Subgroups — #72312643

Order 2048.

At least 6994 conjugacy classes of subgroups; only the 498 normal subgroups are shown.

The subgroup lattice has 278 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72256052	C₂	1	2	yes	maximal, complemented
2	#72255971	C₂	1	2	yes	maximal, complemented
3	#72255830	C₂	1	2	yes	maximal, complemented
4	#72255764	C₂	1	2	yes	maximal, complemented
5	#72234661	C₂	1	1	yes	maximal, characteristic
6	#72219415	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219414	C₂	1	1	yes	maximal, characteristic, complemented
8	#72219310	C₂	1	1	yes	maximal, characteristic, complemented
9	#72219300	C₂	1	1	yes	maximal, characteristic, complemented
10	#72219299	C₂	1	1	yes	maximal, characteristic, complemented
11	#72218306	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#3484	C₂²	1	1	yes	characteristic
13	#7641845	C₄	1	2	yes	complemented
14	C₂×#34980	C₂²	1	1	yes	characteristic
15	#6440460	C₄	1	2	yes	complemented
16	#6440460	C₂²	1	1	yes	characteristic
17	#6440460	C₂²	1	1	yes	characteristic
18	#6437621	C₄	1	2	yes	complemented
19	#6437621	C₂²	1	1	yes	characteristic
20	#6437621	C₂²	1	1	yes	characteristic
21	C₂×#21778	C₂²	1	2	yes	—
22	C₂×#21746	C₄	1	2	yes	complemented
23	C₂²×#1781	C₂²	1	1	yes	characteristic
24	#493879	C₂²	1	2	yes	complemented
25	C₂×#12705	C₂²	1	2	yes	—
26	C₂×#12705	C₂²	1	2	yes	—
27	#493357	C₂²	1	2	yes	complemented
28	#392113	C₂²	1	1	yes	characteristic
29	#392113	C₂²	1	1	yes	characteristic
30	#392113	C₂²	1	1	yes	characteristic
31	#392109	C₂²	1	1	yes	characteristic
32	#382436	C₂²	1	2	yes	—
33	#190147	C₂²	1	2	yes	complemented
34	#188109	C₂²	1	2	yes	complemented
35	C₂×#7316	C₂²	1	2	yes	complemented
36	C₂×#7316	C₂²	1	2	yes	complemented
37	C₂×#7311	C₂²	1	2	yes	complemented
38	C₂×#7311	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	D₄	1	2	yes	—
43	C₂×#3484	D₄	1	1	yes	characteristic
44	C₂×#3484	D₄	1	1	yes	characteristic
45	C₂×#3484	C₂³	1	1	yes	characteristic
46	C₂×#3484	C₂³	1	1	yes	characteristic
47	C₂²×#534	C₂×C₄	1	1	yes	characteristic
48	C₂²×#534	D₄	1	2	yes	—
49	C₂²×#534	C₂³	1	1	yes	characteristic
50	C₂³×#120	C₂×C₄	1	2	yes	—
51	C₂³×#120	C₂×C₄	1	1	yes	characteristic
52	C₂²×#518	D₄	1	1	yes	characteristic
53	#34992	C₂×C₄	1	2	yes	—
54	#34992	C₂×C₄	1	2	yes	—
55	#34980	C₂×C₄	1	2	yes	—
56	#34980	C₂×C₄	1	2	yes	—
57	C₂×#2308	C₂×C₄	1	1	yes	characteristic
58	C₂×#2308	C₂³	1	1	yes	characteristic
59	C₂²×#379	C₂×C₄	1	2	yes	—
60	C₂²×#379	C₂×C₄	1	1	yes	characteristic
61	#24058	D₄	1	8	yes	—
62	#23901	D₄	1	4	yes	—
63	#23900	D₄	1	4	yes	—
64	C₂×#1898	C₂×C₄	1	2	yes	—
65	C₂×#1898	C₂×C₄	1	2	yes	—
66	C₂×#1898	C₂×C₄	1	2	yes	—
67	C₂×#1898	C₂×C₄	1	2	yes	—
68	C₂×#1783	D₄	1	2	yes	complemented
69	C₂×#1783	D₄	1	2	yes	complemented
70	C₂×#1783	D₄	1	2	yes	complemented
71	C₂×#1783	C₂³	1	2	yes	—
72	C₂×#1783	C₂³	1	2	yes	—
73	C₂×#1783	C₂³	1	2	yes	—
74	C₂×#1781	C₂×C₄	1	2	yes	—
75	C₂×#1781	C₂³	1	2	yes	—
76	C₂×#1781	C₂³	1	1	yes	characteristic
77	C₂×#1781	C₂³	1	1	yes	characteristic
78	#20579	D₄	1	2	yes	complemented
79	C₂⁷	C₂²×C₄	1	1	yes	characteristic
80	C₂⁷	#41	1	1	yes	characteristic
81	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	2	yes	—
82	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
84	C₂⁴×D₄	C₂²⋊C₄	1	2	yes	—
85	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
86	C₂⁴×D₄	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
93	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
94	C₂⁴×D₄	#41	1	1	yes	characteristic
95	C₂⁴×D₄	#41	1	1	yes	characteristic
96	C₂⁴×D₄	#41	1	1	yes	characteristic
97	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
98	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
99	#3484	C₂×D₄	1	4	yes	—
100	#3484	C₂×D₄	1	4	yes	—
101	C₂×#534	C₂²⋊C₄	1	2	yes	—
102	C₂×#534	C₂²⋊C₄	1	2	yes	—
103	C₂×#534	C₂×D₄	1	2	yes	—
104	C₂×#534	C₂×D₄	1	2	yes	—
105	C₂²×#123	C₂×D₄	1	2	yes	—
106	C₂²×#120	C₂²⋊C₄	1	2	yes	—
107	C₂²×#120	C₂²⋊C₄	1	2	yes	—
108	C₂²×#120	C₂²×C₄	1	2	yes	—
109	C₂²×#120	C₂²×C₄	1	2	yes	—
110	C₂²×#120	C₂×D₄	1	2	yes	—
111	C₂×#518	C₂²⋊C₄	1	2	yes	—
112	C₂×#518	C₂×D₄	1	1	yes	characteristic
113	C₂×#518	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
115	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
118	C₂²×#100	C₂×D₄	1	4	yes	—
119	C₂×#407	C₂×D₄	1	4	yes	—
120	#2849	C₂²⋊C₄	1	4	yes	complemented
121	#2846	C₂²⋊C₄	1	4	yes	complemented
122	#2622	C₂²⋊C₄	1	8	yes	complemented
123	#2308	C₂×D₄	1	4	yes	—
124	C₂×#379	C₂²×C₄	1	2	yes	—
125	C₂×#379	C₂²×C₄	1	2	yes	—
126	#1898	C₂²⋊C₄	1	8	yes	complemented
127	#1898	C₂²⋊C₄	1	8	yes	complemented
128	#1781	C₂×D₄	1	4	yes	—
129	#1781	C₂×D₄	1	4	yes	—
130	C₂⁶	#99	1	2	yes	—
131	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂⁶	C₄×D₄	1	1	yes	characteristic
135	C₂⁶	C₄×D₄	1	1	yes	characteristic
136	C₂⁶	C₄×D₄	1	1	yes	characteristic
137	C₂⁶	C₄×D₄	1	1	yes	characteristic
138	C₂⁶	C₄×D₄	1	1	yes	characteristic
139	C₂⁶	#121	1	1	yes	characteristic
140	C₂⁶	#121	1	1	yes	characteristic
141	C₂⁶	#123	1	1	yes	characteristic
142	C₂⁶	#123	1	1	yes	characteristic
143	C₂⁶	C₂³×C₄	1	1	yes	characteristic
144	C₂⁶	C₂×#41	1	1	yes	characteristic
145	C₂×H(5,𝔽₂)	#120	1	4	yes	—
146	C₂³×D₄	#99	1	2	yes	—
147	C₂³×D₄	#99	1	2	yes	—
148	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂³×D₄	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
153	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
154	C₂³×D₄	#120	1	1	yes	characteristic
155	C₂³×D₄	#120	1	1	yes	characteristic
156	C₂³×D₄	#121	1	1	yes	characteristic
157	C₂³×D₄	#121	1	1	yes	characteristic
158	C₂³×D₄	#121	1	1	yes	characteristic
159	C₂³×D₄	#121	1	1	yes	characteristic
160	C₂³×D₄	#123	1	1	yes	characteristic
161	C₂³×D₄	#123	1	1	yes	characteristic
162	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
163	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
164	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
165	C₂³×D₄	C₂²×D₄	1	2	yes	—
166	C₂³×D₄	C₂²×D₄	1	2	yes	—
167	C₂³×D₄	C₂×#41	1	1	yes	characteristic
168	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
169	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
170	C₂⁴×C₄	#99	1	2	yes	—
171	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
173	C₂⁴×C₄	#120	1	1	yes	characteristic
174	C₂⁴×C₄	#120	1	1	yes	characteristic
175	C₂⁴×C₄	#121	1	1	yes	characteristic
176	C₂⁴×C₄	#121	1	1	yes	characteristic
177	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
178	#534	#120	1	8	yes	—
179	C₂×#120	#99	1	4	yes	—
180	C₂×#120	#99	1	4	yes	—
181	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
182	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
183	#518	#120	1	4	yes	—
184	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
185	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
186	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
188	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
189	#379	C₂×C₂²⋊C₄	1	4	yes	—
190	#379	C₂×C₂²⋊C₄	1	4	yes	—
191	C₂⁵	#386	1	1	yes	characteristic
192	C₂⁵	#386	1	1	yes	characteristic
193	C₂⁵	#386	1	1	yes	characteristic
194	C₂⁵	#386	1	1	yes	characteristic
195	C₂⁵	#386	1	1	yes	characteristic
196	C₂⁵	#386	1	1	yes	characteristic
197	C₂⁵	C₂×#99	1	2	yes	—
198	C₂⁵	C₂×#99	1	2	yes	—
199	C₂⁵	C₂×#99	1	2	yes	—
200	C₂⁵	C₂×#99	1	2	yes	—
201	C₂⁵	C₂×#99	1	2	yes	—
202	C₂⁵	C₂×#99	1	2	yes	—
203	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
204	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
205	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
206	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂⁵	#518	1	1	yes	characteristic
209	C₂⁵	#518	1	1	yes	characteristic
210	C₂⁵	C₂×#121	1	1	yes	characteristic
211	C₂⁵	C₂×#123	1	1	yes	characteristic
212	C₂⁵	#546	1	2	yes	—
213	C₂⁵	#546	1	2	yes	—
214	C₂²×D₄	#379	1	4	yes	—
215	C₂²×D₄	C₂×#99	1	4	yes	—
216	C₂²×D₄	C₂×#120	1	2	yes	—
217	C₂²×D₄	C₂×#120	1	2	yes	—
218	C₂²×D₄	#534	1	2	yes	—
219	C₂²×D₄	#534	1	2	yes	—
220	C₂²×D₄	D₄²	1	2	yes	—
221	C₂²×D₄	D₄²	1	2	yes	—
222	C₂²×D₄	#546	1	2	yes	—
223	C₂²×D₄	#546	1	2	yes	—
224	C₂³×C₄	#386	1	1	yes	characteristic
225	C₂³×C₄	#386	1	1	yes	characteristic
226	C₂³×C₄	C₂×#99	1	2	yes	—
227	C₂³×C₄	C₂×#99	1	2	yes	—
228	C₂³×C₄	C₂×#120	1	1	yes	characteristic
229	C₂³×C₄	C₂×#121	1	1	yes	characteristic
230	C₂³×C₄	D₄²	1	2	yes	—
231	C₂³×C₄	D₄²	1	2	yes	—
232	C₂×C₂²⋊C₄	#379	1	4	yes	—
233	C₂×C₂²⋊C₄	C₂×#99	1	4	yes	—
234	C₂⁴	#1781	1	4	yes	—
235	C₂⁴	#1898	1	1	yes	characteristic
236	C₂⁴	#1898	1	1	yes	characteristic
237	C₂⁴	#1898	1	1	yes	characteristic
238	C₂⁴	#1898	1	2	yes	—
239	C₂⁴	#1898	1	1	yes	characteristic
240	C₂⁴	#1898	1	2	yes	—
241	C₂⁴	#1913	1	2	yes	—
242	C₂⁴	C₂×#379	1	1	yes	characteristic
243	C₂⁴	C₂×#386	1	1	yes	characteristic
244	C₂⁴	#2308	1	1	yes	characteristic
245	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
246	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
247	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
248	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
249	C₂⁴	#2405	1	2	yes	—
250	C₂⁴	#2410	1	2	yes	—
251	C₂⁴	C₂²×#99	1	2	yes	—
252	C₂⁴	C₂²×#99	1	2	yes	—
253	C₂²×C₄	#1781	1	4	yes	—
254	C₂²×C₄	#1913	1	2	yes	—
255	C₂²×C₄	#2403	1	2	yes	—
256	C₂²×C₄	#2410	1	2	yes	—
257	C₂²×C₄	#3026	1	2	yes	—
258	C₂²×C₄	#3026	1	2	yes	—
259	C₂³	#13867	1	1	yes	characteristic
260	C₂³	C₂×#1781	1	2	yes	—
261	C₂³	#20960	1	1	yes	characteristic
262	C₂³	#20960	1	1	yes	characteristic
263	C₂³	C₂×#1898	1	1	yes	characteristic
264	C₂³	C₂×#1898	1	1	yes	characteristic
265	C₂³	C₂×#1913	1	1	yes	characteristic
266	C₂³	C₂×#1913	1	1	yes	characteristic
267	C₂³	#21746	1	2	yes	—
268	C₂×C₄	#23903	1	4	yes	—
269	C₂²	#201687	1	1	yes	characteristic
270	C₂²	#201687	1	1	yes	characteristic
271	C₂²	#201701	1	1	yes	characteristic
272	C₂²	#201701	1	1	yes	characteristic
273	C₂²	#359428	1	1	yes	characteristic
274	C₂²	#359442	1	1	yes	characteristic
275	C₂²	#359504	1	1	yes	characteristic
276	C₂	#72236815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72236815
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72234661	C₂
LCS	G ▷ C₂⁶ ▷ 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. #72256052	C₂	semidirect
2. #72255971	C₂	semidirect
3. #72255830	C₂	semidirect
4. #72255764	C₂	semidirect
6. #72219415	C₂	semidirect
7. #72219414	C₂	semidirect
8. #72219310	C₂	semidirect
9. #72219300	C₂	semidirect
10. #72219299	C₂	semidirect
11. #72218306	C₂	semidirect
13. #7641845	C₄	semidirect
15. #6440460	C₄	semidirect
18. #6437621	C₄	semidirect
22. C₂×#21746	C₄	semidirect
24. #493879	C₂²	semidirect
27. #493357	C₂²	semidirect
33. #190147	C₂²	semidirect
34. #188109	C₂²	semidirect
35. C₂×#7316	C₂²	semidirect
36. C₂×#7316	C₂²	semidirect
37. C₂×#7311	C₂²	semidirect
38. C₂×#7311	C₂²	semidirect
68. C₂×#1783	D₄	semidirect
69. C₂×#1783	D₄	semidirect
70. C₂×#1783	D₄	semidirect
78. #20579	D₄	semidirect
120. #2849	C₂²⋊C₄	semidirect
121. #2846	C₂²⋊C₄	semidirect
122. #2622	C₂²⋊C₄	semidirect
126. #1898	C₂²⋊C₄	semidirect
127. #1898	C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72256052	2
#72255971	2
#72255830	2
#72255764	2
#72234661	2
#72219415	2
#72219414	2
#72219310	2
#72219300	2
#72219299	2
#72218306	2