Subgroups — #72251595

Order 2048.

At least 7919 conjugacy classes of subgroups; only the 346 normal subgroups are shown.

The subgroup lattice has 246 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251530	C₂	1	1	yes	maximal, characteristic, complemented
2	#72251501	C₂	1	1	yes	maximal, characteristic, complemented
3	#72251475	C₂	1	1	yes	maximal, characteristic, complemented
4	#72251462	C₂	1	1	yes	maximal, characteristic, complemented
5	#72251420	C₂	1	1	yes	maximal, characteristic, complemented
6	#72240419	C₂	1	1	yes	maximal, characteristic, complemented
7	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237402	C₂	1	1	yes	maximal, characteristic, complemented
9	#72236678	C₂	1	1	yes	maximal, characteristic, complemented
10	#72233800	C₂	1	1	yes	maximal, characteristic, complemented
11	#72225641	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#139748	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218215	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218114	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#144382	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	C₂×#60399	C₂²	1	1	yes	characteristic
18	#7642244	C₂²	1	1	yes	characteristic, complemented
19	C₂×#23900	C₂²	1	1	yes	characteristic
20	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
21	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
22	C₂×#23897	C₂²	1	1	yes	characteristic
23	C₂×#23897	C₂²	1	1	yes	characteristic
24	#6437445	C₂²	1	1	yes	characteristic
25	#6437445	C₂²	1	1	yes	characteristic
26	C₂×#21778	C₂²	1	1	yes	characteristic
27	#616961	C₂²	1	1	yes	characteristic, complemented
28	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
29	C₂×#12705	C₂²	1	1	yes	characteristic
30	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
31	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
32	#493370	C₂²	1	1	yes	characteristic, complemented
33	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
34	#382434	C₂²	1	1	yes	characteristic, complemented
35	#359549	C₂²	1	1	yes	characteristic
36	#359549	C₂²	1	1	yes	characteristic, complemented
37	#359549	C₂²	1	1	yes	characteristic, complemented
38	#359509	C₂²	1	1	yes	characteristic, complemented
39	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
40	#188050	C₂²	1	1	yes	characteristic, complemented
41	#187944	C₂²	1	1	yes	characteristic
42	#187944	C₂²	1	1	yes	characteristic, complemented
43	#187944	C₂²	1	1	yes	characteristic, complemented
44	#186835	C₂²	1	1	yes	characteristic, complemented
45	#146475	C₂²	1	1	yes	characteristic, complemented
46	#128033	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
50	#108238	C₂²	1	1	yes	characteristic, complemented
51	C₂×H(7,𝔽₂)	D₄	1	1	yes	characteristic, complemented
52	#62832	D₄	1	2	yes	complemented
53	#62824	D₄	1	2	yes	complemented
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂×#3484	C₂³	1	1	yes	characteristic
57	C₂×#3484	C₂³	1	1	yes	characteristic
58	C₂²×#535	C₂³	1	1	yes	characteristic
59	C₂³×#120	C₂³	1	1	yes	characteristic
60	C₂×#3019	D₄	1	2	yes	complemented
61	#60359	D₄	1	2	yes	complemented
62	C₂×#2883	D₄	1	1	yes	characteristic, complemented
63	C₂×#2883	D₄	1	1	yes	characteristic, complemented
64	C₂×#2883	D₄	1	1	yes	characteristic, complemented
65	C₂×#2846	D₄	1	2	yes	—
66	#34171	C₂³	1	1	yes	characteristic
67	#23907	D₄	1	2	yes	complemented
68	#23906	D₄	1	2	yes	complemented
69	#23897	D₄	1	4	yes	complemented
70	#21778	D₄	1	4	yes	complemented
71	C₂×#1913	D₄	1	2	yes	complemented
72	C₂×#1911	D₄	1	2	yes	complemented
73	C₂×#1898	D₄	1	2	yes	—
74	#20979	D₄	1	2	yes	complemented
75	#20960	D₄	1	2	yes	complemented
76	C₂×#1783	C₂³	1	1	yes	characteristic
77	C₂×#1783	C₂³	1	1	yes	characteristic
78	C₂×#1783	C₂³	1	1	yes	characteristic
79	C₂×#1781	C₂³	1	1	yes	characteristic
80	C₂×#1781	C₂³	1	1	yes	characteristic
81	C₂×#1781	C₂³	1	1	yes	characteristic
82	C₂×#1781	C₂³	1	1	yes	characteristic
83	#14335	C₂³	1	1	yes	characteristic
84	#8534	D₄	1	4	yes	complemented
85	C₂⁷	C₂×D₄	1	1	yes	characteristic
86	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
88	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
89	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
90	C₂³×#41	C₂×D₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
92	#3484	C₂×D₄	1	2	yes	—
93	#3484	C₂×D₄	1	2	yes	—
94	#3484	C₂×D₄	1	2	yes	—
95	#3484	C₂×D₄	1	2	yes	—
96	#3484	C₂×D₄	1	2	yes	—
97	#3484	C₂×D₄	1	2	yes	—
98	#3484	C₂×D₄	1	2	yes	—
99	#3484	#41	1	2	yes	—
100	C₂×#535	C₂×D₄	1	2	yes	—
101	C₂×#534	C₂×D₄	1	2	yes	—
102	C₂²×#127	C₂×D₄	1	1	yes	characteristic
103	C₂²×#127	C₂×D₄	1	1	yes	characteristic
104	C₂²×#124	C₂×D₄	1	1	yes	characteristic
105	C₂²×#124	C₂×D₄	1	1	yes	characteristic
106	C₂×#525	C₂×D₄	1	2	yes	—
107	C₂²×#120	C₂×D₄	1	2	yes	—
108	C₂×#518	C₂×D₄	1	1	yes	characteristic
109	C₂×#518	C₂×D₄	1	1	yes	characteristic
110	C₂×#518	C₂×D₄	1	1	yes	characteristic
111	C₂×#518	C₂×D₄	1	1	yes	characteristic
112	C₂×#518	C₂×D₄	1	1	yes	characteristic
113	C₂×#518	C₂×D₄	1	1	yes	characteristic
114	C₂²×#117	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
116	C₂²×#100	C₂×D₄	1	2	yes	—
117	C₂²×#99	C₂×D₄	1	2	yes	—
118	C₂×#407	C₂×D₄	1	2	yes	—
119	C₂×#379	C₂×D₄	1	2	yes	—
120	#1783	C₂×D₄	1	2	yes	—
121	#1783	C₂×D₄	1	2	yes	—
122	#1783	C₂×D₄	1	2	yes	—
123	#1783	C₂×D₄	1	2	yes	—
124	#1781	C₂×D₄	1	2	yes	—
125	#1438	C₂×D₄	1	2	yes	—
126	C₂⁶	C₂²×D₄	1	1	yes	characteristic
127	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×H(5,𝔽₂)	#120	1	2	yes	—
129	C₂×H(5,𝔽₂)	#120	1	2	yes	—
130	C₂×H(5,𝔽₂)	#120	1	2	yes	complemented
131	C₂×H(5,𝔽₂)	#121	1	2	yes	—
132	C₂²×#41	#120	1	2	yes	—
133	C₂³×Q₈	#120	1	1	yes	characteristic, complemented
134	C₂³×D₄	#120	1	2	yes	—
135	C₂³×D₄	#120	1	1	yes	characteristic, complemented
136	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
137	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
138	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
139	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
140	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
141	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
142	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
143	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₂×#41	1	1	yes	characteristic
145	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
149	#534	#121	1	4	yes	—
150	#534	#127	1	4	yes	—
151	C₂×#120	#120	1	4	yes	—
152	C₂×#120	#121	1	4	yes	—
153	#518	#120	1	2	yes	—
154	#518	#120	1	2	yes	—
155	#518	#120	1	2	yes	complemented
156	#518	#121	1	2	yes	—
157	C₂×#117	#120	1	2	yes	—
158	C₂²×C₂²⋊C₄	#120	1	2	yes	—
159	C₂²×C₄²	#120	1	1	yes	characteristic, complemented
160	C₂²×C₄²	#120	1	1	yes	characteristic, complemented
161	C₂×#100	#120	1	4	yes	complemented
162	#407	#120	1	4	yes	complemented
163	C₂⁵	B(4,𝔽₂)	1	2	yes	—
164	C₂⁵	C₂×#120	1	1	yes	characteristic
165	C₂⁵	C₂×#120	1	1	yes	characteristic
166	C₂⁵	C₂×#121	1	1	yes	characteristic
167	C₂⁵	#534	1	1	yes	characteristic
168	C₂⁵	#534	1	1	yes	characteristic
169	C₂⁵	#535	1	1	yes	characteristic
170	C₂⁵	D₄²	1	1	yes	characteristic
171	C₂⁵	D₄²	1	1	yes	characteristic
172	C₂⁵	D₄²	1	1	yes	characteristic
173	C₂⁵	D₄²	1	1	yes	characteristic
174	C₂⁵	D₄²	1	1	yes	characteristic
175	C₂⁵	D₄²	1	1	yes	characteristic
176	C₂⁵	#546	1	1	yes	characteristic
177	C₂²×Q₈	B(4,𝔽₂)	1	2	yes	complemented
178	C₂²×D₄	#392	1	4	yes	—
179	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
180	C₂²×D₄	B(4,𝔽₂)	1	2	yes	complemented
181	C₂²×D₄	C₂×#120	1	1	yes	characteristic
182	C₂²×D₄	C₂×#120	1	1	yes	characteristic
183	C₂²×D₄	C₂×#120	1	1	yes	characteristic
184	C₂²×D₄	C₂×#120	1	1	yes	characteristic
185	C₂²×D₄	C₂×#121	1	1	yes	characteristic
186	C₂²×D₄	C₂×#121	1	1	yes	characteristic
187	C₂²×D₄	C₂×#127	1	1	yes	characteristic
188	C₂²×D₄	#534	1	1	yes	characteristic
189	C₂²×D₄	#534	1	1	yes	characteristic
190	C₂²×D₄	#535	1	1	yes	characteristic
191	C₂²×D₄	D₄²	1	1	yes	characteristic
192	C₂²×D₄	D₄²	1	1	yes	characteristic
193	C₂²×D₄	D₄²	1	1	yes	characteristic
194	C₂²×D₄	D₄²	1	1	yes	characteristic
195	C₂²×D₄	#546	1	1	yes	characteristic
196	C₂²×D₄	#546	1	1	yes	characteristic
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×#120	1	1	yes	characteristic
201	C₂³×C₄	D₄²	1	1	yes	characteristic
202	C₂³×C₄	D₄²	1	1	yes	characteristic
203	C₂³×C₄	#546	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	#392	1	4	yes	—
205	C₂⁴	#2021	1	4	yes	—
206	C₂⁴	C₂×#392	1	1	yes	characteristic
207	C₂⁴	#2403	1	1	yes	characteristic
208	C₂⁴	#2403	1	1	yes	characteristic
209	C₂⁴	#2403	1	1	yes	characteristic
210	C₂⁴	#2403	1	1	yes	characteristic
211	C₂⁴	#2408	1	1	yes	characteristic
212	C₂⁴	#2408	1	1	yes	characteristic
213	C₂⁴	#2410	1	1	yes	characteristic
214	C₂⁴	#2410	1	1	yes	characteristic
215	C₂⁴	#2433	1	1	yes	characteristic
216	C₂⁴	#2613	1	1	yes	characteristic
217	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
218	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
219	C₂²×C₄	#2021	1	4	yes	—
220	C₂²×C₄	#2410	1	1	yes	characteristic
221	C₂²×C₄	#2410	1	1	yes	characteristic
222	C₂²×C₄	#2433	1	1	yes	characteristic
223	C₂²×C₄	#2846	1	1	yes	characteristic
224	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
225	C₂²×C₄	#3026	1	2	yes	—
226	C₂³	#14715	1	1	yes	characteristic
227	C₂³	C₂×#2021	1	2	yes	—
228	C₂³	#23895	1	1	yes	characteristic
229	C₂³	#23895	1	1	yes	characteristic
230	C₂³	#23900	1	2	yes	—
231	C₂³	#23900	1	1	yes	characteristic
232	C₂³	#23900	1	1	yes	characteristic
233	C₂³	#23900	1	1	yes	characteristic
234	C₂³	#23903	1	1	yes	characteristic
235	C₂×C₄	#24404	1	2	yes	—
236	C₂×C₄	#32898	1	2	yes	—
237	C₂²	#245225	1	1	yes	characteristic
238	C₂²	#245225	1	1	yes	characteristic
239	C₂²	#245246	1	1	yes	characteristic
240	C₂²	#245246	1	1	yes	characteristic
241	C₂²	#394058	1	1	yes	characteristic
242	C₂²	#394064	1	1	yes	characteristic
243	C₂²	#394066	1	1	yes	characteristic
244	C₂	#72225609	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72225609
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ 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. #72251530	C₂	semidirect
2. #72251501	C₂	semidirect
3. #72251475	C₂	semidirect
4. #72251462	C₂	semidirect
5. #72251420	C₂	semidirect
6. #72240419	C₂	semidirect
7. #72238343	C₂	semidirect
8. #72237402	C₂	semidirect
9. #72236678	C₂	semidirect
10. #72233800	C₂	semidirect
11. #72225641	C₂	semidirect
12. C₂×#139748	C₂	semidirect
13. #72218215	C₂	semidirect
14. #72218114	C₂	semidirect
15. C₂×#144382	C₂	semidirect
18. #7642244	C₂²	semidirect
20. C₂×#23900	C₂²	semidirect
21. C₂×#23900	C₂²	semidirect
27. #616961	C₂²	semidirect
28. C₂×#12707	C₂²	semidirect
30. C₂×#12705	C₂²	semidirect
31. C₂×#12705	C₂²	semidirect
32. #493370	C₂²	semidirect
33. C₂×#12687	C₂²	semidirect
34. #382434	C₂²	semidirect
36. #359549	C₂²	semidirect
37. #359549	C₂²	semidirect
38. #359509	C₂²	semidirect
39. C₂×#8547	C₂²	semidirect
40. #188050	C₂²	semidirect
42. #187944	C₂²	semidirect
43. #187944	C₂²	semidirect
44. #186835	C₂²	semidirect
45. #146475	C₂²	semidirect
46. #128033	C₂²	semidirect
47. C₂×#7316	C₂²	semidirect
48. C₂×#7314	C₂²	semidirect
49. C₂×#7311	C₂²	semidirect
50. #108238	C₂²	semidirect
51. C₂×H(7,𝔽₂)	D₄	semidirect
52. #62832	D₄	semidirect
53. #62824	D₄	semidirect
60. C₂×#3019	D₄	semidirect
61. #60359	D₄	semidirect
62. C₂×#2883	D₄	semidirect
63. C₂×#2883	D₄	semidirect
64. C₂×#2883	D₄	semidirect
67. #23907	D₄	semidirect
68. #23906	D₄	semidirect
69. #23897	D₄	semidirect
70. #21778	D₄	semidirect
71. C₂×#1913	D₄	semidirect
72. C₂×#1911	D₄	semidirect
74. #20979	D₄	semidirect
75. #20960	D₄	semidirect
84. #8534	D₄	semidirect
130. C₂×H(5,𝔽₂)	#120	semidirect
133. C₂³×Q₈	#120	semidirect
135. C₂³×D₄	#120	semidirect
155. #518	#120	semidirect
159. C₂²×C₄²	#120	semidirect
160. C₂²×C₄²	#120	semidirect
161. C₂×#100	#120	semidirect
162. #407	#120	semidirect
177. C₂²×Q₈	B(4,𝔽₂)	semidirect
180. C₂²×D₄	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72251530	2
#72251501	2
#72251475	2
#72251462	2
#72251420	2
#72240419	2
#72238343	2
#72237402	2
#72236678	2
#72233800	2
#72225641	2
C₂×#139748	2
#72218215	2
#72218114	2
C₂×#144382	2