Subgroups — #72262449

Order 4096.

At least 14458 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

The subgroup lattice has 231 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262406	C₂	1	1	yes	maximal, characteristic, complemented
2	#72262389	C₂	1	1	yes	maximal, characteristic, complemented
3	#72262373	C₂	1	1	yes	maximal, characteristic, complemented
4	#72262373	C₂	1	1	yes	maximal, characteristic, complemented
5	#72262347	C₂	1	1	yes	maximal, characteristic, complemented
6	#72262338	C₂	1	1	yes	maximal, characteristic, complemented
7	#72262257	C₂	1	1	yes	maximal, characteristic, complemented
8	#72256412	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251651	C₂	1	1	yes	maximal, characteristic, complemented
10	#72251605	C₂²	1	1	yes	characteristic
11	#72251586	C₂²	1	1	yes	characteristic, complemented
12	#72251570	C₂²	1	1	yes	characteristic
13	#72251570	C₂²	1	1	yes	characteristic, complemented
14	#72249450	C₂	1	1	yes	maximal, characteristic, complemented
15	#72249444	C₂²	1	1	yes	characteristic
16	#72238445	C₂	1	1	yes	maximal, characteristic, complemented
17	#72238444	C₂²	1	1	yes	characteristic
18	#72236918	C₂	1	1	yes	maximal, characteristic, complemented
19	#72236915	C₂	1	1	yes	maximal, characteristic, complemented
20	#72236901	C₂²	1	1	yes	characteristic, complemented
21	#72236901	C₂²	1	1	yes	characteristic, complemented
22	#72236898	C₂²	1	1	yes	characteristic
23	#72236898	C₂²	1	1	yes	characteristic
24	#72236875	C₂²	1	1	yes	characteristic, complemented
25	#72236875	C₂²	1	1	yes	characteristic, complemented
26	#72236837	C₂²	1	1	yes	characteristic
27	#72236837	C₂²	1	1	yes	characteristic
28	#72236812	C₂²	1	1	yes	characteristic, complemented
29	C₂×#134045	C₂²	1	1	yes	characteristic, complemented
30	#72235948	C₂	1	1	yes	maximal, characteristic, complemented
31	#72235933	C₂²	1	1	yes	characteristic, complemented
32	#72235932	C₂²	1	1	yes	characteristic
33	#72235909	C₂	1	1	yes	maximal, characteristic, complemented
34	#72235881	C₂²	1	1	yes	characteristic, complemented
35	#72235865	C₂²	1	1	yes	characteristic, complemented
36	#72235865	C₂²	1	1	yes	characteristic, complemented
37	#72235854	C₂²	1	1	yes	characteristic, complemented
38	#72235848	C₂²	1	1	yes	characteristic
39	#72231822	C₂²	1	1	yes	characteristic, complemented
40	#72231810	C₂²	1	1	yes	characteristic
41	#72231810	C₂²	1	1	yes	characteristic
42	#72231810	C₂²	1	1	yes	characteristic
43	#72226038	C₂²	1	1	yes	characteristic
44	#72226038	C₂²	1	1	yes	characteristic
45	#72225873	C₂²	1	1	yes	characteristic, complemented
46	#72225733	C₂²	1	1	yes	characteristic
47	#72219034	C₂²	1	1	yes	characteristic
48	#72219034	C₂²	1	1	yes	characteristic
49	C₂×#245225	C₂²	1	1	yes	characteristic
50	C₂×#245225	C₂²	1	1	yes	characteristic, complemented
51	C₂×#62842	D₄	1	1	yes	characteristic, complemented
52	#7642273	C₂³	1	1	yes	characteristic
53	#7642273	C₂³	1	1	yes	characteristic
54	C₂×#35346	C₂³	1	1	yes	characteristic
55	C₂×#35346	C₂³	1	1	yes	characteristic
56	C₂×#35346	C₂³	1	1	yes	characteristic
57	C₂×#32898	D₄	1	2	yes	—
58	C₂×#20585	D₄	1	1	yes	characteristic, complemented
59	C₂×#20585	D₄	1	1	yes	characteristic, complemented
60	#555894	C₂³	1	1	yes	characteristic
61	#515285	C₂³	1	1	yes	characteristic
62	#417896	D₄	1	2	yes	complemented
63	#417824	D₄	1	2	yes	complemented
64	#382422	C₂³	1	1	yes	characteristic
65	#382422	C₂³	1	1	yes	characteristic
66	#379754	D₄	1	2	yes	complemented
67	#353314	D₄	1	1	yes	characteristic, complemented
68	#249331	D₄	1	2	yes	complemented
69	#249319	D₄	1	2	yes	complemented
70	C₂×#9497	D₄	1	2	yes	complemented
71	C₂×#9495	D₄	1	2	yes	—
72	C₂×#8547	C₂³	1	1	yes	characteristic
73	C₂×#8547	C₂³	1	1	yes	characteristic
74	C₂×#8547	C₂³	1	1	yes	characteristic
75	C₂×#8547	C₂³	1	1	yes	characteristic
76	C₂×#8541	C₂³	1	1	yes	characteristic
77	C₂×#8541	C₂³	1	1	yes	characteristic
78	#185255	D₄	1	2	yes	complemented
79	#185255	D₄	1	2	yes	complemented
80	#106326	D₄	1	2	yes	complemented
81	#106326	D₄	1	2	yes	complemented
82	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
84	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
85	C₂×#3490	C₂×D₄	1	1	yes	characteristic
86	C₂×#3490	C₂×D₄	1	1	yes	characteristic
87	C₂³×#120	C₂⁴	1	1	yes	characteristic
88	C₂²×#518	C₂×D₄	1	1	yes	characteristic
89	#35390	C₂×D₄	1	1	yes	characteristic
90	#35390	C₂×D₄	1	1	yes	characteristic
91	#35390	#41	1	1	yes	characteristic
92	#35390	#41	1	1	yes	characteristic
93	#35367	C₂×D₄	1	1	yes	characteristic
94	#35367	C₂×D₄	1	1	yes	characteristic
95	#35367	#41	1	1	yes	characteristic
96	#35367	#41	1	1	yes	characteristic
97	#35346	C₂×D₄	1	2	yes	—
98	#35346	C₂×D₄	1	2	yes	—
99	C₂×#2403	C₂×D₄	1	2	yes	—
100	C₂×#2403	C₂×D₄	1	2	yes	—
101	#34176	C₂×D₄	1	1	yes	characteristic
102	#34176	C₂×D₄	1	1	yes	characteristic
103	C₂×#2309	C₂×D₄	1	1	yes	characteristic
104	C₂×#2308	C₂×D₄	1	1	yes	characteristic
105	C₂×#2308	C₂×D₄	1	1	yes	characteristic
106	C₂×#2308	C₂×D₄	1	1	yes	characteristic
107	C₂×#2308	C₂×D₄	1	1	yes	characteristic
108	C₂×#2308	C₂×D₄	1	1	yes	characteristic
109	C₂×#1781	C₂×D₄	1	2	yes	—
110	C₂×#1781	C₂×D₄	1	2	yes	—
111	C₂×#1462	C₂×D₄	1	2	yes	—
112	C₂×#1462	C₂×D₄	1	2	yes	—
113	#13868	C₂×D₄	1	2	yes	—
114	#13868	C₂×D₄	1	2	yes	—
115	#10965	C₂×D₄	1	2	yes	—
116	#8547	C₂×D₄	1	2	yes	—
117	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
119	C₂⁴×D₄	#120	1	2	yes	—
120	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
121	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
122	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
123	C₂×#543	#121	1	1	yes	characteristic, complemented
124	C₂×#543	#121	1	1	yes	characteristic, complemented
125	C₂×#534	#121	1	1	yes	characteristic, complemented
126	C₂×#534	#121	1	1	yes	characteristic, complemented
127	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
128	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
129	C₂²×#120	C₂×#41	1	1	yes	characteristic
130	C₂²×#120	C₂×#41	1	1	yes	characteristic
131	C₂²×#120	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂²×#120	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂²×#120	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂²×#120	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²×#117	#120	1	2	yes	—
136	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
137	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
138	C₂³×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
139	#2308	#120	1	2	yes	—
140	C₂×#381	#120	1	2	yes	—
141	C₂×#379	#120	1	2	yes	—
142	#2273	#121	1	1	yes	characteristic, complemented
143	#2273	#121	1	1	yes	characteristic, complemented
144	#2273	#121	1	1	yes	characteristic, complemented
145	#2273	#121	1	1	yes	characteristic, complemented
146	C₂⁶	B(4,𝔽₂)	1	2	yes	—
147	C₂⁶	C₂×#120	1	1	yes	characteristic
148	C₂⁶	#534	1	1	yes	characteristic
149	C₂⁶	#534	1	1	yes	characteristic
150	C₂⁶	#538	1	1	yes	characteristic
151	C₂⁶	D₄²	1	1	yes	characteristic
152	C₂⁶	D₄²	1	1	yes	characteristic
153	C₂⁶	#546	1	1	yes	characteristic
154	C₂⁶	#560	1	1	yes	characteristic
155	C₂³×D₄	C₂×#120	1	1	yes	characteristic
156	C₂³×D₄	#525	1	1	yes	characteristic
157	C₂³×D₄	#534	1	1	yes	characteristic
158	C₂³×D₄	#535	1	1	yes	characteristic
159	C₂³×D₄	D₄²	1	1	yes	characteristic
160	C₂³×D₄	D₄²	1	1	yes	characteristic
161	C₂³×D₄	D₄²	1	1	yes	characteristic
162	C₂³×D₄	D₄²	1	1	yes	characteristic
163	C₂³×D₄	#546	1	1	yes	characteristic
164	C₂³×D₄	#546	1	1	yes	characteristic
165	C₂³×D₄	#546	1	1	yes	characteristic
166	C₂³×D₄	#546	1	1	yes	characteristic
167	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
168	C₂⁴×C₄	C₂×#120	1	1	yes	characteristic
169	C₂⁴×C₄	D₄²	1	1	yes	characteristic
170	C₂⁴×C₄	D₄²	1	1	yes	characteristic
171	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
172	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
173	C₂²×C₂²⋊C₄	#534	1	1	yes	characteristic
174	C₂²×C₂²⋊C₄	#535	1	1	yes	characteristic
175	C₂²×C₂²⋊C₄	#535	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	#538	1	1	yes	characteristic
177	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
179	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
180	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
181	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
182	C₂²×C₂²⋊C₄	#560	1	1	yes	characteristic
183	C₂²×C₄²	B(4,𝔽₂)	1	2	yes	—
184	C₂²×C₄²	B(4,𝔽₂)	1	2	yes	—
185	#379	B(4,𝔽₂)	1	2	yes	—
186	#379	B(4,𝔽₂)	1	2	yes	—
187	C₂⁵	#2403	1	1	yes	characteristic
188	C₂⁵	#2403	1	1	yes	characteristic
189	C₂⁵	#2404	1	1	yes	characteristic
190	C₂⁵	#2410	1	1	yes	characteristic
191	C₂⁵	#2424	1	1	yes	characteristic
192	C₂⁵	#2668	1	1	yes	characteristic
193	C₂⁵	#2668	1	1	yes	characteristic
194	C₂⁵	#2669	1	1	yes	characteristic
195	C₂⁵	#2678	1	1	yes	characteristic
196	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
197	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
198	C₂³×C₄	#2408	1	1	yes	characteristic
199	C₂³×C₄	#2408	1	1	yes	characteristic
200	C₂³×C₄	#2410	1	1	yes	characteristic
201	C₂³×C₄	#2424	1	1	yes	characteristic
202	C₂³×C₄	#2669	1	1	yes	characteristic
203	C₂³×C₄	#2678	1	1	yes	characteristic
204	C₂³×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
205	C₂³×C₄	#3025	1	2	yes	—
206	C₂⁴	#15257	1	1	yes	characteristic
207	C₂⁴	#23895	1	1	yes	characteristic
208	C₂⁴	#23895	1	1	yes	characteristic
209	C₂⁴	#23900	1	1	yes	characteristic
210	C₂⁴	#23900	1	1	yes	characteristic
211	C₂⁴	#23900	1	2	yes	—
212	C₂⁴	#23900	1	1	yes	characteristic
213	C₂⁴	#23902	1	1	yes	characteristic
214	C₂⁴	#23984	1	2	yes	—
215	C₂²×C₄	#24260	1	2	yes	—
216	C₂²×C₄	#24260	1	2	yes	—
217	C₂³	#245225	1	1	yes	characteristic
218	C₂³	#245228	1	1	yes	characteristic
219	C₂³	#245231	1	1	yes	characteristic
220	C₂³	#245233	1	1	yes	characteristic
221	C₂³	#417493	1	1	yes	characteristic
222	C₂³	#417526	1	1	yes	characteristic
223	C₂³	#417532	1	1	yes	characteristic
224	C₂²	#72209844	1	1	yes	characteristic
225	C₂²	#72237376	1	1	yes	characteristic
226	C₂²	#72237498	1	1	yes	characteristic
227	C₂	#72251155	1	1	yes	characteristic
228	C₂	#72251585	1	1	yes	characteristic
229	C₂	#72251674	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72237498
commutator, Frattini, cosocle	C₂³×#120	C₂⁴
LCS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72262406	C₂	semidirect
2. #72262389	C₂	semidirect
3. #72262373	C₂	semidirect
4. #72262373	C₂	semidirect
5. #72262347	C₂	semidirect
6. #72262338	C₂	semidirect
7. #72262257	C₂	semidirect
8. #72256412	C₂	semidirect
9. #72251651	C₂	semidirect
11. #72251586	C₂²	semidirect
13. #72251570	C₂²	semidirect
14. #72249450	C₂	semidirect
16. #72238445	C₂	semidirect
18. #72236918	C₂	semidirect
19. #72236915	C₂	semidirect
20. #72236901	C₂²	semidirect
21. #72236901	C₂²	semidirect
24. #72236875	C₂²	semidirect
25. #72236875	C₂²	semidirect
28. #72236812	C₂²	semidirect
29. C₂×#134045	C₂²	semidirect
30. #72235948	C₂	semidirect
31. #72235933	C₂²	semidirect
33. #72235909	C₂	semidirect
34. #72235881	C₂²	semidirect
35. #72235865	C₂²	semidirect
36. #72235865	C₂²	semidirect
37. #72235854	C₂²	semidirect
39. #72231822	C₂²	semidirect
45. #72225873	C₂²	semidirect
50. C₂×#245225	C₂²	semidirect
51. C₂×#62842	D₄	semidirect
58. C₂×#20585	D₄	semidirect
59. C₂×#20585	D₄	semidirect
62. #417896	D₄	semidirect
63. #417824	D₄	semidirect
66. #379754	D₄	semidirect
67. #353314	D₄	semidirect
68. #249331	D₄	semidirect
69. #249319	D₄	semidirect
70. C₂×#9497	D₄	semidirect
78. #185255	D₄	semidirect
79. #185255	D₄	semidirect
80. #106326	D₄	semidirect
81. #106326	D₄	semidirect
123. C₂×#543	#121	semidirect
124. C₂×#543	#121	semidirect
125. C₂×#534	#121	semidirect
126. C₂×#534	#121	semidirect
142. #2273	#121	semidirect
143. #2273	#121	semidirect
144. #2273	#121	semidirect
145. #2273	#121	semidirect

Maximal subgroups

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

subgroup	index
#72262406	2
#72262389	2
#72262373	2
#72262347	2
#72262338	2
#72262257	2
#72256412	2
#72251651	2
#72249450	2
#72238445	2
#72236918	2
#72236915	2
#72235948	2
#72235909	2