Subgroups — #72318023

Order 4096.

At least 16701 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

The subgroup lattice has 293 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72317935	C₂	1	1	yes	maximal, characteristic, complemented
2	#72317123	C₂	1	1	yes	maximal, characteristic, complemented
3	#72315084	C₂	1	1	yes	maximal, characteristic, complemented
4	#72312328	C₂	1	1	yes	maximal, characteristic, complemented
5	#72312241	C₂	1	1	yes	maximal, characteristic, complemented
6	#72310297	C₂	1	1	yes	maximal, characteristic, complemented
7	#72309583	C₂	1	1	yes	maximal, characteristic, complemented
8	#72309583	C₂	1	1	yes	maximal, characteristic, complemented
9	#72309574	C₂	1	1	yes	maximal, characteristic, complemented
10	#72309472	C₂	1	1	yes	maximal, characteristic, complemented
11	#72309452	C₂	1	1	yes	maximal, characteristic, complemented
12	#72262349	C₂	1	1	yes	maximal, characteristic, complemented
13	#72258993	C₂	1	1	yes	maximal, characteristic, complemented
14	#72251523	C₂	1	1	yes	maximal, characteristic, complemented
15	#72251479	C₂	1	1	yes	maximal, characteristic, complemented
16	#72237398	C₂²	1	1	yes	characteristic
17	#72237398	C₂²	1	1	yes	characteristic
18	#72235847	C₂²	1	1	yes	characteristic
19	#72234668	C₂²	1	1	yes	characteristic
20	#72225667	C₂²	1	1	yes	characteristic
21	#72225650	C₂²	1	1	yes	characteristic
22	#72225650	C₂²	1	1	yes	characteristic
23	#72225619	C₂²	1	1	yes	characteristic
24	#72225617	C₂²	1	1	yes	characteristic
25	#72225617	C₂²	1	1	yes	characteristic
26	#72219418	C₂²	1	1	yes	characteristic
27	#72219415	C₂²	1	1	yes	characteristic
28	#72219414	C₂²	1	1	yes	characteristic
29	#72219414	C₂²	1	1	yes	characteristic
30	#72219300	C₂²	1	1	yes	characteristic
31	#72219300	C₂²	1	1	yes	characteristic
32	#72219257	C₂²	1	1	yes	characteristic
33	#72219251	C₂²	1	1	yes	characteristic
34	#72219248	C₂²	1	1	yes	characteristic
35	#72219243	C₂²	1	1	yes	characteristic
36	#72219237	C₂²	1	1	yes	characteristic
37	#72219228	C₂²	1	1	yes	characteristic
38	#72219228	C₂²	1	1	yes	characteristic
39	#72218498	C₂²	1	1	yes	characteristic
40	#72218498	C₂²	1	1	yes	characteristic
41	#72218455	C₂²	1	1	yes	characteristic
42	#72218455	C₂²	1	1	yes	characteristic
43	#72218306	C₂²	1	1	yes	characteristic
44	#72218198	C₂²	1	1	yes	characteristic
45	#72218198	C₂²	1	1	yes	characteristic
46	#72218180	C₂²	1	1	yes	characteristic
47	#72218180	C₂²	1	1	yes	characteristic
48	#72213862	C₂²	1	1	yes	characteristic
49	#72210555	C₂²	1	1	yes	characteristic
50	#72210511	C₂²	1	1	yes	characteristic
51	C₂²×#3484	C₂³	1	1	yes	characteristic
52	#7630939	C₂³	1	1	yes	characteristic
53	#7630939	C₂³	1	1	yes	characteristic
54	C₂×#34980	C₂³	1	1	yes	characteristic
55	#7626202	D₄	1	1	yes	characteristic, complemented
56	C₂×#32873	D₄	1	1	yes	characteristic, complemented
57	#6440460	D₄	1	1	yes	characteristic
58	#6440460	D₄	1	1	yes	characteristic, complemented
59	C₂×#23895	D₄	1	1	yes	characteristic, complemented
60	#6437621	D₄	1	1	yes	characteristic
61	#6437621	D₄	1	1	yes	characteristic, complemented
62	C₂×#21776	D₄	1	1	yes	characteristic, complemented
63	C₂×#21776	D₄	1	1	yes	characteristic, complemented
64	C₂×#21746	D₄	1	1	yes	characteristic, complemented
65	C₂²×#1781	C₂³	1	1	yes	characteristic
66	C₂²×#1781	C₂³	1	1	yes	characteristic
67	#517411	C₂³	1	1	yes	characteristic
68	#517411	C₂³	1	1	yes	characteristic
69	C₂²×#1438	C₂³	1	1	yes	characteristic
70	#392113	C₂³	1	1	yes	characteristic
71	#392113	C₂³	1	1	yes	characteristic
72	#392109	C₂³	1	1	yes	characteristic
73	#392109	C₂³	1	1	yes	characteristic
74	#392109	C₂³	1	1	yes	characteristic
75	#392109	C₂³	1	1	yes	characteristic
76	#392030	D₄	1	1	yes	characteristic, complemented
77	#392030	D₄	1	1	yes	characteristic, complemented
78	#392030	D₄	1	1	yes	characteristic, complemented
79	#382544	D₄	1	1	yes	characteristic, complemented
80	#382544	D₄	1	1	yes	characteristic, complemented
81	#382434	D₄	1	1	yes	characteristic, complemented
82	#382434	D₄	1	1	yes	characteristic, complemented
83	#373637	D₄	1	1	yes	characteristic, complemented
84	#359811	D₄	1	1	yes	characteristic
85	#359811	D₄	1	1	yes	characteristic, complemented
86	#359549	D₄	1	1	yes	characteristic, complemented
87	#359549	D₄	1	1	yes	characteristic, complemented
88	#359442	D₄	1	1	yes	characteristic
89	#359442	D₄	1	1	yes	characteristic, complemented
90	#355913	D₄	1	1	yes	characteristic, complemented
91	#355913	D₄	1	1	yes	characteristic, complemented
92	#355096	D₄	1	1	yes	characteristic, complemented
93	#355096	D₄	1	1	yes	characteristic, complemented
94	C₂³×H(5,𝔽₂)	#41	1	1	yes	characteristic, complemented
95	C₂⁵×D₄	C₂⁴	1	1	yes	characteristic
96	C₂×#3484	C₂×D₄	1	1	yes	characteristic
97	C₂×#3484	C₂×D₄	1	1	yes	characteristic
98	C₂×#3484	C₂×D₄	1	1	yes	characteristic
99	C₂×#3484	C₂×D₄	1	1	yes	characteristic
100	C₂×#3484	C₂×D₄	1	1	yes	characteristic
101	C₂×#3484	C₂×D₄	1	1	yes	characteristic
102	C₂²×#534	C₂×D₄	1	1	yes	characteristic
103	C₂²×#534	C₂×D₄	1	1	yes	characteristic
104	C₂³×#120	C₂×D₄	1	1	yes	characteristic
105	C₂²×#518	#41	1	1	yes	characteristic, complemented
106	#34171	C₂×D₄	1	1	yes	characteristic
107	#34171	C₂×D₄	1	1	yes	characteristic
108	#34171	C₂×D₄	1	1	yes	characteristic
109	#34171	C₂×D₄	1	1	yes	characteristic
110	#34171	C₂×D₄	1	1	yes	characteristic
111	#34171	C₂×D₄	1	1	yes	characteristic
112	C₂×#2308	C₂×D₄	1	1	yes	characteristic
113	C₂×#2308	C₂×D₄	1	1	yes	characteristic
114	C₂²×#379	C₂×D₄	1	1	yes	characteristic
115	C₂²×#379	C₂×D₄	1	1	yes	characteristic
116	C₂²×#379	C₂×D₄	1	1	yes	characteristic
117	C₂²×#379	C₂×D₄	1	1	yes	characteristic
118	C₂²×#379	C₂×D₄	1	1	yes	characteristic
119	C₂×#1783	#41	1	2	yes	complemented
120	C₂×#1781	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	C₂×#1781	C₂×D₄	1	1	yes	characteristic
124	C₂×#1781	C₂×D₄	1	1	yes	characteristic
125	C₂×#1781	C₂×D₄	1	1	yes	characteristic
126	C₂×#1781	C₂×D₄	1	1	yes	characteristic
127	C₂×#1781	C₂×D₄	1	1	yes	characteristic
128	C₂×#1781	C₂×D₄	1	1	yes	characteristic
129	C₂×#1781	C₂×D₄	1	1	yes	characteristic
130	C₂×#1781	C₂×D₄	1	1	yes	characteristic
131	C₂×#1781	C₂×D₄	1	1	yes	characteristic
132	C₂×#1781	C₂×D₄	1	1	yes	characteristic
133	C₂×#1781	C₂×D₄	1	1	yes	characteristic
134	C₂×#1781	C₂×D₄	1	1	yes	characteristic
135	C₂×#1781	C₂×D₄	1	1	yes	characteristic
136	C₂×#1781	C₂×D₄	1	1	yes	characteristic
137	C₂×#1781	C₂×D₄	1	1	yes	characteristic
138	C₂×#1781	C₂×D₄	1	1	yes	characteristic
139	C₂×#1781	C₂×D₄	1	1	yes	characteristic
140	C₂×#1781	#41	1	2	yes	complemented
141	C₂×#1438	#41	1	2	yes	complemented
142	C₂⁷	C₂²×D₄	1	1	yes	characteristic
143	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
145	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
146	C₂⁴×D₄	#120	1	1	yes	characteristic
147	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
148	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
149	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
150	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
151	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
152	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
153	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
154	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
155	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
156	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
157	C₂⁴×D₄	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂⁴×D₄	2₋¹⁺⁴	1	1	yes	characteristic
159	C₂⁵×C₄	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×#518	#120	1	1	yes	characteristic
161	C₂×#518	#120	1	1	yes	characteristic
162	C₂×#518	#120	1	1	yes	characteristic
163	C₂×#518	#120	1	1	yes	characteristic
164	C₂×#518	#120	1	1	yes	characteristic
165	C₂×#518	#120	1	1	yes	characteristic
166	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
167	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
168	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
169	C₂²×#99	#120	1	1	yes	characteristic
170	C₂²×#99	#120	1	1	yes	characteristic
171	C₂²×#99	#120	1	1	yes	characteristic
172	C₂²×#99	#120	1	1	yes	characteristic
173	C₂²×#99	#120	1	1	yes	characteristic
174	C₂²×#99	#120	1	1	yes	characteristic
175	C₂²×#99	#120	1	1	yes	characteristic
176	C₂²×#99	#120	1	1	yes	characteristic
177	C₂×#379	#120	1	1	yes	characteristic
178	C₂×#379	#120	1	1	yes	characteristic
179	C₂×#379	#120	1	1	yes	characteristic
180	C₂×#379	#120	1	1	yes	characteristic
181	C₂×#379	#120	1	1	yes	characteristic
182	C₂×#379	#120	1	1	yes	characteristic
183	C₂×#379	#120	1	1	yes	characteristic
184	C₂×#379	#120	1	1	yes	characteristic
185	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
186	C₂⁶	C₂×#120	1	1	yes	characteristic
187	C₂⁶	C₂×#120	1	1	yes	characteristic
188	C₂⁶	C₂×#120	1	1	yes	characteristic
189	C₂⁶	#525	1	1	yes	characteristic
190	C₂⁶	#532	1	1	yes	characteristic
191	C₂⁶	#535	1	1	yes	characteristic
192	C₂⁶	#535	1	1	yes	characteristic
193	C₂⁶	#537	1	1	yes	characteristic
194	C₂⁶	#537	1	1	yes	characteristic
195	C₂⁶	#546	1	1	yes	characteristic
196	C₂⁶	#546	1	1	yes	characteristic
197	C₂⁶	#546	1	1	yes	characteristic
198	C₂⁶	#550	1	1	yes	characteristic
199	C₂³×D₄	C₂×#120	1	1	yes	characteristic
200	C₂³×D₄	C₂×#120	1	1	yes	characteristic
201	C₂³×D₄	C₂×#120	1	1	yes	characteristic
202	C₂³×D₄	C₂×#120	1	1	yes	characteristic
203	C₂³×D₄	#525	1	1	yes	characteristic
204	C₂³×D₄	#525	1	1	yes	characteristic
205	C₂³×D₄	#532	1	1	yes	characteristic
206	C₂³×D₄	#532	1	1	yes	characteristic
207	C₂³×D₄	#546	1	1	yes	characteristic
208	C₂³×D₄	#546	1	1	yes	characteristic
209	C₂³×D₄	#546	1	1	yes	characteristic
210	C₂³×D₄	#547	1	1	yes	characteristic
211	C₂³×D₄	#547	1	1	yes	characteristic
212	C₂³×D₄	#547	1	1	yes	characteristic
213	C₂³×D₄	#548	1	1	yes	characteristic
214	C₂³×D₄	#550	1	1	yes	characteristic
215	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
216	C₂⁴×C₄	#535	1	1	yes	characteristic
217	C₂⁴×C₄	#537	1	1	yes	characteristic
218	C₂⁴×C₄	#546	1	1	yes	characteristic
219	C₂⁴×C₄	#546	1	1	yes	characteristic
220	C₂⁴×C₄	#546	1	1	yes	characteristic
221	C₂⁴×C₄	#550	1	1	yes	characteristic
222	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
223	#379	B(4,𝔽₂)	1	2	yes	—
224	#379	B(4,𝔽₂)	1	2	yes	—
225	C₂⁵	#2388	1	1	yes	characteristic
226	C₂⁵	#2404	1	1	yes	characteristic
227	C₂⁵	#2404	1	1	yes	characteristic
228	C₂⁵	#2407	1	1	yes	characteristic
229	C₂⁵	#2407	1	1	yes	characteristic
230	C₂⁵	#2430	1	1	yes	characteristic
231	C₂⁵	#2430	1	1	yes	characteristic
232	C₂⁵	#2431	1	1	yes	characteristic
233	C₂⁵	#2431	1	1	yes	characteristic
234	C₂⁵	#2615	1	1	yes	characteristic
235	C₂⁵	#2846	1	1	yes	characteristic
236	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
237	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
238	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
239	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
240	C₂⁵	#3025	1	1	yes	characteristic
241	C₂⁵	#3025	1	1	yes	characteristic
242	C₂⁵	#3026	1	1	yes	characteristic
243	C₂⁵	#3026	1	1	yes	characteristic
244	C₂⁵	#3026	1	1	yes	characteristic
245	C₂⁵	#3026	1	1	yes	characteristic
246	C₂⁵	#3027	1	1	yes	characteristic
247	C₂⁵	#3027	1	1	yes	characteristic
248	C₂³×C₄	#2404	1	1	yes	characteristic
249	C₂³×C₄	#2407	1	1	yes	characteristic
250	C₂³×C₄	#2430	1	1	yes	characteristic
251	C₂³×C₄	#2431	1	1	yes	characteristic
252	C₂⁴	#14885	1	1	yes	characteristic
253	C₂⁴	#23595	1	1	yes	characteristic
254	C₂⁴	#23597	1	1	yes	characteristic
255	C₂⁴	#23984	1	1	yes	characteristic
256	C₂⁴	#23987	1	1	yes	characteristic
257	C₂⁴	#23996	1	2	yes	—
258	C₂⁴	#24140	1	1	yes	characteristic
259	C₂⁴	#24142	1	1	yes	characteristic
260	C₂⁴	#25563	1	1	yes	characteristic
261	C₂⁴	#25563	1	1	yes	characteristic
262	C₂⁴	#25585	1	1	yes	characteristic
263	C₂⁴	#25587	1	1	yes	characteristic
264	C₂⁴	#29684	1	1	yes	characteristic
265	C₂⁴	#29686	1	1	yes	characteristic
266	C₂⁴	#32898	1	1	yes	characteristic
267	C₂⁴	#32898	1	1	yes	characteristic
268	C₂⁴	#32898	1	1	yes	characteristic
269	C₂⁴	#32898	1	1	yes	characteristic
270	C₂⁴	#32899	1	1	yes	characteristic
271	C₂⁴	#32899	1	1	yes	characteristic
272	C₂²×C₄	#23996	1	2	yes	—
273	C₂³	#347042	1	1	yes	characteristic
274	C₂³	#347043	1	1	yes	characteristic
275	C₂³	#347043	1	1	yes	characteristic
276	C₂³	#347044	1	1	yes	characteristic
277	C₂³	#403939	1	1	yes	characteristic
278	C₂³	#403941	1	1	yes	characteristic
279	C₂³	#403943	1	1	yes	characteristic
280	C₂³	#403943	1	1	yes	characteristic
281	C₂³	#403952	1	1	yes	characteristic
282	C₂³	#404015	1	1	yes	characteristic
283	C₂³	#404015	1	1	yes	characteristic
284	C₂²	#72232805	1	1	yes	characteristic
285	C₂²	#72234956	1	1	yes	characteristic
286	C₂²	#72234956	1	1	yes	characteristic
287	C₂²	#72234958	1	1	yes	characteristic
288	C₂²	#72234958	1	1	yes	characteristic
289	C₂²	#72258395	1	1	yes	characteristic
290	C₂²	#72265945	1	1	yes	characteristic
291	C₂	#72266256	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266256
commutator, Frattini, cosocle	C₂⁵×D₄	C₂⁴
LCS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72317935	C₂	semidirect
2. #72317123	C₂	semidirect
3. #72315084	C₂	semidirect
4. #72312328	C₂	semidirect
5. #72312241	C₂	semidirect
6. #72310297	C₂	semidirect
7. #72309583	C₂	semidirect
8. #72309583	C₂	semidirect
9. #72309574	C₂	semidirect
10. #72309472	C₂	semidirect
11. #72309452	C₂	semidirect
12. #72262349	C₂	semidirect
13. #72258993	C₂	semidirect
14. #72251523	C₂	semidirect
15. #72251479	C₂	semidirect
55. #7626202	D₄	semidirect
56. C₂×#32873	D₄	semidirect
58. #6440460	D₄	semidirect
59. C₂×#23895	D₄	semidirect
61. #6437621	D₄	semidirect
62. C₂×#21776	D₄	semidirect
63. C₂×#21776	D₄	semidirect
64. C₂×#21746	D₄	semidirect
76. #392030	D₄	semidirect
77. #392030	D₄	semidirect
78. #392030	D₄	semidirect
79. #382544	D₄	semidirect
80. #382544	D₄	semidirect
81. #382434	D₄	semidirect
82. #382434	D₄	semidirect
83. #373637	D₄	semidirect
85. #359811	D₄	semidirect
86. #359549	D₄	semidirect
87. #359549	D₄	semidirect
89. #359442	D₄	semidirect
90. #355913	D₄	semidirect
91. #355913	D₄	semidirect
92. #355096	D₄	semidirect
93. #355096	D₄	semidirect
94. C₂³×H(5,𝔽₂)	#41	semidirect
105. C₂²×#518	#41	semidirect
119. C₂×#1783	#41	semidirect
140. C₂×#1781	#41	semidirect
141. C₂×#1438	#41	semidirect

Maximal subgroups

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

subgroup	index
#72317935	2
#72317123	2
#72315084	2
#72312328	2
#72312241	2
#72310297	2
#72309583	2
#72309574	2
#72309472	2
#72309452	2
#72262349	2
#72258993	2
#72251523	2
#72251479	2