋊C₄	1	8	yes	complemented
112	#1783	C₂×D₄	1	2	yes	—
113	#1783	C₂×D₄	1	2	yes	—
114	#1781	C₂²×C₄	1	2	yes	—
115	#1781	C₂²×C₄	1	2	yes	—
116	#1781	C₂×D₄	1	2	yes	—
117	#1781	C₂×D₄	1	2	yes	—
118	#1781	C₂×D₄	1	2	yes	—
119	#1438	C₂²×C₄	1	2	yes	—
120	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂⁶	C₄×D₄	1	2	yes	complemented
122	C₂⁶	C₂²×D₄	1	1	yes	characteristic
123	C₂⁶	C₂×#41	1	1	yes	characteristic
124	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
125	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	complemented
126	C₂×H(5,𝔽₂)	#121	1	2	yes	—
127	C₂³×D₄	#120	1	2	yes	—
128	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
129	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
130	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
131	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
132	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
133	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
134	C₂³×D₄	C₂×#41	1	1	yes	characteristic
135	C₂³×D₄	C₂×#41	1	1	yes	characteristic
136	C₂³×D₄	C₂×#41	1	1	yes	characteristic
137	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
141	#534	C₄×D₄	1	4	yes	—
142	#534	#121	1	4	yes	—
143	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
144	C₂×#120	#123	1	4	yes	—
145	#518	C₄×D₄	1	2	yes	—
146	#518	C₄×D₄	1	2	yes	complemented
147	#518	#121	1	2	yes	—
148	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	complemented
150	C₂²×C₂²⋊C₄	#120	1	2	yes	—
151	C₂×#99	#120	1	4	yes	—
152	#379	C₂×C₂²⋊C₄	1	4	yes	—
153	C₂⁵	B(4,𝔽₂)	1	2	yes	—
154	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
155	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂⁵	C₂×#120	1	1	yes	characteristic
157	C₂⁵	C₂×#123	1	1	yes	characteristic
158	C₂⁵	#525	1	1	yes	characteristic
159	C₂⁵	#525	1	1	yes	characteristic
160	C₂⁵	D₄²	1	1	yes	characteristic
161	C₂⁵	D₄²	1	1	yes	characteristic
162	C₂⁵	#546	1	1	yes	characteristic
163	C₂⁵	#546	1	1	yes	characteristic
164	C₂⁵	#546	1	1	yes	characteristic
165	C₂⁵	#546	1	1	yes	characteristic
166	C₂⁵	#551	1	1	yes	characteristic
167	C₂²×D₄	#386	1	4	yes	—
168	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
169	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×D₄	#518	1	1	yes	characteristic
173	C₂²×D₄	C₂×#120	1	1	yes	characteristic
174	C₂²×D₄	C₂×#121	1	1	yes	characteristic
175	C₂²×D₄	C₂×#121	1	1	yes	characteristic
176	C₂²×D₄	#525	1	1	yes	characteristic
177	C₂²×D₄	#534	1	1	yes	characteristic
178	C₂²×D₄	#538	1	1	yes	characteristic
179	C₂²×D₄	D₄²	1	1	yes	characteristic
180	C₂²×D₄	D₄²	1	1	yes	characteristic
181	C₂²×D₄	D₄²	1	1	yes	characteristic
182	C₂²×D₄	#546	1	1	yes	characteristic
183	C₂²×D₄	#546	1	1	yes	characteristic
184	C₂²×D₄	#551	1	1	yes	characteristic
185	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
186	C₂³×C₄	#518	1	1	yes	characteristic
187	C₂³×C₄	#535	1	1	yes	characteristic
188	C₂³×C₄	#538	1	1	yes	characteristic
189	C₂³×C₄	D₄²	1	1	yes	characteristic
190	C₂³×C₄	#546	1	1	yes	characteristic
191	C₂³×C₄	#546	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	#386	1	4	yes	—
193	C₂⁴	#1889	1	4	yes	—
194	C₂⁴	C₂×#386	1	1	yes	characteristic
195	C₂⁴	C₄×#120	1	1	yes	characteristic
196	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
197	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
198	C₂⁴	#2358	1	1	yes	characteristic
199	C₂⁴	#2403	1	1	yes	characteristic
200	C₂⁴	#2404	1	1	yes	characteristic
201	C₂⁴	#2418	1	1	yes	characteristic
202	C₂⁴	#2418	1	1	yes	characteristic
203	C₂⁴	#2424	1	1	yes	characteristic
204	C₂⁴	#2534	1	1	yes	characteristic
205	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
206	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
207	C₂²×C₄	#1889	1	4	yes	—
208	C₂²×C₄	#2321	1	1	yes	characteristic
209	C₂²×C₄	#2358	1	1	yes	characteristic
210	C₂²×C₄	#2408	1	1	yes	characteristic
211	C₂²×C₄	#2424	1	1	yes	characteristic
212	C₂³	#13967	1	1	yes	characteristic
213	C₂³	C₂×#1889	1	2	yes	—
214	C₂³	#23900	1	2	yes	—
215	C₂³	#23984	1	1	yes	characteristic
216	C₂³	#23984	1	1	yes	characteristic
217	C₂³	#23996	1	1	yes	characteristic
218	C₂³	#23996	1	1	yes	characteristic
219	C₂³	#23996	1	1	yes	characteristic
220	C₂³	#23997	1	1	yes	characteristic
221	C₂²	#251827	1	1	yes	characteristic
222	C₂²	#251837	1	1	yes	characteristic
223	C₂²	#251859	1	1	yes	characteristic
224	C₂²	#251871	1	1	yes	characteristic
225	C₂²	#364410	1	1	yes	characteristic
226	C₂²	#364532	1	1	yes	characteristic
227	C₂²	#364545	1	1	yes	characteristic
228	C₂	#72216344	1	2	yes	—
229	C₂	#72241183	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#251827
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	C₂×#347042	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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. #72312216	C₂	semidirect
3. #72310366	C₂	semidirect
4. #72258334	C₂	semidirect
5. #72246299	C₂	semidirect
6. #72238974	C₂	semidirect
7. #72225743	C₂	semidirect
8. C₂×#139748	C₂	semidirect
9. C₂×#493474	C₂	semidirect
10. C₂×#139716	C₂	semidirect
11. #72219162	C₂	semidirect
12. #72218475	C₂	semidirect
13. #72218317	C₂	semidirect
14. C₂×#139731	C₂	semidirect
21. C₂×#21746	C₂²	semidirect
30. C₂×#12687	C₂²	semidirect
31. #392030	C₂²	semidirect
35. #359501	C₂²	semidirect
36. #355096	C₂²	semidirect
37. #347042	C₄	semidirect
38. C₂×#8547	C₂²	semidirect
42. #146475	C₂²	semidirect
43. C₂×#7304	C₂²	semidirect
44. C₂×#7304	C₂²	semidirect
45. C₂×#7316	C₂²	semidirect
46. C₂×#7314	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7311	C₂²	semidirect
49. C₂×#7310	C₂²	semidirect
50. C₂×#7307	C₂²	semidirect
51. #107130	C₂²	semidirect
52. #62817	C₂×C₄	semidirect
57. C₂×#2883	D₄	semidirect
61. #23900	C₂×C₄	semidirect
63. C₂×#1913	D₄	semidirect
65. #20960	D₄	semidirect
66. C₂×#1828	D₄	semidirect
76. #15947	C₂×C₄	semidirect
78. #12705	D₄	semidirect
79. #12687	D₄	semidirect
80. #8541	C₂×C₄	semidirect
81. #8534	C₂×C₄	semidirect
82. #8534	C₂×C₄	semidirect
109. #2846	C₂²⋊C₄	semidirect
111. #1898	C₂²⋊C₄	semidirect
121. C₂⁶	C₄×D₄	semidirect
125. C₂×H(5,𝔽₂)	C₄×D₄	semidirect
146. #518	C₄×D₄	semidirect
149. C₂²×C₂²⋊C₄	C₄×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
#72312216	2
#72310366	2
#72258334	2
#72246299	2
#72238974	2
#72225743	2
C₂×#139748	2
C₂×#493474	2
C₂×#139716	2
#72219162	2
#72218475	2
#72218317	2
C₂×#139731	2
C₂×#347042	2