Subgroups — #72312516

Order 2048.

At least 6341 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	#72312312	C₂	1	1	yes	maximal, characteristic, complemented
2	#72312218	C₂	1	1	yes	maximal, characteristic, complemented
3	#72309794	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266834	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#139739	C₂	1	1	yes	maximal, characteristic, complemented
6	#72256025	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255830	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251533	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251422	C₂	1	1	yes	maximal, characteristic, complemented
10	#72246450	C₂	1	1	yes	maximal, characteristic, complemented
11	#72246292	C₂	1	1	yes	maximal, characteristic, complemented
12	#72240471	C₂	1	1	yes	maximal, characteristic, complemented
13	#72238342	C₂	1	1	yes	maximal, characteristic
14	#72237634	C₂	1	1	yes	maximal, characteristic, complemented
15	#72236678	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	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	C₂×#23895	C₂²	1	1	yes	characteristic
21	#6437445	C₂²	1	1	yes	characteristic
22	#6374210	C₂²	1	1	yes	characteristic, complemented
23	C₂×#21778	C₂²	1	1	yes	characteristic
24	#6371558	C₂²	1	1	yes	characteristic
25	#763514	C₂²	1	1	yes	characteristic
26	#493879	C₂²	1	1	yes	characteristic, complemented
27	C₂×#12705	C₂²	1	1	yes	characteristic
28	C₂×#12705	C₂²	1	1	yes	characteristic
29	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
30	#493370	C₂²	1	1	yes	characteristic
31	#382544	C₂²	1	1	yes	characteristic, complemented
32	#359811	C₂²	1	1	yes	characteristic, complemented
33	#359558	C₂²	1	1	yes	characteristic
34	#359558	C₂²	1	1	yes	characteristic
35	#359549	C₂²	1	1	yes	characteristic, complemented
36	#359509	C₂²	1	1	yes	characteristic
37	#359509	C₂²	1	1	yes	characteristic
38	#190147	C₂²	1	1	yes	characteristic, complemented
39	#188050	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
49	#108222	C₂²	1	1	yes	characteristic, complemented
50	#107130	C₂²	1	1	yes	characteristic, complemented
51	#62832	D₄	1	2	yes	—
52	#62824	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×#535	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	#20616	D₄	1	2	yes	complemented
67	#20605	D₄	1	2	yes	complemented
68	C₂×#1783	C₂³	1	1	yes	characteristic
69	C₂×#1783	C₂³	1	1	yes	characteristic
70	C₂×#1783	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	#10740	D₄	1	2	yes	complemented
80	#10716	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	C₂³×#41	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×#127	C₂×D₄	1	1	yes	characteristic
96	C₂²×#127	#41	1	1	yes	characteristic
97	C₂²×#124	C₂×D₄	1	1	yes	characteristic
98	C₂²×#124	#41	1	1	yes	characteristic
99	C₂²×#120	C₂×D₄	1	2	yes	—
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	C₂²×#117	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×#100	C₂×D₄	1	2	yes	—
111	C₂²×#99	C₂×D₄	1	2	yes	—
112	C₂×#407	C₂×D₄	1	2	yes	—
113	#2309	C₂×D₄	1	2	yes	—
114	#2280	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	C₂²×D₄	1	1	yes	characteristic
122	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×H(5,𝔽₂)	#121	1	2	yes	—
124	C₂×H(5,𝔽₂)	#121	1	2	yes	—
125	C₂×H(5,𝔽₂)	#127	1	2	yes	—
126	C₂³×Q₈	#121	1	1	yes	characteristic
127	C₂³×D₄	#120	1	2	yes	—
128	C₂³×D₄	#121	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	C₂²×D₄	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	C₂×#41	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	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂⁴×C₄	C₂²×D₄	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	C₂×#117	#120	1	2	yes	—
150	C₂×#117	#120	1	2	yes	—
151	C₂²×C₂²⋊C₄	#120	1	2	yes	—
152	C₂²×C₄²	#121	1	1	yes	characteristic
153	C₂²×C₄²	#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×#121	1	1	yes	characteristic
158	C₂⁵	#534	1	1	yes	characteristic
159	C₂⁵	#534	1	1	yes	characteristic
160	C₂⁵	#538	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	#546	1	1	yes	characteristic
165	C₂⁵	#546	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	C₂×#120	1	1	yes	characteristic
188	C₂³×C₄	C₂×#121	1	1	yes	characteristic
189	C₂³×C₄	#525	1	1	yes	characteristic
190	C₂³×C₄	D₄²	1	1	yes	characteristic
191	C₂³×C₄	D₄²	1	1	yes	characteristic
192	C₂³×C₄	D₄²	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	#2539	1	1	yes	characteristic
205	C₂⁴	#2617	1	1	yes	characteristic
206	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
207	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
208	C₂²×C₄	#2021	1	4	yes	—
209	C₂²×C₄	#2410	1	1	yes	characteristic
210	C₂²×C₄	#2424	1	1	yes	characteristic
211	C₂²×C₄	#2433	1	1	yes	characteristic
212	C₂²×C₄	#2847	1	1	yes	characteristic
213	C₂²×C₄	#3025	1	2	yes	—
214	C₂²×C₄	#3026	1	2	yes	—
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	C₂×C₄	#25773	1	2	yes	—
225	C₂×C₄	#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	#72238342	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. #72312312	C₂	semidirect
2. #72312218	C₂	semidirect
3. #72309794	C₂	semidirect
4. #72266834	C₂	semidirect
5. C₂×#139739	C₂	semidirect
6. #72256025	C₂	semidirect
7. #72255830	C₂	semidirect
8. #72251533	C₂	semidirect
9. #72251422	C₂	semidirect
10. #72246450	C₂	semidirect
11. #72246292	C₂	semidirect
12. #72240471	C₂	semidirect
14. #72237634	C₂	semidirect
15. #72236678	C₂	semidirect
22. #6374210	C₂²	semidirect
26. #493879	C₂²	semidirect
29. C₂×#12705	C₂²	semidirect
31. #382544	C₂²	semidirect
32. #359811	C₂²	semidirect
35. #359549	C₂²	semidirect
38. #190147	C₂²	semidirect
42. C₂×#8453	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. C₂×#7304	C₂²	semidirect
46. C₂×#7316	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7311	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. #20616	D₄	semidirect
67. #20605	D₄	semidirect
75. #20354	D₄	semidirect
76. #20354	D₄	semidirect
79. #10740	D₄	semidirect
80. #10716	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
#72312312	2
#72312218	2
#72309794	2
#72266834	2
C₂×#139739	2
#72256025	2
#72255830	2
#72251533	2
#72251422	2
#72246450	2
#72246292	2
#72240471	2
#72238342	2
#72237634	2
#72236678	2