Subgroups — #72256439

Order 4096.

At least 12985 conjugacy classes of subgroups; only the 537 normal subgroups are shown.

The subgroup lattice has 262 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251868	C₂	1	2	yes	maximal, complemented
2	#72251736	C₂	1	4	yes	maximal, complemented
3	#72238421	C₂	1	2	yes	maximal, complemented
4	#72237563	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237533	C₂²	1	2	yes	—
6	#72237533	C₂²	1	2	yes	complemented
7	#72236862	C₂	1	1	yes	maximal, characteristic, complemented
8	#72236704	C₂²	1	2	yes	complemented
9	#72236557	C₂²	1	2	yes	—
10	#72235124	C₂	1	1	yes	maximal, characteristic
11	#72235121	C₂²	1	1	yes	characteristic
12	#72235115	C₄	1	1	yes	characteristic, complemented
13	#72235111	C₄	1	1	yes	characteristic, complemented
14	#72235088	C₂²	1	1	yes	characteristic
15	#72235082	C₄	1	2	yes	complemented
16	#72235082	C₂²	1	2	yes	—
17	#72235079	C₄	1	1	yes	characteristic, complemented
18	C₂×#7630939	C₂²	1	1	yes	characteristic
19	#72216762	C₂	1	4	yes	maximal, complemented
20	#72214173	C₄	1	2	yes	complemented
21	#72214173	C₂²	1	2	yes	—
22	#72213738	C₂²	1	2	yes	complemented
23	#72213548	C₄	1	1	yes	characteristic, complemented
24	#72213534	C₂²	1	4	yes	complemented
25	#72213519	C₂²	1	4	yes	complemented
26	#72213452	C₂²	1	4	yes	complemented
27	#72213445	C₂²	1	4	yes	complemented
28	#72210555	C₂²	1	2	yes	complemented
29	C₂⁶×D₄	D₄	1	1	yes	characteristic
30	#10586169	C₂×C₄	1	1	yes	characteristic, complemented
31	C₂²×#3484	C₂×C₄	1	1	yes	characteristic
32	C₂²×#3484	C₂×C₄	1	1	yes	characteristic
33	C₂²×#3484	C₂³	1	1	yes	characteristic
34	#10575442	C₂×C₄	1	2	yes	—
35	#10574840	C₂×C₄	1	2	yes	complemented
36	#10574294	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×#36614	C₂×C₄	1	1	yes	characteristic
38	C₂×#36614	D₄	1	1	yes	characteristic
39	C₂×#35222	C₂×C₄	1	1	yes	characteristic
40	C₂×#35222	D₄	1	1	yes	characteristic
41	C₂×#35222	D₄	1	1	yes	characteristic
42	C₂×#35222	D₄	1	1	yes	characteristic
43	C₂×#35222	C₂³	1	1	yes	characteristic
44	C₂×#35222	C₂³	1	1	yes	characteristic
45	C₂×#34171	D₄	1	1	yes	characteristic
46	C₂×#34171	D₄	1	1	yes	characteristic
47	C₂×#34171	D₄	1	1	yes	characteristic
48	#6437616	C₂×C₄	1	2	yes	—
49	#6437616	C₂×C₄	1	2	yes	complemented
50	#6437449	C₂×C₄	1	2	yes	complemented
51	#6437449	C₂³	1	2	yes	—
52	#6374136	C₂×C₄	1	1	yes	characteristic, complemented
53	#6374123	C₂×C₄	1	1	yes	characteristic, complemented
54	#493370	C₂³	1	4	yes	—
55	#449562	D₄	1	8	yes	—
56	#449496	D₄	1	8	yes	—
57	#392132	C₂×C₄	1	2	yes	complemented
58	#382419	C₂×C₄	1	2	yes	—
59	#359513	C₂×C₄	1	2	yes	complemented
60	#359513	C₂³	1	2	yes	—
61	#359507	C₂×C₄	1	2	yes	—
62	#359507	C₂×C₄	1	2	yes	complemented
63	#187811	C₂³	1	4	yes	—
64	#147569	D₄	1	8	yes	complemented
65	C₂³×H(5,𝔽₂)	C₂²×C₄	1	1	yes	characteristic
66	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
67	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
68	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
70	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
71	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
72	C₂⁶×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	#62811	C₂²×C₄	1	2	yes	—
74	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×#535	C₂²×C₄	1	1	yes	characteristic
77	C₂²×#535	C₂²×C₄	1	1	yes	characteristic
78	C₂²×#535	C₂×D₄	1	1	yes	characteristic
79	C₂²×#535	C₂⁴	1	1	yes	characteristic
80	C₂²×#525	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×#525	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×#525	C₂²×C₄	1	1	yes	characteristic
83	C₂²×#525	C₂²×C₄	1	1	yes	characteristic
84	C₂²×#525	C₂²×C₄	1	1	yes	characteristic
85	C₂²×#525	C₂×D₄	1	1	yes	characteristic
86	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×#518	C₂×D₄	1	1	yes	characteristic
89	#60518	C₂²⋊C₄	1	4	yes	complemented
90	C₂×#3025	C₂²⋊C₄	1	4	yes	complemented
91	C₂×#3021	C₂×D₄	1	4	yes	—
92	#60369	C₂²⋊C₄	1	4	yes	complemented
93	#60064	C₂²×C₄	1	2	yes	—
94	#35222	C₂×D₄	1	8	yes	—
95	#35222	C₂×D₄	1	4	yes	—
96	#35222	#41	1	4	yes	—
97	C₂×#2309	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×#2309	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×#2309	C₂×D₄	1	1	yes	characteristic
100	C₂×#2309	C₂×D₄	1	1	yes	characteristic
101	C₂×#2309	C₂×D₄	1	1	yes	characteristic
102	C₂×#2309	C₂×D₄	1	1	yes	characteristic
103	C₂×#2280	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×#2280	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×#2280	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×#2280	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#2280	C₂×D₄	1	1	yes	characteristic
108	C₂×#2280	C₂×D₄	1	1	yes	characteristic
109	C₂×#2127	C₂×D₄	1	8	yes	complemented
110	#20959	C₂²⋊C₄	1	4	yes	complemented
111	#20956	C₂×D₄	1	4	yes	—
112	#20591	C₂²×C₄	1	2	yes	—
113	#20584	C₂²×C₄	1	2	yes	—
114	#10993	C₂²⋊C₄	1	8	yes	complemented
115	#10987	C₂×D₄	1	8	yes	—
116	#10984	C₂²⋊C₄	1	8	yes	complemented
117	C₂⁷	#99	1	1	yes	characteristic
118	C₂×#585	C₂×C₂²⋊C₄	1	2	yes	complemented
119	C₂³×#41	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×#41	C₂³×C₄	1	1	yes	characteristic
121	C₂⁴×D₄	#99	1	2	yes	—
122	C₂⁴×D₄	#99	1	2	yes	—
123	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂⁴×D₄	#120	1	4	yes	—
127	C₂⁴×D₄	#120	1	2	yes	—
128	C₂⁴×D₄	#123	1	2	yes	—
129	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
130	C₂⁵×C₄	#99	1	1	yes	characteristic
131	C₂⁵×C₄	#99	1	1	yes	characteristic
132	C₂⁵×C₄	#99	1	1	yes	characteristic
133	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×#535	C₂×C₂²⋊C₄	1	4	yes	—
137	C₂×#535	C₄×D₄	1	4	yes	—
138	C₂×#535	C₂²×D₄	1	4	yes	—
139	C₂×#535	C₂²×D₄	1	2	yes	—
140	C₂×#535	C₂×#41	1	2	yes	—
141	C₂²×#127	#99	1	2	yes	—
142	C₂²×#127	H(5,𝔽₂)	1	2	yes	—
143	C₂²×#124	#99	1	2	yes	—
144	C₂²×#124	H(5,𝔽₂)	1	2	yes	—
145	C₂×#525	C₂×C₂²⋊C₄	1	4	yes	—
146	C₂×#525	C₄×D₄	1	4	yes	—
147	#3428	C₂×C₂²⋊C₄	1	2	yes	complemented
148	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×#117	C₂²×D₄	1	1	yes	characteristic
154	C₂×#413	C₂×C₂²⋊C₄	1	4	yes	complemented
155	C₂×#410	C₂×C₂²⋊C₄	1	4	yes	complemented
156	#2879	C₂×C₂²⋊C₄	1	4	yes	complemented
157	#2309	#120	1	4	yes	—
158	#2309	#120	1	4	yes	—
159	#2309	#120	1	2	yes	—
160	#2309	#121	1	4	yes	—
161	#2309	#123	1	2	yes	—
162	#2280	#99	1	4	yes	—
163	#2280	#120	1	4	yes	—
164	#2280	#121	1	4	yes	—
165	C₂⁶	C₂×#99	1	1	yes	characteristic
166	C₂⁶	C₂×#99	1	1	yes	characteristic
167	C₂⁶	C₂×#99	1	1	yes	characteristic
168	C₂²×#41	C₂²×C₂²⋊C₄	1	2	yes	—
169	C₂²×#41	C₂×C₄×D₄	1	2	yes	—
170	C₂³×Q₈	C₂×#99	1	1	yes	characteristic
171	C₂³×Q₈	#518	1	1	yes	characteristic
172	C₂³×D₄	AΣL(1,𝔽₁₆)	1	4	yes	—
173	C₂³×D₄	AΣL(1,𝔽₁₆)	1	4	yes	—
174	C₂³×D₄	#379	1	2	yes	—
175	C₂³×D₄	#386	1	2	yes	—
176	C₂³×D₄	C₂×#99	1	2	yes	—
177	C₂³×D₄	C₂×#99	1	1	yes	characteristic
178	C₂³×D₄	C₂×#99	1	2	yes	—
179	C₂³×D₄	C₂×#99	1	1	yes	characteristic
180	C₂³×D₄	C₂×#99	1	1	yes	characteristic
181	C₂³×D₄	#518	1	1	yes	characteristic
182	C₂³×D₄	C₂×#120	1	2	yes	—
183	C₂³×D₄	C₂×#120	1	1	yes	characteristic
184	C₂³×D₄	C₂×#123	1	1	yes	characteristic
185	C₂³×D₄	#534	1	2	yes	—
186	C₂⁴×C₄	#379	1	2	yes	—
187	C₂⁴×C₄	#386	1	2	yes	—
188	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
189	C₂⁴×C₄	C₂×#99	1	2	yes	—
190	C₂⁴×C₄	C₂×#99	1	2	yes	—
191	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
192	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
193	C₂⁴×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×#127	D₄²	1	4	yes	—
195	C₂×#127	#546	1	4	yes	—
196	C₂×#124	D₄²	1	4	yes	—
197	C₂×#124	#546	1	4	yes	—
198	C₂×#117	#379	1	2	yes	—
199	C₂×#117	#379	1	2	yes	—
200	C₂×#117	#386	1	2	yes	—
201	C₂×#117	#386	1	2	yes	—
202	C₂×#117	C₂×#99	1	2	yes	—
203	C₂×#117	C₂×#99	1	2	yes	—
204	C₂×#117	C₂×#120	1	2	yes	—
205	C₂×#117	C₂×#121	1	2	yes	—
206	C₂²×C₄²	C₂×#99	1	2	yes	—
207	C₂²×C₄²	#534	1	2	yes	—
208	C₂⁵	#1781	1	2	yes	—
209	C₂⁵	#1898	1	2	yes	—
210	C₂⁵	C₂²×#99	1	1	yes	characteristic
211	C₂²×Q₈	D₄×C₂²⋊C₄	1	2	yes	—
212	C₂²×D₄	#1781	1	2	yes	—
213	C₂²×D₄	#1781	1	2	yes	—
214	C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
215	C₂²×D₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
216	C₂²×D₄	D₄×C₂²⋊C₄	1	2	yes	—
217	C₂²×D₄	#2403	1	2	yes	—
218	C₂²×D₄	#2405	1	2	yes	—
219	C₂³×C₄	#1781	1	2	yes	—
220	C₂³×C₄	#1898	1	2	yes	—
221	C₂³×C₄	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
222	C₂³×C₄	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
223	C₂³×C₄	#2121	1	2	yes	—
224	C₂³×C₄	#2121	1	2	yes	—
225	C₂³×C₄	C₂×#379	1	1	yes	characteristic
226	C₂³×C₄	C₂×#386	1	1	yes	characteristic
227	C₂³×C₄	#2308	1	1	yes	characteristic
228	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
229	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
230	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
231	C₂×C₄²	#1898	1	4	yes	—
232	C₂×C₄²	#2410	1	4	yes	—
233	C₂⁴	#12705	1	2	yes	—
234	C₂⁴	#12705	1	2	yes	—
235	C₂⁴	#13086	1	2	yes	—
236	C₂⁴	#13086	1	2	yes	—
237	C₂⁴	C₂×#1781	1	1	yes	characteristic
238	C₂⁴	C₂×#1898	1	1	yes	characteristic
239	C₂⁴	#21746	1	1	yes	characteristic
240	C₂×D₄	#12687	1	4	yes	—
241	C₂×D₄	#23895	1	4	yes	—
242	C₂²×C₄	#13086	1	2	yes	—
243	C₂²×C₄	#13086	1	2	yes	—
244	C₂²×C₄	#13867	1	1	yes	characteristic
245	C₂²×C₄	C₂×#1781	1	1	yes	characteristic
246	C₂²×C₄	C₂×#1898	1	1	yes	characteristic
247	C₂²×C₄	#21746	1	1	yes	characteristic
248	C₂²×C₄	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
249	C₂²×C₄	C₂×#2121	1	2	yes	—
250	C₂³	#359428	1	1	yes	characteristic
251	C₂³	C₂×#12705	1	2	yes	—
252	C₂³	C₂×#13086	1	2	yes	—
253	C₂×C₄	#359549	1	2	yes	—
254	C₂×C₄	C₂×#12687	1	1	yes	characteristic
255	C₂×C₄	#493357	1	1	yes	characteristic
256	C₂×C₄	#496225	1	2	yes	—
257	C₂²	#72218069	1	1	yes	characteristic
258	C₂²	#72218180	1	1	yes	characteristic
259	C₂²	#72219294	1	1	yes	characteristic
260	C₂	#72240504	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240504
commutator	C₂³×#41	C₂³×C₄
Frattini, cosocle	C₂²×#535	C₂⁴
prime radical	#72235124	C₂
LCS	G ▷ C₂³×#41 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×#7630939 ◁ G
LECS	G ▷ C₂²×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72251868	C₂	semidirect
2. #72251736	C₂	semidirect
3. #72238421	C₂	semidirect
4. #72237563	C₂	semidirect
6. #72237533	C₂²	semidirect
7. #72236862	C₂	semidirect
8. #72236704	C₂²	semidirect
12. #72235115	C₄	semidirect
13. #72235111	C₄	semidirect
15. #72235082	C₄	semidirect
17. #72235079	C₄	semidirect
19. #72216762	C₂	semidirect
20. #72214173	C₄	semidirect
22. #72213738	C₂²	semidirect
23. #72213548	C₄	semidirect
24. #72213534	C₂²	semidirect
25. #72213519	C₂²	semidirect
26. #72213452	C₂²	semidirect
27. #72213445	C₂²	semidirect
28. #72210555	C₂²	semidirect
30. #10586169	C₂×C₄	semidirect
35. #10574840	C₂×C₄	semidirect
36. #10574294	C₂×C₄	semidirect
49. #6437616	C₂×C₄	semidirect
50. #6437449	C₂×C₄	semidirect
52. #6374136	C₂×C₄	semidirect
53. #6374123	C₂×C₄	semidirect
57. #392132	C₂×C₄	semidirect
59. #359513	C₂×C₄	semidirect
62. #359507	C₂×C₄	semidirect
64. #147569	D₄	semidirect
89. #60518	C₂²⋊C₄	semidirect
90. C₂×#3025	C₂²⋊C₄	semidirect
92. #60369	C₂²⋊C₄	semidirect
109. C₂×#2127	C₂×D₄	semidirect
110. #20959	C₂²⋊C₄	semidirect
114. #10993	C₂²⋊C₄	semidirect
116. #10984	C₂²⋊C₄	semidirect
118. C₂×#585	C₂×C₂²⋊C₄	semidirect
147. #3428	C₂×C₂²⋊C₄	semidirect
154. C₂×#413	C₂×C₂²⋊C₄	semidirect
155. C₂×#410	C₂×C₂²⋊C₄	semidirect
156. #2879	C₂×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72251868	2
#72251736	2
#72238421	2
#72237563	2
#72236862	2
#72235124	2
#72216762	2