Subgroups — #72338765

Order 4096.

Only the 449 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 208 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262550	C₂	1	1	yes	maximal, characteristic
2	#72262540	C₄	1	1	yes	characteristic, complemented
3	#72262540	C₄	1	1	yes	characteristic, complemented
4	#72262500	C₄	1	4	yes	complemented
5	#72238449	C₂	1	2	yes	maximal, complemented
6	#72237864	C₂	1	2	yes	maximal, complemented
7	#72237606	C₂	1	1	yes	maximal, characteristic, complemented
8	#72237385	C₂	1	1	yes	maximal, characteristic, complemented
9	#72234625	C₂²	1	1	yes	characteristic
10	#72234625	C₂²	1	1	yes	characteristic
11	C₂×#10586472	C₄	1	1	yes	characteristic, complemented
12	#72225870	C₂²	1	2	yes	complemented
13	C₂×#188035	C₄	1	1	yes	characteristic, complemented
14	#72225665	C₂²	1	1	yes	characteristic
15	#72219311	C₂²	1	2	yes	complemented
16	#72216698	C₂	1	8	yes	maximal, complemented
17	#72216449	C₂²	1	4	yes	complemented
18	#72216427	C₂²	1	4	yes	complemented
19	#72216323	C₂²	1	4	yes	—
20	#72215671	C₂²	1	8	yes	complemented
21	#72210482	C₂²	1	8	yes	complemented
22	#10586472	C₂×C₄	1	4	yes	—
23	C₂×#62817	C₂×C₄	1	1	yes	characteristic
24	C₂×#62817	C₂×C₄	1	1	yes	characteristic
25	C₂³×D₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×#34980	D₄	1	2	yes	—
27	C₂×#34980	C₂³	1	1	yes	characteristic
28	#6499870	D₄	1	4	yes	—
29	#6499868	C₂×C₄	1	2	yes	—
30	#6499868	C₂×C₄	1	2	yes	—
31	#6499868	D₄	1	2	yes	—
32	#6499867	D₄	1	2	yes	—
33	#6499867	C₂³	1	2	yes	—
34	#6499867	C₂³	1	2	yes	—
35	#6451874	D₄	1	4	yes	—
36	C₂×#21746	C₂×C₄	1	1	yes	characteristic
37	C₂×#21746	C₂×C₄	1	1	yes	characteristic
38	C₂×#21746	D₄	1	2	yes	—
39	#4420586	D₄	1	4	yes	—
40	#663621	C₂×C₄	1	2	yes	—
41	#647656	C₂³	1	2	yes	—
42	#617710	D₄	1	2	yes	—
43	#616960	D₄	1	2	yes	—
44	#556752	C₂×C₄	1	1	yes	characteristic
45	#188035	C₂×C₄	1	4	yes	—
46	#187940	C₂×C₄	1	4	yes	—
47	#187940	C₂×C₄	1	4	yes	—
48	#134033	C₂³	1	4	yes	complemented
49	#115690	C₂³	1	4	yes	complemented
50	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
51	#62817	C₂²⋊C₄	1	2	yes	—
52	#62817	C₂²⋊C₄	1	2	yes	—
53	#62817	C₂²×C₄	1	2	yes	—
54	#62817	C₂²×C₄	1	2	yes	—
55	C₂×#3490	C₂²⋊C₄	1	2	yes	—
56	C₂²×D₄²	C₂²⋊C₄	1	2	yes	—
57	C₂²×D₄²	C₂²×C₄	1	2	yes	—
58	C₂²×#534	C₂×D₄	1	2	yes	—
59	C₂³×#127	C₂²⋊C₄	1	1	yes	characteristic
60	C₂³×#120	C₂²⋊C₄	1	2	yes	—
61	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
63	C₂×#2849	C₂²⋊C₄	1	4	yes	—
64	C₂×#2849	C₂²×C₄	1	1	yes	characteristic
65	C₂×#2849	C₂×D₄	1	1	yes	characteristic
66	C₂×#2846	C₂×D₄	1	1	yes	characteristic
67	C₂×#2846	C₂×D₄	1	4	yes	—
68	C₂×#2846	C₂⁴	1	1	yes	characteristic
69	C₂×#2613	C₂×D₄	1	2	yes	—
70	C₂×#2613	C₂×D₄	1	2	yes	—
71	C₂×#2308	C₂×D₄	1	2	yes	—
72	C₂²×#379	C₂²⋊C₄	1	2	yes	—
73	C₂²×#379	C₂×D₄	1	1	yes	characteristic
74	#23985	C₂²⋊C₄	1	8	yes	—
75	#23984	C₂×D₄	1	8	yes	—
76	#21746	C₂²⋊C₄	1	2	yes	—
77	#21746	C₂²⋊C₄	1	4	yes	—
78	#21746	C₂²⋊C₄	1	2	yes	—
79	#21746	C₂²×C₄	1	2	yes	—
80	#21746	C₂²×C₄	1	2	yes	—
81	#21746	C₂×D₄	1	4	yes	—
82	C₂×#1819	C₂²⋊C₄	1	2	yes	—
83	#15948	C₂²⋊C₄	1	4	yes	—
84	#15947	C₂×D₄	1	4	yes	—
85	#15261	C₂²⋊C₄	1	4	yes	—
86	#15258	C₂×D₄	1	4	yes	—
87	#14657	C₂²⋊C₄	1	2	yes	—
88	#14335	C₂²⋊C₄	1	2	yes	—
89	#14134	C₂²×C₄	1	2	yes	—
90	C₂⁷	#99	1	1	yes	characteristic, complemented
91	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂⁷	C₂²×D₄	1	1	yes	characteristic
93	C₂⁴×D₄	#99	1	2	yes	—
94	C₂⁴×D₄	#99	1	1	yes	characteristic, complemented
95	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂⁴×D₄	#120	1	2	yes	—
97	C₂⁴×D₄	#120	1	2	yes	—
98	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	#3490	#99	1	4	yes	complemented
100	#3490	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×#543	#99	1	2	yes	complemented
102	C₂×#534	#99	1	2	yes	complemented
103	C₂²×#127	#99	1	2	yes	complemented
104	C₂²×#127	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂²×#123	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂²×#123	#120	1	2	yes	—
107	C₂²×#120	#99	1	4	yes	—
108	C₂²×#120	#99	1	2	yes	complemented
109	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×#120	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂²×#120	#120	1	2	yes	—
115	C₂²×#120	C₂³×C₄	1	1	yes	characteristic
116	C₂²×#120	C₂²×D₄	1	2	yes	—
117	C₂³×C₂²⋊C₄	#99	1	2	yes	—
118	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic, complemented
119	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₂²⋊C₄	#120	1	2	yes	—
121	C₂³×C₄²	#99	1	1	yes	characteristic, complemented
122	C₂×#381	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×#381	#120	1	2	yes	—
124	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
126	C₂×#379	#120	1	2	yes	—
127	C₂×#379	#120	1	2	yes	—
128	C₂×#379	C₂²×D₄	1	2	yes	—
129	C₂×#379	C₂²×D₄	1	1	yes	characteristic
130	#1819	C₂×C₂²⋊C₄	1	4	yes	—
131	#1438	C₂×C₂²⋊C₄	1	4	yes	—
132	C₂⁶	AΣL(1,𝔽₁₆)	1	2	yes	complemented
133	C₂⁶	#379	1	1	yes	characteristic
134	C₂⁶	#379	1	1	yes	characteristic
135	C₂⁶	C₂×#99	1	2	yes	—
136	C₂⁶	C₂×#99	1	2	yes	—
137	C₂⁶	C₂×#99	1	1	yes	characteristic
138	C₂⁶	C₂×#99	1	2	yes	—
139	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₂⁶	C₂×#120	1	2	yes	—
141	C₂³×D₄	AΣL(1,𝔽₁₆)	1	4	yes	complemented
142	C₂³×D₄	#379	1	1	yes	characteristic
143	C₂³×D₄	C₂×#99	1	2	yes	—
144	C₂³×D₄	C₂×#99	1	1	yes	characteristic
145	C₂³×D₄	C₂×#99	1	4	yes	—
146	C₂³×D₄	C₂×#99	1	1	yes	characteristic
147	C₂³×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
148	C₂⁴×C₄	AΣL(1,𝔽₁₆)	1	2	yes	complemented
149	C₂⁴×C₄	#379	1	2	yes	—
150	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
151	#543	C₂×#99	1	4	yes	—
152	#534	C₂×#99	1	4	yes	—
153	C₂²×C₂²⋊C₄	AΣL(1,𝔽₁₆)	1	4	yes	complemented
154	C₂²×C₂²⋊C₄	AΣL(1,𝔽₁₆)	1	4	yes	complemented
155	C₂²×C₂²⋊C₄	#379	1	1	yes	characteristic
156	C₂²×C₂²⋊C₄	#379	1	2	yes	—
157	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
158	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
159	C₂²×C₂²⋊C₄	C₂×#99	1	4	yes	—
160	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
161	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
163	C₂²×C₄²	C₂×#99	1	2	yes	—
164	C₂⁵	#1781	1	1	yes	characteristic
165	C₂⁵	#1781	1	1	yes	characteristic
166	C₂⁵	#1781	1	1	yes	characteristic
167	C₂⁵	#1781	1	1	yes	characteristic
168	C₂⁵	#1781	1	1	yes	characteristic
169	C₂⁵	#1781	1	1	yes	characteristic
170	C₂⁵	#1781	1	1	yes	characteristic
171	C₂⁵	#1781	1	2	yes	—
172	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
173	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
174	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
175	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
176	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
177	C₂⁵	C₂×#379	1	1	yes	characteristic
178	C₂⁵	C₂²×#99	1	2	yes	—
179	C₂⁵	C₂²×#99	1	1	yes	characteristic
180	C₂³×C₄	#1781	1	1	yes	characteristic
181	C₂³×C₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
182	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	#1781	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂×AΣL(1,𝔽₁₆)	1	8	yes	—
185	C₂×C₂²⋊C₄	C₂²×#99	1	2	yes	—
186	C₂⁴	#8541	1	1	yes	characteristic
187	C₂⁴	#8541	1	1	yes	characteristic
188	C₂⁴	#8541	1	1	yes	characteristic
189	C₂⁴	#8541	1	1	yes	characteristic
190	C₂⁴	#12687	1	2	yes	—
191	C₂⁴	#12687	1	2	yes	—
192	C₂⁴	#12687	1	2	yes	—
193	C₂⁴	#12687	1	2	yes	—
194	C₂⁴	C₂×#1781	1	1	yes	characteristic
195	C₂⁴	C₂×#1781	1	1	yes	characteristic
196	C₂⁴	C₂×#1781	1	1	yes	characteristic
197	C₂⁴	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
198	C₂⁴	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
199	C₂³	#133822	1	1	yes	characteristic
200	C₂³	#133822	1	1	yes	characteristic
201	C₂³	#133822	1	2	yes	—
202	C₂³	C₂×#8541	1	1	yes	characteristic
203	C₂³	C₂×#12687	1	2	yes	—
204	C₂²	C₂×#133822	1	1	yes	characteristic
205	C₂²	#72240419	1	2	yes	—
206	C₂	#72240424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240424
commutator	C₂²×#120	C₂³×C₄
Frattini, cosocle	C₂×#2846	C₂⁴
prime radical	#72262550	C₂
LCS	G ▷ C₂²×#120 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2846 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72262540	C₄	semidirect
3. #72262540	C₄	semidirect
4. #72262500	C₄	semidirect
5. #72238449	C₂	semidirect
6. #72237864	C₂	semidirect
7. #72237606	C₂	semidirect
8. #72237385	C₂	semidirect
11. C₂×#10586472	C₄	semidirect
12. #72225870	C₂²	semidirect
13. C₂×#188035	C₄	semidirect
15. #72219311	C₂²	semidirect
16. #72216698	C₂	semidirect
17. #72216449	C₂²	semidirect
18. #72216427	C₂²	semidirect
20. #72215671	C₂²	semidirect
21. #72210482	C₂²	semidirect
48. #134033	C₂³	semidirect
49. #115690	C₂³	semidirect
90. C₂⁷	#99	semidirect
94. C₂⁴×D₄	#99	semidirect
99. #3490	#99	semidirect
101. C₂×#543	#99	semidirect
102. C₂×#534	#99	semidirect
103. C₂²×#127	#99	semidirect
108. C₂²×#120	#99	semidirect
118. C₂³×C₂²⋊C₄	#99	semidirect
121. C₂³×C₄²	#99	semidirect
132. C₂⁶	AΣL(1,𝔽₁₆)	semidirect
141. C₂³×D₄	AΣL(1,𝔽₁₆)	semidirect
148. C₂⁴×C₄	AΣL(1,𝔽₁₆)	semidirect
153. C₂²×C₂²⋊C₄	AΣL(1,𝔽₁₆)	semidirect
154. C₂²×C₂²⋊C₄	AΣL(1,𝔽₁₆)	semidirect

Maximal subgroups

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

subgroup	index
#72262550	2
#72238449	2
#72237864	2
#72237606	2
#72237385	2
#72216698	2