Subgroups — #72323213

Order 4096.

At least 20640 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

The subgroup lattice has 297 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72315164	C₂	1	1	yes	maximal, characteristic, complemented
2	#72313018	C₂	1	1	yes	maximal, characteristic, complemented
3	#72312258	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310339	C₂	1	1	yes	maximal, characteristic, complemented
5	#72310337	C₂	1	1	yes	maximal, characteristic, complemented
6	#72310334	C₂	1	1	yes	maximal, characteristic, complemented
7	#72309737	C₂	1	1	yes	maximal, characteristic, complemented
8	#72266073	C₂²	1	1	yes	characteristic
9	#72265757	C₂	1	1	yes	maximal, characteristic, complemented
10	#72262328	C₂	1	1	yes	maximal, characteristic, complemented
11	#72259008	C₂²	1	1	yes	characteristic
12	#72238345	C₂	1	1	yes	maximal, characteristic, complemented
13	#72237400	C₂²	1	1	yes	characteristic
14	#72234661	C₂²	1	1	yes	characteristic
15	#72225729	C₂	1	1	yes	maximal, characteristic, complemented
16	#72225713	C₂²	1	1	yes	characteristic
17	#72225667	C₂²	1	1	yes	characteristic
18	#72225650	C₂²	1	1	yes	characteristic
19	#72225617	C₂²	1	1	yes	characteristic
20	#72219455	C₂	1	1	yes	maximal, characteristic, complemented
21	#72219418	C₂²	1	1	yes	characteristic
22	#72219414	C₂²	1	1	yes	characteristic
23	#72219311	C₂²	1	1	yes	characteristic
24	#72219300	C₂²	1	1	yes	characteristic
25	C₂×#394058	C₂²	1	1	yes	characteristic
26	#72218868	C₂	1	1	yes	maximal, characteristic, complemented
27	#72218847	C₂	1	1	yes	maximal, characteristic, complemented
28	#72218518	C₂²	1	1	yes	characteristic
29	#72218393	C₂	1	1	yes	maximal, characteristic, complemented
30	#72218381	C₂²	1	1	yes	characteristic
31	#72218381	C₂²	1	1	yes	characteristic
32	#72218381	C₂²	1	1	yes	characteristic
33	#72218381	C₂²	1	1	yes	characteristic
34	#72218381	C₂²	1	1	yes	characteristic
35	#72218381	C₂²	1	1	yes	characteristic
36	#72218381	C₂²	1	1	yes	characteristic
37	#72218375	C₂²	1	1	yes	characteristic
38	#72218313	C₂²	1	1	yes	characteristic
39	#72218313	C₂²	1	1	yes	characteristic
40	#72218306	C₂²	1	1	yes	characteristic
41	#72218069	C₂²	1	1	yes	characteristic
42	#72218057	C₂²	1	1	yes	characteristic
43	#72218057	C₂²	1	1	yes	characteristic
44	C₂²×#23900	C₂²	1	1	yes	characteristic
45	#72217789	C₂²	1	1	yes	characteristic
46	#72215858	C₂²	1	1	yes	characteristic
47	#72215179	C₂²	1	1	yes	characteristic
48	#72213862	C₂²	1	1	yes	characteristic
49	#72210555	C₂²	1	1	yes	characteristic
50	#72210555	C₂²	1	1	yes	characteristic
51	C₂²×#3484	C₂³	1	1	yes	characteristic
52	C₂⁴×#120	C₂³	1	1	yes	characteristic
53	C₂²×#2846	D₄	1	1	yes	characteristic, complemented
54	#7630939	C₂³	1	1	yes	characteristic
55	C₂×#34980	C₂³	1	1	yes	characteristic
56	C₂×#32873	D₄	1	1	yes	characteristic, complemented
57	#6440460	D₄	1	1	yes	characteristic, complemented
58	#6437621	D₄	1	1	yes	characteristic, complemented
59	C₂×#23900	D₄	1	1	yes	characteristic, complemented
60	C₂×#23900	D₄	1	1	yes	characteristic, complemented
61	#6374251	D₄	1	1	yes	characteristic, complemented
62	C₂×#21776	D₄	1	1	yes	characteristic, complemented
63	#6374128	D₄	1	1	yes	characteristic, complemented
64	C₂×#21746	D₄	1	1	yes	characteristic, complemented
65	C₂×#21746	D₄	1	1	yes	characteristic, complemented
66	C₂²×#1898	D₄	1	1	yes	characteristic, complemented
67	C₂²×#1781	C₂³	1	1	yes	characteristic
68	C₂²×#1781	C₂³	1	1	yes	characteristic
69	C₂²×#1781	C₂³	1	1	yes	characteristic
70	C₂²×#1781	C₂³	1	1	yes	characteristic
71	C₂×#14679	D₄	1	1	yes	characteristic, complemented
72	#392113	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	#392109	C₂³	1	1	yes	characteristic
77	#392109	C₂³	1	1	yes	characteristic
78	#392109	C₂³	1	1	yes	characteristic
79	#392030	D₄	1	1	yes	characteristic, complemented
80	#392030	D₄	1	1	yes	characteristic, complemented
81	#392030	D₄	1	1	yes	characteristic, complemented
82	#382707	D₄	1	1	yes	characteristic, complemented
83	#382707	D₄	1	1	yes	characteristic, complemented
84	#382544	D₄	1	1	yes	characteristic, complemented
85	#382434	D₄	1	1	yes	characteristic, complemented
86	#382422	D₄	1	1	yes	characteristic, complemented
87	#382422	D₄	1	1	yes	characteristic, complemented
88	#359811	D₄	1	1	yes	characteristic, complemented
89	#359501	D₄	1	1	yes	characteristic, complemented
90	#359501	D₄	1	1	yes	characteristic, complemented
91	#359442	D₄	1	1	yes	characteristic, complemented
92	#355096	D₄	1	1	yes	characteristic, complemented
93	C₂×#10735	D₄	1	1	yes	characteristic, complemented
94	C₂⁸	C₂×D₄	1	1	yes	characteristic
95	C₂³×H(5,𝔽₂)	#41	1	1	yes	characteristic, complemented
96	C₂⁵×D₄	C₂⁴	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	C₂²×#534	C₂×D₄	1	1	yes	characteristic
101	C₂²×#534	C₂×D₄	1	1	yes	characteristic
102	C₂³×#120	C₂×D₄	1	1	yes	characteristic
103	C₂³×#120	C₂×D₄	1	1	yes	characteristic
104	C₂³×#120	C₂×D₄	1	1	yes	characteristic
105	C₂³×#120	C₂×D₄	1	1	yes	characteristic
106	C₂³×#120	C₂×D₄	1	1	yes	characteristic
107	C₂²×#518	#41	1	1	yes	characteristic, complemented
108	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
109	C₂³×#99	#41	1	2	yes	complemented
110	#34171	C₂×D₄	1	1	yes	characteristic
111	#34171	C₂×D₄	1	1	yes	characteristic
112	#34171	C₂×D₄	1	1	yes	characteristic
113	C₂×#2308	C₂×D₄	1	1	yes	characteristic
114	C₂×#2308	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	C₂×#1781	C₂×D₄	1	1	yes	characteristic
119	C₂×#1781	C₂×D₄	1	1	yes	characteristic
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	#14679	#41	1	2	yes	complemented
142	C₂⁷	#120	1	1	yes	characteristic
143	C₂⁷	#120	1	1	yes	characteristic
144	C₂⁷	C₂²×D₄	1	1	yes	characteristic
145	C₂⁷	C₂×#41	1	1	yes	characteristic
146	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
147	C₂⁴×D₄	#120	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	C₂²×D₄	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	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂⁴×D₄	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂⁵×C₄	2₋¹⁺⁴	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	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
165	C₂³×C₂²⋊C₄	#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	C₂²×#99	#120	1	1	yes	characteristic
178	C₂²×#99	#120	1	1	yes	characteristic
179	C₂²×#99	#120	1	1	yes	characteristic
180	C₂²×#99	#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	2	yes	—
186	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
187	C₂⁶	C₂×#120	1	1	yes	characteristic
188	C₂⁶	C₂×#120	1	1	yes	characteristic
189	C₂⁶	C₂×#120	1	1	yes	characteristic
190	C₂⁶	#525	1	1	yes	characteristic
191	C₂⁶	#532	1	1	yes	characteristic
192	C₂⁶	#532	1	1	yes	characteristic
193	C₂⁶	#546	1	1	yes	characteristic
194	C₂⁶	#546	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	#547	1	1	yes	characteristic
199	C₂⁶	#550	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	C₂×#120	1	1	yes	characteristic
204	C₂³×D₄	#525	1	1	yes	characteristic
205	C₂³×D₄	#525	1	1	yes	characteristic
206	C₂³×D₄	#532	1	1	yes	characteristic
207	C₂³×D₄	#535	1	1	yes	characteristic
208	C₂³×D₄	#535	1	1	yes	characteristic
209	C₂³×D₄	#535	1	1	yes	characteristic
210	C₂³×D₄	#546	1	1	yes	characteristic
211	C₂³×D₄	#546	1	1	yes	characteristic
212	C₂³×D₄	#546	1	1	yes	characteristic
213	C₂³×D₄	#546	1	1	yes	characteristic
214	C₂³×D₄	#550	1	1	yes	characteristic
215	C₂³×D₄	#550	1	1	yes	characteristic
216	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
217	C₂⁴×C₄	#537	1	1	yes	characteristic
218	C₂⁴×C₄	#537	1	1	yes	characteristic
219	C₂⁴×C₄	#537	1	1	yes	characteristic
220	C₂⁴×C₄	#547	1	1	yes	characteristic
221	C₂⁴×C₄	#547	1	1	yes	characteristic
222	C₂⁴×C₄	#548	1	1	yes	characteristic
223	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
224	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
225	#379	B(4,𝔽₂)	1	2	yes	—
226	C₂⁵	#2388	1	1	yes	characteristic
227	C₂⁵	#2404	1	1	yes	characteristic
228	C₂⁵	#2404	1	1	yes	characteristic
229	C₂⁵	#2404	1	1	yes	characteristic
230	C₂⁵	#2407	1	1	yes	characteristic
231	C₂⁵	#2430	1	1	yes	characteristic
232	C₂⁵	#2430	1	1	yes	characteristic
233	C₂⁵	#2431	1	1	yes	characteristic
234	C₂⁵	#2431	1	1	yes	characteristic
235	C₂⁵	#2431	1	1	yes	characteristic
236	C₂⁵	#2846	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	C₂×B(4,𝔽₂)	1	1	yes	characteristic
241	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
242	C₂⁵	#3025	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	#3026	1	1	yes	characteristic
247	C₂⁵	#3026	1	1	yes	characteristic
248	C₂⁵	#3027	1	1	yes	characteristic
249	C₂³×C₄	#2407	1	1	yes	characteristic
250	C₂³×C₄	#2407	1	1	yes	characteristic
251	C₂³×C₄	#2430	1	1	yes	characteristic
252	C₂³×C₄	#2615	1	1	yes	characteristic
253	C₂⁴	#14885	1	1	yes	characteristic
254	C₂⁴	#23984	1	1	yes	characteristic
255	C₂⁴	#23984	1	1	yes	characteristic
256	C₂⁴	#23987	1	1	yes	characteristic
257	C₂⁴	#23987	1	1	yes	characteristic
258	C₂⁴	#23996	1	2	yes	—
259	C₂⁴	#23996	1	2	yes	—
260	C₂⁴	#23996	1	2	yes	—
261	C₂⁴	#24140	1	1	yes	characteristic
262	C₂⁴	#24142	1	1	yes	characteristic
263	C₂⁴	#25563	1	1	yes	characteristic
264	C₂⁴	#25563	1	1	yes	characteristic
265	C₂⁴	#25563	1	1	yes	characteristic
266	C₂⁴	#25563	1	1	yes	characteristic
267	C₂⁴	#25585	1	1	yes	characteristic
268	C₂⁴	#25587	1	1	yes	characteristic
269	C₂⁴	#32898	1	1	yes	characteristic
270	C₂⁴	#32898	1	1	yes	characteristic
271	C₂⁴	#32898	1	1	yes	characteristic
272	C₂⁴	#32898	1	1	yes	characteristic
273	C₂⁴	#32898	1	1	yes	characteristic
274	C₂⁴	#32899	1	1	yes	characteristic
275	C₂³	#251827	1	2	yes	—
276	C₂³	#251883	1	2	yes	—
277	C₂³	#347042	1	1	yes	characteristic
278	C₂³	#347043	1	1	yes	characteristic
279	C₂³	#347043	1	1	yes	characteristic
280	C₂³	#347047	1	1	yes	characteristic
281	C₂³	#403941	1	1	yes	characteristic
282	C₂³	#403943	1	1	yes	characteristic
283	C₂³	#403952	1	1	yes	characteristic
284	C₂³	#403952	1	1	yes	characteristic
285	C₂³	#404015	1	1	yes	characteristic
286	C₂³	#404015	1	1	yes	characteristic
287	C₂³	#404015	1	1	yes	characteristic
288	C₂²	#72232934	1	1	yes	characteristic
289	C₂²	#72233110	1	1	yes	characteristic
290	C₂²	#72234956	1	1	yes	characteristic
291	C₂²	#72234958	1	1	yes	characteristic
292	C₂²	#72234958	1	1	yes	characteristic
293	C₂²	#72251466	1	1	yes	characteristic
294	C₂²	#72258386	1	1	yes	characteristic
295	C₂	#72258785	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72258785
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. #72315164	C₂	semidirect
2. #72313018	C₂	semidirect
3. #72312258	C₂	semidirect
4. #72310339	C₂	semidirect
5. #72310337	C₂	semidirect
6. #72310334	C₂	semidirect
7. #72309737	C₂	semidirect
9. #72265757	C₂	semidirect
10. #72262328	C₂	semidirect
12. #72238345	C₂	semidirect
15. #72225729	C₂	semidirect
20. #72219455	C₂	semidirect
26. #72218868	C₂	semidirect
27. #72218847	C₂	semidirect
29. #72218393	C₂	semidirect
53. C₂²×#2846	D₄	semidirect
56. C₂×#32873	D₄	semidirect
57. #6440460	D₄	semidirect
58. #6437621	D₄	semidirect
59. C₂×#23900	D₄	semidirect
60. C₂×#23900	D₄	semidirect
61. #6374251	D₄	semidirect
62. C₂×#21776	D₄	semidirect
63. #6374128	D₄	semidirect
64. C₂×#21746	D₄	semidirect
65. C₂×#21746	D₄	semidirect
66. C₂²×#1898	D₄	semidirect
71. C₂×#14679	D₄	semidirect
79. #392030	D₄	semidirect
80. #392030	D₄	semidirect
81. #392030	D₄	semidirect
82. #382707	D₄	semidirect
83. #382707	D₄	semidirect
84. #382544	D₄	semidirect
85. #382434	D₄	semidirect
86. #382422	D₄	semidirect
87. #382422	D₄	semidirect
88. #359811	D₄	semidirect
89. #359501	D₄	semidirect
90. #359501	D₄	semidirect
91. #359442	D₄	semidirect
92. #355096	D₄	semidirect
93. C₂×#10735	D₄	semidirect
95. C₂³×H(5,𝔽₂)	#41	semidirect
107. C₂²×#518	#41	semidirect
109. C₂³×#99	#41	semidirect
140. C₂×#1781	#41	semidirect
141. #14679	#41	semidirect

Maximal subgroups

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

subgroup	index
#72315164	2
#72313018	2
#72312258	2
#72310339	2
#72310337	2
#72310334	2
#72309737	2
#72265757	2
#72262328	2
#72238345	2
#72225729	2
#72219455	2
#72218868	2
#72218847	2
#72218393	2