Subgroups — #72266225

Order 4096.

At least 13492 conjugacy classes of subgroups; only the 445 normal subgroups are shown.

The subgroup lattice has 229 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266198	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266190	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266167	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266134	C₂	1	1	yes	maximal, characteristic, complemented
5	#72259040	C₂	1	1	yes	maximal, characteristic, complemented
6	#72258223	C₂	1	4	yes	maximal, complemented
7	#72240620	C₂	1	1	yes	maximal, characteristic, complemented
8	#72240548	C₂²	1	2	yes	complemented
9	#72236800	C₂	1	1	yes	maximal, characteristic, complemented
10	#72236528	C₂	1	4	yes	maximal, complemented
11	#72235930	C₂²	1	1	yes	characteristic
12	C₂²×#23895	C₂²	1	2	yes	complemented
13	#72225621	C₂²	1	1	yes	characteristic
14	#72219457	C₂²	1	4	yes	complemented
15	#72219418	C₂²	1	4	yes	complemented
16	#72219377	C₂²	1	1	yes	characteristic
17	#72219377	C₂²	1	1	yes	characteristic
18	#72219307	C₂²	1	1	yes	characteristic
19	C₂×#7631286	C₂²	1	1	yes	characteristic
20	C₂×#7631286	C₂²	1	1	yes	characteristic
21	#72218846	C₂²	1	2	yes	complemented
22	#72218498	C₂²	1	2	yes	complemented
23	#72214832	C₂²	1	4	yes	—
24	#72214832	C₂²	1	2	yes	complemented
25	#72210555	C₂²	1	2	yes	complemented
26	#72210555	C₂²	1	4	yes	complemented
27	C₂⁴×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
28	C₂⁴×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
29	C₂²×#3484	C₂³	1	1	yes	characteristic
30	C₂²×#3484	C₂³	1	2	yes	—
31	C₂³×D₄²	C₂³	1	1	yes	characteristic
32	C₂×#35347	D₄	1	1	yes	characteristic, complemented
33	C₂×#35346	D₄	1	1	yes	characteristic, complemented
34	C₂×#34980	D₄	1	2	yes	complemented
35	C₂²×#2403	C₂³	1	2	yes	—
36	C₂×#34256	D₄	1	2	yes	complemented
37	C₂×#34171	D₄	1	1	yes	characteristic, complemented
38	C₂×#34171	D₄	1	1	yes	characteristic, complemented
39	C₂×#23900	D₄	1	8	yes	complemented
40	C₂×#21746	D₄	1	2	yes	complemented
41	C₂²×#1781	C₂³	1	2	yes	—
42	#517411	C₂³	1	1	yes	characteristic
43	#515362	D₄	1	1	yes	characteristic, complemented
44	#515285	D₄	1	1	yes	characteristic, complemented
45	#392109	C₂³	1	2	yes	—
46	#392109	C₂³	1	2	yes	—
47	#392109	C₂³	1	2	yes	—
48	#382390	D₄	1	2	yes	complemented
49	#359552	D₄	1	8	yes	complemented
50	C₂×#10965	D₄	1	2	yes	complemented
51	C₂×#8546	D₄	1	2	yes	complemented
52	C₂×#8542	D₄	1	2	yes	complemented
53	#154094	D₄	1	2	yes	complemented
54	C₂³×H(5,𝔽₂)	C₂×D₄	1	2	yes	—
55	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
56	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
57	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
58	C₂⁵×D₄	C₂×D₄	1	2	yes	—
59	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
60	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
61	C₂×#3484	C₂×D₄	1	8	yes	—
62	C₂³×#127	C₂×D₄	1	1	yes	characteristic
63	C₂³×#121	C₂×D₄	1	1	yes	characteristic
64	C₂³×#121	C₂×D₄	1	1	yes	characteristic
65	C₂³×#121	C₂×D₄	1	1	yes	characteristic
66	C₂³×#121	#41	1	1	yes	characteristic
67	C₂³×#120	C₂×D₄	1	4	yes	—
68	C₂³×#120	C₂×D₄	1	1	yes	characteristic
69	C₂³×#120	C₂×D₄	1	1	yes	characteristic
70	C₂³×#120	C₂×D₄	1	1	yes	characteristic
71	C₂³×#120	C₂×D₄	1	1	yes	characteristic
72	C₂²×#518	C₂×D₄	1	2	yes	—
73	C₂²×#518	C₂×D₄	1	1	yes	characteristic, complemented
74	C₂³×C₄×D₄	C₂×D₄	1	1	yes	characteristic
75	C₂³×C₄×D₄	#41	1	1	yes	characteristic
76	C₂×#2308	C₂×D₄	1	1	yes	characteristic, complemented
77	C₂×#2308	C₂×D₄	1	1	yes	characteristic, complemented
78	C₂²×#386	C₂×D₄	1	4	yes	—
79	C₂²×#379	C₂×D₄	1	2	yes	—
80	C₂²×#379	C₂×D₄	1	4	yes	complemented
81	C₂×#1781	C₂×D₄	1	4	yes	—
82	C₂×#1781	C₂×D₄	1	8	yes	—
83	C₂×#1781	C₂×D₄	1	4	yes	complemented
84	C₂⁷	#120	1	1	yes	characteristic
85	C₂⁷	#120	1	1	yes	characteristic
86	C₂⁷	#120	1	2	yes	complemented
87	C₂⁷	C₂²×D₄	1	1	yes	characteristic
88	C₂⁷	C₂²×D₄	1	1	yes	characteristic
89	C₂²×H(5,𝔽₂)	#120	1	4	yes	—
90	C₂⁴×D₄	#120	1	1	yes	characteristic
91	C₂⁴×D₄	#120	1	1	yes	characteristic
92	C₂⁴×D₄	#120	1	1	yes	characteristic
93	C₂⁴×D₄	#120	1	1	yes	characteristic
94	C₂⁴×D₄	#120	1	1	yes	characteristic
95	C₂⁴×D₄	#121	1	1	yes	characteristic
96	C₂⁴×D₄	#121	1	1	yes	characteristic
97	C₂⁴×D₄	#121	1	1	yes	characteristic
98	C₂⁴×D₄	#121	1	1	yes	characteristic
99	C₂⁴×D₄	#127	1	1	yes	characteristic
100	C₂⁴×D₄	#127	1	1	yes	characteristic
101	C₂⁴×D₄	#127	1	1	yes	characteristic
102	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
103	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
104	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
105	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
106	C₂⁴×D₄	C₂²×D₄	1	4	yes	—
107	C₂⁴×D₄	H(5,𝔽₂)	1	4	yes	—
108	C₂⁵×C₄	#120	1	1	yes	characteristic
109	C₂⁵×C₄	#120	1	1	yes	characteristic
110	C₂⁵×C₄	#120	1	2	yes	complemented
111	C₂⁵×C₄	#121	1	1	yes	characteristic
112	C₂⁵×C₄	C₂×#41	1	1	yes	characteristic
113	C₂×#518	#120	1	4	yes	—
114	C₂³×C₄⋊C₄	#121	1	1	yes	characteristic
115	C₂³×C₄⋊C₄	#121	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
118	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
119	C₂³×C₂²⋊C₄	#120	1	4	yes	complemented
120	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
121	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
123	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
124	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
125	C₂³×C₄²	#121	1	1	yes	characteristic
126	C₂²×#99	#120	1	8	yes	complemented
127	C₂⁶	#392	1	1	yes	characteristic
128	C₂⁶	#392	1	1	yes	characteristic
129	C₂⁶	#392	1	1	yes	characteristic
130	C₂⁶	B(4,𝔽₂)	1	4	yes	—
131	C₂⁶	B(4,𝔽₂)	1	4	yes	—
132	C₂⁶	C₂×#120	1	1	yes	characteristic
133	C₂⁶	C₂×#120	1	2	yes	—
134	C₂⁶	C₂×#120	1	1	yes	characteristic
135	C₂⁶	C₂×#120	1	1	yes	characteristic
136	C₂⁶	C₂×#121	1	1	yes	characteristic
137	C₂⁶	C₂×#127	1	1	yes	characteristic
138	C₂⁶	#534	1	2	yes	—
139	C₂⁶	D₄²	1	4	yes	—
140	C₂×H(5,𝔽₂)	B(4,𝔽₂)	1	8	yes	complemented
141	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
142	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
143	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
144	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
145	C₂³×D₄	C₂×#120	1	2	yes	—
146	C₂³×D₄	C₂×#120	1	2	yes	—
147	C₂³×D₄	#534	1	2	yes	—
148	C₂³×D₄	#535	1	2	yes	—
149	C₂³×D₄	D₄²	1	4	yes	—
150	C₂³×D₄	D₄²	1	4	yes	—
151	C₂⁴×C₄	#392	1	1	yes	characteristic
152	C₂⁴×C₄	#392	1	1	yes	characteristic
153	C₂⁴×C₄	#392	1	1	yes	characteristic
154	C₂⁴×C₄	#392	1	1	yes	characteristic
155	C₂⁴×C₄	#392	1	1	yes	characteristic
156	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
157	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
158	C₂⁴×C₄	#546	1	4	yes	—
159	C₂⁵	#2021	1	1	yes	characteristic
160	C₂⁵	#2021	1	2	yes	—
161	C₂⁵	#2021	1	2	yes	—
162	C₂⁵	#2021	1	2	yes	—
163	C₂⁵	#2021	1	2	yes	—
164	C₂⁵	#2021	1	1	yes	characteristic
165	C₂⁵	#2021	1	2	yes	—
166	C₂⁵	#2021	1	2	yes	—
167	C₂⁵	C₂×#392	1	1	yes	characteristic
168	C₂⁵	#2403	1	2	yes	—
169	C₂⁵	#2403	1	2	yes	—
170	C₂⁵	#2410	1	2	yes	—
171	C₂⁵	#2433	1	2	yes	—
172	C₂⁵	#2613	1	1	yes	characteristic
173	C₂⁵	#2846	1	1	yes	characteristic
174	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
175	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
176	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
177	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
178	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
179	C₂⁵	#3026	1	2	yes	—
180	C₂²×D₄	C₂×B(4,𝔽₂)	1	4	yes	—
181	C₂²×D₄	#3026	1	4	yes	—
182	C₂³×C₄	#2021	1	2	yes	—
183	C₂³×C₄	#2021	1	2	yes	—
184	C₂³×C₄	#2021	1	2	yes	—
185	C₂³×C₄	#2408	1	2	yes	—
186	C₂³×C₄	#2410	1	2	yes	—
187	C₂⁴	#10336	1	2	yes	—
188	C₂⁴	#10336	1	2	yes	—
189	C₂⁴	#10336	1	4	yes	—
190	C₂⁴	#14715	1	1	yes	characteristic
191	C₂⁴	C₂×#2021	1	1	yes	characteristic
192	C₂⁴	C₂×#2021	1	1	yes	characteristic
193	C₂⁴	C₂×#2021	1	1	yes	characteristic
194	C₂⁴	C₂×#2021	1	1	yes	characteristic
195	C₂⁴	C₂×#2021	1	1	yes	characteristic
196	C₂⁴	#23444	1	1	yes	characteristic
197	C₂⁴	#23895	1	4	yes	—
198	C₂⁴	#23900	1	4	yes	—
199	C₂⁴	#23900	1	4	yes	—
200	C₂⁴	#23900	1	2	yes	—
201	C₂⁴	#23903	1	2	yes	—
202	C₂⁴	#29634	1	2	yes	—
203	C₂⁴	#32898	1	2	yes	—
204	C₂⁴	#32898	1	4	yes	—
205	C₂²×C₄	#24404	1	4	yes	—
206	C₂³	#144378	1	4	yes	complemented
207	C₂³	C₂×#10336	1	1	yes	characteristic
208	C₂³	C₂×#10336	1	1	yes	characteristic
209	C₂³	#233541	1	1	yes	characteristic
210	C₂³	#233541	1	1	yes	characteristic
211	C₂³	#245225	1	4	yes	—
212	C₂³	#245246	1	4	yes	—
213	C₂³	#347042	1	4	yes	—
214	C₂³	#393929	1	1	yes	characteristic
215	C₂³	#394058	1	1	yes	characteristic
216	C₂³	#394058	1	1	yes	characteristic
217	C₂³	#394064	1	2	yes	—
218	C₂³	#394066	1	2	yes	—
219	C₂²	#72210600	1	2	yes	—
220	C₂²	#72214549	1	1	yes	characteristic
221	C₂²	C₂×#144378	1	2	yes	—
222	C₂²	#72219240	1	1	yes	characteristic
223	C₂²	#72225609	1	2	yes	—
224	C₂²	#72225610	1	1	yes	characteristic
225	C₂	#72218868	1	1	yes	characteristic
226	C₂	#72225611	1	1	yes	characteristic
227	C₂	#72258416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72214549
commutator, Frattini, cosocle	C₂⁵×D₄	C₂⁴
LCS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×D₄² ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266198	C₂	semidirect
2. #72266190	C₂	semidirect
3. #72266167	C₂	semidirect
4. #72266134	C₂	semidirect
5. #72259040	C₂	semidirect
6. #72258223	C₂	semidirect
7. #72240620	C₂	semidirect
8. #72240548	C₂²	semidirect
9. #72236800	C₂	semidirect
10. #72236528	C₂	semidirect
12. C₂²×#23895	C₂²	semidirect
14. #72219457	C₂²	semidirect
15. #72219418	C₂²	semidirect
21. #72218846	C₂²	semidirect
22. #72218498	C₂²	semidirect
24. #72214832	C₂²	semidirect
25. #72210555	C₂²	semidirect
26. #72210555	C₂²	semidirect
27. C₂⁴×H(5,𝔽₂)	D₄	semidirect
28. C₂⁴×H(5,𝔽₂)	D₄	semidirect
32. C₂×#35347	D₄	semidirect
33. C₂×#35346	D₄	semidirect
34. C₂×#34980	D₄	semidirect
36. C₂×#34256	D₄	semidirect
37. C₂×#34171	D₄	semidirect
38. C₂×#34171	D₄	semidirect
39. C₂×#23900	D₄	semidirect
40. C₂×#21746	D₄	semidirect
43. #515362	D₄	semidirect
44. #515285	D₄	semidirect
48. #382390	D₄	semidirect
49. #359552	D₄	semidirect
50. C₂×#10965	D₄	semidirect
51. C₂×#8546	D₄	semidirect
52. C₂×#8542	D₄	semidirect
53. #154094	D₄	semidirect
55. C₂³×H(5,𝔽₂)	C₂×D₄	semidirect
73. C₂²×#518	C₂×D₄	semidirect
76. C₂×#2308	C₂×D₄	semidirect
77. C₂×#2308	C₂×D₄	semidirect
80. C₂²×#379	C₂×D₄	semidirect
83. C₂×#1781	C₂×D₄	semidirect
86. C₂⁷	#120	semidirect
110. C₂⁵×C₄	#120	semidirect
119. C₂³×C₂²⋊C₄	#120	semidirect
126. C₂²×#99	#120	semidirect
140. C₂×H(5,𝔽₂)	B(4,𝔽₂)	semidirect
206. C₂³	#144378	semidirect

Maximal subgroups

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

subgroup	index
#72266198	2
#72266190	2
#72266167	2
#72266134	2
#72259040	2
#72258223	2
#72240620	2
#72236800	2
#72236528	2