Subgroups — #72266323

Order 8192.

Only the 320 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 282 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266322	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266321	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266318	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266317	C₂	1	1	yes	maximal, characteristic, complemented
5	#72266312	C₂	1	1	yes	maximal, characteristic, complemented
6	#72266310	C₂²	1	1	yes	characteristic
7	#72266294	C₂	1	1	yes	maximal, characteristic, complemented
8	#72266293	C₂²	1	1	yes	characteristic, complemented
9	#72266290	C₂	1	1	yes	maximal, characteristic, complemented
10	#72266289	C₂²	1	1	yes	characteristic, complemented
11	#72266268	C₂	1	1	yes	maximal, characteristic, complemented
12	#72266268	C₂	1	1	yes	maximal, characteristic, complemented
13	#72266267	C₂²	1	1	yes	characteristic
14	#72266249	C₂	1	1	yes	maximal, characteristic, complemented
15	#72266231	C₂	1	1	yes	maximal, characteristic, complemented
16	#72266225	C₂	1	1	yes	maximal, characteristic, complemented
17	#72266202	C₂²	1	1	yes	characteristic, complemented
18	#72266198	C₂²	1	1	yes	characteristic, complemented
19	#72266190	C₂²	1	1	yes	characteristic, complemented
20	#72266187	C₂	1	1	yes	maximal, characteristic, complemented
21	#72266175	C₂	1	1	yes	maximal, characteristic, complemented
22	#72266167	C₂²	1	1	yes	characteristic
23	#72266167	C₂²	1	1	yes	characteristic
24	#72266159	C₂²	1	1	yes	characteristic, complemented
25	#72266154	C₂²	1	1	yes	characteristic, complemented
26	#72266151	C₂²	1	1	yes	characteristic
27	#72266151	C₂²	1	1	yes	characteristic
28	#72266136	C₂²	1	1	yes	characteristic, complemented
29	#72266134	C₂²	1	1	yes	characteristic, complemented
30	#72266127	C₂	1	1	yes	maximal, characteristic, complemented
31	#72266097	C₂²	1	1	yes	characteristic, complemented
32	#72266086	C₂²	1	1	yes	characteristic, complemented
33	#72266068	C₂²	1	1	yes	characteristic, complemented
34	#72266044	C₂²	1	1	yes	characteristic
35	#72266041	D₄	1	1	yes	characteristic, complemented
36	#72266041	D₄	1	1	yes	characteristic, complemented
37	#72259040	C₂²	1	1	yes	characteristic
38	#72259040	C₂²	1	1	yes	characteristic
39	#72259019	D₄	1	2	yes	complemented
40	#72259016	D₄	1	2	yes	complemented
41	#72259008	D₄	1	1	yes	characteristic, complemented
42	#72258939	C₂²	1	1	yes	characteristic
43	#72258939	C₂²	1	1	yes	characteristic
44	#72258662	C₂²	1	1	yes	characteristic
45	#72258662	C₂²	1	1	yes	characteristic
46	#72258649	D₄	1	1	yes	characteristic, complemented
47	#72258649	D₄	1	1	yes	characteristic, complemented
48	#72258414	C₂²	1	1	yes	characteristic, complemented
49	#72240620	C₂²	1	1	yes	characteristic, complemented
50	#72240619	D₄	1	1	yes	characteristic, complemented
51	#72240619	D₄	1	1	yes	characteristic, complemented
52	#72236821	C₂²	1	1	yes	characteristic
53	#72236821	C₂²	1	1	yes	characteristic
54	#72236816	D₄	1	2	yes	complemented
55	#72236812	D₄	1	2	yes	complemented
56	#72236800	C₂²	1	1	yes	characteristic
57	#72236800	C₂²	1	1	yes	characteristic
58	#72236034	C₂²	1	1	yes	characteristic
59	#72236034	C₂²	1	1	yes	characteristic
60	#72235930	C₂³	1	1	yes	characteristic
61	#72235930	C₂³	1	1	yes	characteristic
62	#72235847	D₄	1	1	yes	characteristic, complemented
63	#72233780	D₄	1	1	yes	characteristic, complemented
64	#72233780	D₄	1	1	yes	characteristic, complemented
65	#72226161	C₂²	1	1	yes	characteristic, complemented
66	#72225904	C₂²	1	1	yes	characteristic, complemented
67	#72225881	C₂³	1	1	yes	characteristic
68	#72225783	D₄	1	1	yes	characteristic, complemented
69	#72225783	D₄	1	1	yes	characteristic, complemented
70	#72225665	C₂³	1	1	yes	characteristic
71	#72225621	C₂³	1	1	yes	characteristic
72	#72225621	C₂³	1	1	yes	characteristic
73	#72219377	C₂³	1	1	yes	characteristic
74	#72219377	C₂³	1	1	yes	characteristic
75	#72219377	C₂³	1	1	yes	characteristic
76	#72219307	C₂³	1	1	yes	characteristic
77	#72219307	C₂³	1	1	yes	characteristic
78	#72219307	C₂³	1	1	yes	characteristic
79	#72219307	C₂³	1	1	yes	characteristic
80	C₂×#7631286	C₂³	1	1	yes	characteristic
81	C₂×#7631286	C₂³	1	1	yes	characteristic
82	#72218854	D₄	1	2	yes	complemented
83	#72218846	D₄	1	2	yes	complemented
84	#72218467	D₄	1	2	yes	complemented
85	#72214832	D₄	1	2	yes	complemented
86	C₂⁴×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂⁴×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
88	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
89	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
90	C₂³×D₄²	C₂⁴	1	1	yes	characteristic
91	C₂×#35347	#41	1	1	yes	characteristic
92	C₂×#35347	#41	1	1	yes	characteristic
93	C₂×#35346	C₂×D₄	1	1	yes	characteristic
94	C₂×#35346	C₂×D₄	1	1	yes	characteristic
95	C₂×#34980	C₂×D₄	1	1	yes	characteristic
96	C₂×#34980	C₂×D₄	1	2	yes	—
97	C₂²×#2403	C₂×D₄	1	2	yes	—
98	C₂²×#2403	C₂×D₄	1	1	yes	characteristic
99	C₂²×#2403	C₂×D₄	1	1	yes	characteristic
100	C₂×#34171	C₂×D₄	1	1	yes	characteristic
101	C₂×#34171	C₂×D₄	1	1	yes	characteristic
102	C₂×#21746	C₂×D₄	1	1	yes	characteristic
103	C₂×#21746	C₂×D₄	1	1	yes	characteristic
104	C₂×#21746	C₂×D₄	1	1	yes	characteristic
105	C₂×#21746	C₂×D₄	1	1	yes	characteristic
106	#556752	C₂×D₄	1	1	yes	characteristic
107	#517411	C₂×D₄	1	1	yes	characteristic
108	#517411	C₂×D₄	1	1	yes	characteristic
109	#517411	C₂×D₄	1	1	yes	characteristic
110	#517411	C₂×D₄	1	1	yes	characteristic
111	#517411	C₂×D₄	1	1	yes	characteristic
112	#517411	C₂×D₄	1	1	yes	characteristic
113	#517411	C₂×D₄	1	1	yes	characteristic
114	#517411	C₂×D₄	1	1	yes	characteristic
115	#515362	#41	1	1	yes	characteristic
116	#515362	#41	1	1	yes	characteristic
117	#515285	C₂×D₄	1	1	yes	characteristic
118	#515285	C₂×D₄	1	1	yes	characteristic
119	#392109	C₂×D₄	1	2	yes	—
120	#392109	C₂×D₄	1	2	yes	—
121	#382422	C₂×D₄	1	2	yes	—
122	#382422	C₂×D₄	1	2	yes	—
123	C₂×#10965	C₂×D₄	1	2	yes	—
124	#154094	C₂×D₄	1	2	yes	—
125	C₂³×H(5,𝔽₂)	#121	1	1	yes	characteristic, complemented
126	C₂³×H(5,𝔽₂)	#121	1	1	yes	characteristic, complemented
127	C₂⁵×D₄	#120	1	1	yes	characteristic
128	C₂⁵×D₄	#120	1	2	yes	—
129	C₂⁵×D₄	C₂²×D₄	1	1	yes	characteristic
130	C₂⁵×D₄	C₂²×D₄	1	1	yes	characteristic
131	C₂⁵×D₄	C₂²×D₄	1	1	yes	characteristic
132	C₂⁵×D₄	C₂²×D₄	1	1	yes	characteristic
133	C₂³×#127	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂³×#121	C₂×#41	1	1	yes	characteristic
135	C₂³×#121	C₂×#41	1	1	yes	characteristic
136	C₂³×#121	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂³×#121	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂³×#120	C₂²×D₄	1	1	yes	characteristic
139	C₂³×#120	C₂²×D₄	1	1	yes	characteristic
140	C₂³×#120	C₂²×D₄	1	1	yes	characteristic
141	C₂³×#120	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²×#518	#121	1	1	yes	characteristic, complemented
143	C₂²×#518	#121	1	1	yes	characteristic, complemented
144	C₂³×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂³×C₄×D₄	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×#2308	#121	1	1	yes	characteristic, complemented
147	C₂×#2308	#121	1	1	yes	characteristic, complemented
148	C₂×#2308	#121	1	1	yes	characteristic, complemented
149	C₂×#2308	#121	1	1	yes	characteristic, complemented
150	C₂²×#379	#120	1	2	yes	—
151	C₂²×#379	#120	1	1	yes	characteristic
152	C₂²×#379	#120	1	1	yes	characteristic
153	C₂²×#379	#120	1	1	yes	characteristic
154	#14679	#120	1	2	yes	—
155	#14679	#120	1	2	yes	—
156	C₂⁷	B(4,𝔽₂)	1	1	yes	characteristic
157	C₂⁷	B(4,𝔽₂)	1	2	yes	complemented
158	C₂⁷	C₂×#120	1	1	yes	characteristic
159	C₂⁷	C₂×#120	1	1	yes	characteristic
160	C₂⁷	C₂×#120	1	1	yes	characteristic
161	C₂⁷	D₄²	1	1	yes	characteristic
162	C₂⁴×D₄	C₂×#121	1	1	yes	characteristic
163	C₂⁴×D₄	C₂×#121	1	1	yes	characteristic
164	C₂⁴×D₄	#534	1	1	yes	characteristic
165	C₂⁴×D₄	#535	1	1	yes	characteristic
166	C₂⁴×D₄	#535	1	1	yes	characteristic
167	C₂⁴×D₄	D₄²	1	1	yes	characteristic
168	C₂⁴×D₄	D₄²	1	1	yes	characteristic
169	C₂⁴×D₄	D₄²	1	1	yes	characteristic
170	C₂⁴×D₄	D₄²	1	1	yes	characteristic
171	C₂⁴×D₄	D₄²	1	1	yes	characteristic
172	C₂⁴×D₄	D₄²	1	1	yes	characteristic
173	C₂⁴×D₄	D₄²	1	1	yes	characteristic
174	C₂⁴×D₄	#546	1	1	yes	characteristic
175	C₂⁴×D₄	#546	1	1	yes	characteristic
176	C₂⁴×D₄	#546	1	1	yes	characteristic
177	C₂⁴×D₄	#546	1	1	yes	characteristic
178	C₂⁵×C₄	B(4,𝔽₂)	1	1	yes	characteristic
179	C₂⁵×C₄	B(4,𝔽₂)	1	2	yes	complemented
180	C₂⁵×C₄	#534	1	1	yes	characteristic
181	C₂⁵×C₄	#534	1	1	yes	characteristic
182	C₂⁵×C₄	#546	1	1	yes	characteristic
183	C₂⁵×C₄	#546	1	1	yes	characteristic
184	C₂³×C₄⋊C₄	#538	1	1	yes	characteristic
185	C₂³×C₄⋊C₄	#538	1	1	yes	characteristic
186	C₂³×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
187	C₂³×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
188	C₂³×C₂²⋊C₄	#525	1	1	yes	characteristic
189	C₂³×C₂²⋊C₄	#534	1	1	yes	characteristic
190	C₂³×C₂²⋊C₄	#535	1	1	yes	characteristic
191	C₂³×C₂²⋊C₄	D₄²	1	1	yes	characteristic
192	C₂³×C₂²⋊C₄	D₄²	1	1	yes	characteristic
193	C₂³×C₂²⋊C₄	#546	1	1	yes	characteristic
194	C₂³×C₂²⋊C₄	#546	1	1	yes	characteristic
195	C₂³×C₂²⋊C₄	#560	1	1	yes	characteristic
196	C₂³×C₄²	#560	1	1	yes	characteristic
197	C₂×#379	B(4,𝔽₂)	1	2	yes	—
198	C₂×#379	B(4,𝔽₂)	1	4	yes	complemented
199	C₂⁶	#2403	1	1	yes	characteristic
200	C₂⁶	#2403	1	1	yes	characteristic
201	C₂⁶	#2404	1	1	yes	characteristic
202	C₂⁶	#2408	1	1	yes	characteristic
203	C₂⁶	#2408	1	1	yes	characteristic
204	C₂⁶	#2410	1	1	yes	characteristic
205	C₂⁶	#2410	1	1	yes	characteristic
206	C₂⁶	#2669	1	1	yes	characteristic
207	C₂⁶	C₂×B(4,𝔽₂)	1	2	yes	—
208	C₂⁶	C₂×B(4,𝔽₂)	1	1	yes	characteristic
209	C₂⁶	C₂×B(4,𝔽₂)	1	1	yes	characteristic
210	C₂⁶	C₂×B(4,𝔽₂)	1	1	yes	characteristic
211	C₂⁶	C₂×B(4,𝔽₂)	1	1	yes	characteristic
212	C₂⁶	C₂×B(4,𝔽₂)	1	2	yes	—
213	C₂⁶	#3025	1	2	yes	—
214	C₂⁶	#3026	1	2	yes	—
215	C₂⁴×C₄	#2424	1	1	yes	characteristic
216	C₂⁴×C₄	#2424	1	1	yes	characteristic
217	C₂⁴×C₄	#2668	1	1	yes	characteristic
218	C₂⁴×C₄	#2668	1	1	yes	characteristic
219	C₂⁴×C₄	#2669	1	1	yes	characteristic
220	C₂⁴×C₄	#2678	1	1	yes	characteristic
221	C₂⁴×C₄	#2678	1	1	yes	characteristic
222	C₂⁵	#15257	1	1	yes	characteristic
223	C₂⁵	#23895	1	1	yes	characteristic
224	C₂⁵	#23895	1	1	yes	characteristic
225	C₂⁵	#23895	1	1	yes	characteristic
226	C₂⁵	#23896	1	1	yes	characteristic
227	C₂⁵	#23900	1	1	yes	characteristic
228	C₂⁵	#23900	1	1	yes	characteristic
229	C₂⁵	#23900	1	1	yes	characteristic
230	C₂⁵	#23900	1	1	yes	characteristic
231	C₂⁵	#23900	1	1	yes	characteristic
232	C₂⁵	#23900	1	1	yes	characteristic
233	C₂⁵	#23902	1	1	yes	characteristic
234	C₂⁵	#23903	1	1	yes	characteristic
235	C₂⁵	#23984	1	1	yes	characteristic
236	C₂⁵	#23984	1	1	yes	characteristic
237	C₂⁵	#23996	1	1	yes	characteristic
238	C₂⁵	#23996	1	1	yes	characteristic
239	C₂⁵	#23996	1	1	yes	characteristic
240	C₂⁵	#23996	1	1	yes	characteristic
241	C₂⁵	#24260	1	2	yes	—
242	C₂⁵	#24260	1	2	yes	—
243	C₂⁵	#24404	1	2	yes	—
244	C₂⁵	#24404	1	2	yes	—
245	C₂⁴	#245225	1	1	yes	characteristic
246	C₂⁴	#245225	1	1	yes	characteristic
247	C₂⁴	#245228	1	1	yes	characteristic
248	C₂⁴	#245229	1	1	yes	characteristic
249	C₂⁴	#245231	1	1	yes	characteristic
250	C₂⁴	#245233	1	1	yes	characteristic
251	C₂⁴	#245254	1	1	yes	characteristic
252	C₂⁴	#245255	1	1	yes	characteristic
253	C₂⁴	#251827	1	1	yes	characteristic
254	C₂⁴	#251828	1	1	yes	characteristic
255	C₂⁴	#251859	1	1	yes	characteristic
256	C₂⁴	#251860	1	1	yes	characteristic
257	C₂⁴	#417493	1	1	yes	characteristic
258	C₂⁴	#417499	1	1	yes	characteristic
259	C₂⁴	#417508	1	1	yes	characteristic
260	C₂⁴	#417526	1	1	yes	characteristic
261	C₂⁴	#417532	1	1	yes	characteristic
262	C₂⁴	#417538	1	1	yes	characteristic
263	C₂⁴	#417542	1	1	yes	characteristic
264	C₂³	B(5,𝔽₂)	1	2	yes	—
265	C₂³	B(5,𝔽₂)	1	2	yes	—
266	C₂³	#72216629	1	1	yes	characteristic
267	C₂³	#72233062	1	1	yes	characteristic
268	C₂³	#72233067	1	1	yes	characteristic
269	C₂³	#72234622	1	1	yes	characteristic
270	C₂³	#72237498	1	1	yes	characteristic
271	C₂³	#72237499	1	1	yes	characteristic
272	C₂³	#72241194	1	1	yes	characteristic
273	C₂²	#72256060	1	1	yes	characteristic
274	C₂²	#72256250	1	1	yes	characteristic
275	C₂²	#72264773	1	1	yes	characteristic
276	C₂²	#72264776	1	1	yes	characteristic
277	C₂²	#72265072	1	1	yes	characteristic
278	C₂	#72265803	1	1	yes	characteristic
279	C₂	#72265816	1	1	yes	characteristic
280	C₂	#72265935	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72265072
commutator, Frattini, cosocle	C₂³×D₄²	C₂⁴
LCS	G ▷ C₂³×D₄² ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×D₄² ◁ G
LECS	G ▷ C₂³×D₄² ▷ C₂⁵ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266322	C₂	semidirect
2. #72266321	C₂	semidirect
3. #72266318	C₂	semidirect
4. #72266317	C₂	semidirect
5. #72266312	C₂	semidirect
7. #72266294	C₂	semidirect
8. #72266293	C₂²	semidirect
9. #72266290	C₂	semidirect
10. #72266289	C₂²	semidirect
11. #72266268	C₂	semidirect
12. #72266268	C₂	semidirect
14. #72266249	C₂	semidirect
15. #72266231	C₂	semidirect
16. #72266225	C₂	semidirect
17. #72266202	C₂²	semidirect
18. #72266198	C₂²	semidirect
19. #72266190	C₂²	semidirect
20. #72266187	C₂	semidirect
21. #72266175	C₂	semidirect
24. #72266159	C₂²	semidirect
25. #72266154	C₂²	semidirect
28. #72266136	C₂²	semidirect
29. #72266134	C₂²	semidirect
30. #72266127	C₂	semidirect
31. #72266097	C₂²	semidirect
32. #72266086	C₂²	semidirect
33. #72266068	C₂²	semidirect
35. #72266041	D₄	semidirect
36. #72266041	D₄	semidirect
39. #72259019	D₄	semidirect
40. #72259016	D₄	semidirect
41. #72259008	D₄	semidirect
46. #72258649	D₄	semidirect
47. #72258649	D₄	semidirect
48. #72258414	C₂²	semidirect
49. #72240620	C₂²	semidirect
50. #72240619	D₄	semidirect
51. #72240619	D₄	semidirect
54. #72236816	D₄	semidirect
55. #72236812	D₄	semidirect
62. #72235847	D₄	semidirect
63. #72233780	D₄	semidirect
64. #72233780	D₄	semidirect
65. #72226161	C₂²	semidirect
66. #72225904	C₂²	semidirect
68. #72225783	D₄	semidirect
69. #72225783	D₄	semidirect
82. #72218854	D₄	semidirect
83. #72218846	D₄	semidirect
84. #72218467	D₄	semidirect
85. #72214832	D₄	semidirect
125. C₂³×H(5,𝔽₂)	#121	semidirect
126. C₂³×H(5,𝔽₂)	#121	semidirect
142. C₂²×#518	#121	semidirect
143. C₂²×#518	#121	semidirect
146. C₂×#2308	#121	semidirect
147. C₂×#2308	#121	semidirect
148. C₂×#2308	#121	semidirect
149. C₂×#2308	#121	semidirect
157. C₂⁷	B(4,𝔽₂)	semidirect
179. C₂⁵×C₄	B(4,𝔽₂)	semidirect
198. C₂×#379	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72266322	2
#72266321	2
#72266318	2
#72266317	2
#72266312	2
#72266294	2
#72266290	2
#72266268	2
#72266249	2
#72266231	2
#72266225	2
#72266187	2
#72266175	2
#72266127	2