Subgroups — #72262287

Order 2048.

At least 6376 conjugacy classes of subgroups; only the 402 normal subgroups are shown.

The subgroup lattice has 250 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72256434	C₂	1	1	yes	maximal, characteristic, complemented
2	#72256430	C₂	1	1	yes	maximal, characteristic, complemented
3	#72256033	C₂	1	2	yes	maximal, complemented
4	#72255971	C₂	1	2	yes	maximal, complemented
5	#72255853	C₂	1	2	yes	maximal, complemented
6	#72255764	C₂	1	2	yes	maximal, complemented
7	#72237351	C₂	1	1	yes	maximal, characteristic, complemented
8	#72235449	C₂	1	1	yes	maximal, characteristic, complemented
9	#72218306	C₂	1	1	yes	maximal, characteristic, complemented
10	#72216073	C₂	1	1	yes	maximal, characteristic, complemented
11	#72213862	C₂	1	1	yes	maximal, characteristic
12	C₂²×#3484	C₂²	1	1	yes	characteristic
13	#7630939	C₂²	1	1	yes	characteristic
14	#6628892	C₄	1	2	yes	complemented
15	#6437621	C₂²	1	2	yes	—
16	C₂×#21778	C₂²	1	1	yes	characteristic, complemented
17	C₂²×#1783	C₂²	1	1	yes	characteristic
18	C₂²×#1783	C₂²	1	1	yes	characteristic
19	C₂×#12705	C₂²	1	2	yes	complemented
20	#493357	C₂²	1	2	yes	—
21	#493357	C₂²	1	2	yes	—
22	#394058	C₄	1	4	yes	complemented
23	#392699	C₄	1	2	yes	complemented
24	#392113	C₂²	1	1	yes	characteristic
25	#392109	C₂²	1	1	yes	characteristic
26	#392109	C₂²	1	1	yes	characteristic
27	#382546	C₂²	1	1	yes	characteristic, complemented
28	#382436	C₂²	1	1	yes	characteristic, complemented
29	#359442	C₂²	1	2	yes	—
30	#355098	C₂²	1	1	yes	characteristic, complemented
31	#188109	C₂²	1	2	yes	—
32	#188109	C₂²	1	2	yes	—
33	#146475	C₂²	1	2	yes	complemented
34	C₂×#7316	C₂²	1	2	yes	complemented
35	C₂×#7316	C₂²	1	2	yes	complemented
36	C₂×#7311	C₂²	1	2	yes	complemented
37	C₂×#7311	C₂²	1	2	yes	complemented
38	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic
39	C₂⁵×D₄	C₂³	1	1	yes	characteristic
40	C₂×#3484	D₄	1	1	yes	characteristic
41	C₂×#3484	D₄	1	1	yes	characteristic
42	C₂×#3484	C₂³	1	1	yes	characteristic
43	C₂²×#534	C₂³	1	2	yes	—
44	C₂³×#120	C₂×C₄	1	2	yes	—
45	C₂²×#518	C₂×C₄	1	1	yes	characteristic
46	C₂²×#407	D₄	1	2	yes	complemented
47	C₂×#2846	C₂×C₄	1	2	yes	—
48	#34980	C₂×C₄	1	4	yes	—
49	#34980	C₂×C₄	1	2	yes	—
50	#34980	D₄	1	2	yes	—
51	#34980	D₄	1	2	yes	—
52	#34171	D₄	1	1	yes	characteristic
53	#34171	D₄	1	1	yes	characteristic
54	#34171	C₂³	1	1	yes	characteristic
55	C₂×#1898	C₂×C₄	1	4	yes	—
56	C₂×#1783	D₄	1	2	yes	complemented
57	C₂×#1783	D₄	1	2	yes	complemented
58	C₂×#1783	D₄	1	2	yes	complemented
59	C₂×#1783	C₂³	1	2	yes	—
60	C₂×#1783	C₂³	1	1	yes	characteristic
61	C₂×#1783	C₂³	1	1	yes	characteristic
62	C₂×#1783	C₂³	1	1	yes	characteristic
63	C₂×#1783	C₂³	1	1	yes	characteristic
64	C₂×#1781	C₂×C₄	1	4	yes	—
65	C₂×#1781	C₂×C₄	1	2	yes	—
66	C₂×#1781	C₂³	1	2	yes	—
67	C₂×#1781	C₂³	1	2	yes	—
68	#14679	C₂×C₄	1	2	yes	—
69	#10735	C₂×C₄	1	4	yes	—
70	#8548	D₄	1	4	yes	complemented
71	#8547	D₄	1	4	yes	complemented
72	C₂⁷	C₂²×C₄	1	1	yes	characteristic
73	C₂⁷	#41	1	1	yes	characteristic
74	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
75	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
76	C₂⁴×D₄	C₂²×C₄	1	2	yes	—
77	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
78	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
79	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
80	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
81	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
83	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
84	C₂⁴×D₄	#41	1	1	yes	characteristic
85	C₂⁴×D₄	#41	1	1	yes	characteristic
86	C₂⁴×D₄	#41	1	1	yes	characteristic
87	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
88	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×#534	C₂²⋊C₄	1	2	yes	—
90	C₂×#534	C₂²⋊C₄	1	2	yes	—
91	C₂×#534	C₂×D₄	1	2	yes	—
92	C₂²×#123	C₂×D₄	1	2	yes	—
93	C₂²×#123	#41	1	2	yes	—
94	C₂²×#120	C₂²⋊C₄	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	#41	1	2	yes	—
99	C₂×#518	C₂×D₄	1	1	yes	characteristic
100	C₂×#518	C₂×D₄	1	1	yes	characteristic
101	C₂×#518	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	C₂²×C₄	1	2	yes	—
103	C₂²×#100	C₂×D₄	1	2	yes	—
104	C₂²×#100	C₂×D₄	1	2	yes	—
105	C₂×#407	C₂×D₄	1	2	yes	—
106	C₂×#407	C₂×D₄	1	2	yes	—
107	#2308	C₂²⋊C₄	1	2	yes	—
108	#2308	C₂²⋊C₄	1	2	yes	—
109	#2308	C₂×D₄	1	2	yes	—
110	C₂×#379	C₂²⋊C₄	1	2	yes	—
111	C₂×#379	C₂²⋊C₄	1	2	yes	—
112	C₂×#379	C₂²×C₄	1	4	yes	—
113	C₂×#379	C₂²×C₄	1	2	yes	—
114	#1783	C₂×D₄	1	4	yes	—
115	#1783	C₂×D₄	1	4	yes	—
116	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂⁶	C₄×D₄	1	1	yes	characteristic
119	C₂⁶	C₄×D₄	1	1	yes	characteristic
120	C₂⁶	C₄×D₄	1	1	yes	characteristic
121	C₂⁶	C₄×D₄	1	1	yes	characteristic
122	C₂⁶	C₄×D₄	1	2	yes	—
123	C₂⁶	C₄×D₄	1	1	yes	characteristic
124	C₂⁶	#121	1	1	yes	characteristic
125	C₂⁶	#121	1	1	yes	characteristic
126	C₂⁶	#123	1	1	yes	characteristic
127	C₂⁶	#123	1	1	yes	characteristic
128	C₂⁶	C₂³×C₄	1	1	yes	characteristic
129	C₂⁶	C₂×#41	1	1	yes	characteristic
130	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
134	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
135	C₂³×D₄	#120	1	1	yes	characteristic
136	C₂³×D₄	#120	1	1	yes	characteristic
137	C₂³×D₄	#121	1	1	yes	characteristic
138	C₂³×D₄	#121	1	1	yes	characteristic
139	C₂³×D₄	#121	1	1	yes	characteristic
140	C₂³×D₄	#121	1	1	yes	characteristic
141	C₂³×D₄	#123	1	1	yes	characteristic
142	C₂³×D₄	#123	1	1	yes	characteristic
143	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
145	C₂³×D₄	C₂²×D₄	1	2	yes	—
146	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
147	C₂³×D₄	C₂×#41	1	1	yes	characteristic
148	C₂³×D₄	C₂×#41	1	2	yes	—
149	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
150	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
151	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂⁴×C₄	C₄×D₄	1	2	yes	—
153	C₂⁴×C₄	#120	1	1	yes	characteristic
154	C₂⁴×C₄	#120	1	1	yes	characteristic
155	C₂⁴×C₄	#121	1	1	yes	characteristic
156	C₂⁴×C₄	#121	1	1	yes	characteristic
157	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
158	C₂×#123	#99	1	4	yes	—
159	C₂×#120	#99	1	4	yes	—
160	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
161	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
162	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	—
164	C₂²×C₂²⋊C₄	C₄×D₄	1	2	yes	—
165	#407	#120	1	4	yes	—
166	C₂⁵	#386	1	1	yes	characteristic
167	C₂⁵	#386	1	1	yes	characteristic
168	C₂⁵	#386	1	1	yes	characteristic
169	C₂⁵	#386	1	1	yes	characteristic
170	C₂⁵	#386	1	1	yes	characteristic
171	C₂⁵	#386	1	1	yes	characteristic
172	C₂⁵	C₂×#99	1	2	yes	—
173	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
174	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂⁵	C₂×C₄×D₄	1	2	yes	—
177	C₂⁵	#518	1	2	yes	—
178	C₂⁵	C₂×#121	1	1	yes	characteristic
179	C₂⁵	C₂×#123	1	1	yes	characteristic
180	C₂⁵	#525	1	2	yes	—
181	C₂⁵	#538	1	2	yes	—
182	C₂²×D₄	C₂×#99	1	4	yes	—
183	C₂²×D₄	D₄²	1	2	yes	—
184	C₂²×D₄	D₄²	1	2	yes	—
185	C₂²×D₄	D₄²	1	2	yes	—
186	C₂²×D₄	#546	1	2	yes	—
187	C₂²×D₄	#546	1	2	yes	—
188	C₂²×D₄	#546	1	2	yes	—
189	C₂²×D₄	#546	1	2	yes	—
190	C₂²×D₄	#551	1	2	yes	—
191	C₂³×C₄	#386	1	1	yes	characteristic
192	C₂³×C₄	#386	1	1	yes	characteristic
193	C₂³×C₄	C₂×#99	1	2	yes	—
194	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
195	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
196	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
197	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
198	C₂³×C₄	C₂×#120	1	1	yes	characteristic
199	C₂³×C₄	C₂×#120	1	1	yes	characteristic
200	C₂³×C₄	C₂×#121	1	1	yes	characteristic
201	C₂³×C₄	#525	1	1	yes	characteristic
202	C₂³×C₄	#534	1	1	yes	characteristic
203	C₂³×C₄	#535	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
205	C₂⁴	#1889	1	1	yes	characteristic
206	C₂⁴	#1889	1	1	yes	characteristic
207	C₂⁴	#1889	1	1	yes	characteristic
208	C₂⁴	#1889	1	1	yes	characteristic
209	C₂⁴	#1898	1	2	yes	—
210	C₂⁴	#1913	1	2	yes	—
211	C₂⁴	C₂×#386	1	1	yes	characteristic
212	C₂⁴	C₄×#120	1	1	yes	characteristic
213	C₂⁴	#2321	1	1	yes	characteristic
214	C₂⁴	D₄×C₂²⋊C₄	1	2	yes	—
215	C₂⁴	#2358	1	2	yes	—
216	C₂⁴	#2418	1	2	yes	—
217	C₂⁴	#2424	1	2	yes	—
218	C₂⁴	C₂²×#99	1	2	yes	—
219	C₂⁴	#2883	1	2	yes	—
220	C₂²×C₄	#1889	1	2	yes	—
221	C₂²×C₄	#1889	1	2	yes	—
222	C₂²×C₄	#1898	1	2	yes	—
223	C₂²×C₄	#1913	1	2	yes	—
224	C₂²×C₄	#2403	1	1	yes	characteristic
225	C₂²×C₄	#2404	1	1	yes	characteristic
226	C₂²×C₄	#2408	1	1	yes	characteristic
227	C₂²×C₄	#2534	1	1	yes	characteristic
228	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
229	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
230	C₂³	#13967	1	1	yes	characteristic
231	C₂³	C₂×#1889	1	1	yes	characteristic
232	C₂³	C₂×#1889	1	1	yes	characteristic
233	C₂³	C₂×#1898	1	1	yes	characteristic
234	C₂³	#20984	1	1	yes	characteristic
235	C₂³	C₂×#1913	1	1	yes	characteristic
236	C₂³	C₂×#1913	1	1	yes	characteristic
237	C₂³	D₄×#99	1	4	yes	—
238	C₂×C₄	#9495	1	4	yes	—
239	C₂×C₄	#23900	1	2	yes	—
240	C₂×C₄	#23984	1	2	yes	—
241	C₂²	C₂×#9495	1	1	yes	characteristic
242	C₂²	#201664	1	1	yes	characteristic
243	C₂²	#201678	1	1	yes	characteristic
244	C₂²	#201678	1	1	yes	characteristic
245	C₂²	#364410	1	1	yes	characteristic
246	C₂²	#364440	1	1	yes	characteristic
247	C₂²	#364471	1	1	yes	characteristic
248	C₂	#72240426	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240426
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72213862	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ 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. #72256434	C₂	semidirect
2. #72256430	C₂	semidirect
3. #72256033	C₂	semidirect
4. #72255971	C₂	semidirect
5. #72255853	C₂	semidirect
6. #72255764	C₂	semidirect
7. #72237351	C₂	semidirect
8. #72235449	C₂	semidirect
9. #72218306	C₂	semidirect
10. #72216073	C₂	semidirect
14. #6628892	C₄	semidirect
16. C₂×#21778	C₂²	semidirect
19. C₂×#12705	C₂²	semidirect
22. #394058	C₄	semidirect
23. #392699	C₄	semidirect
27. #382546	C₂²	semidirect
28. #382436	C₂²	semidirect
30. #355098	C₂²	semidirect
33. #146475	C₂²	semidirect
34. C₂×#7316	C₂²	semidirect
35. C₂×#7316	C₂²	semidirect
36. C₂×#7311	C₂²	semidirect
37. C₂×#7311	C₂²	semidirect
46. C₂²×#407	D₄	semidirect
56. C₂×#1783	D₄	semidirect
57. C₂×#1783	D₄	semidirect
58. C₂×#1783	D₄	semidirect
70. #8548	D₄	semidirect
71. #8547	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72256434	2
#72256430	2
#72256033	2
#72255971	2
#72255853	2
#72255764	2
#72237351	2
#72235449	2
#72218306	2
#72216073	2
#72213862	2