Subgroups — #72310377

Order 2048.

At least 6411 conjugacy classes of subgroups; only the 452 normal subgroups are shown.

The subgroup lattice has 265 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310366	C₂	1	2	yes	maximal, complemented
2	#72310358	C₂	1	2	yes	maximal, complemented
3	#72258368	C₂	1	2	yes	maximal, complemented
4	C₂×#139739	C₂	1	2	yes	maximal, complemented
5	#72219299	C₂	1	1	yes	maximal, characteristic, complemented
6	#72219251	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219243	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#8534	C₂	1	1	yes	maximal, characteristic, complemented
9	#72218313	C₂	1	1	yes	maximal, characteristic, complemented
10	#72218069	C₂	1	1	yes	maximal, characteristic, complemented
11	#72213862	C₂	1	1	yes	maximal, characteristic
12	C₂²×#3484	C₂²	1	1	yes	characteristic
13	#7630939	C₂²	1	1	yes	characteristic
14	C₂×#23895	C₄	1	2	yes	complemented
15	C₂×#23895	C₂²	1	1	yes	characteristic
16	C₂×#23895	C₂²	1	1	yes	characteristic
17	#6437621	C₄	1	2	yes	complemented
18	C₂×#21778	C₂²	1	2	yes	—
19	C₂²×#1781	C₂²	1	1	yes	characteristic
20	C₂²×#1438	C₂²	1	1	yes	characteristic
21	C₂×#12705	C₂²	1	2	yes	complemented
22	#392113	C₂²	1	1	yes	characteristic
23	#392113	C₂²	1	1	yes	characteristic
24	#392109	C₂²	1	1	yes	characteristic
25	#382436	C₂²	1	2	yes	—
26	#359549	C₄	1	2	yes	complemented
27	#359549	C₂²	1	1	yes	characteristic
28	#359549	C₂²	1	1	yes	characteristic
29	#359442	C₄	1	2	yes	complemented
30	C₂×#8534	C₂²	1	2	yes	—
31	C₂×#8534	C₂²	1	2	yes	—
32	C₂×#8453	C₂²	1	2	yes	complemented
33	#146475	C₂²	1	2	yes	complemented
34	C₂×#7304	C₂²	1	2	yes	complemented
35	C₂×#7304	C₂²	1	2	yes	complemented
36	C₂×#7314	C₂²	1	2	yes	complemented
37	C₂×#7314	C₂²	1	2	yes	complemented
38	#107130	C₂²	1	2	yes	complemented
39	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic
40	C₂⁵×D₄	C₂³	1	1	yes	characteristic
41	C₂×#3484	C₂×C₄	1	2	yes	—
42	C₂×#3484	C₂×C₄	1	1	yes	characteristic
43	C₂×#3484	C₂×C₄	1	1	yes	characteristic
44	C₂×#3484	D₄	1	2	yes	—
45	C₂×#3484	D₄	1	1	yes	characteristic
46	C₂×#3484	D₄	1	1	yes	characteristic
47	C₂×#3484	C₂³	1	1	yes	characteristic
48	C₂×#3484	C₂³	1	1	yes	characteristic
49	C₂×#3484	C₂³	1	1	yes	characteristic
50	C₂²×#534	C₂×C₄	1	2	yes	—
51	C₂³×#120	D₄	1	2	yes	—
52	C₂²×#518	D₄	1	1	yes	characteristic
53	C₂×#3026	C₂×C₄	1	2	yes	complemented
54	C₂×#3026	C₂×C₄	1	2	yes	complemented
55	#34171	C₂×C₄	1	1	yes	characteristic
56	#34171	C₂×C₄	1	1	yes	characteristic
57	#34171	C₂³	1	1	yes	characteristic
58	C₂×#1913	C₂×C₄	1	2	yes	complemented
59	C₂×#1913	C₂×C₄	1	2	yes	complemented
60	C₂×#1913	C₂×C₄	1	2	yes	complemented
61	C₂×#1913	C₂×C₄	1	2	yes	complemented
62	#20960	C₂×C₄	1	2	yes	complemented
63	#20960	C₂×C₄	1	2	yes	complemented
64	C₂×#1783	D₄	1	2	yes	complemented
65	C₂×#1783	D₄	1	2	yes	complemented
66	C₂×#1783	D₄	1	2	yes	complemented
67	C₂×#1783	C₂³	1	2	yes	—
68	C₂×#1783	C₂³	1	2	yes	—
69	C₂×#1781	C₂×C₄	1	2	yes	—
70	C₂×#1781	C₂×C₄	1	2	yes	—
71	C₂×#1781	C₂³	1	2	yes	—
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	#20579	D₄	1	2	yes	complemented
75	C₂×#1438	C₂³	1	2	yes	—
76	#8534	D₄	1	8	yes	complemented
77	C₂⁷	C₂×D₄	1	1	yes	characteristic
78	C₂⁷	#41	1	1	yes	characteristic
79	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
80	C₂²×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
81	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
82	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
84	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
85	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
86	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
87	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
88	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
89	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
90	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
92	C₂⁴×D₄	#41	1	1	yes	characteristic
93	C₂⁴×D₄	#41	1	1	yes	characteristic
94	C₂⁴×D₄	#41	1	1	yes	characteristic
95	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
96	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
97	#3484	C₂×D₄	1	4	yes	—
98	#3484	#41	1	4	yes	—
99	C₂×#534	C₂²⋊C₄	1	2	yes	—
100	C₂×#534	C₂²⋊C₄	1	2	yes	—
101	C₂×#534	C₂²×C₄	1	2	yes	—
102	C₂×#534	C₂²×C₄	1	2	yes	—
103	C₂²×#120	C₂²⋊C₄	1	2	yes	—
104	C₂²×#120	C₂²⋊C₄	1	2	yes	—
105	C₂²×#120	C₂×D₄	1	2	yes	—
106	C₂²×#120	C₂×D₄	1	2	yes	—
107	C₂×#518	C₂²⋊C₄	1	2	yes	—
108	C₂×#518	C₂²×C₄	1	1	yes	characteristic
109	C₂×#518	C₂²×C₄	1	1	yes	characteristic
110	C₂×#518	C₂²×C₄	1	1	yes	characteristic
111	C₂×#518	C₂×D₄	1	1	yes	characteristic
112	C₂×#518	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
114	C₂²×#100	C₂×D₄	1	4	yes	—
115	C₂²×#99	C₂²×C₄	1	2	yes	—
116	C₂²×#99	C₂²×C₄	1	2	yes	—
117	C₂×#407	C₂×D₄	1	4	yes	—
118	#1781	C₂×D₄	1	4	yes	—
119	#1438	C₂×D₄	1	4	yes	—
120	C₂⁶	#99	1	2	yes	—
121	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂⁶	C₄×D₄	1	1	yes	characteristic
124	C₂⁶	C₄×D₄	1	1	yes	characteristic
125	C₂⁶	#121	1	1	yes	characteristic
126	C₂⁶	#121	1	1	yes	characteristic
127	C₂⁶	#121	1	1	yes	characteristic
128	C₂⁶	#121	1	1	yes	characteristic
129	C₂⁶	#121	1	1	yes	characteristic
130	C₂⁶	#123	1	1	yes	characteristic
131	C₂⁶	#123	1	1	yes	characteristic
132	C₂⁶	C₂²×D₄	1	1	yes	characteristic
133	C₂⁶	C₂×#41	1	1	yes	characteristic
134	C₂×H(5,𝔽₂)	#121	1	4	yes	—
135	C₂³×D₄	#99	1	2	yes	—
136	C₂³×D₄	#99	1	2	yes	—
137	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
141	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
142	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
143	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
145	C₂³×D₄	#120	1	1	yes	characteristic
146	C₂³×D₄	#120	1	1	yes	characteristic
147	C₂³×D₄	#121	1	1	yes	characteristic
148	C₂³×D₄	#121	1	1	yes	characteristic
149	C₂³×D₄	#123	1	1	yes	characteristic
150	C₂³×D₄	#123	1	1	yes	characteristic
151	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
152	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
153	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
154	C₂³×D₄	C₂²×D₄	1	2	yes	—
155	C₂³×D₄	C₂×#41	1	1	yes	characteristic
156	C₂³×D₄	C₂×#41	1	2	yes	—
157	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
158	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
159	C₂⁴×C₄	#99	1	2	yes	—
160	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
161	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
162	C₂⁴×C₄	#120	1	1	yes	characteristic
163	C₂⁴×C₄	#120	1	1	yes	characteristic
164	C₂⁴×C₄	#121	1	1	yes	characteristic
165	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
166	#534	C₄×D₄	1	8	yes	—
167	C₂×#120	#99	1	8	yes	—
168	C₂×#120	#120	1	4	yes	—
169	C₂×#120	#123	1	4	yes	—
170	#518	#121	1	4	yes	—
171	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
172	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
173	C₂⁵	#386	1	1	yes	characteristic
174	C₂⁵	#386	1	1	yes	characteristic
175	C₂⁵	#386	1	1	yes	characteristic
176	C₂⁵	#386	1	1	yes	characteristic
177	C₂⁵	#386	1	1	yes	characteristic
178	C₂⁵	#386	1	1	yes	characteristic
179	C₂⁵	C₂×#99	1	2	yes	—
180	C₂⁵	C₂×#99	1	2	yes	—
181	C₂⁵	C₂×#99	1	2	yes	—
182	C₂⁵	C₂×#99	1	4	yes	—
183	C₂⁵	C₂×#99	1	2	yes	—
184	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
186	C₂⁵	C₂×#120	1	1	yes	characteristic
187	C₂⁵	C₂×#121	1	1	yes	characteristic
188	C₂⁵	C₂×#121	1	1	yes	characteristic
189	C₂⁵	C₂×#123	1	1	yes	characteristic
190	C₂⁵	C₂×#123	1	1	yes	characteristic
191	C₂⁵	#534	1	1	yes	characteristic
192	C₂⁵	#534	1	1	yes	characteristic
193	C₂⁵	#546	1	2	yes	—
194	C₂⁵	#551	1	2	yes	—
195	C₂²×D₄	#386	1	4	yes	—
196	C₂²×D₄	C₂×#99	1	4	yes	—
197	C₂²×D₄	C₂×C₄×D₄	1	2	yes	—
198	C₂²×D₄	#518	1	2	yes	—
199	C₂²×D₄	C₂×#121	1	2	yes	—
200	C₂²×D₄	#534	1	2	yes	—
201	C₂²×D₄	D₄²	1	2	yes	—
202	C₂²×D₄	#546	1	2	yes	—
203	C₂²×D₄	#546	1	2	yes	—
204	C₂²×D₄	#551	1	2	yes	—
205	C₂³×C₄	#386	1	1	yes	characteristic
206	C₂³×C₄	#386	1	1	yes	characteristic
207	C₂³×C₄	C₂×#99	1	2	yes	—
208	C₂³×C₄	C₂×#99	1	2	yes	—
209	C₂³×C₄	B(4,𝔽₂)	1	4	yes	—
210	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
211	C₂³×C₄	C₂×#120	1	1	yes	characteristic
212	C₂³×C₄	D₄²	1	2	yes	—
213	C₂³×C₄	#546	1	2	yes	—
214	C₂×C₂²⋊C₄	#386	1	4	yes	—
215	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
216	C₂⁴	#1889	1	2	yes	—
217	C₂⁴	#1898	1	1	yes	characteristic
218	C₂⁴	#1898	1	2	yes	—
219	C₂⁴	#1898	1	2	yes	—
220	C₂⁴	#1898	1	2	yes	—
221	C₂⁴	#1898	1	1	yes	characteristic
222	C₂⁴	#1898	1	4	yes	—
223	C₂⁴	#1913	1	2	yes	—
224	C₂⁴	#1913	1	2	yes	—
225	C₂⁴	C₂×#386	1	1	yes	characteristic
226	C₂⁴	C₂×#386	1	1	yes	characteristic
227	C₂⁴	#2308	1	1	yes	characteristic
228	C₂⁴	#2358	1	2	yes	—
229	C₂⁴	#2410	1	1	yes	characteristic
230	C₂⁴	#2410	1	1	yes	characteristic
231	C₂⁴	#2412	1	1	yes	characteristic
232	C₂⁴	#2412	1	2	yes	—
233	C₂⁴	#2412	1	1	yes	characteristic
234	C₂⁴	C₂²×#99	1	2	yes	—
235	C₂⁴	C₂²×#99	1	2	yes	—
236	C₂²×C₄	#1889	1	2	yes	—
237	C₂²×C₄	#1898	1	4	yes	—
238	C₂²×C₄	#2358	1	2	yes	—
239	C₂²×C₄	#2410	1	2	yes	—
240	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
241	C₂²×C₄	#3026	1	2	yes	—
242	C₂³	#9495	1	2	yes	—
243	C₂³	#9495	1	2	yes	—
244	C₂³	#14134	1	1	yes	characteristic
245	C₂³	C₂×#1889	1	1	yes	characteristic
246	C₂³	#20960	1	1	yes	characteristic
247	C₂³	C₂×#1898	1	1	yes	characteristic
248	C₂³	C₂×#1898	1	2	yes	—
249	C₂³	C₂×#1898	1	1	yes	characteristic
250	C₂³	C₂×#1913	1	1	yes	characteristic
251	C₂³	C₂×#1913	1	1	yes	characteristic
252	C₂³	#21746	1	2	yes	—
253	C₂×C₄	#24404	1	4	yes	—
254	C₂²	#139739	1	4	yes	complemented
255	C₂²	C₂×#9495	1	1	yes	characteristic
256	C₂²	#201664	1	1	yes	characteristic
257	C₂²	#201687	1	1	yes	characteristic
258	C₂²	#201701	1	1	yes	characteristic
259	C₂²	#374022	1	1	yes	characteristic
260	C₂²	#374038	1	1	yes	characteristic
261	C₂²	#374058	1	1	yes	characteristic
262	C₂	#72240425	1	1	yes	characteristic
263	C₂	C₂×#139739	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#9495
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72213862	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁵×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. #72310366	C₂	semidirect
2. #72310358	C₂	semidirect
3. #72258368	C₂	semidirect
4. C₂×#139739	C₂	semidirect
5. #72219299	C₂	semidirect
6. #72219251	C₂	semidirect
7. #72219243	C₂	semidirect
8. C₂²×#8534	C₂	semidirect
9. #72218313	C₂	semidirect
10. #72218069	C₂	semidirect
14. C₂×#23895	C₄	semidirect
17. #6437621	C₄	semidirect
21. C₂×#12705	C₂²	semidirect
26. #359549	C₄	semidirect
29. #359442	C₄	semidirect
32. C₂×#8453	C₂²	semidirect
33. #146475	C₂²	semidirect
34. C₂×#7304	C₂²	semidirect
35. C₂×#7304	C₂²	semidirect
36. C₂×#7314	C₂²	semidirect
37. C₂×#7314	C₂²	semidirect
38. #107130	C₂²	semidirect
53. C₂×#3026	C₂×C₄	semidirect
54. C₂×#3026	C₂×C₄	semidirect
58. C₂×#1913	C₂×C₄	semidirect
59. C₂×#1913	C₂×C₄	semidirect
60. C₂×#1913	C₂×C₄	semidirect
61. C₂×#1913	C₂×C₄	semidirect
62. #20960	C₂×C₄	semidirect
63. #20960	C₂×C₄	semidirect
64. C₂×#1783	D₄	semidirect
65. C₂×#1783	D₄	semidirect
66. C₂×#1783	D₄	semidirect
74. #20579	D₄	semidirect
76. #8534	D₄	semidirect
254. C₂²	#139739	semidirect

Maximal subgroups

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

subgroup	index
#72310366	2
#72310358	2
#72258368	2
C₂×#139739	2
#72219299	2
#72219251	2
#72219243	2
C₂²×#8534	2
#72218313	2
#72218069	2
#72213862	2