Subgroups — #72266317

Order 4096.

At least 13349 conjugacy classes of subgroups; only the 437 normal subgroups are shown.

The subgroup lattice has 228 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266316	C₂	1	4	yes	maximal, complemented
2	#72266315	C₂	1	4	yes	maximal, complemented
3	#72266293	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266154	C₂	1	1	yes	maximal, characteristic, complemented
5	#72259040	C₂	1	1	yes	maximal, characteristic, complemented
6	#72259019	C₂²	1	2	yes	complemented
7	#72259016	C₂²	1	2	yes	complemented
8	#72240620	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238594	C₂²	1	4	yes	—
10	#72237035	C₂²	1	4	yes	complemented
11	#72236800	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236754	C₂²	1	2	yes	complemented
13	#72236034	C₂	1	1	yes	maximal, characteristic, complemented
14	#72235930	C₂²	1	1	yes	characteristic
15	#72235865	C₂²	1	2	yes	complemented
16	#72231822	C₂²	1	2	yes	complemented
17	#72226161	C₂	1	1	yes	maximal, characteristic, complemented
18	#72225994	C₂²	1	4	yes	complemented
19	#72225881	C₂²	1	1	yes	characteristic
20	#72225621	C₂²	1	1	yes	characteristic
21	#72219377	C₂²	1	1	yes	characteristic
22	#72219307	C₂²	1	1	yes	characteristic
23	C₂×#7631286	C₂²	1	1	yes	characteristic
24	C₂×#7631286	C₂²	1	1	yes	characteristic
25	C₂×#131967	C₂²	1	4	yes	complemented
26	C₂×#245225	C₂²	1	2	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	D₄	1	1	yes	characteristic, complemented
30	C₂³×D₄²	C₂³	1	1	yes	characteristic
31	C₂×#35347	D₄	1	2	yes	complemented
32	C₂×#35347	D₄	1	1	yes	characteristic, complemented
33	C₂×#35346	C₂³	1	1	yes	characteristic
34	C₂×#35346	C₂³	1	2	yes	—
35	C₂×#34980	C₂³	1	2	yes	—
36	C₂²×#2403	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×#21746	D₄	1	2	yes	complemented
40	#517411	D₄	1	1	yes	characteristic, complemented
41	#515362	D₄	1	1	yes	characteristic, complemented
42	#515285	C₂³	1	1	yes	characteristic
43	#449501	D₄	1	8	yes	complemented
44	#449475	D₄	1	8	yes	complemented
45	#392245	D₄	1	2	yes	complemented
46	#382422	D₄	1	2	yes	complemented
47	#382422	C₂³	1	2	yes	—
48	C₂×#10965	D₄	1	2	yes	complemented
49	C₂×#10965	C₂³	1	2	yes	—
50	C₂×#10761	D₄	1	2	yes	complemented
51	C₂×#8547	C₂³	1	2	yes	—
52	C₂×#8546	D₄	1	2	yes	complemented
53	C₂×#8541	C₂³	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
61	C₂³×#127	C₂×D₄	1	1	yes	characteristic
62	C₂³×#121	C₂×D₄	1	1	yes	characteristic
63	C₂³×#121	C₂×D₄	1	1	yes	characteristic
64	C₂³×#121	#41	1	1	yes	characteristic
65	C₂³×#121	#41	1	1	yes	characteristic
66	C₂³×#120	C₂×D₄	1	1	yes	characteristic
67	C₂³×#120	C₂×D₄	1	1	yes	characteristic
68	C₂³×#120	C₂×D₄	1	1	yes	characteristic
69	C₂³×#120	C₂⁴	1	1	yes	characteristic
70	C₂²×#518	C₂×D₄	1	1	yes	characteristic, complemented
71	C₂³×C₄×D₄	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₄×D₄	C₂×D₄	1	1	yes	characteristic
73	C₂×#2849	C₂×D₄	1	4	yes	—
74	C₂×#2846	C₂×D₄	1	4	yes	—
75	#35346	C₂×D₄	1	8	yes	—
76	C₂×#2308	C₂×D₄	1	2	yes	—
77	C₂×#2308	C₂×D₄	1	1	yes	characteristic, complemented
78	C₂×#2308	C₂×D₄	1	1	yes	characteristic, complemented
79	C₂²×#379	C₂×D₄	1	2	yes	—
80	C₂×#1819	C₂×D₄	1	4	yes	—
81	C₂×#1781	C₂×D₄	1	4	yes	complemented
82	C₂²×#342	C₂×D₄	1	4	yes	complemented
83	#10965	C₂×D₄	1	8	yes	—
84	C₂⁷	#120	1	1	yes	characteristic
85	C₂⁷	#120	1	1	yes	characteristic
86	C₂⁷	#120	1	1	yes	characteristic
87	C₂⁷	#120	1	2	yes	complemented
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	#120	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	#121	1	1	yes	characteristic
100	C₂⁴×D₄	#121	1	1	yes	characteristic
101	C₂⁴×D₄	#121	1	1	yes	characteristic
102	C₂⁴×D₄	#127	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	C₂⁵×C₄	#120	1	1	yes	characteristic
107	C₂⁵×C₄	#120	1	1	yes	characteristic
108	C₂⁵×C₄	#120	1	2	yes	complemented
109	C₂⁵×C₄	#121	1	1	yes	characteristic
110	C₂⁵×C₄	#121	1	1	yes	characteristic
111	C₂²×#120	C₂²×D₄	1	4	yes	—
112	C₂²×#120	H(5,𝔽₂)	1	4	yes	—
113	C₂³×C₄⋊C₄	#121	1	1	yes	characteristic
114	C₂³×C₄⋊C₄	#121	1	1	yes	characteristic
115	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
118	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
119	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
120	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
121	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
123	C₂³×C₄²	#127	1	1	yes	characteristic
124	#2308	#120	1	4	yes	—
125	C₂×#379	#120	1	4	yes	complemented
126	C₂×#342	#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	#392	1	1	yes	characteristic
131	C₂⁶	B(4,𝔽₂)	1	4	yes	—
132	C₂⁶	B(4,𝔽₂)	1	4	yes	—
133	C₂⁶	C₂×#120	1	1	yes	characteristic
134	C₂⁶	C₂×#120	1	2	yes	—
135	C₂⁶	C₂×#120	1	1	yes	characteristic
136	C₂⁶	C₂×#120	1	1	yes	characteristic
137	C₂⁶	C₂×#121	1	1	yes	characteristic
138	C₂⁶	#534	1	2	yes	—
139	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
140	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
141	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
142	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
143	C₂³×D₄	C₂×#120	1	2	yes	—
144	C₂³×D₄	#534	1	2	yes	—
145	C₂³×D₄	D₄²	1	4	yes	—
146	C₂³×D₄	D₄²	1	4	yes	—
147	C₂⁴×C₄	#392	1	1	yes	characteristic
148	C₂⁴×C₄	#392	1	1	yes	characteristic
149	C₂⁴×C₄	#392	1	1	yes	characteristic
150	C₂⁴×C₄	#392	1	1	yes	characteristic
151	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
152	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
153	C₂⁴×C₄	C₂×#127	1	1	yes	characteristic
154	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
155	C₂²×C₂²⋊C₄	#535	1	2	yes	—
156	C₂²×C₂²⋊C₄	D₄²	1	4	yes	—
157	C₂²×C₂²⋊C₄	#546	1	4	yes	—
158	C₂⁵	#2021	1	1	yes	characteristic
159	C₂⁵	#2021	1	2	yes	—
160	C₂⁵	#2021	1	2	yes	—
161	C₂⁵	#2021	1	1	yes	characteristic
162	C₂⁵	#2021	1	2	yes	—
163	C₂⁵	#2021	1	2	yes	—
164	C₂⁵	#2021	1	2	yes	—
165	C₂⁵	#2021	1	2	yes	—
166	C₂⁵	C₂×#392	1	1	yes	characteristic
167	C₂⁵	#2403	1	2	yes	—
168	C₂⁵	#2403	1	2	yes	—
169	C₂⁵	#2410	1	2	yes	—
170	C₂⁵	#2613	1	1	yes	characteristic
171	C₂⁵	#2846	1	1	yes	characteristic
172	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
173	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
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	#3026	1	2	yes	—
178	C₂²×D₄	#2196	1	4	yes	—
179	C₂²×D₄	#2199	1	4	yes	—
180	C₂³×C₄	#2021	1	2	yes	—
181	C₂³×C₄	#2021	1	2	yes	—
182	C₂³×C₄	#2021	1	2	yes	—
183	C₂³×C₄	#2408	1	2	yes	—
184	C₂³×C₄	#2410	1	2	yes	—
185	C₂³×C₄	#2433	1	2	yes	—
186	C₂⁴	#10336	1	2	yes	—
187	C₂⁴	#10336	1	2	yes	—
188	C₂⁴	#10336	1	4	yes	—
189	C₂⁴	#13343	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	#23900	1	4	yes	—
198	C₂⁴	#23900	1	4	yes	—
199	C₂⁴	#29634	1	2	yes	—
200	C₂⁴	#32898	1	2	yes	—
201	C₂⁴	C₂×#2196	1	2	yes	—
202	C₂⁴	C₂×#2199	1	2	yes	—
203	C₂⁴	#33551	1	4	yes	—
204	C₂²×C₄	#13343	1	4	yes	—
205	C₂³	#144366	1	4	yes	—
206	C₂³	C₂×#10336	1	1	yes	characteristic
207	C₂³	C₂×#10336	1	1	yes	characteristic
208	C₂³	#233541	1	1	yes	characteristic
209	C₂³	#233541	1	1	yes	characteristic
210	C₂³	#393929	1	1	yes	characteristic
211	C₂³	#394058	1	1	yes	characteristic
212	C₂³	#394058	1	1	yes	characteristic
213	C₂³	C₂×#13343	1	2	yes	—
214	C₂³	#498886	1	2	yes	—
215	C₂³	#499787	1	4	yes	—
216	C₂³	#499795	1	4	yes	—
217	C₂³	#511790	1	4	yes	—
218	C₂²	#72214549	1	1	yes	characteristic
219	C₂²	C₂×#144366	1	1	yes	characteristic
220	C₂²	#72218121	1	2	yes	—
221	C₂²	#72218838	1	1	yes	characteristic
222	C₂²	#72225750	1	2	yes	—
223	C₂²	#72225751	1	2	yes	—
224	C₂	#72218841	1	1	yes	characteristic
225	C₂	#72225753	1	1	yes	characteristic
226	C₂	#72258416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72214549
commutator, Frattini, cosocle	C₂³×#120	C₂⁴
LCS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×D₄² ◁ G
LECS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266316	C₂	semidirect
2. #72266315	C₂	semidirect
3. #72266293	C₂	semidirect
4. #72266154	C₂	semidirect
5. #72259040	C₂	semidirect
6. #72259019	C₂²	semidirect
7. #72259016	C₂²	semidirect
8. #72240620	C₂	semidirect
10. #72237035	C₂²	semidirect
11. #72236800	C₂	semidirect
12. #72236754	C₂²	semidirect
13. #72236034	C₂	semidirect
15. #72235865	C₂²	semidirect
16. #72231822	C₂²	semidirect
17. #72226161	C₂	semidirect
18. #72225994	C₂²	semidirect
25. C₂×#131967	C₂²	semidirect
26. C₂×#245225	C₂²	semidirect
27. C₂⁴×H(5,𝔽₂)	D₄	semidirect
28. C₂⁴×H(5,𝔽₂)	D₄	semidirect
29. C₂²×#3484	D₄	semidirect
31. C₂×#35347	D₄	semidirect
32. C₂×#35347	D₄	semidirect
36. C₂²×#2403	D₄	semidirect
37. C₂×#34171	D₄	semidirect
38. C₂×#34171	D₄	semidirect
39. C₂×#21746	D₄	semidirect
40. #517411	D₄	semidirect
41. #515362	D₄	semidirect
43. #449501	D₄	semidirect
44. #449475	D₄	semidirect
45. #392245	D₄	semidirect
46. #382422	D₄	semidirect
48. C₂×#10965	D₄	semidirect
50. C₂×#10761	D₄	semidirect
52. C₂×#8546	D₄	semidirect
55. C₂³×H(5,𝔽₂)	C₂×D₄	semidirect
70. C₂²×#518	C₂×D₄	semidirect
77. C₂×#2308	C₂×D₄	semidirect
78. C₂×#2308	C₂×D₄	semidirect
81. C₂×#1781	C₂×D₄	semidirect
82. C₂²×#342	C₂×D₄	semidirect
87. C₂⁷	#120	semidirect
108. C₂⁵×C₄	#120	semidirect
125. C₂×#379	#120	semidirect
126. C₂×#342	#120	semidirect

Maximal subgroups

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

subgroup	index
#72266316	2
#72266315	2
#72266293	2
#72266154	2
#72259040	2
#72240620	2
#72236800	2
#72236034	2
#72226161	2