Subgroups — #72321647

Order 4096.

At least 12389 conjugacy classes of subgroups; only the 265 normal subgroups are shown.

The subgroup lattice has 263 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72321572	C₂	1	1	yes	maximal, characteristic, complemented
2	#72321539	C₂	1	1	yes	maximal, characteristic, complemented
3	#72321499	C₂	1	1	yes	maximal, characteristic, complemented
4	#72319229	C₂	1	1	yes	maximal, characteristic, complemented
5	#72319196	C₂²	1	1	yes	characteristic
6	#72310595	C₂	1	1	yes	maximal, characteristic, complemented
7	#72237973	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237970	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237965	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237939	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237938	C₂²	1	1	yes	characteristic
12	#72237937	C₂	1	1	yes	maximal, characteristic, complemented
13	#72237935	C₂²	1	1	yes	characteristic
14	#72237880	C₂	1	1	yes	maximal, characteristic, complemented
15	#72237879	C₂²	1	1	yes	characteristic
16	#72237878	C₂²	1	1	yes	characteristic
17	#72237875	C₂²	1	1	yes	characteristic
18	#72237875	C₂²	1	1	yes	characteristic
19	#72237875	C₂²	1	1	yes	characteristic
20	#72237872	C₂²	1	1	yes	characteristic
21	#72237870	C₂²	1	1	yes	characteristic
22	#72237750	C₂	1	1	yes	maximal, characteristic, complemented
23	#72237749	C₂²	1	1	yes	characteristic
24	#72237748	C₂	1	1	yes	maximal, characteristic, complemented
25	#72237747	C₂²	1	1	yes	characteristic
26	#72237747	C₂²	1	1	yes	characteristic
27	#72237747	C₂²	1	1	yes	characteristic
28	#72237746	C₂²	1	1	yes	characteristic
29	#72237626	C₂	1	1	yes	maximal, characteristic, complemented
30	#72237625	C₂²	1	1	yes	characteristic
31	#72237624	C₂²	1	1	yes	characteristic
32	#72237619	C₂	1	1	yes	maximal, characteristic, complemented
33	#72237618	C₂²	1	1	yes	characteristic
34	#72237616	C₂²	1	1	yes	characteristic
35	#72237616	C₂²	1	1	yes	characteristic
36	#72237616	C₂²	1	1	yes	characteristic
37	#72237594	C₂²	1	1	yes	characteristic
38	#72237478	C₂²	1	1	yes	characteristic
39	#72237425	C₂²	1	1	yes	characteristic
40	#72237387	C₂²	1	1	yes	characteristic
41	#72234712	C₂²	1	1	yes	characteristic
42	#72234681	C₂²	1	1	yes	characteristic
43	#72234597	C₂²	1	1	yes	characteristic
44	#72232634	C₂²	1	1	yes	characteristic
45	#72231851	C₂²	1	1	yes	characteristic
46	#72231809	C₂²	1	1	yes	characteristic
47	#72231809	C₂²	1	1	yes	characteristic
48	#72231809	C₂²	1	1	yes	characteristic
49	#72225654	C₂²	1	1	yes	characteristic
50	#72219340	C₂²	1	1	yes	characteristic
51	C₂×#46996	C₂³	1	1	yes	characteristic
52	#7630944	D₄	1	1	yes	characteristic, complemented
53	C₂×#34981	C₂³	1	1	yes	characteristic
54	#7626202	D₄	1	1	yes	characteristic, complemented
55	#6451874	D₄	1	1	yes	characteristic, complemented
56	#6451874	C₂³	1	1	yes	characteristic
57	#6437621	D₄	1	1	yes	characteristic
58	C₂×#23900	D₄	1	1	yes	characteristic
59	#6374210	C₂³	1	1	yes	characteristic
60	C₂×#21746	D₄	1	1	yes	characteristic, complemented
61	#576189	C₂³	1	1	yes	characteristic
62	#480250	C₂³	1	1	yes	characteristic
63	#480250	C₂³	1	1	yes	characteristic
64	#480250	C₂³	1	1	yes	characteristic
65	#480250	C₂³	1	1	yes	characteristic
66	#453946	D₄	1	1	yes	characteristic, complemented
67	#453905	D₄	1	1	yes	characteristic, complemented
68	#449090	D₄	1	1	yes	characteristic, complemented
69	#439378	D₄	1	1	yes	characteristic, complemented
70	#439378	D₄	1	1	yes	characteristic, complemented
71	#436464	D₄	1	1	yes	characteristic, complemented
72	#432839	D₄	1	1	yes	characteristic, complemented
73	#418911	D₄	1	1	yes	characteristic, complemented
74	#392207	D₄	1	1	yes	characteristic, complemented
75	#391639	D₄	1	1	yes	characteristic, complemented
76	#391095	D₄	1	1	yes	characteristic, complemented
77	#388990	D₄	1	1	yes	characteristic, complemented
78	#388001	D₄	1	1	yes	characteristic, complemented
79	#388001	D₄	1	1	yes	characteristic, complemented
80	#382707	C₂³	1	1	yes	characteristic
81	#378118	D₄	1	1	yes	characteristic, complemented
82	#359921	D₄	1	1	yes	characteristic, complemented
83	#359921	D₄	1	1	yes	characteristic, complemented
84	#359705	D₄	1	1	yes	characteristic, complemented
85	#359705	C₂³	1	1	yes	characteristic
86	#359705	C₂³	1	1	yes	characteristic
87	#359705	C₂³	1	1	yes	characteristic
88	#359501	D₄	1	1	yes	characteristic
89	#359442	D₄	1	1	yes	characteristic
90	#357285	D₄	1	1	yes	characteristic, complemented
91	#356055	D₄	1	1	yes	characteristic, complemented
92	#190147	C₂³	1	1	yes	characteristic
93	#190116	C₂³	1	1	yes	characteristic
94	C₂×#3490	C₂×D₄	1	1	yes	characteristic
95	C₂×#3484	C₂×D₄	1	1	yes	characteristic
96	C₂²×#534	C₂×D₄	1	1	yes	characteristic
97	C₂³×#120	C₂×D₄	1	1	yes	characteristic
98	C₂³×#120	C₂×D₄	1	1	yes	characteristic
99	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
100	C₂×#2840	#41	1	1	yes	characteristic, complemented
101	C₂×#2403	#41	1	1	yes	characteristic, complemented
102	#34176	C₂×D₄	1	1	yes	characteristic
103	#34176	C₂×D₄	1	1	yes	characteristic
104	#34176	C₂×D₄	1	1	yes	characteristic
105	#34171	C₂×D₄	1	1	yes	characteristic
106	C₂×#2315	C₂×D₄	1	1	yes	characteristic
107	C₂×#2308	C₂×D₄	1	1	yes	characteristic
108	C₂×#2308	C₂⁴	1	1	yes	characteristic
109	#23468	C₂×D₄	1	1	yes	characteristic
110	#23468	C₂×D₄	1	1	yes	characteristic
111	#23444	C₂×D₄	1	1	yes	characteristic
112	#23444	C₂×D₄	1	1	yes	characteristic
113	#20689	C₂×D₄	1	1	yes	characteristic
114	#20689	C₂×D₄	1	1	yes	characteristic
115	C₂×#1819	C₂×D₄	1	1	yes	characteristic
116	C₂×#1819	C₂×D₄	1	1	yes	characteristic
117	C₂×#1819	C₂×D₄	1	1	yes	characteristic
118	C₂×#1819	C₂×D₄	1	1	yes	characteristic
119	#20585	C₂×D₄	1	1	yes	characteristic
120	#20585	C₂×D₄	1	1	yes	characteristic
121	C₂×#1781	C₂×D₄	1	1	yes	characteristic
122	C₂×#1781	C₂×D₄	1	1	yes	characteristic
123	#13828	#41	1	1	yes	characteristic, complemented
124	#13807	#41	1	1	yes	characteristic, complemented
125	#11426	#41	1	2	yes	complemented
126	#11166	C₂×D₄	1	1	yes	characteristic
127	#11166	C₂×D₄	1	1	yes	characteristic
128	#11166	C₂×D₄	1	1	yes	characteristic
129	#11166	C₂×D₄	1	1	yes	characteristic
130	#10938	C₂×D₄	1	1	yes	characteristic
131	#10938	C₂×D₄	1	1	yes	characteristic
132	#10938	C₂×D₄	1	1	yes	characteristic
133	#10938	C₂×D₄	1	1	yes	characteristic
134	#10938	C₂×D₄	1	1	yes	characteristic
135	#10938	C₂×D₄	1	1	yes	characteristic
136	#10938	C₂×D₄	1	1	yes	characteristic
137	#10938	C₂×D₄	1	1	yes	characteristic
138	#10938	C₂×D₄	1	1	yes	characteristic
139	#10938	C₂×D₄	1	1	yes	characteristic
140	#10938	C₂×D₄	1	1	yes	characteristic
141	#10938	C₂×D₄	1	1	yes	characteristic
142	#8747	#41	1	2	yes	complemented
143	C₂⁷	#120	1	1	yes	characteristic
144	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
145	C₂⁴×D₄	#120	1	1	yes	characteristic
146	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
147	C₂×#543	#120	1	1	yes	characteristic
148	C₂×#534	#120	1	1	yes	characteristic
149	C₂×#518	#120	1	1	yes	characteristic
150	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
151	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
152	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
153	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
154	C₂²×#99	#120	1	1	yes	characteristic
155	C₂²×#99	#120	1	1	yes	characteristic
156	C₂²×#99	#120	1	1	yes	characteristic
157	C₂²×#99	#120	1	1	yes	characteristic
158	#2308	#120	1	1	yes	characteristic
159	#2308	#120	1	1	yes	characteristic
160	#2308	C₂²×D₄	1	1	yes	characteristic
161	#2308	C₂²×D₄	1	1	yes	characteristic
162	#2308	C₂²×D₄	1	1	yes	characteristic
163	#2308	C₂²×D₄	1	1	yes	characteristic
164	#2308	C₂×#41	1	1	yes	characteristic
165	#2308	C₂×#41	1	1	yes	characteristic
166	#2308	H(5,𝔽₂)	1	1	yes	characteristic
167	#2308	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂×#381	C₂×#41	1	1	yes	characteristic
169	C₂×#381	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×#379	C₂×#41	1	1	yes	characteristic
171	C₂×#379	H(5,𝔽₂)	1	1	yes	characteristic
172	#2273	#120	1	1	yes	characteristic
173	#2273	#120	1	1	yes	characteristic
174	#2273	#120	1	1	yes	characteristic
175	#2273	#120	1	1	yes	characteristic
176	#2273	#120	1	1	yes	characteristic
177	#2273	#120	1	1	yes	characteristic
178	C₂×#342	#120	1	1	yes	characteristic
179	C₂×#342	#120	1	1	yes	characteristic
180	C₂×#342	#120	1	1	yes	characteristic
181	C₂×#342	#120	1	1	yes	characteristic
182	C₂×#342	#120	1	1	yes	characteristic
183	C₂×#342	#120	1	1	yes	characteristic
184	C₂×#342	#120	1	1	yes	characteristic
185	C₂×#342	#120	1	1	yes	characteristic
186	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
187	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
188	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
189	C₂⁶	C₂×#120	1	1	yes	characteristic
190	C₂⁶	#546	1	1	yes	characteristic
191	C₂³×D₄	C₂×#120	1	1	yes	characteristic
192	C₂³×D₄	C₂×#120	1	1	yes	characteristic
193	C₂³×D₄	#546	1	1	yes	characteristic
194	C₂³×D₄	#547	1	1	yes	characteristic
195	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
196	C₂⁴×C₄	#546	1	1	yes	characteristic
197	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
198	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
199	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
200	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
201	C₂²×C₂²⋊C₄	#525	1	1	yes	characteristic
202	C₂²×C₂²⋊C₄	#532	1	1	yes	characteristic
203	C₂²×C₂²⋊C₄	#535	1	1	yes	characteristic
204	C₂²×C₂²⋊C₄	#537	1	1	yes	characteristic
205	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
206	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
207	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
208	C₂²×C₂²⋊C₄	#547	1	1	yes	characteristic
209	#381	#532	1	1	yes	characteristic
210	#381	#532	1	1	yes	characteristic
211	#381	#535	1	1	yes	characteristic
212	#381	#537	1	1	yes	characteristic
213	#381	#546	1	1	yes	characteristic
214	#381	#547	1	1	yes	characteristic
215	#381	#550	1	1	yes	characteristic
216	#381	#550	1	1	yes	characteristic
217	#379	#525	1	1	yes	characteristic
218	#379	#525	1	1	yes	characteristic
219	#379	#535	1	1	yes	characteristic
220	#379	#537	1	1	yes	characteristic
221	#379	#546	1	1	yes	characteristic
222	#379	#546	1	1	yes	characteristic
223	#379	#548	1	1	yes	characteristic
224	#379	#550	1	1	yes	characteristic
225	C₂⁵	#2404	1	1	yes	characteristic
226	C₂⁵	#2404	1	1	yes	characteristic
227	C₂⁵	#2407	1	1	yes	characteristic
228	C₂⁵	#2430	1	1	yes	characteristic
229	C₂⁵	#2846	1	1	yes	characteristic
230	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
231	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
232	C₂⁵	#3026	1	1	yes	characteristic
233	C₂⁵	#3026	1	1	yes	characteristic
234	C₂³×C₄	#2430	1	1	yes	characteristic
235	C₂³×C₄	#2431	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	#2388	1	1	yes	characteristic
237	C₂×C₂²⋊C₄	#2404	1	1	yes	characteristic
238	C₂×C₂²⋊C₄	#2407	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	#2407	1	1	yes	characteristic
240	C₂×C₂²⋊C₄	#2430	1	1	yes	characteristic
241	C₂×C₂²⋊C₄	#2431	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	#2431	1	1	yes	characteristic
243	C₂×C₂²⋊C₄	#2615	1	1	yes	characteristic
244	C₂⁴	#14885	1	1	yes	characteristic
245	C₂⁴	#23984	1	1	yes	characteristic
246	C₂⁴	#23987	1	1	yes	characteristic
247	C₂⁴	#25563	1	1	yes	characteristic
248	C₂⁴	#25563	1	1	yes	characteristic
249	C₂⁴	#32898	1	1	yes	characteristic
250	C₂⁴	#32898	1	1	yes	characteristic
251	C₂³	#403952	1	1	yes	characteristic
252	C₂³	#404012	1	1	yes	characteristic
253	C₂³	#404013	1	1	yes	characteristic
254	C₂³	#511790	1	1	yes	characteristic
255	C₂³	#511790	1	1	yes	characteristic
256	C₂³	#511791	1	1	yes	characteristic
257	C₂³	#511793	1	1	yes	characteristic
258	C₂²	#72233141	1	1	yes	characteristic
259	C₂²	#72240122	1	1	yes	characteristic
260	C₂²	#72255873	1	1	yes	characteristic
261	C₂	#72266275	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266275
commutator, Frattini, cosocle	C₂×#2308	C₂⁴
LCS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34981 ◁ G
LECS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72321572	C₂	semidirect
2. #72321539	C₂	semidirect
3. #72321499	C₂	semidirect
4. #72319229	C₂	semidirect
6. #72310595	C₂	semidirect
7. #72237973	C₂	semidirect
8. #72237970	C₂	semidirect
9. #72237965	C₂	semidirect
10. #72237939	C₂	semidirect
12. #72237937	C₂	semidirect
14. #72237880	C₂	semidirect
22. #72237750	C₂	semidirect
24. #72237748	C₂	semidirect
29. #72237626	C₂	semidirect
32. #72237619	C₂	semidirect
52. #7630944	D₄	semidirect
54. #7626202	D₄	semidirect
55. #6451874	D₄	semidirect
60. C₂×#21746	D₄	semidirect
66. #453946	D₄	semidirect
67. #453905	D₄	semidirect
68. #449090	D₄	semidirect
69. #439378	D₄	semidirect
70. #439378	D₄	semidirect
71. #436464	D₄	semidirect
72. #432839	D₄	semidirect
73. #418911	D₄	semidirect
74. #392207	D₄	semidirect
75. #391639	D₄	semidirect
76. #391095	D₄	semidirect
77. #388990	D₄	semidirect
78. #388001	D₄	semidirect
79. #388001	D₄	semidirect
81. #378118	D₄	semidirect
82. #359921	D₄	semidirect
83. #359921	D₄	semidirect
84. #359705	D₄	semidirect
90. #357285	D₄	semidirect
91. #356055	D₄	semidirect
100. C₂×#2840	#41	semidirect
101. C₂×#2403	#41	semidirect
123. #13828	#41	semidirect
124. #13807	#41	semidirect
125. #11426	#41	semidirect
142. #8747	#41	semidirect

Maximal subgroups

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

subgroup	index
#72321572	2
#72321539	2
#72321499	2
#72319229	2
#72310595	2
#72237973	2
#72237970	2
#72237965	2
#72237939	2
#72237937	2
#72237880	2
#72237750	2
#72237748	2
#72237626	2
#72237619	2