Subgroups — #72265788

Order 2048.

At least 6357 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

The subgroup lattice has 213 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72265398	C₂	1	1	yes	maximal, characteristic
2	#72225897	C₂	1	1	yes	maximal, characteristic, complemented
3	#72218854	C₂	1	1	yes	maximal, characteristic, complemented
4	#72218498	C₂	1	1	yes	maximal, characteristic, complemented
5	#72218467	C₂	1	1	yes	maximal, characteristic
6	#72218306	C₂	1	1	yes	maximal, characteristic
7	#72218198	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#3484	C₂²	1	1	yes	characteristic
9	#7625680	C₄	1	1	yes	characteristic, complemented
10	C₂×#20585	C₄	1	1	yes	characteristic, complemented
11	#517411	C₂²	1	1	yes	characteristic
12	#517411	C₂²	1	1	yes	characteristic
13	C₂×#12705	C₄	1	4	yes	complemented
14	#392113	C₂²	1	1	yes	characteristic
15	#392109	C₂²	1	1	yes	characteristic
16	#392109	C₂²	1	1	yes	characteristic
17	#353408	C₄	1	1	yes	characteristic, complemented
18	#353408	C₄	1	1	yes	characteristic, complemented
19	C₂×#8534	C₄	1	4	yes	complemented
20	#154094	C₂²	1	1	yes	characteristic
21	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic
22	C₂⁵×D₄	C₂³	1	1	yes	characteristic
23	C₂×#3484	C₂×C₄	1	2	yes	—
24	C₂×#3484	C₂×C₄	1	2	yes	—
25	C₂×#3484	D₄	1	2	yes	—
26	C₂×#3484	Q₈	1	2	yes	—
27	C₂²×#534	D₄	1	1	yes	characteristic
28	C₂²×#534	D₄	1	1	yes	characteristic
29	C₂³×#120	D₄	1	1	yes	characteristic
30	C₂³×#120	D₄	1	1	yes	characteristic
31	C₂²×#518	C₂×C₄	1	1	yes	characteristic
32	C₂×#2308	C₂×C₄	1	1	yes	characteristic
33	C₂×#2308	C₂×C₄	1	1	yes	characteristic
34	C₂×#2308	C₂×C₄	1	1	yes	characteristic
35	C₂×#2308	C₂×C₄	1	1	yes	characteristic
36	C₂×#1783	C₂×C₄	1	2	yes	complemented
37	C₂×#1781	C₂×C₄	1	2	yes	complemented
38	C₂×#1781	C₂×C₄	1	2	yes	complemented
39	#14681	D₄	1	2	yes	—
40	#14679	D₄	1	2	yes	—
41	#14335	D₄	1	2	yes	—
42	#14335	D₄	1	2	yes	—
43	C₂×#1438	C₂×C₄	1	2	yes	complemented
44	C₂⁷	C₂×D₄	1	1	yes	characteristic
45	C₂⁷	C₂×D₄	1	1	yes	characteristic
46	C₂²×H(5,𝔽₂)	C₄²	1	2	yes	—
47	C₂²×H(5,𝔽₂)	C₄⋊C₄	1	2	yes	—
48	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
50	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
51	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
52	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
53	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
54	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
55	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
56	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
57	C₂⁴×D₄	C₂×Q₈	1	1	yes	characteristic
58	C₂⁴×D₄	#41	1	1	yes	characteristic
59	C₂⁴×D₄	#41	1	1	yes	characteristic
60	C₂⁴×D₄	#41	1	1	yes	characteristic
61	C₂⁴×D₄	#41	1	1	yes	characteristic
62	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×#534	C₂²⋊C₄	1	2	yes	—
64	C₂×#534	C₂²⋊C₄	1	2	yes	—
65	C₂×#534	C₄⋊C₄	1	2	yes	—
66	C₂×#534	C₄⋊C₄	1	2	yes	—
67	C₂²×#120	C₂²⋊C₄	1	2	yes	—
68	C₂²×#120	C₂²⋊C₄	1	2	yes	—
69	C₂²×#120	C₄⋊C₄	1	2	yes	—
70	C₂²×#120	C₄⋊C₄	1	2	yes	—
71	C₂×#518	C₄²	1	2	yes	—
72	C₂×#518	C₄⋊C₄	1	2	yes	—
73	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×#381	C₂²⋊C₄	1	2	yes	—
76	C₂×#381	C₂²⋊C₄	1	2	yes	—
77	C₂×#379	C₂²⋊C₄	1	2	yes	—
78	C₂×#379	C₂²⋊C₄	1	2	yes	—
79	C₂⁶	#99	1	1	yes	characteristic
80	C₂⁶	#99	1	1	yes	characteristic
81	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂⁶	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂⁶	C₄×D₄	1	1	yes	characteristic
86	C₂⁶	C₄×D₄	1	1	yes	characteristic
87	C₂⁶	#120	1	1	yes	characteristic
88	C₂⁶	#120	1	1	yes	characteristic
89	C₂⁶	#120	1	1	yes	characteristic
90	C₂⁶	#121	1	1	yes	characteristic
91	C₂⁶	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂⁶	#123	1	1	yes	characteristic
93	C₂⁶	#123	1	1	yes	characteristic
94	C₂³×D₄	#95	1	2	yes	—
95	C₂³×D₄	#95	1	2	yes	—
96	C₂³×D₄	#99	1	2	yes	—
97	C₂³×D₄	#99	1	2	yes	—
98	C₂³×D₄	C₂×C₄²	1	1	yes	characteristic
99	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×D₄	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×D₄	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×D₄	#117	1	1	yes	characteristic
103	C₂³×D₄	#117	1	1	yes	characteristic
104	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
105	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
106	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
107	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
108	C₂³×D₄	#121	1	1	yes	characteristic
109	C₂³×D₄	#121	1	1	yes	characteristic
110	C₂³×D₄	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂³×D₄	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂³×D₄	#123	1	1	yes	characteristic
113	C₂³×D₄	#123	1	1	yes	characteristic
114	C₂⁴×C₄	#99	1	1	yes	characteristic
115	C₂⁴×C₄	#99	1	1	yes	characteristic
116	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
118	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
119	C₂⁴×C₄	#120	1	1	yes	characteristic
120	C₂⁴×C₄	#121	1	1	yes	characteristic
121	C₂⁴×C₄	C₂²⋊Q₈	1	1	yes	characteristic
122	#534	#99	1	8	yes	—
123	C₂×#120	#99	1	8	yes	—
124	C₂²×C₂²⋊C₄	#95	1	2	yes	—
125	C₂²×C₂²⋊C₄	#95	1	2	yes	—
126	C₂²×C₂²⋊C₄	#99	1	4	yes	complemented
127	C₂×#95	#99	1	4	yes	complemented
128	C₂⁵	#342	1	2	yes	—
129	C₂⁵	#342	1	2	yes	—
130	C₂⁵	C₂×#95	1	1	yes	characteristic
131	C₂⁵	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂⁵	#379	1	1	yes	characteristic
133	C₂⁵	#380	1	1	yes	characteristic
134	C₂⁵	#381	1	1	yes	characteristic
135	C₂⁵	#385	1	1	yes	characteristic
136	C₂⁵	#385	1	1	yes	characteristic
137	C₂⁵	#385	1	1	yes	characteristic
138	C₂⁵	#386	1	1	yes	characteristic
139	C₂⁵	#386	1	1	yes	characteristic
140	C₂⁵	#386	1	1	yes	characteristic
141	C₂⁵	C₂×#99	1	1	yes	characteristic
142	C₂⁵	C₂×#99	1	1	yes	characteristic
143	C₂⁵	C₂×#99	1	1	yes	characteristic
144	C₂⁵	C₂×#99	1	1	yes	characteristic
145	C₂⁵	C₂×#99	1	2	yes	—
146	C₂⁵	C₂×#99	1	4	yes	—
147	C₂⁵	#410	1	2	yes	—
148	C₂²×D₄	#342	1	4	yes	—
149	C₂²×D₄	#342	1	4	yes	—
150	C₂²×D₄	C₂×#99	1	4	yes	—
151	C₂³×C₄	#342	1	2	yes	—
152	C₂³×C₄	#342	1	2	yes	—
153	C₂³×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₂³×C₄	#380	1	1	yes	characteristic
155	C₂³×C₄	#385	1	1	yes	characteristic
156	C₂³×C₄	#386	1	1	yes	characteristic
157	C₂³×C₄	C₂×#99	1	2	yes	—
158	C₂³×C₄	#410	1	2	yes	—
159	C₂×C₂²⋊C₄	#342	1	4	yes	—
160	C₂×C₂²⋊C₄	#342	1	4	yes	—
161	C₂⁴	#1304	1	4	yes	complemented
162	C₂⁴	C₂²⋊#95	1	1	yes	characteristic
163	C₂⁴	C₂×#342	1	1	yes	characteristic
164	C₂⁴	C₂×#342	1	1	yes	characteristic
165	C₂⁴	C₂×#342	1	2	yes	—
166	C₂⁴	C₂×#342	1	2	yes	—
167	C₂⁴	C₄×#99	1	2	yes	—
168	C₂⁴	#1781	1	1	yes	characteristic
169	C₂⁴	#1781	1	1	yes	characteristic
170	C₂⁴	#1781	1	1	yes	characteristic
171	C₂⁴	#1781	1	1	yes	characteristic
172	C₂⁴	#1782	1	1	yes	characteristic
173	C₂⁴	#1782	1	1	yes	characteristic
174	C₂⁴	#1799	1	2	yes	—
175	C₂⁴	#1819	1	1	yes	characteristic
176	C₂⁴	#1819	1	1	yes	characteristic
177	C₂⁴	#1828	1	2	yes	—
178	C₂⁴	#1828	1	2	yes	—
179	C₂⁴	#1898	1	2	yes	—
180	C₂⁴	#1898	1	2	yes	—
181	C₂⁴	#1898	1	2	yes	—
182	C₂²×C₄	#1304	1	4	yes	complemented
183	C₂²×C₄	C₄×#99	1	2	yes	—
184	C₂²×C₄	#1799	1	2	yes	—
185	C₂²×C₄	#1898	1	2	yes	—
186	C₂³	C₂×#1304	1	2	yes	—
187	C₂³	#8101	1	2	yes	—
188	C₂³	#8541	1	1	yes	characteristic
189	C₂³	#8542	1	1	yes	characteristic
190	C₂³	#8542	1	1	yes	characteristic
191	C₂³	#8543	1	1	yes	characteristic
192	C₂³	#8746	1	2	yes	—
193	C₂³	#8747	1	2	yes	—
194	C₂³	#10725	1	1	yes	characteristic
195	C₂³	#10735	1	1	yes	characteristic
196	C₂³	#10735	1	1	yes	characteristic
197	C₂³	#10761	1	1	yes	characteristic
198	C₂³	#10761	1	1	yes	characteristic
199	C₂³	#10768	1	1	yes	characteristic
200	C₂³	#10768	1	1	yes	characteristic
201	C₂²	C₂²⋊#1304	1	1	yes	characteristic
202	C₂²	#105808	1	1	yes	characteristic
203	C₂²	#106132	1	1	yes	characteristic
204	C₂²	#106654	1	1	yes	characteristic
205	C₂²	#106654	1	1	yes	characteristic
206	C₂²	#106654	1	1	yes	characteristic
207	C₂²	#106654	1	1	yes	characteristic
208	C₂²	#133839	1	2	yes	—
209	C₂	#72219423	1	1	yes	characteristic
210	C₂	#72246292	1	1	yes	characteristic
211	C₂	#72251550	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#106132
commutator	C₂³×D₄	C₂×C₄²
Frattini, cosocle	C₂⁵×D₄	C₂³
prime radical	C₂²×#3484	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72225897	C₂	semidirect
3. #72218854	C₂	semidirect
4. #72218498	C₂	semidirect
7. #72218198	C₂	semidirect
9. #7625680	C₄	semidirect
10. C₂×#20585	C₄	semidirect
13. C₂×#12705	C₄	semidirect
17. #353408	C₄	semidirect
18. #353408	C₄	semidirect
19. C₂×#8534	C₄	semidirect
36. C₂×#1783	C₂×C₄	semidirect
37. C₂×#1781	C₂×C₄	semidirect
38. C₂×#1781	C₂×C₄	semidirect
43. C₂×#1438	C₂×C₄	semidirect
126. C₂²×C₂²⋊C₄	#99	semidirect
127. C₂×#95	#99	semidirect
161. C₂⁴	#1304	semidirect
182. C₂²×C₄	#1304	semidirect

Maximal subgroups

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

subgroup	index
#72265398	2
#72225897	2
#72218854	2
#72218498	2
#72218467	2
#72218306	2
#72218198	2