Subgroups — #72310363

Order 2048.

At least 6039 conjugacy classes of subgroups; only the 358 normal subgroups are shown.

The subgroup lattice has 258 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310358	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310322	C₂	1	1	yes	maximal, characteristic, complemented
3	#72309555	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266864	C₂	1	1	yes	maximal, characteristic, complemented
5	#72266366	C₂	1	1	yes	maximal, characteristic, complemented
6	#72258097	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255777	C₂	1	1	yes	maximal, characteristic, complemented
8	#72255772	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251552	C₂	1	1	yes	maximal, characteristic, complemented
10	#72238443	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237476	C₂	1	1	yes	maximal, characteristic, complemented
12	#72237374	C₂	1	1	yes	maximal, characteristic, complemented
13	#72234953	C₂	1	1	yes	maximal, characteristic
14	#72219160	C₂	1	1	yes	maximal, characteristic, complemented
15	#72219159	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586404	C₂²	1	1	yes	characteristic
17	#6451863	C₄	1	2	yes	complemented
18	#6437629	C₄	1	2	yes	complemented
19	C₂×#23895	C₂²	1	1	yes	characteristic
20	#6437621	C₂²	1	1	yes	characteristic
21	#6431795	C₂²	1	1	yes	characteristic
22	#6374128	C₂²	1	1	yes	characteristic
23	#552670	C₂²	1	1	yes	characteristic
24	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
25	C₂×#12705	C₂²	1	1	yes	characteristic
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
28	#493357	C₂²	1	1	yes	characteristic
29	#493357	C₂²	1	1	yes	characteristic
30	#382436	C₂²	1	1	yes	characteristic
31	#382434	C₂²	1	1	yes	characteristic
32	#382434	C₂²	1	1	yes	characteristic
33	#363969	C₄	1	2	yes	complemented
34	#359549	C₂²	1	1	yes	characteristic
35	#359528	C₄	1	2	yes	complemented
36	#359442	C₂²	1	1	yes	characteristic
37	#187944	C₂²	1	1	yes	characteristic
38	#187944	C₂²	1	1	yes	characteristic
39	C₂×#8534	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
42	#186835	C₂²	1	1	yes	characteristic, complemented
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	#128033	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
51	#128009	C₂²	1	1	yes	characteristic, complemented
52	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
53	#107130	C₂²	1	1	yes	characteristic, complemented
54	#107114	C₂²	1	1	yes	characteristic, complemented
55	#62844	C₂×C₄	1	2	yes	—
56	#62844	D₄	1	2	yes	—
57	#62842	C₂×C₄	1	2	yes	—
58	#62842	C₂×C₄	1	1	yes	characteristic, complemented
59	#62842	D₄	1	2	yes	—
60	C₂×#3490	D₄	1	2	yes	—
61	C₂×#3484	D₄	1	2	yes	—
62	C₂×#3484	C₂³	1	1	yes	characteristic
63	C₂×#3484	C₂³	1	1	yes	characteristic
64	C₂×#3484	C₂³	1	1	yes	characteristic
65	C₂²×#534	C₂³	1	1	yes	characteristic
66	C₂×#3026	C₂×C₄	1	2	yes	—
67	C₂×#3026	C₂×C₄	1	2	yes	complemented
68	C₂×#2883	C₂×C₄	1	1	yes	characteristic, complemented
69	C₂×#2883	C₂×C₄	1	1	yes	characteristic, complemented
70	#34171	C₂³	1	1	yes	characteristic
71	#21747	D₄	1	2	yes	complemented
72	#21746	D₄	1	2	yes	complemented
73	C₂×#1913	C₂×C₄	1	2	yes	—
74	C₂×#1913	C₂×C₄	1	2	yes	—
75	C₂×#1913	C₂×C₄	1	2	yes	complemented
76	C₂×#1913	C₂×C₄	1	2	yes	complemented
77	#20960	C₂×C₄	1	2	yes	—
78	#20960	C₂×C₄	1	2	yes	complemented
79	C₂×#1828	C₂×C₄	1	2	yes	complemented
80	C₂×#1783	C₂³	1	1	yes	characteristic
81	C₂×#1783	C₂³	1	1	yes	characteristic
82	#20585	C₂×C₄	1	2	yes	complemented
83	C₂×#1781	C₂³	1	1	yes	characteristic
84	C₂×#1781	C₂³	1	1	yes	characteristic
85	C₂×#1781	C₂³	1	1	yes	characteristic
86	C₂×#1781	C₂³	1	1	yes	characteristic
87	C₂×#1781	C₂³	1	1	yes	characteristic
88	C₂×#1781	C₂³	1	1	yes	characteristic
89	#20354	C₂×C₄	1	1	yes	characteristic, complemented
90	#14658	D₄	1	1	yes	characteristic, complemented
91	#14657	D₄	1	1	yes	characteristic, complemented
92	#14335	C₂³	1	1	yes	characteristic
93	#14167	D₄	1	1	yes	characteristic, complemented
94	#14134	D₄	1	1	yes	characteristic, complemented
95	C₂×#1438	C₂³	1	1	yes	characteristic
96	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
97	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
98	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
99	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
100	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
101	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
102	C₂×#543	C₂²⋊C₄	1	2	yes	—
103	C₂×#534	C₂²⋊C₄	1	2	yes	—
104	C₂×#534	C₂²⋊C₄	1	2	yes	—
105	C₂×#534	C₂²⋊C₄	1	2	yes	—
106	C₂×#534	C₂²×C₄	1	2	yes	—
107	C₂×#534	C₂²×C₄	1	2	yes	—
108	C₂×#534	C₂×D₄	1	2	yes	—
109	C₂×#534	C₂×D₄	1	2	yes	—
110	C₂×#534	#41	1	2	yes	—
111	C₂×#534	#41	1	2	yes	—
112	C₂²×#123	C₂×D₄	1	1	yes	characteristic
113	C₂²×#123	C₂×D₄	1	1	yes	characteristic
114	C₂²×#120	C₂×D₄	1	2	yes	—
115	C₂²×#120	C₂×D₄	1	2	yes	—
116	C₂²×#120	C₂×D₄	1	1	yes	characteristic
117	C₂²×#120	C₂×D₄	1	1	yes	characteristic
118	C₂×#518	C₂²⋊C₄	1	2	yes	—
119	C₂×#518	C₂²⋊C₄	1	2	yes	—
120	C₂×#518	C₂²⋊C₄	1	2	yes	—
121	C₂×#518	C₂²×C₄	1	1	yes	characteristic
122	C₂×#518	C₂²×C₄	1	1	yes	characteristic
123	C₂×#518	C₂²×C₄	1	1	yes	characteristic
124	C₂×#518	C₂²×C₄	1	1	yes	characteristic
125	C₂×#518	C₂×D₄	1	1	yes	characteristic
126	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
127	C₂²×#99	C₂²×C₄	1	2	yes	—
128	C₂²×#99	C₂²×C₄	1	2	yes	—
129	#1781	C₂×D₄	1	2	yes	—
130	#1781	C₂×D₄	1	2	yes	—
131	#1438	C₂×D₄	1	1	yes	characteristic
132	#1438	C₂×D₄	1	1	yes	characteristic
133	#1438	C₂×D₄	1	1	yes	characteristic
134	#1438	C₂×D₄	1	1	yes	characteristic
135	C₂⁶	#120	1	1	yes	characteristic
136	C₂⁶	C₂²×D₄	1	1	yes	characteristic
137	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
139	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
141	C₂³×D₄	#123	1	2	yes	—
142	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
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	C₂×#41	1	1	yes	characteristic
147	C₂³×D₄	C₂×#41	1	1	yes	characteristic
148	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂³×D₄	H(5,𝔽₂)	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	C₂⁴×C₄	#120	1	1	yes	characteristic
155	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×#123	#99	1	4	yes	complemented
157	C₂×#123	#99	1	4	yes	complemented
158	C₂×#123	C₄×D₄	1	2	yes	—
159	C₂×#123	C₄×D₄	1	2	yes	—
160	C₂×#123	#121	1	2	yes	—
161	C₂×#123	#121	1	2	yes	—
162	C₂×#120	#99	1	4	yes	complemented
163	C₂×#120	#99	1	4	yes	complemented
164	C₂×#120	C₄×D₄	1	2	yes	—
165	C₂×#120	C₄×D₄	1	2	yes	—
166	C₂×#120	#121	1	2	yes	—
167	C₂×#120	#121	1	2	yes	—
168	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
171	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
172	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
173	C₂²×C₂²⋊C₄	#123	1	2	yes	—
174	C₂⁵	#386	1	2	yes	—
175	C₂⁵	C₂×#99	1	2	yes	—
176	C₂⁵	C₂×#99	1	2	yes	—
177	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
178	C₂⁵	#518	1	1	yes	characteristic
179	C₂⁵	C₂×#120	1	1	yes	characteristic
180	C₂⁵	C₂×#123	1	1	yes	characteristic
181	C₂⁵	#534	1	1	yes	characteristic
182	C₂⁵	#534	1	1	yes	characteristic
183	C₂⁵	#534	1	1	yes	characteristic
184	C₂⁵	#534	1	1	yes	characteristic
185	C₂⁵	#534	1	1	yes	characteristic
186	C₂⁵	#534	1	1	yes	characteristic
187	C₂⁵	#535	1	1	yes	characteristic
188	C₂⁵	#538	1	1	yes	characteristic
189	C₂⁵	#538	1	1	yes	characteristic
190	C₂²×D₄	C₂×#99	1	4	yes	—
191	C₂²×D₄	C₂×#99	1	4	yes	—
192	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂²×D₄	#518	1	1	yes	characteristic
195	C₂²×D₄	#518	1	1	yes	characteristic
196	C₂²×D₄	C₂×#121	1	1	yes	characteristic
197	C₂²×D₄	C₂×#121	1	1	yes	characteristic
198	C₂²×D₄	#534	1	1	yes	characteristic
199	C₂²×D₄	#534	1	1	yes	characteristic
200	C₂²×D₄	#534	1	1	yes	characteristic
201	C₂²×D₄	#534	1	1	yes	characteristic
202	C₂²×D₄	#535	1	1	yes	characteristic
203	C₂²×D₄	#535	1	1	yes	characteristic
204	C₂²×D₄	#538	1	1	yes	characteristic
205	C₂²×D₄	#538	1	1	yes	characteristic
206	C₂²×D₄	#538	1	1	yes	characteristic
207	C₂²×D₄	#538	1	1	yes	characteristic
208	C₂³×C₄	#386	1	2	yes	—
209	C₂³×C₄	C₂×#99	1	2	yes	—
210	C₂³×C₄	C₂×#99	1	2	yes	—
211	C₂³×C₄	#518	1	1	yes	characteristic
212	C₂³×C₄	#534	1	1	yes	characteristic
213	C₂³×C₄	#534	1	1	yes	characteristic
214	C₂³×C₄	#535	1	1	yes	characteristic
215	C₂³×C₄	#538	1	1	yes	characteristic
216	C₂³×C₄	#538	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	#386	1	2	yes	—
218	C₂×C₂²⋊C₄	#386	1	2	yes	—
219	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
220	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
221	C₂⁴	#1898	1	4	yes	—
222	C₂⁴	C₂×#386	1	1	yes	characteristic
223	C₂⁴	#2308	1	1	yes	characteristic
224	C₂⁴	#2308	1	1	yes	characteristic
225	C₂⁴	#2321	1	1	yes	characteristic
226	C₂⁴	#2321	1	1	yes	characteristic
227	C₂⁴	#2613	1	1	yes	characteristic
228	C₂⁴	#2613	1	1	yes	characteristic
229	C₂⁴	#2617	1	1	yes	characteristic
230	C₂⁴	#2617	1	1	yes	characteristic
231	C₂⁴	#2622	1	1	yes	characteristic
232	C₂⁴	#2622	1	1	yes	characteristic
233	C₂⁴	C₂²×#99	1	2	yes	—
234	C₂⁴	C₂²×#99	1	2	yes	—
235	C₂²×C₄	#1898	1	4	yes	—
236	C₂²×C₄	#2321	1	1	yes	characteristic
237	C₂²×C₄	#2321	1	1	yes	characteristic
238	C₂²×C₄	#2617	1	1	yes	characteristic
239	C₂²×C₄	#2617	1	1	yes	characteristic
240	C₂³	#14255	1	1	yes	characteristic
241	C₂³	C₂×#1898	1	2	yes	—
242	C₂³	#21746	1	2	yes	—
243	C₂³	#21746	1	1	yes	characteristic
244	C₂³	#21746	1	1	yes	characteristic
245	C₂³	#21776	1	1	yes	characteristic
246	C₂³	#21776	1	1	yes	characteristic
247	C₂³	#21776	1	1	yes	characteristic
248	C₂³	#21777	1	1	yes	characteristic
249	C₂²	#213852	1	1	yes	characteristic
250	C₂²	#213854	1	1	yes	characteristic
251	C₂²	#213860	1	1	yes	characteristic
252	C₂²	#213864	1	1	yes	characteristic
253	C₂²	#379723	1	1	yes	characteristic
254	C₂²	#379806	1	1	yes	characteristic
255	C₂²	#379810	1	1	yes	characteristic
256	C₂	#72266146	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266146
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72234953	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ 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. #72310358	C₂	semidirect
2. #72310322	C₂	semidirect
3. #72309555	C₂	semidirect
4. #72266864	C₂	semidirect
5. #72266366	C₂	semidirect
6. #72258097	C₂	semidirect
7. #72255777	C₂	semidirect
8. #72255772	C₂	semidirect
9. #72251552	C₂	semidirect
10. #72238443	C₂	semidirect
11. #72237476	C₂	semidirect
12. #72237374	C₂	semidirect
14. #72219160	C₂	semidirect
15. #72219159	C₂	semidirect
17. #6451863	C₄	semidirect
18. #6437629	C₄	semidirect
24. C₂×#12707	C₂²	semidirect
27. C₂×#12705	C₂²	semidirect
33. #363969	C₄	semidirect
35. #359528	C₄	semidirect
41. C₂×#8534	C₂²	semidirect
42. #186835	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. #128033	C₂²	semidirect
46. C₂×#7304	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7314	C₂²	semidirect
49. C₂×#7310	C₂²	semidirect
50. C₂×#7310	C₂²	semidirect
51. #128009	C₂²	semidirect
52. C₂×#7307	C₂²	semidirect
53. #107130	C₂²	semidirect
54. #107114	C₂²	semidirect
58. #62842	C₂×C₄	semidirect
67. C₂×#3026	C₂×C₄	semidirect
68. C₂×#2883	C₂×C₄	semidirect
69. C₂×#2883	C₂×C₄	semidirect
71. #21747	D₄	semidirect
72. #21746	D₄	semidirect
75. C₂×#1913	C₂×C₄	semidirect
76. C₂×#1913	C₂×C₄	semidirect
78. #20960	C₂×C₄	semidirect
79. C₂×#1828	C₂×C₄	semidirect
82. #20585	C₂×C₄	semidirect
89. #20354	C₂×C₄	semidirect
90. #14658	D₄	semidirect
91. #14657	D₄	semidirect
93. #14167	D₄	semidirect
94. #14134	D₄	semidirect
156. C₂×#123	#99	semidirect
157. C₂×#123	#99	semidirect
162. C₂×#120	#99	semidirect
163. C₂×#120	#99	semidirect

Maximal subgroups

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

subgroup	index
#72310358	2
#72310322	2
#72309555	2
#72266864	2
#72266366	2
#72258097	2
#72255777	2
#72255772	2
#72251552	2
#72238443	2
#72237476	2
#72237374	2
#72234953	2
#72219160	2
#72219159	2