Subgroups — #72251479

Order 2048.

At least 6391 conjugacy classes of subgroups; only the 402 normal subgroups are shown.

The subgroup lattice has 245 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251476	C₂	1	2	yes	maximal, complemented
2	#72251475	C₂	1	2	yes	maximal, complemented
3	#72251473	C₂	1	2	yes	maximal, complemented
4	#72251471	C₂	1	2	yes	maximal, complemented
5	#72237398	C₂	1	1	yes	maximal, characteristic, complemented
6	#72225617	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219415	C₂	1	1	yes	maximal, characteristic, complemented
8	#72218455	C₂	1	1	yes	maximal, characteristic, complemented
9	#72218306	C₂	1	1	yes	maximal, characteristic, complemented
10	#72218198	C₂	1	1	yes	maximal, characteristic, complemented
11	#72210511	C₂	1	1	yes	maximal, characteristic
12	C₂²×#3484	C₂²	1	1	yes	characteristic
13	#7630939	C₂²	1	1	yes	characteristic
14	#6628867	C₄	1	2	yes	complemented
15	C₂×#23897	C₂²	1	2	yes	—
16	C₂×#21776	C₂²	1	1	yes	characteristic, complemented
17	C₂²×#1781	C₂²	1	1	yes	characteristic
18	#517411	C₂²	1	1	yes	characteristic
19	C₂×#12705	C₂²	1	2	yes	—
20	C₂×#12705	C₂²	1	2	yes	—
21	#393975	C₄	1	4	yes	complemented
22	#392696	C₄	1	2	yes	complemented
23	#392113	C₂²	1	1	yes	characteristic
24	#392113	C₂²	1	1	yes	characteristic
25	#392109	C₂²	1	1	yes	characteristic
26	#382544	C₂²	1	1	yes	characteristic, complemented
27	#382434	C₂²	1	1	yes	characteristic, complemented
28	#359558	C₂²	1	2	yes	—
29	#355096	C₂²	1	1	yes	characteristic, complemented
30	#187944	C₂²	1	2	yes	—
31	#187944	C₂²	1	2	yes	—
32	#186835	C₂²	1	2	yes	complemented
33	#128033	C₂²	1	2	yes	complemented
34	#128033	C₂²	1	2	yes	complemented
35	C₂×#7311	C₂²	1	2	yes	complemented
36	C₂×#7311	C₂²	1	2	yes	complemented
37	#108238	C₂²	1	2	yes	complemented
38	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂⁵×D₄	C₂³	1	1	yes	characteristic
40	C₂×#3484	C₂×C₄	1	2	yes	complemented
41	C₂×#3484	D₄	1	1	yes	characteristic
42	C₂×#3484	D₄	1	1	yes	characteristic
43	C₂×#3484	C₂³	1	2	yes	—
44	C₂×#3484	C₂³	1	1	yes	characteristic
45	C₂²×#518	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#3019	C₂×C₄	1	4	yes	complemented
47	#36614	D₄	1	2	yes	—
48	#35222	C₂×C₄	1	2	yes	—
49	#35222	C₂×C₄	1	2	yes	—
50	#35222	D₄	1	2	yes	—
51	#34171	D₄	1	1	yes	characteristic
52	#34171	D₄	1	1	yes	characteristic
53	#34171	C₂³	1	1	yes	characteristic
54	C₂²×#379	D₄	1	2	yes	—
55	C₂×#1911	C₂×C₄	1	4	yes	complemented
56	#20616	C₂×C₄	1	4	yes	complemented
57	C₂×#1783	C₂×C₄	1	2	yes	complemented
58	C₂×#1783	C₂×C₄	1	2	yes	complemented
59	C₂×#1783	C₂³	1	2	yes	—
60	C₂×#1783	C₂³	1	2	yes	—
61	C₂×#1781	D₄	1	2	yes	—
62	C₂×#1781	D₄	1	2	yes	—
63	C₂×#1781	D₄	1	2	yes	—
64	C₂×#1781	C₂³	1	1	yes	characteristic
65	C₂×#1781	C₂³	1	1	yes	characteristic
66	C₂×#1781	C₂³	1	1	yes	characteristic
67	C₂×#1781	C₂³	1	1	yes	characteristic
68	#14335	C₂³	1	2	yes	—
69	#10740	C₂×C₄	1	4	yes	complemented
70	#8534	D₄	1	8	yes	complemented
71	C₂⁷	C₂×D₄	1	1	yes	characteristic
72	C₂⁷	#41	1	1	yes	characteristic
73	C₂²×H(5,𝔽₂)	C₂²×C₄	1	2	yes	—
74	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
75	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
76	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
77	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
78	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
79	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
80	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
81	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
83	C₂⁴×D₄	#41	1	1	yes	characteristic
84	C₂⁴×D₄	#41	1	1	yes	characteristic
85	C₂⁴×D₄	#41	1	1	yes	characteristic
86	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
87	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
88	#3484	C₂×D₄	1	4	yes	—
89	#3484	#41	1	4	yes	—
90	C₂×#535	C₂²⋊C₄	1	2	yes	—
91	C₂×#535	C₂²⋊C₄	1	2	yes	—
92	C₂×#535	C₂²×C₄	1	2	yes	—
93	C₂×#525	C₂²⋊C₄	1	2	yes	—
94	C₂×#525	C₂²⋊C₄	1	2	yes	—
95	C₂×#525	C₂×D₄	1	2	yes	—
96	C₂×#518	C₂²×C₄	1	2	yes	—
97	C₂×#518	C₂×D₄	1	1	yes	characteristic
98	C₂×#518	C₂×D₄	1	1	yes	characteristic
99	C₂×#518	C₂×D₄	1	1	yes	characteristic
100	C₂²×#100	C₂²×C₄	1	4	yes	—
101	C₂²×#99	C₂×D₄	1	2	yes	—
102	C₂²×#99	C₂×D₄	1	2	yes	—
103	#2309	C₂²⋊C₄	1	2	yes	—
104	#2309	C₂²⋊C₄	1	2	yes	—
105	#2309	C₂²×C₄	1	2	yes	—
106	#2280	C₂²⋊C₄	1	2	yes	—
107	#2280	C₂²⋊C₄	1	2	yes	—
108	#2280	C₂×D₄	1	2	yes	—
109	C₂×#379	C₂×D₄	1	2	yes	—
110	C₂×#379	C₂×D₄	1	2	yes	—
111	#1781	C₂×D₄	1	4	yes	—
112	#1438	C₂×D₄	1	4	yes	—
113	C₂⁶	C₄×D₄	1	1	yes	characteristic
114	C₂⁶	C₄×D₄	1	1	yes	characteristic
115	C₂⁶	C₄×D₄	1	1	yes	characteristic
116	C₂⁶	C₄×D₄	1	1	yes	characteristic
117	C₂⁶	#120	1	1	yes	characteristic
118	C₂⁶	#120	1	1	yes	characteristic
119	C₂⁶	#121	1	1	yes	characteristic
120	C₂⁶	#121	1	1	yes	characteristic
121	C₂⁶	#121	1	1	yes	characteristic
122	C₂⁶	#123	1	1	yes	characteristic
123	C₂⁶	#123	1	1	yes	characteristic
124	C₂⁶	C₂²×D₄	1	1	yes	characteristic
125	C₂⁶	C₂×#41	1	1	yes	characteristic
126	C₂×H(5,𝔽₂)	C₄×D₄	1	4	yes	complemented
127	C₂²×#41	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
131	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
132	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
133	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
134	C₂³×D₄	#120	1	1	yes	characteristic
135	C₂³×D₄	#120	1	1	yes	characteristic
136	C₂³×D₄	#121	1	1	yes	characteristic
137	C₂³×D₄	#121	1	1	yes	characteristic
138	C₂³×D₄	#123	1	1	yes	characteristic
139	C₂³×D₄	#123	1	1	yes	characteristic
140	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
141	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
142	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
143	C₂³×D₄	C₂²×D₄	1	2	yes	—
144	C₂³×D₄	C₂×#41	1	1	yes	characteristic
145	C₂³×D₄	C₂×#41	1	2	yes	—
146	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
147	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
148	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂⁴×C₄	#121	1	1	yes	characteristic
151	C₂⁴×C₄	#121	1	1	yes	characteristic
152	C₂⁴×C₄	#121	1	1	yes	characteristic
153	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
154	C₂×#127	#99	1	4	yes	complemented
155	C₂×#124	#99	1	4	yes	complemented
156	#518	C₄×D₄	1	4	yes	complemented
157	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂×#117	C₂×C₂²⋊C₄	1	2	yes	—
160	#379	#120	1	4	yes	—
161	C₂⁵	#386	1	1	yes	characteristic
162	C₂⁵	#386	1	1	yes	characteristic
163	C₂⁵	#386	1	1	yes	characteristic
164	C₂⁵	#386	1	1	yes	characteristic
165	C₂⁵	#386	1	1	yes	characteristic
166	C₂⁵	#386	1	1	yes	characteristic
167	C₂⁵	B(4,𝔽₂)	1	2	yes	—
168	C₂⁵	B(4,𝔽₂)	1	2	yes	—
169	C₂⁵	B(4,𝔽₂)	1	2	yes	—
170	C₂⁵	B(4,𝔽₂)	1	2	yes	—
171	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂⁵	C₂×#120	1	1	yes	characteristic
174	C₂⁵	C₂×#120	1	1	yes	characteristic
175	C₂⁵	C₂×#121	1	1	yes	characteristic
176	C₂⁵	C₂×#123	1	1	yes	characteristic
177	C₂⁵	#525	1	1	yes	characteristic
178	C₂⁵	#534	1	1	yes	characteristic
179	C₂⁵	#535	1	1	yes	characteristic
180	C₂⁵	#546	1	2	yes	—
181	C₂⁵	#551	1	2	yes	—
182	C₂²×Q₈	C₂×#99	1	2	yes	complemented
183	C₂²×D₄	C₂×#99	1	4	yes	—
184	C₂²×D₄	C₂×#99	1	2	yes	complemented
185	C₂²×D₄	C₂×C₄×D₄	1	2	yes	—
186	C₂²×D₄	#518	1	2	yes	—
187	C₂²×D₄	#525	1	2	yes	—
188	C₂²×D₄	#538	1	2	yes	—
189	C₂²×D₄	D₄²	1	2	yes	—
190	C₂²×D₄	D₄²	1	2	yes	—
191	C₂²×D₄	#546	1	2	yes	—
192	C₂²×D₄	#546	1	2	yes	—
193	C₂³×C₄	#386	1	1	yes	characteristic
194	C₂³×C₄	#386	1	1	yes	characteristic
195	C₂³×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₄	C₂×#121	1	1	yes	characteristic
197	C₂³×C₄	D₄²	1	2	yes	—
198	C₂³×C₄	#546	1	2	yes	—
199	C₂×C₄²	C₂×#99	1	4	yes	complemented
200	C₂⁴	#1889	1	1	yes	characteristic
201	C₂⁴	#1889	1	1	yes	characteristic
202	C₂⁴	#1889	1	1	yes	characteristic
203	C₂⁴	#1889	1	2	yes	—
204	C₂⁴	#1889	1	1	yes	characteristic
205	C₂⁴	#1889	1	2	yes	—
206	C₂⁴	#1898	1	2	yes	—
207	C₂⁴	#1898	1	2	yes	—
208	C₂⁴	#1913	1	2	yes	—
209	C₂⁴	C₂×#386	1	1	yes	characteristic
210	C₂⁴	C₄×#120	1	1	yes	characteristic
211	C₂⁴	#2321	1	1	yes	characteristic
212	C₂⁴	#2358	1	2	yes	—
213	C₂⁴	#2403	1	1	yes	characteristic
214	C₂⁴	#2404	1	1	yes	characteristic
215	C₂⁴	#2408	1	1	yes	characteristic
216	C₂⁴	#2418	1	2	yes	—
217	C₂⁴	#2534	1	1	yes	characteristic
218	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
219	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
220	C₂²×C₄	#1913	1	2	yes	—
221	C₂²×C₄	D₄×C₂²⋊C₄	1	2	yes	—
222	C₂²×C₄	#2424	1	2	yes	—
223	C₂²×C₄	C₂²×#99	1	2	yes	—
224	C₂²×C₄	#2883	1	2	yes	—
225	C₂³	#9495	1	4	yes	—
226	C₂³	#13967	1	1	yes	characteristic
227	C₂³	C₂×#1889	1	1	yes	characteristic
228	C₂³	C₂×#1889	1	1	yes	characteristic
229	C₂³	C₂×#1898	1	1	yes	characteristic
230	C₂³	#20984	1	1	yes	characteristic
231	C₂³	C₂×#1913	1	1	yes	characteristic
232	C₂³	C₂×#1913	1	1	yes	characteristic
233	C₂³	#23900	1	2	yes	—
234	C₂³	#23984	1	2	yes	—
235	C₂×C₄	D₄×#99	1	4	yes	—
236	C₂²	C₂×#9495	1	1	yes	characteristic
237	C₂²	#201664	1	1	yes	characteristic
238	C₂²	#201678	1	1	yes	characteristic
239	C₂²	#201678	1	1	yes	characteristic
240	C₂²	#364410	1	1	yes	characteristic
241	C₂²	#364440	1	1	yes	characteristic
242	C₂²	#364471	1	1	yes	characteristic
243	C₂	#72240426	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240426
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72210511	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ 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. #72251476	C₂	semidirect
2. #72251475	C₂	semidirect
3. #72251473	C₂	semidirect
4. #72251471	C₂	semidirect
5. #72237398	C₂	semidirect
6. #72225617	C₂	semidirect
7. #72219415	C₂	semidirect
8. #72218455	C₂	semidirect
9. #72218306	C₂	semidirect
10. #72218198	C₂	semidirect
14. #6628867	C₄	semidirect
16. C₂×#21776	C₂²	semidirect
21. #393975	C₄	semidirect
22. #392696	C₄	semidirect
26. #382544	C₂²	semidirect
27. #382434	C₂²	semidirect
29. #355096	C₂²	semidirect
32. #186835	C₂²	semidirect
33. #128033	C₂²	semidirect
34. #128033	C₂²	semidirect
35. C₂×#7311	C₂²	semidirect
36. C₂×#7311	C₂²	semidirect
37. #108238	C₂²	semidirect
38. C₂³×H(5,𝔽₂)	C₂×C₄	semidirect
40. C₂×#3484	C₂×C₄	semidirect
45. C₂²×#518	C₂×C₄	semidirect
46. C₂×#3019	C₂×C₄	semidirect
55. C₂×#1911	C₂×C₄	semidirect
56. #20616	C₂×C₄	semidirect
57. C₂×#1783	C₂×C₄	semidirect
58. C₂×#1783	C₂×C₄	semidirect
69. #10740	C₂×C₄	semidirect
70. #8534	D₄	semidirect
126. C₂×H(5,𝔽₂)	C₄×D₄	semidirect
154. C₂×#127	#99	semidirect
155. C₂×#124	#99	semidirect
156. #518	C₄×D₄	semidirect
182. C₂²×Q₈	C₂×#99	semidirect
184. C₂²×D₄	C₂×#99	semidirect
199. C₂×C₄²	C₂×#99	semidirect

Maximal subgroups

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

subgroup	index
#72251476	2
#72251475	2
#72251473	2
#72251471	2
#72237398	2
#72225617	2
#72219415	2
#72218455	2
#72218306	2
#72218198	2
#72210511	2