Subgroups — C₂×#72216490 #72337609

Order 4096.

At least 16890 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

The subgroup lattice has 215 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#72216484	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#72216427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72216481	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#72215671	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#72210587	C₂	1	1	yes	maximal, characteristic
6	C₂×#139718	C₄	1	1	yes	characteristic, complemented
7	C₂×#346446	C₄	1	1	yes	characteristic, complemented
8	C₂×#139716	C₄	1	1	yes	characteristic, complemented
9	C₂×#647656	C₂²	1	1	yes	characteristic
10	C₂×#72210489	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#72210482	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#134033	C₂²	1	1	yes	characteristic
13	C₂×#133822	C₂²	1	1	yes	characteristic
14	C₂×#115690	C₂²	1	1	yes	characteristic
15	C₂×#115690	C₂²	1	1	yes	characteristic
16	C₂×#115690	C₂²	1	1	yes	characteristic
17	C₂×#6499867	C₂²	1	1	yes	characteristic
18	#72216490	C₂	1	8	yes	maximal, direct product factor
19	#72216484	C₂²	1	4	yes	complemented
20	#72216481	C₂²	1	4	yes	complemented
21	#72216427	C₂²	1	4	yes	complemented
22	#72215671	C₂²	1	4	yes	complemented
23	C₂×#347042	C₄	1	1	yes	characteristic, complemented
24	#72210587	C₄	1	4	yes	complemented
25	#72210587	C₂²	1	4	yes	—
26	#72210489	C₂²	1	4	yes	complemented
27	#72210482	C₂²	1	4	yes	complemented
28	C₂×#62817	C₂×C₄	1	1	yes	characteristic, complemented
29	C₂³×D₄²	D₄	1	1	yes	characteristic
30	C₂²×#2846	C₂³	1	1	yes	characteristic
31	#6499867	C₂×C₄	1	2	yes	complemented
32	#6499867	C₂³	1	2	yes	—
33	C₂×#21746	C₂×C₄	1	1	yes	characteristic, complemented
34	#647656	C₂×C₄	1	2	yes	—
35	#647656	D₄	1	2	yes	—
36	#647656	D₄	1	2	yes	—
37	#647656	C₂³	1	2	yes	—
38	C₂×#15947	C₂×C₄	1	1	yes	characteristic
39	C₂×#15947	D₄	1	1	yes	characteristic
40	C₂×#15258	C₂×C₄	1	1	yes	characteristic
41	C₂×#15258	D₄	1	1	yes	characteristic
42	C₂×#14679	D₄	1	2	yes	complemented
43	C₂×#14134	D₄	1	1	yes	characteristic
44	#347042	C₂×C₄	1	4	yes	complemented
45	#346446	C₂×C₄	1	4	yes	complemented
46	C₂×#8547	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×#8541	C₂×C₄	1	1	yes	characteristic, complemented
48	C₂×#8101	D₄	1	2	yes	complemented
49	#139718	C₂×C₄	1	4	yes	complemented
50	#139716	C₂×C₄	1	4	yes	complemented
51	#134033	C₂³	1	2	yes	—
52	#133822	C₂×C₄	1	2	yes	complemented
53	#133822	C₂³	1	2	yes	—
54	#115690	D₄	1	2	yes	complemented
55	#115690	D₄	1	2	yes	complemented
56	#115690	C₂³	1	2	yes	—
57	#115690	C₂³	1	2	yes	—
58	#115690	C₂³	1	2	yes	—
59	C₂⁸	C₂×D₄	1	1	yes	characteristic
60	#62817	C₂²×C₄	1	2	yes	complemented
61	C₂²×D₄²	C₂²⋊C₄	1	2	yes	—
62	C₂²×D₄²	C₂×D₄	1	2	yes	—
63	C₂³×#127	C₂²⋊C₄	1	1	yes	characteristic
64	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
65	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
66	C₂³×#120	C₂×D₄	1	1	yes	characteristic
67	C₂³×#120	C₂×D₄	1	1	yes	characteristic
68	C₂×#2846	C₂²×C₄	1	1	yes	characteristic
69	C₂×#2846	C₂×D₄	1	1	yes	characteristic
70	C₂×#2846	C₂×D₄	1	1	yes	characteristic
71	C₂×#2846	C₂×D₄	1	1	yes	characteristic
72	C₂×#2846	C₂×D₄	1	1	yes	characteristic
73	C₂×#2846	#41	1	1	yes	characteristic
74	C₂×#2846	#41	1	1	yes	characteristic
75	C₂×#2846	C₂⁴	1	1	yes	characteristic
76	C₂²×#379	C₂×D₄	1	1	yes	characteristic
77	C₂²×#379	#41	1	1	yes	characteristic
78	C₂²×#379	#41	1	1	yes	characteristic
79	#21746	C₂²×C₄	1	2	yes	complemented
80	#15947	C₂²⋊C₄	1	2	yes	—
81	#15947	C₂²⋊C₄	1	2	yes	—
82	#15947	C₂²×C₄	1	2	yes	—
83	#15947	C₂×D₄	1	2	yes	—
84	#15258	C₂²⋊C₄	1	2	yes	—
85	#15258	C₂²⋊C₄	1	2	yes	—
86	#15258	C₂²×C₄	1	2	yes	—
87	#15258	C₂×D₄	1	2	yes	—
88	#14679	C₂×D₄	1	4	yes	complemented
89	#14134	C₂²⋊C₄	1	2	yes	—
90	#14134	C₂×D₄	1	2	yes	—
91	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic
93	#8547	C₂²×C₄	1	2	yes	complemented
94	#8541	C₂²×C₄	1	2	yes	complemented
95	#8101	C₂×D₄	1	4	yes	complemented
96	C₂⁷	#99	1	1	yes	characteristic, complemented
97	C₂⁷	C₄×D₄	1	1	yes	characteristic
98	C₂⁷	C₄×D₄	1	1	yes	characteristic
99	C₂⁷	#120	1	1	yes	characteristic
100	C₂⁷	#120	1	1	yes	characteristic
101	C₂⁷	#121	1	1	yes	characteristic
102	C₂⁷	#121	1	1	yes	characteristic
103	C₂⁷	C₂²×D₄	1	1	yes	characteristic
104	C₂⁴×D₄	#99	1	1	yes	characteristic, complemented
105	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×#127	#99	1	2	yes	complemented
107	C₂²×#127	C₂×C₂²⋊C₄	1	2	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	C₄×D₄	1	1	yes	characteristic
114	C₂²×#120	C₄×D₄	1	1	yes	characteristic
115	C₂²×#120	#120	1	1	yes	characteristic
116	C₂²×#120	#121	1	1	yes	characteristic
117	C₂²×#120	#121	1	1	yes	characteristic
118	C₂²×#120	#123	1	1	yes	characteristic
119	C₂²×#120	C₂³×C₄	1	1	yes	characteristic
120	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
121	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic, complemented
123	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
124	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
125	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
126	C₂³×C₄²	#99	1	1	yes	characteristic, complemented
127	C₂×#379	C₄×D₄	1	1	yes	characteristic
128	C₂×#379	C₄×D₄	1	1	yes	characteristic
129	C₂×#379	C₄×D₄	1	1	yes	characteristic
130	C₂×#379	#120	1	1	yes	characteristic
131	C₂×#379	#121	1	1	yes	characteristic
132	C₂×#379	#121	1	1	yes	characteristic
133	C₂×#379	#121	1	1	yes	characteristic
134	C₂×#379	#123	1	1	yes	characteristic
135	C₂×#379	#123	1	1	yes	characteristic
136	C₂×#379	C₂²×D₄	1	1	yes	characteristic
137	C₂×#379	C₂×#41	1	1	yes	characteristic
138	C₂×#379	C₂×#41	1	1	yes	characteristic
139	#1438	C₂×C₂²⋊C₄	1	2	yes	—
140	#1438	C₂×C₂²⋊C₄	1	2	yes	—
141	C₂⁶	#386	1	1	yes	characteristic
142	C₂⁶	#386	1	1	yes	characteristic
143	C₂⁶	#386	1	1	yes	characteristic
144	C₂⁶	#386	1	1	yes	characteristic
145	C₂⁶	C₂×#99	1	1	yes	characteristic
146	C₂⁶	C₂×#99	1	2	yes	complemented
147	C₂⁶	B(4,𝔽₂)	1	2	yes	—
148	C₂⁶	B(4,𝔽₂)	1	2	yes	—
149	C₂⁶	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂⁶	C₂×#120	1	1	yes	characteristic
151	C₂⁶	C₂×#121	1	1	yes	characteristic
152	C₂³×D₄	#386	1	1	yes	characteristic
153	C₂³×D₄	C₂×#99	1	1	yes	characteristic
154	C₂³×D₄	C₂×#99	1	1	yes	characteristic
155	C₂³×D₄	C₂×#99	1	2	yes	complemented
156	C₂³×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
157	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
159	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
160	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
161	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	complemented
162	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
164	C₂²×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
165	C₂²×C₄²	C₂×#99	1	2	yes	complemented
166	C₂⁵	#1889	1	1	yes	characteristic
167	C₂⁵	#1889	1	1	yes	characteristic
168	C₂⁵	#1889	1	2	yes	—
169	C₂⁵	#1898	1	1	yes	characteristic
170	C₂⁵	#1898	1	1	yes	characteristic
171	C₂⁵	#1898	1	1	yes	characteristic
172	C₂⁵	#1913	1	1	yes	characteristic
173	C₂⁵	#1913	1	1	yes	characteristic
174	C₂⁵	#2196	1	2	yes	—
175	C₂⁵	#2196	1	2	yes	—
176	C₂⁵	#2197	1	2	yes	—
177	C₂⁵	#2197	1	2	yes	—
178	C₂⁵	C₂×#386	1	1	yes	characteristic
179	C₂⁵	C₂²×#99	1	1	yes	characteristic
180	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
181	C₂³×C₄	#1898	1	1	yes	characteristic
182	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
183	C₂⁴	#9495	1	1	yes	characteristic
184	C₂⁴	#9495	1	1	yes	characteristic
185	C₂⁴	#9495	1	1	yes	characteristic
186	C₂⁴	#9495	1	1	yes	characteristic
187	C₂⁴	#13041	1	2	yes	—
188	C₂⁴	#13041	1	2	yes	—
189	C₂⁴	#13041	1	2	yes	—
190	C₂⁴	#13041	1	2	yes	—
191	C₂⁴	C₂×#1889	1	1	yes	characteristic
192	C₂⁴	C₂×#1898	1	1	yes	characteristic
193	C₂⁴	C₂×#1913	1	1	yes	characteristic
194	C₂⁴	C₂×#2196	1	2	yes	—
195	C₂⁴	C₂×#2197	1	2	yes	—
196	C₂³	#139700	1	1	yes	characteristic
197	C₂³	#139700	1	2	yes	—
198	C₂³	#139700	1	1	yes	characteristic
199	C₂³	#139743	1	1	yes	characteristic
200	C₂³	#139743	1	1	yes	characteristic
201	C₂³	#139750	1	1	yes	characteristic
202	C₂³	#139750	1	1	yes	characteristic
203	C₂³	C₂×#9495	1	1	yes	characteristic
204	C₂³	C₂×#13041	1	2	yes	—
205	C₂²	#72216488	1	1	yes	characteristic
206	C₂²	#72216488	1	1	yes	characteristic
207	C₂²	#72216488	1	1	yes	characteristic
208	C₂²	#72216488	1	1	yes	characteristic
209	C₂²	C₂×#139743	1	1	yes	characteristic
210	C₂²	C₂×#139750	1	1	yes	characteristic
211	C₂²	C₂×#139700	1	1	yes	characteristic
212	C₂	#72216490	1	2	yes	direct product factor
213	C₂	C₂×#72216488	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72216488
commutator	C₂²×#120	C₂³×C₄
Frattini, cosocle	C₂×#2846	C₂⁴
prime radical	C₂×#72210587	C₂
LCS	G ▷ C₂²×#120 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁶ ◁ C₂²×#2846 ◁ 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
1. C₂×#72216484	C₂	semidirect
2. C₂×#72216427	C₂	semidirect
3. C₂×#72216481	C₂	semidirect
4. C₂×#72215671	C₂	semidirect
6. C₂×#139718	C₄	semidirect
7. C₂×#346446	C₄	semidirect
8. C₂×#139716	C₄	semidirect
10. C₂×#72210489	C₂	semidirect
11. C₂×#72210482	C₂	semidirect
18. #72216490	C₂	direct (N × H)
19. #72216484	C₂²	semidirect
20. #72216481	C₂²	semidirect
21. #72216427	C₂²	semidirect
22. #72215671	C₂²	semidirect
23. C₂×#347042	C₄	semidirect
24. #72210587	C₄	semidirect
26. #72210489	C₂²	semidirect
27. #72210482	C₂²	semidirect
28. C₂×#62817	C₂×C₄	semidirect
31. #6499867	C₂×C₄	semidirect
33. C₂×#21746	C₂×C₄	semidirect
42. C₂×#14679	D₄	semidirect
44. #347042	C₂×C₄	semidirect
45. #346446	C₂×C₄	semidirect
46. C₂×#8547	C₂×C₄	semidirect
47. C₂×#8541	C₂×C₄	semidirect
48. C₂×#8101	D₄	semidirect
49. #139718	C₂×C₄	semidirect
50. #139716	C₂×C₄	semidirect
52. #133822	C₂×C₄	semidirect
54. #115690	D₄	semidirect
55. #115690	D₄	semidirect
60. #62817	C₂²×C₄	semidirect
79. #21746	C₂²×C₄	semidirect
88. #14679	C₂×D₄	semidirect
93. #8547	C₂²×C₄	semidirect
94. #8541	C₂²×C₄	semidirect
95. #8101	C₂×D₄	semidirect
96. C₂⁷	#99	semidirect
104. C₂⁴×D₄	#99	semidirect
106. C₂²×#127	#99	semidirect
108. C₂²×#120	#99	semidirect
122. C₂³×C₂²⋊C₄	#99	semidirect
126. C₂³×C₄²	#99	semidirect
146. C₂⁶	C₂×#99	semidirect
155. C₂³×D₄	C₂×#99	semidirect
161. C₂²×C₂²⋊C₄	C₂×#99	semidirect
165. C₂²×C₄²	C₂×#99	semidirect
212. C₂	#72216490	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#72216484	2
C₂×#72216427	2
C₂×#72216481	2
C₂×#72215671	2
C₂×#72210587	2
C₂×#72210489	2
C₂×#72210482	2
#72216490	2

Subgroups — C2×#72216490 #72337609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72216490 #72337609