Subgroups — #72266824

Order 2048.

At least 6401 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

The subgroup lattice has 235 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72264754	C₂	1	1	yes	maximal, characteristic, complemented
2	#72258983	C₂	1	1	yes	maximal, characteristic, complemented
3	#72255863	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255838	C₂	1	1	yes	maximal, characteristic, complemented
5	#72246292	C₂	1	1	yes	maximal, characteristic, complemented
6	#72240471	C₂	1	1	yes	maximal, characteristic, complemented
7	#72238609	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238351	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237633	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237473	C₂	1	1	yes	maximal, characteristic, complemented
11	#72235130	C₂	1	1	yes	maximal, characteristic
12	#72234631	C₂	1	1	yes	maximal, characteristic, complemented
13	#72234613	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219163	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#139743	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586404	C₂²	1	1	yes	characteristic
17	#6440460	C₂²	1	1	yes	characteristic
18	#6440460	C₂²	1	1	yes	characteristic
19	C₂×#23895	C₂²	1	1	yes	characteristic
20	#6437621	C₂²	1	1	yes	characteristic
21	C₂×#23897	C₂²	1	1	yes	characteristic
22	#6374210	C₂²	1	1	yes	characteristic, complemented
23	C₂×#21776	C₂²	1	1	yes	characteristic
24	#6371558	C₂²	1	1	yes	characteristic
25	#763512	C₂²	1	1	yes	characteristic
26	#493879	C₂²	1	1	yes	characteristic, complemented
27	C₂×#12707	C₂²	1	1	yes	characteristic
28	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
29	C₂×#12705	C₂²	1	1	yes	characteristic
30	#493357	C₂²	1	1	yes	characteristic
31	#382544	C₂²	1	1	yes	characteristic, complemented
32	#359811	C₂²	1	1	yes	characteristic, complemented
33	#359549	C₂²	1	1	yes	characteristic
34	#359549	C₂²	1	1	yes	characteristic
35	#359549	C₂²	1	1	yes	characteristic, complemented
36	#359442	C₂²	1	1	yes	characteristic
37	#359442	C₂²	1	1	yes	characteristic
38	#190116	C₂²	1	1	yes	characteristic, complemented
39	#188049	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
49	#108222	C₂²	1	1	yes	characteristic, complemented
50	#107130	C₂²	1	1	yes	characteristic, complemented
51	#62820	D₄	1	2	yes	—
52	#62817	D₄	1	2	yes	—
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂²×#534	C₂³	1	1	yes	characteristic
57	#62652	D₄	1	1	yes	characteristic, complemented
58	C₂×#3026	D₄	1	2	yes	—
59	C₂×#2883	D₄	1	1	yes	characteristic, complemented
60	#34992	D₄	1	2	yes	complemented
61	#34171	C₂³	1	1	yes	characteristic
62	#34171	C₂³	1	1	yes	characteristic
63	C₂×#2308	C₂³	1	1	yes	characteristic
64	C₂×#1913	D₄	1	2	yes	—
65	C₂×#1898	D₄	1	2	yes	complemented
66	#20609	D₄	1	2	yes	complemented
67	C₂×#1783	C₂³	1	1	yes	characteristic
68	C₂×#1781	D₄	1	2	yes	complemented
69	C₂×#1781	C₂³	1	1	yes	characteristic
70	C₂×#1781	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	C₂×#1781	C₂³	1	1	yes	characteristic
75	#20354	D₄	1	1	yes	characteristic, complemented
76	#20354	D₄	1	1	yes	characteristic, complemented
77	C₂×#1438	C₂³	1	1	yes	characteristic
78	#12705	D₄	1	4	yes	—
79	#10754	D₄	1	2	yes	complemented
80	#10735	D₄	1	2	yes	complemented
81	#8534	D₄	1	2	yes	complemented
82	#8534	D₄	1	2	yes	complemented
83	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
84	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
85	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
86	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
87	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
88	#3484	C₂×D₄	1	2	yes	—
89	#3484	C₂×D₄	1	2	yes	—
90	#3484	C₂×D₄	1	2	yes	—
91	#3484	C₂×D₄	1	2	yes	—
92	#3484	#41	1	2	yes	—
93	#3484	#41	1	2	yes	—
94	C₂×#534	C₂×D₄	1	2	yes	—
95	C₂²×#123	C₂×D₄	1	1	yes	characteristic
96	C₂²×#123	#41	1	1	yes	characteristic
97	C₂²×#120	C₂×D₄	1	2	yes	—
98	C₂²×#120	C₂×D₄	1	1	yes	characteristic
99	C₂²×#120	#41	1	1	yes	characteristic
100	C₂×#518	C₂×D₄	1	1	yes	characteristic
101	C₂×#518	C₂×D₄	1	1	yes	characteristic
102	C₂×#518	C₂×D₄	1	1	yes	characteristic
103	C₂×#518	C₂×D₄	1	1	yes	characteristic
104	C₂×#518	C₂×D₄	1	1	yes	characteristic
105	C₂×#518	C₂×D₄	1	1	yes	characteristic
106	C₂×#518	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
109	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
110	C₂²×#99	C₂×D₄	1	2	yes	—
111	C₂²×#99	C₂×D₄	1	2	yes	—
112	#2308	C₂×D₄	1	2	yes	—
113	C₂×#379	C₂×D₄	1	2	yes	—
114	C₂×#379	C₂×D₄	1	2	yes	—
115	C₂×#379	C₂×D₄	1	2	yes	—
116	#1783	C₂×D₄	1	2	yes	—
117	#1781	C₂×D₄	1	2	yes	—
118	#1781	C₂×D₄	1	2	yes	—
119	#1438	C₂×D₄	1	1	yes	characteristic
120	#1438	C₂×D₄	1	1	yes	characteristic
121	C₂⁶	#121	1	1	yes	characteristic
122	C₂⁶	C₂²×D₄	1	1	yes	characteristic
123	C₂⁶	C₂²×D₄	1	1	yes	characteristic
124	C₂×H(5,𝔽₂)	#121	1	2	yes	—
125	C₂×H(5,𝔽₂)	#121	1	2	yes	—
126	C₂×H(5,𝔽₂)	#127	1	2	yes	—
127	C₂³×D₄	#120	1	2	yes	—
128	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
129	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
130	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
131	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
132	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
133	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
134	C₂³×D₄	C₂×#41	1	1	yes	characteristic
135	C₂³×D₄	C₂×#41	1	1	yes	characteristic
136	C₂³×D₄	C₂×#41	1	1	yes	characteristic
137	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂⁴×C₄	#121	1	1	yes	characteristic
141	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
142	#534	#120	1	4	yes	—
143	#534	#121	1	4	yes	—
144	C₂×#120	#120	1	4	yes	—
145	C₂×#120	#121	1	4	yes	—
146	#518	#121	1	2	yes	—
147	#518	#121	1	2	yes	—
148	#518	#127	1	2	yes	—
149	C₂²×C₂²⋊C₄	#120	1	2	yes	—
150	C₂²×C₂²⋊C₄	#120	1	2	yes	—
151	C₂²×C₂²⋊C₄	#120	1	2	yes	—
152	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
154	C₂⁵	B(4,𝔽₂)	1	2	yes	—
155	C₂⁵	C₂×#120	1	1	yes	characteristic
156	C₂⁵	C₂×#120	1	1	yes	characteristic
157	C₂⁵	C₂×#120	1	1	yes	characteristic
158	C₂⁵	C₂×#121	1	1	yes	characteristic
159	C₂⁵	C₂×#121	1	1	yes	characteristic
160	C₂⁵	#525	1	1	yes	characteristic
161	C₂⁵	D₄²	1	1	yes	characteristic
162	C₂⁵	D₄²	1	1	yes	characteristic
163	C₂⁵	D₄²	1	1	yes	characteristic
164	C₂⁵	D₄²	1	1	yes	characteristic
165	C₂⁵	D₄²	1	1	yes	characteristic
166	C₂⁵	#546	1	1	yes	characteristic
167	C₂⁵	#546	1	1	yes	characteristic
168	C₂²×D₄	#392	1	4	yes	—
169	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
170	C₂²×D₄	C₂×#120	1	1	yes	characteristic
171	C₂²×D₄	C₂×#121	1	1	yes	characteristic
172	C₂²×D₄	C₂×#121	1	1	yes	characteristic
173	C₂²×D₄	C₂×#121	1	1	yes	characteristic
174	C₂²×D₄	#525	1	1	yes	characteristic
175	C₂²×D₄	#525	1	1	yes	characteristic
176	C₂²×D₄	C₂×#127	1	1	yes	characteristic
177	C₂²×D₄	#534	1	1	yes	characteristic
178	C₂²×D₄	#535	1	1	yes	characteristic
179	C₂²×D₄	#538	1	1	yes	characteristic
180	C₂²×D₄	D₄²	1	1	yes	characteristic
181	C₂²×D₄	D₄²	1	1	yes	characteristic
182	C₂²×D₄	#546	1	1	yes	characteristic
183	C₂²×D₄	#546	1	1	yes	characteristic
184	C₂²×D₄	#546	1	1	yes	characteristic
185	C₂²×D₄	#546	1	1	yes	characteristic
186	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
187	C₂³×C₄	#534	1	1	yes	characteristic
188	C₂³×C₄	#534	1	1	yes	characteristic
189	C₂³×C₄	#538	1	1	yes	characteristic
190	C₂³×C₄	D₄²	1	1	yes	characteristic
191	C₂³×C₄	#546	1	1	yes	characteristic
192	C₂³×C₄	#546	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	#392	1	4	yes	—
194	C₂⁴	#2021	1	4	yes	—
195	C₂⁴	C₂×#392	1	1	yes	characteristic
196	C₂⁴	#2403	1	1	yes	characteristic
197	C₂⁴	#2404	1	1	yes	characteristic
198	C₂⁴	#2410	1	1	yes	characteristic
199	C₂⁴	#2410	1	1	yes	characteristic
200	C₂⁴	#2410	1	1	yes	characteristic
201	C₂⁴	#2424	1	1	yes	characteristic
202	C₂⁴	#2424	1	1	yes	characteristic
203	C₂⁴	#2424	1	1	yes	characteristic
204	C₂⁴	#2433	1	1	yes	characteristic
205	C₂⁴	#2847	1	1	yes	characteristic
206	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
207	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
208	C₂⁴	#3025	1	2	yes	—
209	C₂⁴	#3026	1	2	yes	—
210	C₂²×C₄	#2021	1	4	yes	—
211	C₂²×C₄	#2410	1	1	yes	characteristic
212	C₂²×C₄	#2424	1	1	yes	characteristic
213	C₂²×C₄	#2539	1	1	yes	characteristic
214	C₂²×C₄	#2617	1	1	yes	characteristic
215	C₂³	#14716	1	1	yes	characteristic
216	C₂³	C₂×#2021	1	2	yes	—
217	C₂³	#23895	1	1	yes	characteristic
218	C₂³	#23896	1	1	yes	characteristic
219	C₂³	#23900	1	1	yes	characteristic
220	C₂³	#23900	1	1	yes	characteristic
221	C₂³	#23902	1	1	yes	characteristic
222	C₂³	#23903	1	1	yes	characteristic
223	C₂³	#23984	1	2	yes	—
224	C₂³	#25773	1	2	yes	—
225	C₂³	#32928	1	2	yes	—
226	C₂²	#245227	1	1	yes	characteristic
227	C₂²	#245231	1	1	yes	characteristic
228	C₂²	#245246	1	1	yes	characteristic
229	C₂²	#245254	1	1	yes	characteristic
230	C₂²	#394167	1	1	yes	characteristic
231	C₂²	#394173	1	1	yes	characteristic
232	C₂²	#394176	1	1	yes	characteristic
233	C₂	#72265996	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72265996
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72235130	C₂
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. #72264754	C₂	semidirect
2. #72258983	C₂	semidirect
3. #72255863	C₂	semidirect
4. #72255838	C₂	semidirect
5. #72246292	C₂	semidirect
6. #72240471	C₂	semidirect
7. #72238609	C₂	semidirect
8. #72238351	C₂	semidirect
9. #72237633	C₂	semidirect
10. #72237473	C₂	semidirect
12. #72234631	C₂	semidirect
13. #72234613	C₂	semidirect
14. #72219163	C₂	semidirect
15. C₂×#139743	C₂	semidirect
22. #6374210	C₂²	semidirect
26. #493879	C₂²	semidirect
28. C₂×#12707	C₂²	semidirect
31. #382544	C₂²	semidirect
32. #359811	C₂²	semidirect
35. #359549	C₂²	semidirect
38. #190116	C₂²	semidirect
42. C₂×#8534	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. C₂×#7316	C₂²	semidirect
46. C₂×#7310	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7307	C₂²	semidirect
49. #108222	C₂²	semidirect
50. #107130	C₂²	semidirect
57. #62652	D₄	semidirect
59. C₂×#2883	D₄	semidirect
60. #34992	D₄	semidirect
65. C₂×#1898	D₄	semidirect
66. #20609	D₄	semidirect
68. C₂×#1781	D₄	semidirect
75. #20354	D₄	semidirect
76. #20354	D₄	semidirect
79. #10754	D₄	semidirect
80. #10735	D₄	semidirect
81. #8534	D₄	semidirect
82. #8534	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72264754	2
#72258983	2
#72255863	2
#72255838	2
#72246292	2
#72240471	2
#72238609	2
#72238351	2
#72237633	2
#72237473	2
#72235130	2
#72234631	2
#72234613	2
#72219163	2
C₂×#139743	2