Subgroups — #72337810

Order 2048.

At least 6287 conjugacy classes of subgroups; only the 336 normal subgroups are shown.

The subgroup lattice has 200 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72337743	C₂	1	4	yes	maximal, complemented
2	#72323552	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310531	C₂	1	1	yes	maximal, characteristic, complemented
4	#72231822	C₂	1	1	yes	maximal, characteristic, complemented
5	#72227706	C₂	1	4	yes	maximal, complemented
6	#72227421	C₂	1	1	yes	maximal, characteristic, complemented
7	#72226038	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#134033	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#449475	C₂	1	1	yes	maximal, characteristic
10	C₂³×B(4,𝔽₂)	C₄	1	2	yes	complemented
11	C₂²×#2846	C₂²	1	1	yes	characteristic
12	C₂×#35346	C₂²	1	1	yes	characteristic
13	#449475	C₄	1	4	yes	complemented
14	#449475	C₂²	1	2	yes	—
15	#449475	C₂²	1	2	yes	—
16	#391692	C₂²	1	2	yes	complemented
17	#382422	C₂²	1	1	yes	characteristic
18	#382422	C₂²	1	1	yes	characteristic
19	#374022	C₂²	1	2	yes	complemented
20	C₂×#10991	C₄	1	2	yes	complemented
21	C₂×#10965	C₂²	1	1	yes	characteristic
22	C₂×#8547	C₂²	1	1	yes	characteristic
23	C₂×#8547	C₂²	1	1	yes	characteristic
24	#134033	C₂²	1	2	yes	complemented
25	#134033	C₂²	1	2	yes	complemented
26	#133278	C₂²	1	4	yes	complemented
27	#131967	C₂²	1	4	yes	complemented
28	#93177	C₂²	1	4	yes	complemented
29	#93148	C₂²	1	4	yes	complemented
30	C₂⁸	D₄	1	1	yes	characteristic
31	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic, complemented
32	C₂³×#120	C₂×C₄	1	2	yes	—
33	C₂³×#120	C₂³	1	1	yes	characteristic
34	C₂²×B(4,𝔽₂)	C₂×C₄	1	4	yes	—
35	C₂²×B(4,𝔽₂)	C₂×C₄	1	4	yes	complemented
36	C₂³×#99	C₂×C₄	1	2	yes	complemented
37	C₂×#2846	C₂×C₄	1	2	yes	—
38	C₂×#2846	D₄	1	2	yes	—
39	C₂×#2846	D₄	1	1	yes	characteristic
40	C₂×#2846	D₄	1	1	yes	characteristic
41	C₂×#2846	C₂³	1	1	yes	characteristic
42	C₂×#2846	C₂³	1	1	yes	characteristic
43	#35346	C₂×C₄	1	2	yes	complemented
44	#35346	C₂³	1	2	yes	—
45	C₂×#2308	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂²×#379	D₄	1	2	yes	—
47	C₂²×#379	D₄	1	1	yes	characteristic
48	C₂×#1819	D₄	1	2	yes	complemented
49	C₂×#1781	D₄	1	2	yes	complemented
50	#10991	C₂×C₄	1	4	yes	—
51	#10991	C₂×C₄	1	4	yes	complemented
52	#10965	C₂×C₄	1	2	yes	complemented
53	#10965	D₄	1	2	yes	complemented
54	#10965	D₄	1	2	yes	complemented
55	#10965	C₂³	1	2	yes	—
56	#10965	C₂³	1	2	yes	—
57	#10965	C₂³	1	2	yes	—
58	#8547	C₂³	1	2	yes	—
59	#8547	C₂³	1	2	yes	—
60	C₂⁷	C₂²⋊C₄	1	2	yes	—
61	C₂⁷	C₂²⋊C₄	1	2	yes	—
62	C₂⁷	C₂×D₄	1	1	yes	characteristic
63	C₂⁷	C₂×D₄	1	1	yes	characteristic
64	C₂⁷	C₂×D₄	1	1	yes	characteristic
65	C₂²×H(5,𝔽₂)	C₂²×C₄	1	2	yes	—
66	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
67	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
68	C₂⁴×D₄	#41	1	1	yes	characteristic
69	C₂²×#120	C₂²⋊C₄	1	2	yes	—
70	C₂²×#120	C₂²⋊C₄	1	2	yes	—
71	C₂²×#120	C₂²×C₄	1	2	yes	—
72	C₂²×#120	C₂²×C₄	1	2	yes	—
73	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
74	C₂²×#120	C₂×D₄	1	1	yes	characteristic
75	C₂²×#120	C₂×D₄	1	1	yes	characteristic
76	C₂²×#120	C₂×D₄	1	1	yes	characteristic
77	C₂²×#120	C₂×D₄	1	1	yes	characteristic
78	C₂²×#120	#41	1	1	yes	characteristic
79	C₂²×#120	#41	1	1	yes	characteristic
80	C₂²×#120	C₂⁴	1	1	yes	characteristic
81	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
82	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
84	C₂³×C₂²⋊C₄	#41	1	1	yes	characteristic
85	C₂²×#99	C₂²×C₄	1	4	yes	—
86	C₂²×#99	C₂×D₄	1	4	yes	complemented
87	#2308	C₂²×C₄	1	2	yes	—
88	C₂×#379	C₂²⋊C₄	1	2	yes	—
89	C₂×#379	C₂²⋊C₄	1	2	yes	—
90	C₂×#379	C₂²⋊C₄	1	2	yes	—
91	C₂×#379	C₂×D₄	1	1	yes	characteristic
92	C₂×#379	C₂×D₄	1	1	yes	characteristic
93	C₂×#379	C₂×D₄	1	2	yes	—
94	C₂×#379	C₂×D₄	1	2	yes	—
95	C₂×#342	C₂×D₄	1	4	yes	complemented
96	C₂⁶	#99	1	2	yes	—
97	C₂⁶	#99	1	2	yes	—
98	C₂⁶	#99	1	2	yes	—
99	C₂⁶	#99	1	2	yes	—
100	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂⁶	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂⁶	C₄×D₄	1	1	yes	characteristic
105	C₂⁶	#120	1	1	yes	characteristic
106	C₂⁶	#120	1	1	yes	characteristic
107	C₂⁶	#123	1	1	yes	characteristic
108	C₂⁶	#123	1	1	yes	characteristic
109	C₂⁶	C₂²×D₄	1	1	yes	characteristic
110	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
112	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
113	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
114	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
115	C₂³×D₄	#120	1	1	yes	characteristic
116	C₂³×D₄	#121	1	1	yes	characteristic
117	C₂³×D₄	#121	1	1	yes	characteristic
118	C₂³×D₄	#123	1	1	yes	characteristic
119	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
120	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
121	C₂³×D₄	C₂×#41	1	1	yes	characteristic
122	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
123	C₂⁴×C₄	#121	1	1	yes	characteristic
124	C₂⁴×C₄	#121	1	1	yes	characteristic
125	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
129	C₂²×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
130	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
131	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
133	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
134	C₂²×C₂²⋊C₄	#121	1	1	yes	characteristic
135	C₂²×C₂²⋊C₄	#123	1	1	yes	characteristic
136	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
137	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₂²⋊C₄	C₂×#41	1	1	yes	characteristic
139	C₂⁵	AΣL(1,𝔽₁₆)	1	8	yes	complemented
140	C₂⁵	#386	1	1	yes	characteristic
141	C₂⁵	#386	1	1	yes	characteristic
142	C₂⁵	#386	1	1	yes	characteristic
143	C₂⁵	#386	1	1	yes	characteristic
144	C₂⁵	#386	1	1	yes	characteristic
145	C₂⁵	C₂×#99	1	2	yes	—
146	C₂⁵	C₂×#99	1	2	yes	—
147	C₂⁵	C₂×#99	1	2	yes	—
148	C₂⁵	C₂×#99	1	2	yes	—
149	C₂⁵	C₂×#99	1	2	yes	—
150	C₂⁵	C₂×#99	1	2	yes	—
151	C₂⁵	B(4,𝔽₂)	1	2	yes	—
152	C₂⁵	B(4,𝔽₂)	1	2	yes	—
153	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
154	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂⁵	C₂×#120	1	1	yes	characteristic
156	C₂⁵	C₂×#123	1	1	yes	characteristic
157	C₂³×C₄	#386	1	1	yes	characteristic
158	C₂³×C₄	#386	1	1	yes	characteristic
159	C₂³×C₄	#386	1	1	yes	characteristic
160	C₂³×C₄	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂³×C₄	C₂×#121	1	1	yes	characteristic
162	C₂³×C₄	C₂×#121	1	1	yes	characteristic
163	C₂⁴	#1889	1	1	yes	characteristic
164	C₂⁴	#1889	1	1	yes	characteristic
165	C₂⁴	#1889	1	2	yes	—
166	C₂⁴	#1898	1	1	yes	characteristic
167	C₂⁴	#1898	1	2	yes	—
168	C₂⁴	#1898	1	1	yes	characteristic
169	C₂⁴	#1913	1	1	yes	characteristic
170	C₂⁴	#1913	1	1	yes	characteristic
171	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
172	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
173	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	4	yes	—
174	C₂⁴	#2121	1	4	yes	—
175	C₂⁴	C₂×#386	1	1	yes	characteristic
176	C₂⁴	C₂²×#99	1	2	yes	—
177	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
178	C₂³	#9495	1	1	yes	characteristic
179	C₂³	#9495	1	1	yes	characteristic
180	C₂³	#9495	1	1	yes	characteristic
181	C₂³	#9495	1	1	yes	characteristic
182	C₂³	#13086	1	4	yes	—
183	C₂³	#13086	1	4	yes	—
184	C₂³	C₂×#1889	1	1	yes	characteristic
185	C₂³	C₂×#1898	1	1	yes	characteristic
186	C₂³	C₂×#1913	1	1	yes	characteristic
187	C₂³	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
188	C₂³	C₂×#2121	1	2	yes	—
189	C₂³	#33551	1	4	yes	—
190	C₂²	#139716	1	2	yes	—
191	C₂²	#139716	1	2	yes	—
192	C₂²	#139731	1	2	yes	—
193	C₂²	C₂×#9495	1	1	yes	characteristic
194	C₂²	#494575	1	2	yes	—
195	C₂²	C₂×#13086	1	2	yes	—
196	C₂	C₂×#139731	1	1	yes	characteristic
197	C₂	C₂×#139716	1	1	yes	characteristic
198	C₂	#72237037	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#9495
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#120	C₂⁴
prime radical	C₂×#449475	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂²×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72337743	C₂	semidirect
2. #72323552	C₂	semidirect
3. #72310531	C₂	semidirect
4. #72231822	C₂	semidirect
5. #72227706	C₂	semidirect
6. #72227421	C₂	semidirect
7. #72226038	C₂	semidirect
8. C₂×#134033	C₂	semidirect
10. C₂³×B(4,𝔽₂)	C₄	semidirect
13. #449475	C₄	semidirect
16. #391692	C₂²	semidirect
19. #374022	C₂²	semidirect
20. C₂×#10991	C₄	semidirect
24. #134033	C₂²	semidirect
25. #134033	C₂²	semidirect
26. #133278	C₂²	semidirect
27. #131967	C₂²	semidirect
28. #93177	C₂²	semidirect
29. #93148	C₂²	semidirect
31. C₂³×H(5,𝔽₂)	C₂×C₄	semidirect
35. C₂²×B(4,𝔽₂)	C₂×C₄	semidirect
36. C₂³×#99	C₂×C₄	semidirect
43. #35346	C₂×C₄	semidirect
45. C₂×#2308	C₂×C₄	semidirect
48. C₂×#1819	D₄	semidirect
49. C₂×#1781	D₄	semidirect
51. #10991	C₂×C₄	semidirect
52. #10965	C₂×C₄	semidirect
53. #10965	D₄	semidirect
54. #10965	D₄	semidirect
86. C₂²×#99	C₂×D₄	semidirect
95. C₂×#342	C₂×D₄	semidirect
139. C₂⁵	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
#72337743	2
#72323552	2
#72310531	2
#72231822	2
#72227706	2
#72227421	2
#72226038	2
C₂×#134033	2
C₂×#449475	2