Subgroups — #72266275

Order 2048.

At least 6940 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

The subgroup lattice has 259 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266274	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266273	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266264	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266171	C₂	1	1	yes	maximal, characteristic, complemented
5	#72235928	C₂	1	1	yes	maximal, characteristic, complemented
6	#72235926	C₂	1	1	yes	maximal, characteristic, complemented
7	#72235924	C₂	1	1	yes	maximal, characteristic, complemented
8	#72235923	C₂	1	1	yes	maximal, characteristic, complemented
9	#72235919	C₂	1	1	yes	maximal, characteristic, complemented
10	#72235868	C₂	1	1	yes	maximal, characteristic, complemented
11	#72235820	C₂	1	1	yes	maximal, characteristic, complemented
12	#72235732	C₂	1	1	yes	maximal, characteristic, complemented
13	#72235729	C₂	1	1	yes	maximal, characteristic, complemented
14	#72235704	C₂	1	1	yes	maximal, characteristic, complemented
15	#72235690	C₂	1	1	yes	maximal, characteristic, complemented
16	#7642350	C₂²	1	1	yes	characteristic
17	#6451874	C₂²	1	1	yes	characteristic
18	#6346594	C₂²	1	1	yes	characteristic
19	#4893962	C₂²	1	1	yes	characteristic
20	#4335643	C₂²	1	1	yes	characteristic
21	#635987	C₂²	1	1	yes	characteristic
22	#618316	C₂²	1	1	yes	characteristic
23	#587363	C₂²	1	1	yes	characteristic
24	#576160	C₂²	1	1	yes	characteristic
25	#515362	C₂²	1	1	yes	characteristic
26	#482549	C₂²	1	1	yes	characteristic
27	#482549	C₂²	1	1	yes	characteristic
28	#480250	C₂²	1	1	yes	characteristic
29	#480250	C₂²	1	1	yes	characteristic
30	#359705	C₂²	1	1	yes	characteristic
31	#201462	C₂²	1	1	yes	characteristic
32	C₂×#8549	C₂²	1	1	yes	characteristic
33	C₂×#8542	C₂²	1	1	yes	characteristic
34	#187945	C₂²	1	1	yes	characteristic
35	#185255	C₂²	1	1	yes	characteristic
36	#185255	C₂²	1	1	yes	characteristic
37	#185255	C₂²	1	1	yes	characteristic
38	#184796	C₂²	1	1	yes	characteristic
39	#128157	C₂²	1	1	yes	characteristic
40	#128157	C₂²	1	1	yes	characteristic
41	#127215	C₂²	1	1	yes	characteristic
42	#127215	C₂²	1	1	yes	characteristic
43	#127215	C₂²	1	1	yes	characteristic
44	#127215	C₂²	1	1	yes	characteristic
45	#127215	C₂²	1	1	yes	characteristic
46	#127215	C₂²	1	1	yes	characteristic
47	#107115	C₂²	1	1	yes	characteristic
48	#106326	C₂²	1	1	yes	characteristic
49	#106326	C₂²	1	1	yes	characteristic
50	#106326	C₂²	1	1	yes	characteristic
51	C₂×#3490	C₂³	1	1	yes	characteristic
52	C₂²×#525	D₄	1	1	yes	characteristic, complemented
53	C₂³×#121	C₂³	1	1	yes	characteristic
54	C₂²×#518	D₄	1	1	yes	characteristic, complemented
55	C₂×#2404	D₄	1	1	yes	characteristic, complemented
56	C₂×#2404	C₂³	1	1	yes	characteristic
57	C₂×#2403	D₄	1	1	yes	characteristic
58	C₂×#2403	D₄	1	1	yes	characteristic
59	#34176	C₂³	1	1	yes	characteristic
60	C₂×#2308	D₄	1	1	yes	characteristic, complemented
61	C₂×#2308	C₂³	1	1	yes	characteristic
62	C₂×#1819	C₂³	1	1	yes	characteristic
63	C₂×#1819	C₂³	1	1	yes	characteristic
64	C₂×#1462	C₂³	1	1	yes	characteristic
65	C₂×#1462	C₂³	1	1	yes	characteristic
66	C₂×#1462	C₂³	1	1	yes	characteristic
67	C₂×#1462	C₂³	1	1	yes	characteristic
68	#16247	D₄	1	1	yes	characteristic, complemented
69	#16211	D₄	1	1	yes	characteristic, complemented
70	#15930	D₄	1	1	yes	characteristic, complemented
71	#15636	D₄	1	1	yes	characteristic, complemented
72	#15636	D₄	1	1	yes	characteristic, complemented
73	#15561	D₄	1	1	yes	characteristic, complemented
74	#15467	D₄	1	1	yes	characteristic, complemented
75	#15267	D₄	1	1	yes	characteristic, complemented
76	#14681	D₄	1	1	yes	characteristic, complemented
77	#14649	D₄	1	1	yes	characteristic, complemented
78	#14609	D₄	1	1	yes	characteristic, complemented
79	#14533	D₄	1	1	yes	characteristic, complemented
80	#14507	D₄	1	1	yes	characteristic, complemented
81	#14507	D₄	1	1	yes	characteristic, complemented
82	#14341	C₂³	1	1	yes	characteristic
83	#14207	D₄	1	1	yes	characteristic, complemented
84	#13870	D₄	1	1	yes	characteristic, complemented
85	#13870	D₄	1	1	yes	characteristic, complemented
86	#13868	D₄	1	1	yes	characteristic, complemented
87	#13868	C₂³	1	1	yes	characteristic
88	#13868	C₂³	1	1	yes	characteristic
89	#13868	C₂³	1	1	yes	characteristic
90	#13867	D₄	1	1	yes	characteristic
91	#13867	D₄	1	1	yes	characteristic
92	#13827	D₄	1	1	yes	characteristic, complemented
93	C₂²⋊C₄²	D₄	1	1	yes	characteristic, complemented
94	C₂³×#41	C₂×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
96	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
97	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
98	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
99	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂×#543	#41	1	1	yes	characteristic, complemented
101	C₂×#534	#41	1	1	yes	characteristic, complemented
102	C₂²×#117	C₂×D₄	1	1	yes	characteristic
103	C₂²×#117	C₂×D₄	1	1	yes	characteristic
104	C₂²×#117	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
108	C₂³×C₂²⋊C₄	C₂⁴	1	1	yes	characteristic
109	C₂×#393	C₂×D₄	1	1	yes	characteristic
110	C₂×#393	C₂×D₄	1	1	yes	characteristic
111	C₂×#392	C₂×D₄	1	1	yes	characteristic
112	C₂×#392	C₂×D₄	1	1	yes	characteristic
113	C₂×#381	C₂×D₄	1	1	yes	characteristic
114	C₂×#381	C₂×D₄	1	1	yes	characteristic
115	C₂×#381	C₂×D₄	1	1	yes	characteristic
116	C₂×#381	C₂×D₄	1	1	yes	characteristic
117	C₂×#381	C₂×D₄	1	1	yes	characteristic
118	C₂×#381	C₂×D₄	1	1	yes	characteristic
119	C₂×#379	C₂×D₄	1	1	yes	characteristic
120	C₂×#379	C₂×D₄	1	1	yes	characteristic
121	C₂×#379	C₂×D₄	1	1	yes	characteristic
122	C₂×#379	C₂×D₄	1	1	yes	characteristic
123	#2273	#41	1	2	yes	complemented
124	#1819	#41	1	2	yes	complemented
125	C₂×#323	#41	1	2	yes	complemented
126	#1440	C₂×D₄	1	1	yes	characteristic
127	#1440	C₂×D₄	1	1	yes	characteristic
128	#1440	C₂×D₄	1	1	yes	characteristic
129	#1440	C₂×D₄	1	1	yes	characteristic
130	#1437	C₂×D₄	1	1	yes	characteristic
131	#1437	C₂×D₄	1	1	yes	characteristic
132	#1437	C₂×D₄	1	1	yes	characteristic
133	#1437	C₂×D₄	1	1	yes	characteristic
134	#1437	C₂×D₄	1	1	yes	characteristic
135	#1437	C₂×D₄	1	1	yes	characteristic
136	#1437	C₂×D₄	1	1	yes	characteristic
137	#1437	C₂×D₄	1	1	yes	characteristic
138	#1437	C₂×D₄	1	1	yes	characteristic
139	#1437	C₂×D₄	1	1	yes	characteristic
140	#1437	C₂×D₄	1	1	yes	characteristic
141	#1437	C₂×D₄	1	1	yes	characteristic
142	C₂⁶	#120	1	1	yes	characteristic
143	C₂⁶	#120	1	1	yes	characteristic
144	C₂⁶	#120	1	1	yes	characteristic
145	C₂⁶	C₂²×D₄	1	1	yes	characteristic
146	C₂³×Q₈	#120	1	1	yes	characteristic
147	C₂³×D₄	#120	1	1	yes	characteristic
148	C₂⁴×C₄	#120	1	1	yes	characteristic
149	C₂⁴×C₄	#120	1	1	yes	characteristic
150	C₂⁴×C₄	#120	1	1	yes	characteristic
151	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
154	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
156	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
157	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
159	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
161	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
162	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
163	C₂²×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
164	C₂²×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
165	C₂²×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
166	C₂²×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
167	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂²×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂²×C₂²⋊C₄	2₋¹⁺⁴	1	1	yes	characteristic
171	C₂²×C₄²	#120	1	1	yes	characteristic
172	C₂²×C₄²	#120	1	1	yes	characteristic
173	C₂²×C₄²	#120	1	1	yes	characteristic
174	C₂²×C₄²	#120	1	1	yes	characteristic
175	C₂²×C₄²	#120	1	1	yes	characteristic
176	C₂²×C₄²	#120	1	1	yes	characteristic
177	C₂×#95	#120	1	1	yes	characteristic
178	C₂×#95	#120	1	1	yes	characteristic
179	C₂×#95	#120	1	1	yes	characteristic
180	C₂×#95	#120	1	1	yes	characteristic
181	C₂×#95	#120	1	1	yes	characteristic
182	C₂×#95	#120	1	1	yes	characteristic
183	C₂×#95	#120	1	1	yes	characteristic
184	C₂×#95	#120	1	1	yes	characteristic
185	C₂⁵	B(4,𝔽₂)	1	2	yes	—
186	C₂⁵	B(4,𝔽₂)	1	2	yes	—
187	C₂⁵	C₂×#120	1	1	yes	characteristic
188	C₂⁵	C₂×#120	1	1	yes	characteristic
189	C₂⁵	C₂×#120	1	1	yes	characteristic
190	C₂⁵	#546	1	1	yes	characteristic
191	C₂⁵	#546	1	1	yes	characteristic
192	C₂⁵	#546	1	1	yes	characteristic
193	C₂⁵	#547	1	1	yes	characteristic
194	C₂³×C₄	C₂×#120	1	1	yes	characteristic
195	C₂³×C₄	C₂×#120	1	1	yes	characteristic
196	C₂³×C₄	C₂×#120	1	1	yes	characteristic
197	C₂³×C₄	C₂×#120	1	1	yes	characteristic
198	C₂³×C₄	#525	1	1	yes	characteristic
199	C₂³×C₄	#532	1	1	yes	characteristic
200	C₂³×C₄	#535	1	1	yes	characteristic
201	C₂³×C₄	#537	1	1	yes	characteristic
202	C₂³×C₄	#546	1	1	yes	characteristic
203	C₂³×C₄	#546	1	1	yes	characteristic
204	C₂³×C₄	#546	1	1	yes	characteristic
205	C₂³×C₄	#547	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	#525	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	#525	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	#532	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	#532	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	#535	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	#535	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	#537	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	#537	1	1	yes	characteristic
214	C₂×C₂²⋊C₄	#546	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	#546	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	#546	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	#547	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	#548	1	1	yes	characteristic
219	C₂×C₂²⋊C₄	#550	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	#550	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	#550	1	1	yes	characteristic
222	C₂⁴	#2404	1	1	yes	characteristic
223	C₂⁴	#2404	1	1	yes	characteristic
224	C₂⁴	#2407	1	1	yes	characteristic
225	C₂⁴	#2430	1	1	yes	characteristic
226	C₂⁴	#2430	1	1	yes	characteristic
227	C₂⁴	#2431	1	1	yes	characteristic
228	C₂⁴	#2846	1	1	yes	characteristic
229	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
230	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
231	C₂⁴	#3026	1	1	yes	characteristic
232	C₂⁴	#3026	1	1	yes	characteristic
233	C₂²×C₄	#2388	1	1	yes	characteristic
234	C₂²×C₄	#2404	1	1	yes	characteristic
235	C₂²×C₄	#2407	1	1	yes	characteristic
236	C₂²×C₄	#2407	1	1	yes	characteristic
237	C₂²×C₄	#2430	1	1	yes	characteristic
238	C₂²×C₄	#2431	1	1	yes	characteristic
239	C₂²×C₄	#2431	1	1	yes	characteristic
240	C₂²×C₄	#2615	1	1	yes	characteristic
241	C₂³	#14885	1	1	yes	characteristic
242	C₂³	#23984	1	1	yes	characteristic
243	C₂³	#23987	1	1	yes	characteristic
244	C₂³	#25563	1	1	yes	characteristic
245	C₂³	#25563	1	1	yes	characteristic
246	C₂³	#32898	1	1	yes	characteristic
247	C₂³	#32898	1	1	yes	characteristic
248	C₂²	#403952	1	1	yes	characteristic
249	C₂²	#404012	1	1	yes	characteristic
250	C₂²	#404013	1	1	yes	characteristic
251	C₂²	#511790	1	1	yes	characteristic
252	C₂²	#511790	1	1	yes	characteristic
253	C₂²	#511791	1	1	yes	characteristic
254	C₂²	#511793	1	1	yes	characteristic
255	C₂	#72233141	1	1	yes	characteristic
256	C₂	#72240122	1	1	yes	characteristic
257	C₂	#72255873	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#403952
commutator, Frattini, cosocle	C₂³×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#121 ◁ G
LECS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266274	C₂	semidirect
2. #72266273	C₂	semidirect
3. #72266264	C₂	semidirect
4. #72266171	C₂	semidirect
5. #72235928	C₂	semidirect
6. #72235926	C₂	semidirect
7. #72235924	C₂	semidirect
8. #72235923	C₂	semidirect
9. #72235919	C₂	semidirect
10. #72235868	C₂	semidirect
11. #72235820	C₂	semidirect
12. #72235732	C₂	semidirect
13. #72235729	C₂	semidirect
14. #72235704	C₂	semidirect
15. #72235690	C₂	semidirect
52. C₂²×#525	D₄	semidirect
54. C₂²×#518	D₄	semidirect
55. C₂×#2404	D₄	semidirect
60. C₂×#2308	D₄	semidirect
68. #16247	D₄	semidirect
69. #16211	D₄	semidirect
70. #15930	D₄	semidirect
71. #15636	D₄	semidirect
72. #15636	D₄	semidirect
73. #15561	D₄	semidirect
74. #15467	D₄	semidirect
75. #15267	D₄	semidirect
76. #14681	D₄	semidirect
77. #14649	D₄	semidirect
78. #14609	D₄	semidirect
79. #14533	D₄	semidirect
80. #14507	D₄	semidirect
81. #14507	D₄	semidirect
83. #14207	D₄	semidirect
84. #13870	D₄	semidirect
85. #13870	D₄	semidirect
86. #13868	D₄	semidirect
92. #13827	D₄	semidirect
93. C₂²⋊C₄²	D₄	semidirect
100. C₂×#543	#41	semidirect
101. C₂×#534	#41	semidirect
123. #2273	#41	semidirect
124. #1819	#41	semidirect
125. C₂×#323	#41	semidirect

Maximal subgroups

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

subgroup	index
#72266274	2
#72266273	2
#72266264	2
#72266171	2
#72235928	2
#72235926	2
#72235924	2
#72235923	2
#72235919	2
#72235868	2
#72235820	2
#72235732	2
#72235729	2
#72235704	2
#72235690	2