Subgroups — #72314996

Order 2048.

At least 6070 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

The subgroup lattice has 255 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72314739	C₂	1	1	yes	maximal, characteristic, complemented
2	#72314734	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266835	C₂	1	1	yes	maximal, characteristic
4	#72266378	C₂	1	1	yes	maximal, characteristic, complemented
5	#72260562	C₂	1	1	yes	maximal, characteristic, complemented
6	#72258334	C₂	1	1	yes	maximal, characteristic, complemented
7	#72258019	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251528	C₂	1	1	yes	maximal, characteristic, complemented
9	#72246287	C₂	1	1	yes	maximal, characteristic, complemented
10	#72246286	C₂	1	1	yes	maximal, characteristic, complemented
11	#72238613	C₂	1	1	yes	maximal, characteristic, complemented
12	#72238349	C₂	1	1	yes	maximal, characteristic, complemented
13	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
14	#72237638	C₂	1	1	yes	maximal, characteristic, complemented
15	#72237473	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	#6437621	C₂²	1	1	yes	characteristic
18	#6437621	C₂²	1	1	yes	characteristic
19	#6374251	C₂²	1	1	yes	characteristic, complemented
20	C₂×#21776	C₂²	1	1	yes	characteristic
21	C₂×#21776	C₂²	1	1	yes	characteristic
22	#763333	C₂²	1	1	yes	characteristic
23	C₂×#12707	C₂²	1	1	yes	characteristic
24	C₂×#12707	C₂²	1	1	yes	characteristic
25	C₂×#12705	C₂²	1	1	yes	characteristic
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	#493370	C₂²	1	1	yes	characteristic
28	#493370	C₂²	1	1	yes	characteristic
29	#493370	C₂²	1	1	yes	characteristic
30	#493357	C₂²	1	1	yes	characteristic, complemented
31	#392030	C₂²	1	1	yes	characteristic, complemented
32	#359549	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic
34	#359509	C₂²	1	1	yes	characteristic
35	#359442	C₂²	1	1	yes	characteristic, complemented
36	#355096	C₂²	1	1	yes	characteristic, complemented
37	#347057	C₄	1	2	yes	complemented
38	#347056	C₄	1	2	yes	complemented
39	#347047	C₄	1	2	yes	complemented
40	#347045	C₄	1	2	yes	complemented
41	#188050	C₂²	1	1	yes	characteristic
42	#188049	C₂²	1	1	yes	characteristic, complemented
43	C₂×#8534	C₂²	1	1	yes	characteristic
44	C₂×#8534	C₂²	1	1	yes	characteristic
45	#146479	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7304	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×#7310	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
51	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
52	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
53	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
54	#108222	C₂²	1	1	yes	characteristic, complemented
55	#62832	C₂×C₄	1	2	yes	—
56	#62824	C₂×C₄	1	2	yes	—
57	C₂×#3484	C₂³	1	1	yes	characteristic
58	C₂×#3484	C₂³	1	1	yes	characteristic
59	C₂²×#535	C₂³	1	1	yes	characteristic
60	C₂²×#534	C₂³	1	1	yes	characteristic
61	C₂×#2883	D₄	1	2	yes	complemented
62	#34980	D₄	1	2	yes	—
63	#34171	C₂³	1	1	yes	characteristic
64	#23901	C₂×C₄	1	2	yes	—
65	#23901	C₂×C₄	1	2	yes	complemented
66	#23901	D₄	1	2	yes	—
67	#23900	C₂×C₄	1	2	yes	—
68	#23900	C₂×C₄	1	2	yes	complemented
69	#23900	D₄	1	2	yes	—
70	C₂×#1913	D₄	1	2	yes	complemented
71	C₂×#1898	D₄	1	2	yes	—
72	#20960	D₄	1	2	yes	complemented
73	C₂×#1828	D₄	1	2	yes	complemented
74	C₂×#1783	C₂³	1	1	yes	characteristic
75	C₂×#1781	C₂³	1	1	yes	characteristic
76	C₂×#1781	C₂³	1	1	yes	characteristic
77	C₂×#1781	C₂³	1	1	yes	characteristic
78	C₂×#1781	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	#15262	C₂×C₄	1	1	yes	characteristic, complemented
84	#15261	C₂×C₄	1	1	yes	characteristic, complemented
85	#15259	C₂×C₄	1	1	yes	characteristic, complemented
86	#15258	C₂×C₄	1	1	yes	characteristic, complemented
87	C₂×#1438	C₂³	1	1	yes	characteristic
88	#12707	D₄	1	4	yes	complemented
89	#12694	D₄	1	2	yes	complemented
90	#12693	D₄	1	2	yes	complemented
91	#8546	C₂×C₄	1	2	yes	complemented
92	#8544	C₂×C₄	1	2	yes	complemented
93	#8534	C₂×C₄	1	2	yes	—
94	#8534	C₂×C₄	1	2	yes	—
95	#8534	C₂×C₄	1	2	yes	complemented
96	#8534	C₂×C₄	1	2	yes	complemented
97	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
98	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
99	C₂³×#41	#41	1	1	yes	characteristic
100	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
102	#3484	C₂²×C₄	1	2	yes	—
103	#3484	C₂²×C₄	1	2	yes	—
104	#3484	C₂×D₄	1	2	yes	—
105	#3484	C₂×D₄	1	2	yes	—
106	#3484	#41	1	2	yes	—
107	#3484	#41	1	2	yes	—
108	C₂×#534	C₂×D₄	1	2	yes	—
109	C₂²×#127	C₂²×C₄	1	1	yes	characteristic
110	C₂²×#127	C₂×D₄	1	1	yes	characteristic
111	C₂²×#124	C₂²×C₄	1	1	yes	characteristic
112	C₂²×#124	C₂×D₄	1	1	yes	characteristic
113	C₂²×#123	C₂²×C₄	1	1	yes	characteristic
114	C₂²×#123	C₂×D₄	1	1	yes	characteristic
115	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
116	C₂²×#120	C₂×D₄	1	1	yes	characteristic
117	C₂²×#120	C₂×D₄	1	2	yes	—
118	C₂×#518	C₂×D₄	1	1	yes	characteristic
119	C₂×#518	C₂×D₄	1	1	yes	characteristic
120	C₂×#518	C₂×D₄	1	1	yes	characteristic
121	C₂×#518	#41	1	1	yes	characteristic
122	C₂²×#117	#41	1	1	yes	characteristic
123	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
124	C₂²×#99	C₂×D₄	1	2	yes	—
125	C₂²×#99	C₂×D₄	1	2	yes	—
126	C₂²×#99	C₂×D₄	1	2	yes	—
127	#2849	C₂²⋊C₄	1	4	yes	complemented
128	#2846	C₂²⋊C₄	1	4	yes	complemented
129	C₂×#379	C₂×D₄	1	2	yes	—
130	#1898	C₂²⋊C₄	1	4	yes	complemented
131	#1898	C₂²⋊C₄	1	4	yes	complemented
132	#1781	C₂²×C₄	1	2	yes	—
133	#1781	C₂²×C₄	1	2	yes	—
134	#1781	C₂×D₄	1	2	yes	—
135	#1781	C₂×D₄	1	2	yes	—
136	#1781	C₂×D₄	1	2	yes	—
137	#1781	C₂×D₄	1	2	yes	—
138	#1781	C₂×D₄	1	2	yes	—
139	#1438	C₂²×C₄	1	1	yes	characteristic
140	#1438	C₂²×C₄	1	1	yes	characteristic
141	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂⁶	C₂²×D₄	1	1	yes	characteristic
143	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
145	C₂×H(5,𝔽₂)	C₄×D₄	1	2	yes	—
146	C₂×H(5,𝔽₂)	#121	1	2	yes	—
147	C₂³×Q₈	C₄×D₄	1	1	yes	characteristic
148	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
149	C₂³×D₄	#120	1	2	yes	—
150	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
151	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
152	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
153	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
154	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
155	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
156	C₂³×D₄	C₂×#41	1	1	yes	characteristic
157	C₂³×D₄	C₂×#41	1	1	yes	characteristic
158	C₂³×D₄	C₂×#41	1	1	yes	characteristic
159	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
164	#534	C₄×D₄	1	4	yes	—
165	#534	#121	1	4	yes	—
166	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
167	C₂×#120	#123	1	4	yes	—
168	#518	C₄×D₄	1	2	yes	—
169	#518	C₄×D₄	1	2	yes	—
170	#518	#121	1	2	yes	—
171	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₂²⋊C₄	#120	1	2	yes	—
174	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
175	C₂²×C₄²	C₄×D₄	1	1	yes	characteristic
176	C₂×#99	#120	1	4	yes	—
177	#379	C₂×C₂²⋊C₄	1	4	yes	—
178	C₂⁵	B(4,𝔽₂)	1	2	yes	—
179	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
180	C₂⁵	#518	1	1	yes	characteristic
181	C₂⁵	C₂×#120	1	1	yes	characteristic
182	C₂⁵	C₂×#123	1	1	yes	characteristic
183	C₂⁵	#535	1	1	yes	characteristic
184	C₂⁵	#538	1	1	yes	characteristic
185	C₂⁵	D₄²	1	1	yes	characteristic
186	C₂⁵	D₄²	1	1	yes	characteristic
187	C₂⁵	D₄²	1	1	yes	characteristic
188	C₂⁵	#546	1	1	yes	characteristic
189	C₂⁵	#546	1	1	yes	characteristic
190	C₂⁵	#546	1	1	yes	characteristic
191	C₂⁵	#546	1	1	yes	characteristic
192	C₂²×D₄	#386	1	4	yes	—
193	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
194	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
195	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂²×D₄	#518	1	1	yes	characteristic
198	C₂²×D₄	C₂×#120	1	1	yes	characteristic
199	C₂²×D₄	C₂×#121	1	1	yes	characteristic
200	C₂²×D₄	C₂×#121	1	1	yes	characteristic
201	C₂²×D₄	#525	1	1	yes	characteristic
202	C₂²×D₄	#534	1	1	yes	characteristic
203	C₂²×D₄	#538	1	1	yes	characteristic
204	C₂²×D₄	D₄²	1	1	yes	characteristic
205	C₂²×D₄	D₄²	1	1	yes	characteristic
206	C₂²×D₄	D₄²	1	1	yes	characteristic
207	C₂²×D₄	#546	1	1	yes	characteristic
208	C₂²×D₄	#546	1	1	yes	characteristic
209	C₂²×D₄	#551	1	1	yes	characteristic
210	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
211	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂³×C₄	#525	1	1	yes	characteristic
213	C₂³×C₄	#525	1	1	yes	characteristic
214	C₂³×C₄	#546	1	1	yes	characteristic
215	C₂³×C₄	#546	1	1	yes	characteristic
216	C₂³×C₄	#551	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	#386	1	4	yes	—
218	C₂⁴	#1889	1	4	yes	—
219	C₂⁴	C₂×#386	1	1	yes	characteristic
220	C₂⁴	#2321	1	1	yes	characteristic
221	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
222	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
223	C₂⁴	#2358	1	1	yes	characteristic
224	C₂⁴	#2403	1	1	yes	characteristic
225	C₂⁴	#2404	1	1	yes	characteristic
226	C₂⁴	#2408	1	1	yes	characteristic
227	C₂⁴	#2418	1	1	yes	characteristic
228	C₂⁴	#2418	1	1	yes	characteristic
229	C₂⁴	#2424	1	1	yes	characteristic
230	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
231	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
232	C₂²×C₄	#1889	1	4	yes	—
233	C₂²×C₄	C₄×#120	1	1	yes	characteristic
234	C₂²×C₄	#2358	1	1	yes	characteristic
235	C₂²×C₄	#2424	1	1	yes	characteristic
236	C₂²×C₄	#2534	1	1	yes	characteristic
237	C₂³	#13967	1	1	yes	characteristic
238	C₂³	C₂×#1889	1	2	yes	—
239	C₂³	#23900	1	2	yes	—
240	C₂³	#23984	1	1	yes	characteristic
241	C₂³	#23984	1	1	yes	characteristic
242	C₂³	#23996	1	1	yes	characteristic
243	C₂³	#23996	1	1	yes	characteristic
244	C₂³	#23996	1	1	yes	characteristic
245	C₂³	#23997	1	1	yes	characteristic
246	C₂²	#251843	1	1	yes	characteristic
247	C₂²	#251854	1	1	yes	characteristic
248	C₂²	#251859	1	1	yes	characteristic
249	C₂²	#251877	1	1	yes	characteristic
250	C₂²	#364410	1	1	yes	characteristic
251	C₂²	#364532	1	1	yes	characteristic
252	C₂²	#364545	1	1	yes	characteristic
253	C₂	#72241284	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241284
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72266835	C₂
LCS	G ▷ C₂³×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. #72314739	C₂	semidirect
2. #72314734	C₂	semidirect
4. #72266378	C₂	semidirect
5. #72260562	C₂	semidirect
6. #72258334	C₂	semidirect
7. #72258019	C₂	semidirect
8. #72251528	C₂	semidirect
9. #72246287	C₂	semidirect
10. #72246286	C₂	semidirect
11. #72238613	C₂	semidirect
12. #72238349	C₂	semidirect
13. #72238343	C₂	semidirect
14. #72237638	C₂	semidirect
15. #72237473	C₂	semidirect
19. #6374251	C₂²	semidirect
30. #493357	C₂²	semidirect
31. #392030	C₂²	semidirect
35. #359442	C₂²	semidirect
36. #355096	C₂²	semidirect
37. #347057	C₄	semidirect
38. #347056	C₄	semidirect
39. #347047	C₄	semidirect
40. #347045	C₄	semidirect
42. #188049	C₂²	semidirect
45. #146479	C₂²	semidirect
46. C₂×#7304	C₂²	semidirect
47. C₂×#7316	C₂²	semidirect
48. C₂×#7314	C₂²	semidirect
49. C₂×#7310	C₂²	semidirect
50. C₂×#7310	C₂²	semidirect
51. C₂×#7310	C₂²	semidirect
52. C₂×#7307	C₂²	semidirect
53. C₂×#7307	C₂²	semidirect
54. #108222	C₂²	semidirect
61. C₂×#2883	D₄	semidirect
65. #23901	C₂×C₄	semidirect
68. #23900	C₂×C₄	semidirect
70. C₂×#1913	D₄	semidirect
72. #20960	D₄	semidirect
73. C₂×#1828	D₄	semidirect
83. #15262	C₂×C₄	semidirect
84. #15261	C₂×C₄	semidirect
85. #15259	C₂×C₄	semidirect
86. #15258	C₂×C₄	semidirect
88. #12707	D₄	semidirect
89. #12694	D₄	semidirect
90. #12693	D₄	semidirect
91. #8546	C₂×C₄	semidirect
92. #8544	C₂×C₄	semidirect
95. #8534	C₂×C₄	semidirect
96. #8534	C₂×C₄	semidirect
127. #2849	C₂²⋊C₄	semidirect
128. #2846	C₂²⋊C₄	semidirect
130. #1898	C₂²⋊C₄	semidirect
131. #1898	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72314739	2
#72314734	2
#72266835	2
#72266378	2
#72260562	2
#72258334	2
#72258019	2
#72251528	2
#72246287	2
#72246286	2
#72238613	2
#72238349	2
#72238343	2
#72237638	2
#72237473	2