Subgroups — #72238670

Order 4096.

At least 11002 conjugacy classes of subgroups; only the 325 normal subgroups are shown.

The subgroup lattice has 199 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72238644	C₂	1	1	yes	maximal, characteristic, complemented
2	#72238615	C₂	1	1	yes	maximal, characteristic, complemented
3	#72237936	C₂	1	4	yes	maximal, complemented
4	#72237935	C₂²	1	2	yes	complemented
5	#72237907	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237885	C₂	1	4	yes	maximal, complemented
7	#72237881	C₂²	1	2	yes	complemented
8	#72237879	C₂²	1	4	yes	complemented
9	#72237871	C₂²	1	4	yes	—
10	#72237864	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237596	C₂	1	1	yes	maximal, characteristic, complemented
12	#72237594	C₂²	1	4	yes	complemented
13	#72237594	C₂²	1	4	yes	complemented
14	#72237368	C₂	1	1	yes	maximal, characteristic, complemented
15	#72237362	C₂²	1	2	yes	complemented
16	#72235718	C₂²	1	1	yes	characteristic
17	#72234717	C₂	1	1	yes	maximal, characteristic, complemented
18	#72234716	C₂²	1	1	yes	characteristic
19	#72234712	C₂²	1	2	yes	complemented
20	#72234712	C₂²	1	2	yes	complemented
21	#72234625	C₂²	1	1	yes	characteristic
22	#72232556	C₂²	1	2	yes	complemented
23	#72225870	C₂²	1	1	yes	characteristic
24	#72225713	C₂²	1	1	yes	characteristic
25	#72224136	C₂²	1	1	yes	characteristic
26	#72219311	C₂²	1	1	yes	characteristic
27	#10586404	D₄	1	1	yes	characteristic
28	C₂×#62817	D₄	1	1	yes	characteristic
29	C₂×#47031	C₂³	1	1	yes	characteristic
30	C₂×#34980	C₂³	1	1	yes	characteristic
31	#6451874	D₄	1	2	yes	—
32	#6451874	C₂³	1	2	yes	—
33	#6451874	C₂³	1	2	yes	—
34	C₂×#23984	D₄	1	2	yes	—
35	#6374251	D₄	1	1	yes	characteristic
36	#6374128	D₄	1	1	yes	characteristic
37	#6374128	D₄	1	1	yes	characteristic
38	C₂×#21746	D₄	1	1	yes	characteristic
39	#5025772	D₄	1	1	yes	characteristic, complemented
40	#5020747	D₄	1	1	yes	characteristic, complemented
41	#4435227	D₄	1	1	yes	characteristic, complemented
42	#686193	D₄	1	1	yes	characteristic, complemented
43	#576189	C₂³	1	1	yes	characteristic
44	#480250	C₂³	1	2	yes	—
45	#480250	C₂³	1	2	yes	—
46	#479303	D₄	1	4	yes	complemented
47	#477178	D₄	1	4	yes	complemented
48	#477178	D₄	1	4	yes	complemented
49	#477008	D₄	1	4	yes	complemented
50	#392109	D₄	1	2	yes	—
51	#382707	C₂³	1	2	yes	—
52	#190116	C₂³	1	2	yes	—
53	#134037	D₄	1	2	yes	complemented
54	#134036	D₄	1	2	yes	complemented
55	#134033	D₄	1	2	yes	complemented
56	#134033	D₄	1	2	yes	complemented
57	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
58	C₂×#3490	C₂×D₄	1	2	yes	—
59	C₂×#3490	C₂×D₄	1	2	yes	—
60	C₂²×#534	C₂×D₄	1	1	yes	characteristic
61	C₂²×#534	C₂×D₄	1	1	yes	characteristic
62	C₂³×#120	C₂×D₄	1	1	yes	characteristic
63	C₂×#2815	C₂×D₄	1	1	yes	characteristic, complemented
64	C₂×#2849	C₂×D₄	1	1	yes	characteristic
65	C₂×#2849	#41	1	1	yes	characteristic
66	C₂×#2846	C₂×D₄	1	1	yes	characteristic
67	C₂×#2846	C₂×D₄	1	1	yes	characteristic
68	C₂×#2607	C₂×D₄	1	1	yes	characteristic, complemented
69	C₂×#2613	C₂×D₄	1	1	yes	characteristic
70	C₂×#2613	C₂×D₄	1	1	yes	characteristic
71	C₂×#2613	C₂×D₄	1	1	yes	characteristic
72	C₂×#2613	#41	1	1	yes	characteristic
73	C₂×#2308	C₂×D₄	1	1	yes	characteristic
74	C₂×#2308	C₂⁴	1	1	yes	characteristic
75	C₂²×#379	C₂×D₄	1	1	yes	characteristic
76	#23468	C₂×D₄	1	4	yes	—
77	#23468	C₂×D₄	1	4	yes	—
78	#23444	C₂×D₄	1	4	yes	—
79	#23444	C₂×D₄	1	4	yes	—
80	C₂×#1819	C₂×D₄	1	2	yes	—
81	C₂×#1819	C₂×D₄	1	2	yes	—
82	C₂×#1781	C₂×D₄	1	2	yes	—
83	C₂×#1781	C₂×D₄	1	2	yes	—
84	#13828	C₂×D₄	1	1	yes	characteristic, complemented
85	#13807	C₂×D₄	1	1	yes	characteristic, complemented
86	#11499	C₂×D₄	1	4	yes	complemented
87	#11499	C₂×D₄	1	4	yes	complemented
88	#11166	C₂×D₄	1	4	yes	—
89	#11166	C₂×D₄	1	4	yes	—
90	C₂⁷	#120	1	1	yes	characteristic
91	C₂⁷	#127	1	1	yes	characteristic
92	C₂⁴×D₄	#120	1	1	yes	characteristic
93	C₂⁴×D₄	#120	1	1	yes	characteristic
94	C₂⁴×D₄	#121	1	1	yes	characteristic
95	C₂⁴×D₄	#127	1	1	yes	characteristic
96	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
97	C₂⁵×C₄	#121	1	1	yes	characteristic
98	C₂×#543	#120	1	2	yes	—
99	C₂×#543	#120	1	2	yes	—
100	C₂×#534	#120	1	2	yes	—
101	C₂×#534	#120	1	2	yes	—
102	C₂²×#123	#120	1	1	yes	characteristic
103	C₂²×#123	#121	1	1	yes	characteristic
104	C₂²×#123	#121	1	1	yes	characteristic
105	C₂²×#123	#121	1	1	yes	characteristic
106	C₂²×#120	#120	1	1	yes	characteristic
107	C₂²×#120	#120	1	1	yes	characteristic
108	C₂²×#120	#120	1	1	yes	characteristic
109	C₂²×#120	#121	1	1	yes	characteristic
110	C₂²×#120	#121	1	1	yes	characteristic
111	C₂²×#120	#121	1	1	yes	characteristic
112	C₂²×#120	#127	1	1	yes	characteristic
113	C₂²×#120	#127	1	1	yes	characteristic
114	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
115	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
116	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
117	C₂²×#99	#120	1	2	yes	—
118	C₂²×#99	#120	1	4	yes	—
119	C₂²×#99	#120	1	2	yes	—
120	#2308	C₂²×D₄	1	4	yes	—
121	#2308	H(5,𝔽₂)	1	4	yes	—
122	C₂×#381	#120	1	1	yes	characteristic
123	C₂×#381	#121	1	1	yes	characteristic
124	C₂×#381	C₂²×D₄	1	1	yes	characteristic
125	C₂×#381	C₂×#41	1	1	yes	characteristic
126	C₂×#379	#120	1	1	yes	characteristic
127	C₂×#379	#120	1	1	yes	characteristic
128	C₂×#379	#120	1	1	yes	characteristic
129	C₂×#379	#121	1	1	yes	characteristic
130	C₂×#379	#121	1	1	yes	characteristic
131	C₂×#379	#121	1	1	yes	characteristic
132	C₂×#379	C₂²×D₄	1	1	yes	characteristic
133	C₂×#379	C₂²×D₄	1	1	yes	characteristic
134	C₂×#342	#120	1	4	yes	—
135	C₂×#342	#120	1	4	yes	—
136	C₂⁶	#392	1	1	yes	characteristic
137	C₂⁶	#392	1	1	yes	characteristic
138	C₂⁶	#392	1	1	yes	characteristic
139	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
140	C₂⁶	C₂×#120	1	1	yes	characteristic
141	C₂⁶	C₂×#127	1	1	yes	characteristic
142	C₂³×D₄	#392	1	1	yes	characteristic
143	C₂³×D₄	#392	1	1	yes	characteristic
144	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
145	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
146	C₂³×D₄	C₂×#120	1	2	yes	—
147	C₂³×D₄	#535	1	2	yes	—
148	C₂⁴×C₄	#392	1	1	yes	characteristic
149	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
150	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
151	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
152	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
154	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
155	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
156	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
157	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
159	C₂²×C₂²⋊C₄	#534	1	2	yes	—
160	C₂²×C₂²⋊C₄	#534	1	2	yes	—
161	#381	D₄²	1	4	yes	—
162	#381	#546	1	4	yes	—
163	#379	D₄²	1	4	yes	—
164	#379	D₄²	1	4	yes	—
165	C₂⁵	#2021	1	1	yes	characteristic
166	C₂⁵	#2021	1	1	yes	characteristic
167	C₂⁵	C₂×#392	1	1	yes	characteristic
168	C₂⁵	#2433	1	2	yes	—
169	C₂⁵	#2613	1	1	yes	characteristic
170	C₂⁵	#2846	1	1	yes	characteristic
171	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
172	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
173	C₂⁵	#3026	1	1	yes	characteristic
174	C₂⁵	#3026	1	1	yes	characteristic
175	C₂³×C₄	#2021	1	1	yes	characteristic
176	C₂³×C₄	#2021	1	1	yes	characteristic
177	C₂³×C₄	#2408	1	2	yes	—
178	C₂×C₂²⋊C₄	#2403	1	2	yes	—
179	C₂×C₂²⋊C₄	#2403	1	2	yes	—
180	C₂×C₂²⋊C₄	#2410	1	2	yes	—
181	C₂×C₂²⋊C₄	#2410	1	2	yes	—
182	C₂⁴	#14715	1	1	yes	characteristic
183	C₂⁴	C₂×#2021	1	1	yes	characteristic
184	C₂⁴	#23444	1	1	yes	characteristic
185	C₂⁴	#29634	1	1	yes	characteristic
186	C₂⁴	#29634	1	1	yes	characteristic
187	C₂⁴	#32898	1	1	yes	characteristic
188	C₂⁴	#32898	1	1	yes	characteristic
189	C₂³	#393897	1	2	yes	—
190	C₂³	#393929	1	1	yes	characteristic
191	C₂³	#511790	1	1	yes	characteristic
192	C₂³	#511790	1	2	yes	—
193	C₂³	#511790	1	1	yes	characteristic
194	C₂²	#72218114	1	1	yes	characteristic
195	C₂²	#72218114	1	1	yes	characteristic
196	C₂²	#72235174	1	1	yes	characteristic
197	C₂	#72235939	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72235939
commutator, Frattini, cosocle	C₂×#2308	C₂⁴
LCS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72238644	C₂	semidirect
2. #72238615	C₂	semidirect
3. #72237936	C₂	semidirect
4. #72237935	C₂²	semidirect
5. #72237907	C₂	semidirect
6. #72237885	C₂	semidirect
7. #72237881	C₂²	semidirect
8. #72237879	C₂²	semidirect
10. #72237864	C₂	semidirect
11. #72237596	C₂	semidirect
12. #72237594	C₂²	semidirect
13. #72237594	C₂²	semidirect
14. #72237368	C₂	semidirect
15. #72237362	C₂²	semidirect
17. #72234717	C₂	semidirect
19. #72234712	C₂²	semidirect
20. #72234712	C₂²	semidirect
22. #72232556	C₂²	semidirect
39. #5025772	D₄	semidirect
40. #5020747	D₄	semidirect
41. #4435227	D₄	semidirect
42. #686193	D₄	semidirect
46. #479303	D₄	semidirect
47. #477178	D₄	semidirect
48. #477178	D₄	semidirect
49. #477008	D₄	semidirect
53. #134037	D₄	semidirect
54. #134036	D₄	semidirect
55. #134033	D₄	semidirect
56. #134033	D₄	semidirect
63. C₂×#2815	C₂×D₄	semidirect
68. C₂×#2607	C₂×D₄	semidirect
84. #13828	C₂×D₄	semidirect
85. #13807	C₂×D₄	semidirect
86. #11499	C₂×D₄	semidirect
87. #11499	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72238644	2
#72238615	2
#72237936	2
#72237907	2
#72237885	2
#72237864	2
#72237596	2
#72237368	2
#72234717	2