Subgroups — #72265838

Order 2048.

At least 7219 conjugacy classes of subgroups; only the 352 normal subgroups are shown.

The subgroup lattice has 256 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72265789	C₂	1	1	yes	maximal, characteristic, complemented
2	#72241271	C₂	1	1	yes	maximal, characteristic, complemented
3	#72237045	C₂	1	1	yes	maximal, characteristic, complemented
4	#72237037	C₂	1	1	yes	maximal, characteristic, complemented
5	#72233800	C₂	1	1	yes	maximal, characteristic, complemented
6	#72232048	C₂	1	1	yes	maximal, characteristic, complemented
7	#72225610	C₂	1	1	yes	maximal, characteristic, complemented
8	#72219040	C₂	1	1	yes	maximal, characteristic, complemented
9	#72218980	C₂	1	1	yes	maximal, characteristic, complemented
10	#72218840	C₂	1	1	yes	maximal, characteristic, complemented
11	#72218838	C₂	1	1	yes	maximal, characteristic, complemented
12	#72218215	C₂	1	1	yes	maximal, characteristic, complemented
13	#72218206	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#133822	C₂	1	1	yes	maximal, characteristic, complemented
15	#72217730	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#3532	C₂²	1	1	yes	characteristic
17	#7642244	C₂²	1	1	yes	characteristic, complemented
18	C₂²×#2846	C₂²	1	1	yes	characteristic
19	#6437621	C₂²	1	1	yes	characteristic, complemented
20	C₂×#23900	C₂²	1	1	yes	characteristic
21	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
22	#6437445	C₂²	1	1	yes	characteristic, complemented
23	#616961	C₂²	1	1	yes	characteristic
24	#616961	C₂²	1	1	yes	characteristic
25	#616960	C₂²	1	1	yes	characteristic
26	#616960	C₂²	1	1	yes	characteristic
27	#556752	C₂²	1	1	yes	characteristic
28	#493370	C₂²	1	1	yes	characteristic, complemented
29	#493370	C₂²	1	1	yes	characteristic, complemented
30	#493357	C₂²	1	1	yes	characteristic, complemented
31	#493357	C₂²	1	1	yes	characteristic, complemented
32	C₂×#12687	C₂²	1	1	yes	characteristic
33	C₂×#12687	C₂²	1	1	yes	characteristic
34	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
35	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
36	#382422	C₂²	1	1	yes	characteristic, complemented
37	#359509	C₂²	1	1	yes	characteristic, complemented
38	#359501	C₂²	1	1	yes	characteristic
39	#359501	C₂²	1	1	yes	characteristic, complemented
40	#359442	C₂²	1	1	yes	characteristic, complemented
41	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
42	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
43	#187944	C₂²	1	1	yes	characteristic, complemented
44	#186835	C₂²	1	1	yes	characteristic, complemented
45	#128033	C₂²	1	1	yes	characteristic, complemented
46	#128033	C₂²	1	1	yes	characteristic, complemented
47	#128009	C₂²	1	1	yes	characteristic, complemented
48	#128009	C₂²	1	1	yes	characteristic, complemented
49	#108238	C₂²	1	1	yes	characteristic, complemented
50	#107114	C₂²	1	1	yes	characteristic, complemented
51	C₂⁸	D₄	1	1	yes	characteristic, complemented
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂²×#551	D₄	1	1	yes	characteristic, complemented
54	C₂²×D₄²	D₄	1	1	yes	characteristic, complemented
55	C₂²×H(3,𝔽₄)	D₄	1	1	yes	characteristic, complemented
56	C₂²×#560	D₄	1	1	yes	characteristic, complemented
57	C₂²×#535	C₂³	1	1	yes	characteristic
58	C₂²×#534	C₂³	1	1	yes	characteristic
59	C₂³×#120	C₂³	1	1	yes	characteristic
60	C₂³×#120	C₂³	1	1	yes	characteristic
61	C₂³×#120	C₂³	1	1	yes	characteristic
62	C₂²×B(4,𝔽₂)	D₄	1	2	yes	complemented
63	C₂×#2404	D₄	1	2	yes	complemented
64	C₂×#2403	D₄	1	2	yes	complemented
65	#34171	C₂³	1	1	yes	characteristic
66	C₂²×#379	D₄	1	1	yes	characteristic, complemented
67	C₂²×#379	D₄	1	1	yes	characteristic, complemented
68	C₂²×#379	D₄	1	1	yes	characteristic, complemented
69	C₂×#1889	D₄	1	2	yes	complemented
70	C₂×#1889	D₄	1	4	yes	complemented
71	C₂×#1783	C₂³	1	1	yes	characteristic
72	C₂×#1783	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	C₂×#1781	C₂³	1	1	yes	characteristic
75	C₂×#1781	C₂³	1	1	yes	characteristic
76	C₂×#1781	C₂³	1	1	yes	characteristic
77	#15948	D₄	1	2	yes	complemented
78	#15947	D₄	1	2	yes	complemented
79	#15259	D₄	1	2	yes	complemented
80	#15258	D₄	1	2	yes	complemented
81	#14657	D₄	1	2	yes	complemented
82	#14335	C₂³	1	1	yes	characteristic
83	#14335	C₂³	1	1	yes	characteristic
84	#14134	D₄	1	2	yes	complemented
85	#13868	D₄	1	2	yes	complemented
86	#13867	D₄	1	2	yes	complemented
87	#12687	D₄	1	4	yes	complemented
88	C₂⁷	C₂×D₄	1	1	yes	characteristic
89	C₂⁷	C₂×D₄	1	1	yes	characteristic
90	C₂⁷	C₂×D₄	1	1	yes	characteristic
91	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
92	C₂³×#41	C₂×D₄	1	1	yes	characteristic
93	C₂⁴×D₄	C₂×D₄	1	2	yes	—
94	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
96	C₂²×#127	C₂×D₄	1	1	yes	characteristic
97	C₂²×#127	C₂×D₄	1	1	yes	characteristic
98	C₂²×#124	C₂×D₄	1	1	yes	characteristic
99	C₂²×#124	C₂×D₄	1	1	yes	characteristic
100	C₂²×#123	C₂×D₄	1	1	yes	characteristic
101	C₂²×#123	C₂×D₄	1	1	yes	characteristic
102	C₂²×#120	C₂×D₄	1	2	yes	—
103	C₂²×#120	C₂×D₄	1	2	yes	—
104	C₂²×#120	C₂×D₄	1	2	yes	—
105	C₂²×#120	C₂×D₄	1	2	yes	—
106	C₂²×#120	C₂×D₄	1	1	yes	characteristic
107	C₂²×#120	C₂×D₄	1	1	yes	characteristic
108	C₂²×#120	C₂×D₄	1	1	yes	characteristic
109	C₂²×#120	C₂×D₄	1	1	yes	characteristic
110	C₂²×#120	C₂×D₄	1	1	yes	characteristic
111	C₂²×#120	C₂×D₄	1	1	yes	characteristic
112	C₂²×#120	C₂×D₄	1	2	yes	—
113	C₂²×#120	#41	1	2	yes	—
114	C₂×#518	C₂×D₄	1	1	yes	characteristic, complemented
115	C₂×#518	C₂×D₄	1	2	yes	complemented
116	C₂²×#117	C₂×D₄	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
118	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
119	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
120	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
122	C₂²×#99	C₂×D₄	1	2	yes	complemented
123	C₂²×#99	C₂×D₄	1	2	yes	complemented
124	C₂×#381	C₂×D₄	1	2	yes	—
125	C₂×#379	C₂×D₄	1	2	yes	—
126	#1783	C₂×D₄	1	2	yes	complemented
127	#1781	C₂×D₄	1	2	yes	complemented
128	#1438	C₂×D₄	1	2	yes	—
129	#1438	C₂×D₄	1	2	yes	—
130	#1438	C₂×D₄	1	2	yes	—
131	#1438	C₂×D₄	1	2	yes	—
132	C₂⁶	#120	1	1	yes	characteristic
133	C₂⁶	#120	1	2	yes	—
134	C₂⁶	#120	1	1	yes	characteristic
135	C₂⁶	#120	1	2	yes	—
136	C₂⁶	#120	1	2	yes	—
137	C₂⁶	#120	1	1	yes	characteristic, complemented
138	C₂⁶	#121	1	2	yes	—
139	C₂⁶	C₂²×D₄	1	1	yes	characteristic
140	C₂⁶	C₂²×D₄	1	1	yes	characteristic
141	C₂³×Q₈	#120	1	1	yes	characteristic, complemented
142	C₂³×D₄	#120	1	1	yes	characteristic, complemented
143	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
145	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
146	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
147	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
148	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
149	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
150	C₂³×D₄	C₂×#41	1	1	yes	characteristic
151	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂⁴×C₄	#120	1	2	yes	—
156	C₂⁴×C₄	#120	1	1	yes	characteristic, complemented
157	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
158	C₂×#120	#120	1	2	yes	—
159	C₂×#120	#120	1	2	yes	—
160	C₂×#120	#121	1	2	yes	—
161	C₂×#120	#121	1	2	yes	—
162	C₂×#120	#121	1	2	yes	—
163	C₂×#120	#121	1	2	yes	—
164	C₂×#120	#127	1	2	yes	—
165	C₂×#120	#127	1	2	yes	—
166	C₂²×C₂²⋊C₄	#120	1	2	yes	—
167	C₂²×C₂²⋊C₄	#120	1	2	yes	—
168	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
169	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
170	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic, complemented
171	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic, complemented
172	C₂²×C₂²⋊C₄	#120	1	4	yes	complemented
173	C₂²×C₂²⋊C₄	#121	1	2	yes	—
174	C₂²×C₄²	#120	1	1	yes	characteristic, complemented
175	C₂²×C₄²	#120	1	1	yes	characteristic, complemented
176	C₂×#95	#120	1	4	yes	complemented
177	C₂⁵	#392	1	2	yes	—
178	C₂⁵	#392	1	2	yes	—
179	C₂⁵	B(4,𝔽₂)	1	4	yes	—
180	C₂⁵	B(4,𝔽₂)	1	8	yes	—
181	C₂⁵	C₂×#120	1	1	yes	characteristic
182	C₂⁵	C₂×#120	1	1	yes	characteristic
183	C₂⁵	C₂×#120	1	1	yes	characteristic
184	C₂⁵	C₂×#120	1	1	yes	characteristic
185	C₂⁵	C₂×#120	1	1	yes	characteristic
186	C₂⁵	C₂×#120	1	1	yes	characteristic
187	C₂⁵	C₂×#121	1	1	yes	characteristic
188	C₂⁵	#534	1	1	yes	characteristic
189	C₂⁵	#534	1	1	yes	characteristic
190	C₂⁵	D₄²	1	1	yes	characteristic
191	C₂⁵	D₄²	1	1	yes	characteristic
192	C₂⁵	D₄²	1	1	yes	characteristic
193	C₂⁵	#546	1	1	yes	characteristic
194	C₂²×D₄	C₂×#120	1	1	yes	characteristic
195	C₂²×D₄	C₂×#121	1	1	yes	characteristic
196	C₂²×D₄	C₂×#121	1	1	yes	characteristic
197	C₂²×D₄	C₂×#127	1	1	yes	characteristic
198	C₂²×D₄	#534	1	1	yes	characteristic
199	C₂²×D₄	#534	1	1	yes	characteristic
200	C₂²×D₄	#535	1	1	yes	characteristic
201	C₂²×D₄	#535	1	1	yes	characteristic
202	C₂²×D₄	D₄²	1	1	yes	characteristic
203	C₂²×D₄	D₄²	1	1	yes	characteristic
204	C₂²×D₄	D₄²	1	1	yes	characteristic
205	C₂²×D₄	D₄²	1	1	yes	characteristic
206	C₂²×D₄	D₄²	1	1	yes	characteristic
207	C₂²×D₄	D₄²	1	1	yes	characteristic
208	C₂²×D₄	#546	1	1	yes	characteristic
209	C₂²×D₄	#546	1	1	yes	characteristic
210	C₂³×C₄	C₂×#120	1	1	yes	characteristic
211	C₂³×C₄	C₂×#120	1	1	yes	characteristic
212	C₂³×C₄	D₄²	1	1	yes	characteristic
213	C₂³×C₄	D₄²	1	1	yes	characteristic
214	C₂³×C₄	D₄²	1	1	yes	characteristic
215	C₂³×C₄	#546	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	#392	1	2	yes	—
217	C₂×C₂²⋊C₄	#392	1	2	yes	—
218	C₂⁴	#2021	1	4	yes	—
219	C₂⁴	#2021	1	4	yes	—
220	C₂⁴	C₂×#392	1	1	yes	characteristic
221	C₂⁴	#2403	1	1	yes	characteristic
222	C₂⁴	#2403	1	1	yes	characteristic
223	C₂⁴	#2403	1	1	yes	characteristic
224	C₂⁴	#2408	1	1	yes	characteristic
225	C₂⁴	#2410	1	1	yes	characteristic
226	C₂⁴	#2410	1	1	yes	characteristic
227	C₂⁴	#2410	1	1	yes	characteristic
228	C₂⁴	#2433	1	1	yes	characteristic
229	C₂⁴	#2613	1	1	yes	characteristic
230	C₂⁴	#2846	1	1	yes	characteristic
231	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
232	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
233	C₂⁴	#3026	1	2	yes	—
234	C₂²×C₄	#2403	1	1	yes	characteristic
235	C₂²×C₄	#2408	1	1	yes	characteristic
236	C₂²×C₄	#2410	1	1	yes	characteristic
237	C₂²×C₄	#2433	1	1	yes	characteristic
238	C₂³	#14715	1	1	yes	characteristic
239	C₂³	C₂×#2021	1	2	yes	—
240	C₂³	#23895	1	2	yes	—
241	C₂³	#23900	1	1	yes	characteristic
242	C₂³	#23900	1	2	yes	—
243	C₂³	#23900	1	2	yes	—
244	C₂³	#23903	1	1	yes	characteristic
245	C₂³	#24404	1	2	yes	—
246	C₂³	#32898	1	2	yes	—
247	C₂²	#245225	1	2	yes	—
248	C₂²	#245246	1	2	yes	—
249	C₂²	#394058	1	1	yes	characteristic
250	C₂²	#394064	1	1	yes	characteristic
251	C₂²	#394066	1	1	yes	characteristic
252	C₂²	#511790	1	4	yes	—
253	C₂	#72218099	1	2	yes	—
254	C₂	#72225609	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#394066
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72265789	C₂	semidirect
2. #72241271	C₂	semidirect
3. #72237045	C₂	semidirect
4. #72237037	C₂	semidirect
5. #72233800	C₂	semidirect
6. #72232048	C₂	semidirect
7. #72225610	C₂	semidirect
8. #72219040	C₂	semidirect
9. #72218980	C₂	semidirect
10. #72218840	C₂	semidirect
11. #72218838	C₂	semidirect
12. #72218215	C₂	semidirect
13. #72218206	C₂	semidirect
14. C₂×#133822	C₂	semidirect
15. #72217730	C₂	semidirect
17. #7642244	C₂²	semidirect
19. #6437621	C₂²	semidirect
21. C₂×#23900	C₂²	semidirect
22. #6437445	C₂²	semidirect
28. #493370	C₂²	semidirect
29. #493370	C₂²	semidirect
30. #493357	C₂²	semidirect
31. #493357	C₂²	semidirect
34. C₂×#12687	C₂²	semidirect
35. C₂×#12687	C₂²	semidirect
36. #382422	C₂²	semidirect
37. #359509	C₂²	semidirect
39. #359501	C₂²	semidirect
40. #359442	C₂²	semidirect
41. C₂×#8547	C₂²	semidirect
42. C₂×#8541	C₂²	semidirect
43. #187944	C₂²	semidirect
44. #186835	C₂²	semidirect
45. #128033	C₂²	semidirect
46. #128033	C₂²	semidirect
47. #128009	C₂²	semidirect
48. #128009	C₂²	semidirect
49. #108238	C₂²	semidirect
50. #107114	C₂²	semidirect
51. C₂⁸	D₄	semidirect
53. C₂²×#551	D₄	semidirect
54. C₂²×D₄²	D₄	semidirect
55. C₂²×H(3,𝔽₄)	D₄	semidirect
56. C₂²×#560	D₄	semidirect
62. C₂²×B(4,𝔽₂)	D₄	semidirect
63. C₂×#2404	D₄	semidirect
64. C₂×#2403	D₄	semidirect
66. C₂²×#379	D₄	semidirect
67. C₂²×#379	D₄	semidirect
68. C₂²×#379	D₄	semidirect
69. C₂×#1889	D₄	semidirect
70. C₂×#1889	D₄	semidirect
77. #15948	D₄	semidirect
78. #15947	D₄	semidirect
79. #15259	D₄	semidirect
80. #15258	D₄	semidirect
81. #14657	D₄	semidirect
84. #14134	D₄	semidirect
85. #13868	D₄	semidirect
86. #13867	D₄	semidirect
87. #12687	D₄	semidirect
91. C₂²×H(5,𝔽₂)	C₂×D₄	semidirect
114. C₂×#518	C₂×D₄	semidirect
115. C₂×#518	C₂×D₄	semidirect
122. C₂²×#99	C₂×D₄	semidirect
123. C₂²×#99	C₂×D₄	semidirect
126. #1783	C₂×D₄	semidirect
127. #1781	C₂×D₄	semidirect
137. C₂⁶	#120	semidirect
141. C₂³×Q₈	#120	semidirect
142. C₂³×D₄	#120	semidirect
156. C₂⁴×C₄	#120	semidirect
170. C₂²×C₂²⋊C₄	#120	semidirect
171. C₂²×C₂²⋊C₄	#120	semidirect
172. C₂²×C₂²⋊C₄	#120	semidirect
174. C₂²×C₄²	#120	semidirect
175. C₂²×C₄²	#120	semidirect
176. C₂×#95	#120	semidirect

Maximal subgroups

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

subgroup	index
#72265789	2
#72241271	2
#72237045	2
#72237037	2
#72233800	2
#72232048	2
#72225610	2
#72219040	2
#72218980	2
#72218840	2
#72218838	2
#72218215	2
#72218206	2
C₂×#133822	2
#72217730	2